text:
\section{Introduction}

Flavour Changing Neutral currents mediated through $b\to s\ell^+\ell^-$ transitions are an ideal laboratory to search for physics beyond the Standard Model (SM).
Their parametrically suppressed amplitudes, due to the GIM mechanism, render them a powerful probe of the intrinsic properties of the Standard Model itself, but also of the New Physics (NP) ones, should it be present.

A key strategy in the study of this type of transitions consists in measuring a large number of observables in $b$-hadrons decays and seeing if patterns emerge.
In the last decade, $B\to K^{(*)}\ell^+\ell^-$ decays received a lot of attention from both the theory and experimental communities. Indeed, the hadronic transitions $B\to K^{(*)}$ have been calculated using Lattice QCD techniques mostly at high $q^2$, where $q^2$ is the invariant mass of the dilepton system in the final state \cite{Horgan:2013hoa,Bouchard:2013eph,Horgan:2015vla,Bailey:2015dka,Parrott:2022dnu}, and also by means of other non-perturbative techniques such as Light-Cone Sum Rules \cite{Bharucha:2015bzk,Gubernari:2018wyi}. Experimentally, several measurements have been performed, from total branching fractions to  angular distributions \cite{LHCb:2020gog,LHCb:2020lmf,LHCb:2013zuf,ATLAS:2018gqc,CMS:2020oqb}. Furthermore, some tensions have been observed when comparing $B\to K^{(*)}\mu^+\mu^-$ and $B\to K^{(*)}e^+ e^-$ decays, at the level of $2.5-3.1$ standard deviations\cite{LHCb:2021lvy,LHCb:2019hip,LHCb:2017avl,LHCb:2014vgu}. In light of this, it is essential to assess $b$-baryon modes to expand our scope of understanding.\\
While the underlying quark-level transition remains the same with respect to mesons, decays of baryons have a different spin structure.
They, therefore, provide the possibility of studying a richer pattern of observables.
From a theoretical point of view, studies of local contributions to the ground state $\Lambda_b\to\Lambda$ transition have already been performed, both using Lattice QCD \cite{Detmold:2016pkz} and dispersive analysis \cite{Blake:2022vfl}, and measurements of $\Lambda_b\to\Lambda \mu^+\mu^-$ angular distribution are available in Ref.~\cite{LHCb:2018jna}.
However, from an experimental point of view, these decays are challenging to reconstruct because of the long lifetime {$\cal O$}(100 ps) of the weakly decaying $\Lambda$ particle \cite{LHCb:2018jna}.
This is not the case of $\Lambda^*$ excited states, such as the $\Lambda(1520)$, which decays strongly.
The difficulty in the treatment of $\Lambda^*$ excited states arises from the presence of many overlapping resonant states that need to be disentangled experimentally. Despite this challenge, the $\Lambda(1520)$ resonance has the advantage of being fairly narrow \cite{ParticleDataGroup:2020ssz} and produced in non-negligible quantities\cite{LHCb:2015yax}.

In this work, we perform an analysis of $\Lambda_b\to\Lambda(1520)$  local form factors.
This transition has already been studied using various techniques, which, however, are often applicable only in specific kinematic regions \cite{Descotes-Genon:2019dbw,Meinel:2021mdj,Meinel:2020owd,Bordone:2021bop,Mott:2011cx,Das:2020cpv}.
Up to now, no thorough study of an interpolation between the latter results that is valid in the whole kinematic region has been provided.
We address this precise issue, providing for the first time a parametrisation based on the analyticity of the form factors, viable in the entire phase space. In other words, our aim is to perform a pilot study to control the extrapolation uncertainties associated with local form factors.
To fix the coefficients of the form factors expansion, we use the results in Refs.~\cite{Meinel:2020owd,Meinel:2021mdj}, valid at low recoil (high $q^2$), endpoint relations at low and large recoil (low $q^2$), dispersive bounds and relations from Soft-Collinear Effective Theory.
We use our findings to perform numerical studies, providing the values for some observables that can be measured in the near future at the LHCb experiment \cite{Amhis:2020phx}.

This paper is organised as follows: in \sec{sec:parametrisation} we propose a form factors parametrisation based on the conformal mapping onto the complex $z$ plane, and we derive for the first time dispersive bounds for the $\Lambda_b\to\Lambda(1520)$ transition; in \sec{sec:fit} we discuss the fit of our parametrisation onto the available data; in \sec{sec:pheno}, we provide predictions for several observables; the conclusions of this work are drawn in \sec{sec:conclusions}.

\section{Theory framework}
\label{sec:parametrisation}
The $\Lambda_b\to\Lambda(1520)$ transition is characterised by $14$ hadronic form factors. We denote $\Lambda(1520)\equiv\Lambda^*$ in the remainder of this work. The complete basis reads (using the conventions in Ref.~\cite{Meinel:2021mdj}):
\begin{align}
 \nonumber \langle \Lambda^*(k,s_{\Lambda^*}) | \bar{s} \,\gamma^\mu\, b | \Lambda_b(p,s_{\Lambda_b}) \rangle =\,&+
 \bar{u}_\lambda(k,s_{\Lambda^*})\bigg[ f_0(q^2)\, \frac{m_{\Lambda^*}}{s_+} (m_{\Lambda_{b}}-m_{\Lambda^*})\frac{q^\mu}{q^2} p^\lambda \\
 \nonumber & + f_+(q^2) \frac{m_{\Lambda^*}}{s_-} \frac{m_{\Lambda_{b}}+m_{\Lambda^*}}{s_+}\left( p^\mu + k^{\mu} - (m_{\Lambda_{b}}^2-m_{\Lambda^*}^2)\frac{q^\mu}{q^2}  \right)p^\lambda \\
 &+ f_\perp(q^2) \frac{m_{\Lambda^*}}{s_-} \left(\gamma^\mu - \frac{2m_{\Lambda^*}}{s_+} p^\mu - \frac{2 m_{\Lambda_{b}}}{s_+} k^{ \mu} \right)p^\lambda\nonumber\\
 &+ f_{\perp^\prime}(q^2) \frac{m_{\Lambda^*}}{s_-} \left(\gamma^\mu p^\lambda + \frac{2(m_{\Lambda_{b}} k^\mu+m_{\Lambda^*} p^\mu)}{s_+} p^\lambda - \frac{2 k^\mu}{m_{\Lambda^*}} p^{\lambda} \right. \nonumber\\
 &+ \left. \frac{s_-}{m_{\Lambda^*}}g^{\lambda\mu} \right) \bigg] u_{\Lambda_b}(p,s_{\Lambda_b}), \label{eq:HMEL1}\\
 \nonumber \langle \Lambda^*(k,s_{\Lambda^*}) | \bar{s} \,\gamma^\mu\gamma_5\, b | \Lambda_b(p,s_{\Lambda_b}) \rangle =\,&
 -\bar{u}_\lambda(k,s_{\Lambda^*}) \:\gamma_5 \bigg[ g_0(q^2)\:\frac{m_{\Lambda^*}}{s_-} (m_{\Lambda_{b}}+m_{\Lambda^*})\frac{q^\mu}{q^2}p^\lambda \\
 \nonumber & + g_+(q^2)\frac{m_{\Lambda^*}}{s_+}\frac{m_{\Lambda_{b}}-m_{\Lambda^*}}{s_-}\left( p^\mu + k^{\mu} - (m_{\Lambda_{b}}^2-m_{\Lambda^*}^2)\frac{q^\mu}{q^2}  \right)p^\lambda \nonumber\\
 &+ g_\perp(q^2) \frac{m_{\Lambda^*}}{s_+}\left(\gamma^\mu + \frac{2m_{\Lambda^*}}{s_-} p^\mu - \frac{2 m_{\Lambda_{b}}}{s_-} k^{\mu} \right)p^\lambda\nonumber\\
 &+ g_{\perp^\prime}(q^2)\frac{m_{\Lambda^*}}{s_+} \left(\gamma^\mu p^\lambda + \frac{2(m_{\Lambda_{b}} k^\mu-m_{\Lambda^*} p^\mu)}{s_-} p^\lambda + \frac{2 k^\mu}{m_{\Lambda^*}} p^{\lambda} \right. \nonumber \\
 &- \left. \frac{s_+}{m_{\Lambda^*}}g^{\lambda\mu} \right) \bigg]  u_{\Lambda_b}(p,s_{\Lambda_b})\,, \label{eq:HMEL2}\\
\langle\Lambda^*(k,s_{\Lambda^*})|\bar{s} i \sigma^{\mu\nu} q_\nu b|\Lambda_b(p,s_{\Lambda_b})\rangle= \, & - \frac{m_{\Lambda^*}}{s_-}\bar{u}_{\lambda}(k, s_\Lambda) \Big\{h_+(q^2)\frac{q^2 (p^\mu+k^{ \mu})-(m_{\Lambda_{b}}^2-m_{\Lambda^*}^2)q^\mu}{s_{+}}p^\lambda \nonumber\\
&+h_\perp (q^2)(m_{\Lambda_{b}}+m_{\Lambda^*})\Big[\gamma^\mu-2\frac{m_{\Lambda_{b}} k^\mu+m_{\Lambda^*} p^\mu}{s_+} \Big]p^\lambda \nonumber\\
&+ h_{\perp^\prime}(m_{\Lambda_{b}}+m_{\Lambda^*})\Big[\gamma^\mu p^\lambda+2\frac{m_{\Lambda_{b}} k^\mu+m_{\Lambda^*} p^\mu}{s_+}p^\lambda-2\frac{k^\mu}{m_{\Lambda^*}}p^\lambda \nonumber\\
&+\frac{s_-}{m_{\Lambda^*}}g^{\mu\lambda} \Big]\Big\} u_{\Lambda_b}(p, s_{\Lambda_b})\, \label{eq:HMEL3} \\
\langle\Lambda^*(k,s_{\Lambda^*})|\bar{s} i \sigma^{\mu\nu}q_\nu \gamma_5 b|\Lambda_b(p,s_{\Lambda_b})\rangle= \, & -\frac{m_{\Lambda^*}}{s_+} \bar{u}_{\lambda}(k, s_\Lambda)\gamma_5 \Big\{\tilde{h}_+ \frac{q^2 (p^\mu+k^{ \nu})-(m_{\Lambda_{b}}^2-m_{\Lambda^*}^2)q^\nu}{s_{-}}p^\lambda\nonumber\\
&+\tilde{h}_\perp(q^2)(m_{\Lambda_{b}}-m_{\Lambda^*}) \Big[\gamma^\mu-2\frac{m_{\Lambda_{b}} k^\mu-m_{\Lambda^*} p^\mu}{s_-}\Big]p^\lambda\nonumber\\
&+\tilde{h}_{\perp^\prime}(m_{\Lambda_{b}}-m_{\Lambda^*}) \Big[\gamma^\mu p^\lambda+2\frac{m_{\Lambda_{b}} k^\mu-m_{\Lambda^*} p^\mu}{s_-}p^\lambda+2 \frac{k^\mu}{m_{\Lambda^*}}p^\lambda\nonumber\\
&-\frac{s_+}{m_{\Lambda^*}}g^{\lambda\mu}\Big]\Big\} u_{\Lambda_b}(p, s_{\Lambda_b})\,, \label{eq:HMEL3}
\end{align}
where $u_\lambda$ is the $3/2^-$ projection of a Rarita-Schwinger object \cite{Falk:1991nq}, and $\{p,s_{\Lambda_b}\}$ and $\{k,s_{\Lambda^*}\}$ are the momenta and spin of the $\Lambda_b$ and $\Lambda^*$, respectively.
For convenience, we also define the momentum transfer $q = p-k$, and $s_\pm = (m_{\Lambda_{b}}\pmm_{\Lambda^*})-q^2$.\\
The matrix elements that we listed above obey endpoint relations at $q^2 = q^2_\mathrm{max} = (m_{\Lambda_{b}} - m_{\Lambda^*})^2$ and $q^2=0$.
In our basis, these relations read\cite{Hiller:2021zth,Descotes-Genon:2019dbw,Meinel:2021mdj,Papucci:2021pmj}:
\begin{align}
    f_\perp(q^2_\mathrm{max}) =&~ - f_{\perp^\prime}(q^2_\mathrm{max})\,, \label{eq:fperpq2max} \\
    f_+(q^2_\mathrm{max})     =&~ 2 \, \frac{m_{\Lambda_{b}}-m_{\Lambda^*}}{m_{\Lambda_{b}}+m_{\Lambda^*}}f_{\perp^\prime}(q^2_\mathrm{max})\,, \\
    g_0(q^2_\mathrm{max})     =&~ 0 \,, \\
    g_+(q^2_\mathrm{max})     =&~ g_\perp(q^2_\mathrm{max})-g_{\perp^\prime}(q^2_\mathrm{max})\,, \label{eq:gpatq2max} \\
    h_\perp(q^2_\mathrm{max}) =&~ -h_{\perp^\prime}(q^2_\mathrm{max})\,, \\
    h_+(q^2_\mathrm{max})     =&~ 2 \, \frac{m_{\Lambda_{b}}+m_{\Lambda^*}}{m_{\Lambda_{b}}-m_{\Lambda^*}}h_{\perp^\prime}(q^2_\mathrm{max})\,, \\
    \tilde{h}_+(q^2_\mathrm{max}) =&~ \tilde h_{\perp}(q^2_\mathrm{max})- \tilde h_{\perp^\prime}(q^2_\mathrm{max})\,,\label{eq:htPlusq2max}
\end{align}
and
 \begin{align}
    f_0(0) =& \left(\frac{m_{\Lambda_{b}}+m_{\Lambda^*}}{m_{\Lambda_{b}}-m_{\Lambda^*}}\right)^2f_+(0)\,,  \label{eq:f0at0}\\
    g_0(0) =& \left(\frac{m_{\Lambda_{b}}-m_{\Lambda^*}}{m_{\Lambda_{b}}+m_{\Lambda^*}}\right)^2g_+(0)\,, \label{eq:g0at0}\\
    \tilde{h}_\perp(0)=& \left(\frac{m_{\Lambda_{b}}+m_{\Lambda^*}}{m_{\Lambda_{b}}-m_{\Lambda^*}}\right)^2 h_\perp(0)\,, \\
    -\tilde{h}_{\perp^\prime}(0)=& \left(\frac{m_{\Lambda_{b}}+m_{\Lambda^*}}{m_{\Lambda_{b}}-m_{\Lambda^*}}\right)^2 h_{\perp^\prime}(0)\,. \label{eq:htppat0}
\end{align}
In the following, we discuss the parametrisation that we adopt for the form factors and how they are subject to bounds from dispersive analysis.

\subsection{Form factors parametrisation}
The analytic properties of the form factors suggest that it is convenient to parametrise them in terms of the complex variable $z$, that is defined by the conformal transformation
\begin{equation}
z(q^2, t_0) = \frac{\sqrt{t_+-q^2}-\sqrt{t_+-t_0}}{\sqrt{t_+-q^2}+\sqrt{t_+-t_0}}\,,
\end{equation}
which maps the $q^2$ axis onto the complex $z$ plane, with the advantage that $|z|\leq1$.
We set the parameter $t_+$ to be the two-body threshold production for $\bar{s}b$ states, which is $t_+ = (m_B+m_K)^2$.
We also choose $t_0 = t_+ - \sqrt{t_+^2 - t_+ t_-} = 9.865\, \text{GeV}^2$, with $t_-=(m_B-m_K)^2$, to lower the maximum allowed values for $z$ in the semileptonic region to $|z_\text{max}|\sim 0.09$. For convenience, we further introduce the quantities $t_{\pm}^b = (m_{\Lambda_{b}} \pm m_{\Lambda^*})^2$. \\
This setup has been used to work out the so-called BGL parametrisation \cite{Boyd:1994tt,Boyd:1995cf}, whose success lies in the model independency of the parametrisation itself.
However, for transitions in which $t_+$ does not coincide with the threshold productions of the hadrons involved, this formalism needs to be slightly revisited in order to account for the fact that the region where $q^2>t_+$ is mapped only on an arc of the unit circle described by $|z|=1$. This formalism has been introduced in Refs.~\cite{Blake:2022vfl,Gubernari:2020eft}, and we closely follow that approach. This is illustrated in \fig{fig:z_plane}. Hence, we parametrise the form factors as
\begin{equation}
f_i(z) = \frac{1}{P_{f_i}(z)\phi_{f_i}(z)}\sum_{n=0}^N a^{f_i}_n \, p_n(z)\,,
\label{eq:z_exp}
\end{equation}
where $N$ is the truncation order of the series, and $p_n$ are orthonormal polynomials on an arc of the unit circle, as described in Appendix \ref{app:orthonormal_polynomials}. As usual, $P_{f_i}(z)$ is the so-called Blaschke factor, which captures the contribution of sub-threshold resonances, and reads
\begin{equation}
    P_{f_i}(z)= \prod_j \frac{z - z(m_j^2, t_0)}{1 - z \, z^*(m_j^2, t_0)}
\end{equation}
where $m_j$ are the masses of the sub-threshold resonances listed in \Table{tab:resonances} and $z^*$ is the complex conjugate of $z$. Finally, the functions $\phi_{f_i}$ are the so-called outer functions, for which we give an explicit representation in the next Section.
The advantage of this setup is that it allows to apply dispersive techniques to bound the form factors coefficients in \eq{eq:z_exp}, as illustrated in the next Section.
\begin{figure}
    \centering
    \includegraphics{figures/figure}
    \caption{Illustration of the complex $q^2$ and $z$ plane. The green region represents the semileptonic region, the red crosses possible sub-threshold resonances, and the orange contours the branch cut. The purple line(arc) in the left(right) panel arises from $t_+\neq t^b_+$. In blue we indicate the relevant $q^2$ points.}
    \label{fig:z_plane}
\end{figure}

\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{ c c c c c}
\toprule
Form factor & $J^P$ & Resonance & $m_\text{res}^j [\mathrm{GeV}]$ & $\schi{J}{\Gamma}{\mathrm{OPE}}\times 10^2$ \\
\midrule
$g_0$& $0^-$ & $B_s$ & $5.367$\, \cite{ParticleDataGroup:2020ssz} & $1.57$\\
$g_{+}$, $g_{\perp}$, $g_{\perp^\prime}$ & $1^+$ & $B_{s,1}$ & $5.750$\, \cite{Lang:2015hza} & $1.13/m_b^2$\\
$f_0$& $0^+$ & $B_{s,0}$ & $5.711$\, \cite{Lang:2015hza} & $1.42$\\
$f_+$, $f_{\perp}$, $f_{\perp^\prime}$& $1^-$ & $B_s^*$ & $5.416$\, \cite{ParticleDataGroup:2020ssz} & $1.20/m_b^2$\\
$h_{+}$, $h_{\perp}$, $h_{\perp^\prime}$& $1^-$ & $B_s^*$ & $5.416$\, \cite{ParticleDataGroup:2020ssz} & $0.803/m_b^2$\\
$\tilde{h}_{+}$\, $\tilde{h}_{\perp}$, $\tilde{h}_{\perp^\prime}$& $1^+$ & $B_{s,1}$ & $5.750$\, \cite{Lang:2015hza} & $0.748/m_b^2$\\
\toprule
\end{tabular}
\caption{Parameters for the form factor expansion. The reference value for the $b$-quark mass is $m_b=4.2\,\text{GeV}$.}
\label{tab:resonances}
\end{center}
\end{table}

\subsection{Dispersive Bounds}

The main idea is to use the dispersive analysis to extract bounds on the form factors parameters \cite{Caprini:2019osi,Boyd:1994tt,Boyd:1995cf}. In this, we follow closely the approach of Ref.~\cite{Blake:2022vfl}. The starting point is the two-point correlation function:
\begin{align}
    \Pi^{\mu\nu}_{\Gamma}(q)
        & \equiv i \int d^4x\, e^{iq\cdot x} \langle 0 | \mathcal{T}[J_\Gamma^\mu(x), J_\Gamma^{\dagger,\nu}(0)] | 0\rangle
         = \left(\frac{q^\mu q^\nu}{q^2} - g^{\mu\nu}\right) \Pi_{1}^\Gamma(q^2) + \frac{q^\mu q^\nu}{q^2} \Pi_{0}^\Gamma(q^2)
        \,,
\end{align}
where $\Gamma = V,A,T,T5$ for vector, axial-vector, tensor and pseudo-tensor currents, respectively. We note that  $\Pi_{0}^\Gamma(q^2)=0$ for tensor and pseudo-tensor currents. We introduce the projectors:
\begin{equation}
    \mathcal{P}^0_{\mu\nu} = \frac{q_\mu q_\nu}{q^2}\,, \qquad \mathrm{and} \qquad \mathcal{P}^1_{\mu\nu} = \frac{1}{3}\left(\frac{q_\mu q_\nu}{q^2}-g_{\mu\nu}\right)\,,
\end{equation}
such that
\begin{equation}
    \mathcal{P}^0_{\mu\nu}\,\Pi^{\mu\nu}_{\Gamma}(q) = \Pi_{0}^\Gamma(q^2)\,, \qquad \mathrm{and} \qquad \mathcal{P}^1_{\mu\nu}\,\Pi^{\mu\nu}_{\Gamma}(q) = \Pi_{1}^\Gamma(q^2)\,.
\end{equation}
To evaluate the bounds, we are actually interested in extracting the discontinuity of $\Pi^{\mu\nu}_\Gamma$. Therefore, we define the susceptibility
\begin{equation}
    \chi_\Gamma^J (Q^2) = \frac{1}{n!}\left(\frac{d}{dQ^2}\right)^n \Pi_\Gamma^J(Q^2) =\frac{1}{2\pi i}\int_0^\infty dt \,\frac{\mathrm{ Disc}[\Pi_V^J(t)]}{(t-Q^2)^{n+1}}\,,
    \label{eq:chi}
\end{equation}
where $n$ is the number of subtractions and might vary for the different currents. The bounds are obtained using quark-hadron duality, which implies
\begin{equation}
    \schi{J}{\Gamma}{\mathrm{OPE}} =  \schi{J}{\Gamma}{\mathrm{hadr}}\,,
    \label{eq:QHD}
\end{equation}
where the left-hand side (l.h.s.) is the perturbative calculation obtained in the Operator-Product-Expansion (OPE). The results for each current and angular momentum have been calculated in Ref.~\cite{Bharucha:2010im} and include $\mathcal{O}(\alpha_s)$ and $\mathcal{O}(\Lambda_\mathrm{QCD}/m_b)$ corrections, and are reported in \Table{tab:resonances}. In the right-hand side (r.h.s.) of \eq{eq:QHD}, one uses the spectral representation to introduce a sum over possible states mediated by an hadronic $\bar{s}b$ state. This yields:
\begin{equation}
    \mathrm{Disc}[\Pi_V^J(t)]= i \sum_{\mathrm{spin}}\int \mathrm{d}\rho (2\pi^4)\delta^{(4)}(q-\sum_{i=1}^n p_i)\mathcal{P}_{\mu\nu}^J\langle 0|J^\mu_\Gamma|H_{\bar{s}b}(p_1,\dots p_n)\rangle\langle H_{b\bar{s}}(p_1,\dots p_n)|J_\Gamma^{\nu\dagger}|0\rangle\,,
\end{equation}
that is plugged in \eq{eq:chi} to obtain $\schi{J}{\Gamma}{\mathrm{hadr}}$. We can use the number of states in $H_{\bar{s}b}(p_1,\dots p_n)$ to further decompose $\schi{J}{\Gamma}{\mathrm{hadr}}$, finding
\begin{equation}
    \schi{J}{\Gamma}{\mathrm{hadr}} = \schi{J}{\Gamma}{\mathrm{1pt}} +  \schi{J}{\Gamma}{\mathrm{2pt}} + \dots\,,
    \label{eq:chihad_decomp}
\end{equation}
where the first term in the r.h.s. of \eq{eq:chihad_decomp} encodes the one particle contribution, the second term the two-particle contribution, and the ellipses stand for higher multiplicity states. The one-particle contribution accounts for the $B_s$ and $B_s^*$ states. For each current and angular momentum state, we find
\begin{align}
    \schi{1}{V}{\mathrm{1pt}} =& ~ \frac{m_{B_s^*}^2 f_{B_{s}^*}^2}{(m_{B_s^*}^2-Q^2)^{n+1}}\,, & \schi{0}{V}{\mathrm{1pt}} =& ~ \frac{m_{B_{s,0}^*}^2 f_{B_{s,0}^*}^2}{(m_{B_{s,0}^*}^2-Q^2)^{n+1}}\,, \\
    \schi{1}{A}{\mathrm{1pt}} =& ~ \frac{m_{B_{s,1}^*}^2 f_{B_{s,1}^*}^2}{(m_{B_{s,1}^*}^2-Q^2)^{n+1}}\,, & \schi{0}{A}{\mathrm{1pt}} =& ~ \frac{m_{B_{s}}^2 f_{B_{s}}^2}{(m_{B_{s}}^2-Q^2)^{n+1}}\,, \\
    \schi{1}{T}{\mathrm{1pt}} =& ~ \frac{m_{B_{s}^*}^4 (f_{B_{s,1}^*}^T)^2}{(m_{B_{s}^*}^2-Q^2)^{n+1}}\,, & \schi{1}{T5}{\mathrm{1pt}} =& ~ \frac{m_{B_{s,1}}^4 (f_{B_{s,1}}^T)^2}{(m_{B_{s}}^2-Q^2)^{n+1}}\,,
\end{align}
in agreement with Ref.~\cite{Blake:2022vfl}. \\
The two-particle contribution in principle contains an infinite sum over two-body hadronic states (e.g. $\bar{B}K$, $\bar{B}K^*$, $\bar{B}_s\phi$, etc.).
It is far beyond the scope of this work to perform a global analysis of $b\to s$ transition; therefore, in the following, we only consider $H_{\bar{s}b}(p_1,\dots p_n)=\Lambda_b(p_1)\bar\Lambda^*(-p_2)$.
With this choice, we find
\begin{align}
\label{eq:dispersive_bounds_V0}
    \schi{0}{V}{\mathrm{2pt}} =&\, \frac{1}{48\pi^2}\int_{t_+^b}^\infty dt \frac{\sqrt{s_+ s_-^3}}{t^2(t-Q^2)^{n+1}}(m_{\Lambda_{b}}-m_{\Lambda^*})^2|f_0|^2  \,, \\
    \schi{1}{V}{\mathrm{2pt}} =&\, \frac{1}{144\pi^2} \int_{t_+^b}^\infty dt \frac{\sqrt{s_+^3 s_-}}{t^2(t-Q^2)^{n+1}}\left[(m_{\Lambda_{b}}+m_{\Lambda^*})^2 |f_+|^2+2t|f_\perp|^2+6t|f_{\perp^\prime}|^2\right] \,, \\
    \schi{0}{A}{\mathrm{2pt}} =&\,\frac{1}{48\pi^2} \int_{t_+^b}^\infty dt \frac{\sqrt{s_- s_+^3}}{t^2(t-Q^2)^{n+1}}(m_{\Lambda_{b}}+m_{\Lambda^*})^2|g_0|^2  \,, \\
    \schi{1}{A}{\mathrm{2pt}} =&\, \frac{1}{144\pi^2} \int_{t_+^b}^\infty dt \frac{\sqrt{s_-^3 s_+}}{t^2(t-Q^2)^{n+1}}\left[(m_{\Lambda_{b}}-m_{\Lambda^*})^2 |g_+|^2+2t|g_\perp|^2+6t|g_{\perp^\prime}|^2\right]  \,, \\
    \schi{1}{T}{\mathrm{2pt}} =&\,  \frac{1}{144\pi^2} \int_{t_+^b}^\infty dt \frac{\sqrt{s_+^3 s_-}}{t (t-Q^2)^{n+1}}\left[t |h_+|^2+2(m_{\Lambda_{b}}+m_{\Lambda^*})^2|h_\perp|^2+6(m_{\Lambda_{b}}+m_{\Lambda^*})^2|h_{\perp^\prime}|^2\right] \,, \\
\label{eq:dispersive_bounds_T51}
    \schi{1}{T5}{\mathrm{2pt}} =&\, \frac{1}{144\pi^2} \int_{t_+^b}^\infty dt \frac{\sqrt{s_-^3 s_+}}{t (t-Q^2)^{n+1}}\left[t |\tilde{h}_+|^2+2(m_{\Lambda_{b}}-m_{\Lambda^*})^2|\tilde{h}_\perp|^2+6(m_{\Lambda_{b}}-m_{\Lambda^*})^2|\tilde{h}_{\perp^\prime}|^2\right]  \,.
\end{align}
where $s_\pm = t_\pm^b - t$. The parameter $Q^2$ is chosen to ensure that the two-point function is highly virtual \cite{Boyd:1997kz}. In this context this means $Q^2-(m_b+m_s)^2\ll-(m_b+m_s)\Lambda_\mathrm{QCD}$, that is well respected if $Q^2=0$. Hence, we set $Q^2=0$ in the remainder of this work. \\
In order to formulate the bounds, we still need to parametrise the outer functions. Using the results in Refs.~\cite{Blake:2022vfl,Caprini:2019osi}, we have
\begin{equation}
\begin{aligned}
    \phi_{f_i} (z)&= \frac{(t_+^b)^{A/2}(t_-^b)^{B/2}(1-z)^{n+g-e/2-f/2-3/2}(1+z)^{1/2}}{\sqrt{16 d \,\pi^2 \schi{J}{\Gamma}{\mathrm{OPE}}}}\\
    &\times\left[\frac{-1}{t_0 (1 + z)^2 - 2 t_+ (1 + z^2) - 2 (1 - z^2) \sqrt{t_+} \sqrt{t_+ - t_0}}\right]^{(n+g)/2} \\
    &\times\left[- (1-z)^2 t_-^b-(1+z)^2 t_0+2t_+(1+z^2)+2(1-z^2)\sqrt{t_+ - t_-^b} \sqrt{t_+ - t_0}\right]^{e/4} \\ &\times\left[(1-z)^2 t_+^b-(1+z)^2 t_0+4 t_+z\right]^{f/4}\sqrt{4(t_+-t_0)}\,,
\end{aligned}
\end{equation}
where the coefficients $\{A,B,d,e,f,g,n\}$ are given in \Table{tab:coeff_phi}. Our representation of the outer functions is not unique, but their modulus square is fixed by \eqs{eq:dispersive_bounds_V0}{eq:dispersive_bounds_T51}. Any modification that preserves the analytic properties of $\phi_{f_i}$ and conserves $|\phi_{f_i}|^2$ is valid.\\
The relations in \eq{eq:chihad_decomp} are never exactly satisfied since we are unable to calculate all perturbative orders in the OPE, and we cannot sum over all possible hadronic states in the spectral representation. Taking this limitation into account, we obtain the inequality
\begin{equation}
    \frac{\schi{J}{\Gamma}{\mathrm{hadr}}}{\schi{J}{\Gamma}{\mathrm{OPE}}} < 1\,,
\end{equation}
that, after imposing our previous findings, yields:
\begin{equation}
\label{eq:bounds_1}
\begin{aligned}
    &\sum_{i=0}^N |a^{f_0}_i|^2 <1-\frac{\schi{0}{V}{\mathrm{1pt}}}{\schi{0}{V}{\mathrm{OPE}}}\,, \qquad \sum_{i=0}^N (|a_i^{f_+}|^2+|a_i^{f_\perp}|^2+|a_i^{f_{\perp^\prime}}|^2) <1-\frac{\schi{1}{V}{\mathrm{1pt}}}{\schi{1}{V}{\mathrm{OPE}}}\,,\\
    &\sum_{i=0}^N |a^{g_0}_i|^2 <1-\frac{\schi{0}{A}{\mathrm{1pt}}}{\schi{0}{A}{\mathrm{OPE}}}\,, \qquad \sum_{i=0}^N (|a_i^{g_+}|^2+|a_i^{g_\perp}|^2+|a_i^{g_{\perp^\prime}}|^2) <1-\frac{\schi{1}{A}{\mathrm{1pt}}}{\schi{1}{A}{\mathrm{OPE}}}\,, \\
    &\sum_{i=0}^N (|a_i^{h_+}|^2+|a_i^{h_\perp}|^2+|a_i^{h_{\perp^\prime}}|^2) <1-\frac{\schi{1}{T}{\mathrm{1pt}}}{\schi{1}{T}{\mathrm{OPE}}}\,, \\
    &\sum_{i=0}^N (|a_i^{\tilde{h}_+}|^2+|a_i^{\tilde{h}_\perp}|^2+|a_i^{\tilde{h}_{\perp^\prime}}|^2) <1-\frac{\schi{1}{T5}{\mathrm{1pt}}}{\schi{1}{T5}{\mathrm{OPE}}}\,.
\end{aligned}
\end{equation}
The above relations are usually addressed as dispersive bounds. The l.h.s. of these inequalities corresponds to the relative saturation of the bound due to two-particle contributions. In principle, one could add to \eq{eq:bounds_1} contributions from further two-body states and, in general, higher multiplicity states. This would render the bounds on the form factor coefficients that we are interested in extracting even stronger.

\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.1}
\begin{tabular}{ c c c c c c c c }
\toprule
Form factor & $A$ &  $B$ & $d$ &  $e$ &  $f$ & $g$ & $n$ \\
\midrule
$f_0$ & $0$ & $1$ & $6$ & $3$ & $1$ & $3$ & $1$  \\
$f_+$ & $1$ & $0$ & $18$ & $1$ & $3$ & $3$ & $2$ \\
$f_\perp$ & $0$ & $0$ & $9$ & $1$ & $3$ & $2$ & $2$ \\
$f_{\perp^\prime}$ & $0$ & $0$ & $3$ & $1$ & $3$ & $2$ & $2$ \\
$g_0$ & $1$ & $0$ & $6$ & $1$ & $3$ & $3$ & $1$ \\
$g_+$ & $0$ & $1$ & $18$ & $3$ & $1$ & $3$ & $2$ \\
$g_\perp$ & $0$ & $0$ & $9$ & $3$ & $1$ & $2$ & $2$ \\
$g_{\perp^\prime}$ & $0$ & $0$ & $3$ & $3$ & $1$ & $2$ & $2$ \\
$h_+$ & $0$ & $0$ & $18$ & $1$ & $3$ & $1$ & $3$ \\
$h_\perp$ & $1$ & $0$ & $9$ & $1$ & $3$ & $2$ & $3$ \\
$h_{\perp^\prime}$ & $1$ & $0$ & $3$ & $1$ & $3$ & $2$ & $3$ \\
$\tilde{h}_+$ & $0$ & $0$ & $18$ & $3$ & $1$ & $1$ & $3$  \\
$\tilde{h}_\perp$ & $0$ & $1$ & $9$ & $3$ & $1$ & $2$ & $3$ \\
$\tilde{h}_{\perp^\prime}$ & $0$ & $1$ & $3$ & $3$ & $1$ & $2$ & $3$ \\
\toprule
\end{tabular}
\caption{Coefficients involved in the outer functions.}
\label{tab:coeff_phi}
\end{center}
\end{table}

\section{Fit Results}
\label{sec:fit}
We extract the coefficients in \eq{eq:z_exp} by fitting them against a set of data. They include
\begin{description}
    \item[Lattice QCD:] we use the results of Ref.~\cite{Meinel:2021mdj} to extract pseudo-points close to the position of the lattice points, namely at $q^2= 16.2975\,\mathrm{GeV}^2$ and $q^2=16.5537\,\mathrm{GeV}^2$. Two points per form factor are available at very low-recoil, yielding 28 pseudo-points. However, the results in Ref.~\cite{Meinel:2021mdj} take already into account endpoint relations at $q^2_\mathrm{max}$, effectively reducing the number of independent points to $21$.
    \item[Dispersive Bounds:] we apply the dispersive bounds in \eq{eq:bounds_1}. To account for the uncertainty in the computation of $\schi{J}{\Gamma}{\mathrm{OPE}}$, we implement the same approach of Ref.~\cite{Bordone:2019vic}, with a penalty function
    \begin{equation}
        -2 \log P(r)=
        \begin{cases}
            0 & \mathrm{if}\, r<1\,, \\
            \frac{(r-1)^2}{\sigma^2} & \mathrm{otherwise}\,,
        \end{cases}
    \end{equation}
    where $r$ is the saturation of the dispersive bounds, and we use  $\sigma=10\%$. This matches the precision of the OPE calculation in Ref.~\cite{Bharucha:2010im}.
    \item[Endpoint relations:] we apply endpoint relations at $q^2=0$ as in \eqs{eq:f0at0}{eq:htppat0} and at $q^2_\mathrm{max}$ as in \eqs{eq:fperpq2max}{eq:htPlusq2max}. Practically, these relations lower the number of free parameters, and seven of them are already accounted for in the lattice QCD extrapolation. We stress that endpoint relations are derived from fundamental properties of the hadronic amplitudes and are therefore independent of any chosen form factors model.
    \item[SCET Relations:] we use the fact that at large recoil Soft-Collinear Effective Theory (SCET) predicts the form factors to be either zero or proportional to a single soft form factor at leading order in $\alpha_s/\pi$ and leading power in $\Lambda_\mathrm{QCD}/m_b$ \cite{Boer:2014kda,Feldmann:2011xf,Mannel:2011xg,Descotes-Genon:2019dbw}.
    These relations are needed due to the lack of computation either on the lattice or in Sum-Rule approaches at large recoil.
    We apply these relations at $q^2=0$ with an uncertainty of $0.2$ that accounts for the unknown next-to-leading order and next-to-leading power corrections.
    In principle, this uncertainty is strongly correlated among all form factors.
    However, the correlations are impossible to guess without explicit computation of higher-order effects in the SCET expansion.
    To circumvent this problem, we restrict ourselves to the following set of relations:
    \begin{equation}
    \begin{aligned}
        f_{\perp'}(0) &=\, 0 \pm 0.2 \,, &  g_{\perp'}(0) &=\, 0 \pm 0.2 \,, &
        h_{\perp'}(0) &=\, 0 \pm 0.2 \,, \\
        \tilde{h}_{\perp'}(0) &=\, 0 \pm 0.2  \,, & f_+(0) / f_\perp(0) &=\, 1 \pm 0.2 \,, &
        f_\perp(0) / g_0(0) &=\, 1 \pm 0.2 \,, \\
        g_\perp(0) / g_+(0) &=\, 1 \pm 0.2 \,, &  h_+(0) / h_\perp(0) &=\, 1 \pm 0.2 \,, & f_+(0) / h_+(0) &=\, 1 \pm 0.2\,,
    \end{aligned}
    \end{equation}
    and we treat the uncertainties as uncorrelated. This corresponds to a conservative assumption, since correlated uncertainties would restrict the available parameter space.
\end{description}

In summary, we have $21$ constraints from Lattice QCD calculations, $9$ constraints from SCET relations, and $6$ non-linear constraints from the dispersive bounds.
We test three fit scenarios, by fixing in \eq{eq:z_exp} $N=\{1,2,3\}$.
These three scenarios have $17$, $31$ and $45$ free parameters, respectively, which implies that the $N=\{2,3\}$ cases are under-constrained.
However, the presence of the dispersive bounds ensures bounded posterior distributions despite the appearance of blind directions.

\begin{table}[t]
    \centering
    \begin{tabular}{c c}
    \toprule
        $m_{\Lambda_{b}}$ & $5.620\, \mathrm{GeV}$  \\
        $m_{\Lambda^*}$ & $1.520\, \mathrm{GeV}$  \\
        $m_B$ & $5.279\, \mathrm{GeV}$  \\
        $m_K$ & $0.494\, \mathrm{GeV}$  \\
        $m_{K^*}$ & $0.896\, \mathrm{GeV}$  \\
        $\alpha_\mathrm{EM}$ & $1/133$ \\
        $\tau_{\Lambda_b}$ & 1.471 ps\\
    \bottomrule
    \end{tabular}
    \caption{Numerical inputs of our parametrisation. All inputs are taken from the PDG \cite{ParticleDataGroup:2020ssz}, apart from the $\Lambda_b$ lifetime \cite{Amhis:2020phx}.}
    \label{tab:constants}
\end{table}

All the numerical studies are performed using the version 1.0.4 of the \texttt{EOS} software \cite{EOS:paper,EOS:repo}.
We use \texttt{EOS} default values for the physical constants, which include the values in \Table{tab:constants}.
Posterior samples are drawn using a Markov chain Monte-Carlo method based on the Metropolis-Hastings algorithm \cite{Metropolis:1953am, Hastings:1970aa}.
Since the posterior-predictive samples for the parameters of the $z$ expansion are usually non-Gaussian distributed, we provide an ancillary notebook that can be used to reproduce them.
Running the notebook requires only an installation of \texttt{EOS} v1.0.4 or greater\footnote{See \url{https://eos.github.io/doc/installation.html} for installation instructions.}.

In the following, we discuss our findings.
The results are summarized in Table \ref{tab:fit_summary} and drawn in \figs{fig:fitres1}{fig:fitres2}.
The blue and green bands are obtained by fixing in \eq{eq:z_exp} $N=\{1,2\}$ and we added the $N=3$ scenario in red dashed lines.
These fit scenarios have different characteristics, which we describe below.

\begin{table}[t]
    \centering
    \begin{tabular}{lccccc}
    \toprule
        \multirow{2}{*}{$N$} & \multirow{2}{*}{parameters} & \multicolumn{4}{c}{$-2 \log \mathcal{L}$} \\
        & & Lattice & SCET & Bounds & Global \\
    \midrule
        1 & 17 & 48.4 & 2.6 & 0.1 & 51.1\\
        2 & 31 & 23.6 & 0.3 & 0.1 & 24.0\\
        3 & 45 & 19.6 & 0.2 & 0.1 & 19.9\\
    \bottomrule
    \end{tabular}
    \caption{Summary of the fit results for the different scenarios.
    The number of constraints is the same for all the scenarios, namely 30 linear constraints, and 6 non-linear ones.
    Since the bounds provide non-linear constraints on the parameters, the corresponding test statistic is not known.
    For the SCET relations and the lattice data, the contribution to the likelihood $\mathcal{L}$ can be interpreted as a $\chi^2$.}
    \label{tab:fit_summary}
\end{table}

\begin{description}
\item[The $N=1$ scenario.] This parametrisation resembles the one used for the continuum extrapolation of the lattice QCD results in that we have the same number of free parameters. In fact, a fit to the lattice QCD data only, without endpoint relations at low $q^2$, yields å$\chi^2=0$ at the best-fit point. However, when adding endpoint relations at low $q^2$, SCET relations and dispersive bounds, the fit quality drops to a $p$-value of $10^{-6}$. This is mainly due to the incompatibility of lattice QCD data and endpoint relations at low $q^2$ in this scenario. This is expected since lattice QCD data are only available at very high $q^2$.
It is also evident from \figs{fig:fitres1}{fig:fitres2} that the $N=1$ scenario massively underestimates the uncertainties at large recoil for some of the form factors.
Concerning the saturations of the two-particle contributions to the dispersive bounds, they all peak between 5 and 10\%, except the one associated with the pseudo-tensor current which saturates the inequality.

\item[The $N=2$ scenario.] This scenario is characterised by a number of free parameters larger than the number of constraints.
Hence, we are unable to provide a global value for the fit quality. The fit to lattice QCD data and the full set of endpoint relations, yields $\chi^2=0$ at the best-fit point. However, the latter violates the dispersive bounds massively.
Performing then the fit including all constraints, namely adding SCET relations and the dispersive bounds, we obtain in the minimum $\chi^2 = 24$. However, this cannot be interpreted with a $p$-value, since we have a negative number of degrees of freedom. It is anyway useful to mention that the $\chi^2$ at the best-fit point associated with the constraints from lattice QCD drops from $\sim 48.4$ in the $N=1$ scenario to $23.6$ in the $N=2$ case.
Since the number of constraints due to the lattice data is 21, this is equivalent to a satisfactory, \emph{individual} $p$-value of $31\%$ for the lattice points only for $N=2$.
Since the other constraints are equally satisfied, we conclude that the $N=2$ scenario shows an overall better agreement with the data.
In this case, having more free parameters than constraints makes the dispersive bound essential and, therefore, leads to larger relative saturations.

\item[The $N=3$ scenario.] This model has even a larger number of free parameters, allowing a perfect fit to lattice QCD data, the full set of endpoint relations and SCET relations. Also in this case, the fit results, however, massively violate the dispersive bounds.
Adding the latter as constraints, we obtain a \emph{individual} $p$-value for the lattice points of $55\%$.
As visible in \figs{fig:fitres1}{fig:fitres2}, the 68\% C.L. bands overlaps perfectly with the one for $N=2$ results.
This means that the uncertainties are saturated due to the fact that the free parameters are all constrained by the dispersive bounds.
We expect this property to be equally verified by $N>3$ scenarios. Concerning the saturations, similar conclusions as for the $N=2$ case apply.
\end{description}

\begin{figure}
\hspace{-10mm}
    \begin{tabular}{ccc}
        \includegraphics[width=.35\textwidth]{figures/f_0}  &
        \includegraphics[width=.35\textwidth]{figures/f_plus.pdf} &
        \includegraphics[width=.35\textwidth]{figures/f_perp.pdf}
        \\
        \includegraphics[width=.35\textwidth]{figures/f_perpP.pdf}&
        \includegraphics[width=.35\textwidth]{figures/g_0}  &
        \includegraphics[width=.35\textwidth]{figures/g_plus.pdf}  \\[1.25em]
        \includegraphics[width=.35\textwidth]{figures/g_perp.pdf} &
        \includegraphics[width=.35\textwidth]{figures/g_perpP.pdf}\\[1.25em]
    \end{tabular}
    \caption{Fit results for the vector and axial-vector $\Lambda_b\to\Lambda(1520)$ form factors. The blue and green bands are the 68\% C.L. region, and the solid line shows the median of the distribution for $N=1$ and $N=2$, respectively. The dashed red lines contain the 68\% C.L. region for the $N=3$ scenario.
    }
    \label{fig:fitres1}
\end{figure}

\begin{figure}[H]
\hspace{-10mm}
    \begin{tabular}{ccc}
        \includegraphics[width=.35\textwidth]{figures/h_plus.pdf} &
        \includegraphics[width=.35\textwidth]{figures/h_perp.pdf} &
        \includegraphics[width=.35\textwidth]{figures/h_perpP.pdf}
        \\[1.25em]
        \includegraphics[width=.35\textwidth]{figures/ht_plus.pdf} &
        \includegraphics[width=.35\textwidth]{figures/ht_perp.pdf} &
        \includegraphics[width=.35\textwidth]{figures/ht_perpP.pdf}
        \\[1.25em]
    \end{tabular}
    \caption{Fit results for the tensor and pseudo-tensor $\Lambda_b\to\Lambda(1520)$ form factors. The blue and green bands are the 68\% C.L. region, and the solid line shows the median of the distribution for $N=1$ and $N=2$, respectively. The dashed red lines contain the 68\% C.L. region for the $N=3$ scenario.
    }
    \label{fig:fitres2}
\end{figure}

Comparing our findings, we conclude that none of the scenarios is fully satisfactory, because a tension between lattice QCD data and the dispersive bounds remains.
Our results clearly indicate that $N=1$ scenario is ill-suited for an extrapolation of the lattice QCD results to small $q^2$.
For $N>1$ however, the negative number of degrees of freedom ensures the uncertainties to be completely driven by the dispersive bounds.
The latter, therefore, already accounts for the truncation error of the parametrisation and is resistant to any residual model uncertainty.
Despite the residual tension, we can thus set $N=2$ as our nominal fit for the phenomenology applications and, with the current data, considering higher extrapolation orders will not modify our results nor the attached uncertainties.

We emphasize that our approach is systematically improvable by computations of the form factors in the large-recoil region.
For example, future lattice simulations that extend to lower $q^2$ values will provide means to extract more precise information on the higher order coefficients, possibly also removing the aforementioned current tension.
Since the $N=2,\,3$ scenarios show a saturation of the dispersive bound, a global analysis of all $b\to s$ transitions would help reduce the uncertainties on all the form factors.

\section{Phenomenology}
\label{sec:pheno}
The effective Hamiltonian describing $b\to s\ell^+\ell^-$ transitions at the scale $\mu=m_b$ reads \cite{Buchalla:1995vs}:
\begin{equation}
    \mathcal{H}(b\to s\ell^+\ell^-)=-\frac{4G_F}{\sqrt{2}}V_{tb}V^*_{ts}\sum_{i=1}^{10} C_i\mathcal{O}_i\,,
    \label{eq:Heffbsll}
\end{equation}
where the relevant operators for our discussion are the four quarks operators:
\begin{align}
    \mathcal{O}_1 =& (\bar{c}\gamma_\mu T^a P_L c)(\bar{s} \gamma_\mu T^a P_{L} b) & \mathcal{O}_2 =& (\bar{c}\gamma_\mu P_L c)(\bar{s} \gamma_\mu P_{L} b)\,,
\end{align}
with $C_1 =1.010$ and $C_2=-0.291$ \cite{Gorbahn:2004my,Bobeth:1999ww}, and the semileptonic and dipole operators:
\begin{align}
    \mathcal{O}_9^{(\prime)} =&\, \frac{e^2}{16\pi^2}(\bar{s} \gamma_\mu P_{L(R)} b)(\bar{\ell}\gamma^\mu\ell)\,, &  \mathcal{O}_{10}^{(\prime)} =&\, \frac{e^2}{16\pi^2}(\bar{s} \gamma_\mu P_{L(R)} b)(\bar{\ell}\gamma^\mu\gamma_5\ell)\,, \\[0.5em]
    \mathcal{O}^{(\prime)}_7=&\, \frac{e\,m_b}{16\pi^2}(\bar{s}\sigma_{\mu\nu}P_{R(L)} b)F^{\mu\nu}\,,
\end{align}
where in the SM $C_9\simeq 4.2$, $C_9\simeq -4.3$, $C_7\simeq-0.29$, and $C_7^\prime \sim m_s/m_b \approx 0$, while $C_9^\prime = C_{10}^\prime =0$ \cite{Gorbahn:2004my,Bobeth:1999ww}.
The operators $\mathcal{O}_{1-6}$ mix perturbatively onto $\mathcal{O}_9$ and $\mathcal{O}_7$, even though $C_{3-6}\sim \mathcal{O}(10^{-2})$ only generate small effects.
The mixing has been calculated at leading logarithmic power, and is accounted for by the shift $C_9\to C_9^\mathrm{eff}(q^2)$ and $C_7\to C_7^\mathrm{eff}(q^2)$ \cite{Buchalla:1995vs,Chetyrkin:1996vx}.
The operators $\mathcal{O}_{1,2}$ are also responsible for non-factorisable long distance contributions in $b\to s\ell^+\ell^-$ transitions.
In the literature, these long-distance effects have been studied in different setups \cite{Gubernari:2022hxn,Capdevila:2017ert,Khodjamirian:2010vf,Bobeth:2017vxj}.
A detailed analysis of these effects is beyond the scope of this work.
Therefore, we restrain ourselves from providing any prediction where non-factorisable long-distance effects dominate, namely the $c\bar{c}$ resonant region.\\
The effective Hamiltonian in \eq{eq:Heffbsll} is used to derive several observables for $\Lambda_b\to\Lambda^*\ell^+\ell^-$ decays. We refer to Ref.~\cite{Descotes-Genon:2019dbw} for the full expression of the 4-fold differential distribution for $\Lambda_b\to\Lambda^*(\to K p)\ell^+\ell^-$. Notice that the formulae provided in Ref.~\cite{Descotes-Genon:2019dbw} account for massless leptons only. Hence, our predictions are valid only in the case $\ell=e\,,\mu$. For the very same reason, we don't provide numerical predictions in the region $q^2<0.3\,\mathrm{GeV}^2$. In the following, we always integrate on the $\Lambda^*\to K p$ variables, in the hypothesis of a narrow $\Lambda^*$, such that the remaining distribution for $\Lambda_b\to\Lambda^*\ell^+\ell^-$ decays is
\begin{equation}
    \frac{d\Gamma}{dq^2 d\cos\theta_\ell} = a + b \cos\theta_\ell + c \cos^2\theta_\ell\,,
\end{equation}
where $\theta_\ell$ is the helicity angle of the negative-charged lepton in the dilepton rest-frame, and
\begin{equation}
\begin{aligned}
    a = &\; \frac{1}{2}(L_{1ss}+2 L_{2ss}+L_{3ss})\,, & b = &\; \frac{1}{2}(L_{1c}+2L_{2c}) \,,\\
    c = &\; \frac{1}{2}(L_{1cc}-L_{1ss}+2L_{2cc}-2L_{2ss}-L_{3ss})  \,,
\end{aligned}
\end{equation}
and the coefficients $L_i$ are listed in Ref.~\cite{Descotes-Genon:2019dbw}.
With these, we define the following differential and integrated observables:
\begin{equation}
\begin{aligned}
    \frac{d\mathcal{B}(q^2)}{dq^2}=&\,2\,\tau_{\Lambda_b}\left(a+\frac{c}{3}\right)\,,   &
    \mathcal{B}[q_\mathrm{min}^2,q_\mathrm{max}^2]=&\,2\tau_{\Lambda_b}\int_{q^2_\mathrm{min}}^{q^2_\mathrm{max}}dq^2\left(a+\frac{c}{3}\right)\,,  \\ \mathcal{A}^\ell_\mathrm{FB}(q^2) =&\,\frac{b}{2\left(a+\frac{c}{3}\right)}\,,
     &  \mathcal{A}^\ell_\mathrm{FB}[q_\mathrm{min}^2,q_\mathrm{max}^2] =&\,\frac{\int_{q^2_\mathrm{min}}^{q^2_\mathrm{max}}dq^2 b}{2\int_{q^2_\mathrm{min}}^{q^2_\mathrm{max}}dq^2\left(a+\frac{c}{3}\right)} \,, \\
   S_{1cc}(q^2)=&\,\frac{L_{1cc}+\bar{L}_{1cc}}{\frac{d\Gamma}{dq^2}+\frac{d\bar\Gamma}{dq^2}}\,,  &  S_{1cc}[q^2_\mathrm{min},q^2_\mathrm{max}]=&\,\frac{\int_{q^2_\mathrm{min}}^{q^2_\mathrm{max}}dq^2\left(L_{1cc}+\bar{L}_{1cc}\right)}{\int_{q^2_\mathrm{min}}^{q^2_\mathrm{max}} dq^2 \left(\frac{d\Gamma}{dq^2}+\frac{d\bar\Gamma}{dq^2}\right)}\,,
\end{aligned}
\label{eq:observables}
\end{equation}
where the bar denotes CP-conjugated quantities, and
\begin{equation}
    \frac{d\Gamma}{dq^2} = \frac{1}{\tau_{\Lambda_b}}\frac{d\mathcal{B}(q^2)}{dq^2}
\end{equation}
The observables above are chosen because they are the most promising ones to be measured at LHCb \cite{Amhis:2020phx}.\\

We use now the results in \sec{sec:fit} to predict the above observables. We stress that we use as nominal fit results for the form factors parameters the $N=2$ scenario.
We plot differential distributions in \fig{fig:phenores}.
As already mentioned, we veto the $c\bar{c}$ resonant region ($8\,\mathrm{GeV}^2<q^2<15\,\mathrm{GeV}^2$), but we show for completeness the naive extrapolation in the resonances region with dashed lines. \\
In the left upper panel of \fig{fig:phenores}, we show the differential branching ratio. We compare our results with the literature \cite{Descotes-Genon:2019dbw,Mott:2011cx,Das:2020cpv} and find some discrepancies.
We notice that the order of magnitude of our prediction of the differential branching ratio differs, in the entire $q^2$ range, from the one quoted in Ref.\cite{Descotes-Genon:2019dbw}.
We then compare with the most recent measurement of $\mathcal{B}(\Lambda_b\to p K \mu^+\mu^-)$ in Ref.\cite{LHCb:2019efc}, finding a good agreement between the order of magnitude of the latter and our predictions, even though the various $\Lambda^* \to p K$ resonances are not separated.
Furthermore, we also see a difference in the branching ratio as a function of $q^2$ at very large recoil. We trace back this difference to the different form factor parametrisation.
We verified explicitly that the Quark Model predictions in their current form violate SCET relations at low $q^2$ as well as the dispersive bounds, even of more than an order of magnitude for some currents.
Without a reasonable estimation of the uncertainties and correlations of the Quark Model parameters, it is impossible to extract more meaningful information.
Also, for some of the form factors, lattice QCD results and Quark Model ones are not in good agreement close to zero-recoil.\\
Finally, we observe that the lack of theory points for the form factors at large recoil results in large uncertainties.
We stress that due to this, the shape of the differential branching ratio in the central $q^2$ region could be qualitatively different when new lattice data or QCD Sum Rule-based computation will be available.
This model dependency is less evident in the differential forward-backwards asymmetry, right panel of \fig{fig:phenores}.
We note that, differently than in the case of $B\to K^*\ell^+\ell^-$ and $\Lambda_b\to\Lambda\ell^+\ell^-$, $\mathcal{A}^\ell_\mathrm{FB}$ for $\Lambda_b\to\Lambda^*\ell^+\ell^-$ shows two zero-crossings, one at low and the other at high-recoil. The one at large recoil has to be understood in the same way as for $B\to K^*\ell^+\ell^-$ and $\Lambda_b\to \Lambda\ell^+\ell^-$, namely due to a combination of $C_7$ and $C_9$ that vanishes for some $q^2$ value. The zero crossing at low recoil is instead of a different nature, since it is due to the zero-crossing in $f_\perp$. This can for example be more easily seen in the Heavy Quark Limit \cite{Descotes-Genon:2019dbw,Amhis:2020phx}, where, at leading order in $\alpha_s$ and leading power in $1/m_b$, $\mathcal{A}^\ell_\mathrm{FB}$ is proportional to $f_\perp$.
We stress that this behaviour is also present in predictions based on the Quark Model \cite{Descotes-Genon:2019dbw,Amhis:2020phx,Mott:2011cx}.
The zero-crossing in $f_\perp$ seems to be a peculiarity of several $1/2^+\to 3/2^-$ transitions \cite{Meinel:2021mdj,Mott:2011cx}, and therefore, any further theory data or experimental measurement of such modes could help to confirm or refute this property. For convenience, we also provide integrated observables in \Table{tab:predictions}.
\begin{figure}[h]
    \centering
    \includegraphics[width=.4\textwidth]{figures/LbToLll_BR.pdf}
    \includegraphics[width=.4\textwidth]{figures/LbToLll_AFB.pdf} \\
    \includegraphics[width=.4\textwidth]{figures/LbToLll_S1cc.pdf}
    \caption{Plots of the differential branching fraction (upper left plot), differential forward-backwards lepton asymmetry (upper right plot) and differential $S_{1cc}$ (lower plot). The green bands are the 68\% C.L. region, and the full line is the median of the distribution in the nominal fit. We use dashed lines in the vetoed resonant region.
    }
    \label{fig:phenores}
\end{figure}

With our setup we can also predict $\mathcal{B}(\Lambda_b\to\Lambda^*\gamma)$. We have
\begin{equation}
\begin{aligned}
    \mathcal{B}(\Lambda_b\to\Lambda^*\gamma)=\,&\tau_{\Lambda_b}\frac{G_F^2 |V_{tb}V^*_{ts}|^2 \alpha_\mathrm{em}m_b^2}{384\pi^4 m_{\Lambda_{b}}^3}(m_{\Lambda_{b}}^2-m_{\Lambda^*}^2)\\
    \times&\bigg[|C_7+C_7^\prime|^2(m_{\Lambda_{b}}+m_{\Lambda^*})^4(|h_\perp(0)|^2+3|h_{\perp^\prime}(0)|^2 ) \\
    +&|C_7-C_7^\prime|^2(m_{\Lambda_{b}}-m_{\Lambda^*})^4(|\tilde{h}_{\perp}(0)|^2+3|\tilde{h}_{\perp^\prime}(0)|^2)\bigg]\,.
    \end{aligned}
\end{equation}
If we restrict ourselves to the SM, we find:
\begin{equation}
    \mathcal{B}(\Lambda_b\to\Lambda^*\gamma) = (3.5^{+2.7}_{-1.6}) \times 10^{-5} \,.
    \label{eq:pred_LbtoLstgamma}
\end{equation}
Even though the large uncertainties forbid us to obtain a precise prediction, we notice that $\mathcal{B}(\Lambda_b\to\Lambda^*\gamma)$ has the same order of magnitude of LHCb measurement of $\mathcal{B}(\Lambda_b\to\Lambda\gamma)$ \cite{LHCb:2019wwi}. This can be tested by measurements of the $\Lambda_b\to pK\gamma$ resonant spectrum which can be performed at the LHCb experiment\cite{Albrecht:2020azd}.
\begin{table}[h]
    \centering
    \begin{tabular}{c c c }
    \toprule
        $q^2$ bin ($\mathrm{GeV}^2$) & $\mathcal{B}[q^2_\mathrm{min},q^2_\mathrm{max}]\times 10^8$ & $\mathcal{A}^\ell_\mathrm{FB}[q^2_\mathrm{min},q^2_\mathrm{max}] \times 10^{2} $ \\
        \midrule
        $[0.3,1.1]$ & $ 4.7^{+2.4}_{-2.0}$  &  $10.2^{+2.1}_{-2.0}$ \\[0.5ex]
        $[1.1,3]$   & $ 8.1^{+3.8}_{-2.9}$  &  $ 6.0^{+2.1}_{-3.8}$ \\[0.5ex]
        $[3,6]$     & $12.8^{+4.5}_{-3.5}$  &  $-7.9^{+2.6}_{-2.1}$ \\[0.5ex]
        $[1.1,6]$   & $20.8^{+4.5}_{-8.4}$  &  $-2.6^{+2.0}_{-2.8}$ \\[0.5ex]
        $[15,q^2_\mathrm{max}]$ & $0.60^{+0.05}_{-0.03}$ & $-17.8^{+1.5}_{-2.3}$ \\
    \bottomrule
    \end{tabular}
    \caption{Predictions of the integrated branching ratio and forward-backwards lepton asymmetry in various $q^2$ bins.}
    \label{tab:predictions}
\end{table}

\section{Conclusions}
\label{sec:conclusions}
In this work, we present a parametrisation of $\Lambda_b\to\Lambda(1520)$ local form factors which respects the analytical properties and the dispersive bounds associated with local $b\to s$ currents. With this setup, we are able to evaluate for the first time dispersive bounds for the form factor coefficients. We determine the form factor parameters through fits to theory data from Lattice QCD, with constraints from endpoint relations, SCET relations, and dispersive bounds. \\
We study the consequences of expanding the form factors up to different orders, finding that the minimal viable scenario is a second-order expansion, and additional orders does not add any information. Hence, we use this fit scenario as the nominal one.
Our results can be easily reproduced using \texttt{EOS} and the python notebook provided as an ancillary file.\\
We provide predictions for differential and integrated branching ratio and leptonic forward-backward asymmetry for $\Lambda_b\to\Lambda(1520)\ell^+\ell^-$ decays, where $\ell$ is a light lepton, and also an update for $\mathcal{B}(\Lambda_b\to\Lambda(1520)\gamma)$.
We stress that our approach suffers from large uncertainties at low $q^2$, due to the lack of calculations of the form factors in that region.
It is, hence, of the utmost importance to focus on that to improve predictions of $\Lambda_b\to\Lambda(1520)$ form factors.

\section*{Acknowledgements}
We thank Claudia Cornella, Carla Marin Benito, Stefan Meinel, Mart\'{i}n Novoa-Brunet, Gumaro Rendon, Danny van Dyk and Felicia Volle for useful discussions.

The work of MR is supported by the DFG within the Emmy Noether Programme under grant DY-130/1-1
and the Sino-German Collaborative Research Center TRR110 ``Symmetries and the Emergence of Structure in QCD''
(DFG Project-ID 196253076, NSFC Grant No. 12070131001, TRR 110). MR is grateful to the CERN Theory Division for the hospitality during parts of this work.

redpajama_set_name: RedPajamaArXiv


text:
Vietnam requests aid from Japan for mega railway, refinery projects
Vietnam has asked Japan for support in building a high-speed railway that runs along the country's length, the government said late on Friday.
The proposal was made at a meeting between Vietnamese Prime Minister Pham Minh Chinh and Japanese Finance Minister Shunichi Suzuki in Hanoi earlier in the day, the government said in a statement.
Japan is the largest source of official development aid to Vietnam, and is the Southeast Asian country's third largest source of foreign direct investment.
Vietnam is studying the possibility to build the 1,545 km (960 miles) railway with a possible price tag of up to $64.8 billion, according to state media.
At the meeting, Chinh also asked Japan for help to "restructure stakes" at Nghi Son Refinery and Petrochemical, Vietnam's largest oil refinery, the government said, without elaborating.
The $9-billion refinery is 35.1% owned by Japan's Idemitsu Kosan Co, 35.1% by Kuwait Petroleum, 25.1% by Vietnam's state oil firm PetroVietnam and 4.7% by Mitsui Chemicals Inc .
The 200,000-barrel-per-day refinery early last year cut its run rate following a disagreement between shareholders about financing for crude. Idemitsu Kosan said at the time it had no plan to give fresh financial aid to the refinery.
Source: https://tuoitrenews.vn/news/society/20230115/vietnam-requests-aid-from-japan-for-mega-railway-refinery-projects/70942.html
Babysitter in Ho Chi Minh City arrested for allegedly abusing 6-month-old boy
Mischievous monkey in Ho Chi Minh City anesthetized, sent to rescue station
Business3 mins ago
Ho Chi Minh City Expects to Attract $7.4 Billion in FDI in 2023
Sci-tech-environment1 hour ago
Celebrating the Tết traditions of Việt Nam's three regions in Hà Nội

redpajama_set_name: RedPajamaCommonCrawl


Curious to know if timing wise a turkey hunt in your fair state about the 10th of May is good? Too early? Too late?
I'd like to plan a "last minute" trip for this year or get all my "turkeys" in a row for next year.
We have a very limited population up here and it take years to get a tag.
Depends on where in MT you are going but everything is late this year. Still snowing here in NW MT and turkey hunting has been very slow. Just a guess but late might be the best this year.
Chad, thanks for the reply. I'll be hunting outside of Heron. I thought with the late spring the timing might be good so I'm going, I leave on the 9th.


text:
naar nederlandse site
home | top 100 | per country | per type | background | search
Al-Qala'a al-Gabal (Citadel of the Mountain, Al Muqattam)
Top 100: position 611	 Sources top 100: position 720
Total score: 194 points	 Rating visitors: 4.5 of 4 visitors.
Total score now: 483 points	 Rating visitors now: 7.6 of 12 visitors.
Country: Egypt Founder: Al-Malik al-Nasir Salah al-Din al-Ayyubi (also known as Saladin, r. AD 1169–93).
Building structure: Donjon in curtain wall
Place: Cairo Castle type: City castle
Era: 12-th century Condition: Intact
Description: The Citadel of Salah al-Din al-Ayyubi, known also as Citadel of the Mountain (Qala'a al-Jabal), is one of the most important landmarks of Cairo and amongst the most imposing military citadels to be built in the Middle Ages. Having assumed power in Egypt S???
Remarkable: THe largest tower in the CItadel is Burg al-Muqattam, built in 16th century. It is 24 m in diameter and designed to withstand artillery attacks.
482. Hrad Landštejn	( Czechia)
The content of this website and the pictures taken by Robert Rongen are licensed under a Creative Commons Attribution-Noncommercial 3.0 Unported License.
© Other pictures: Copyright reserved by photographer marked on the picture.


text:
Pedri reflects on his tenure as Luzerne County Director
By Randy D. Rooker	 Last updated Jun 24, 2021
While still officially in office for two more weeks, Luzerne County Director C. David Pedri publicly said goodbye at his last County Council meeting on Tuesday.
"What a privilege it has been to serve in this capacity," said Pedri, who has accepted a supervisory position from the Luzerne Foundation. "Having the opportunity to work in my home country, where I choose to raise my children, has been the highlight of my career. "
When he took up the post of second permanent county director under national self-government "five years ago in the short and long term," Pedri said he indicated that the county was moving into the second. phase of the personalized government structure that came into effect in 2012.
The first was former manager Robert Lawton and the initial 11-member board "bringing us through the crisis" of implementing the rule changes at home, he said.
During his tenure, Pedri said he worked with the board to fix "our financial house", end deficits and achieve the highest credit score in county history – an A- from Standard. & Poor's.
Job creation is "through the roof," as employers "die for workers," he said.
Real estate is being ripped off in response, Pedri said. As proof, he later said five people separately approached his family to make offers to buy them as they recently cleaned the residence of a deceased relative in Hazleton, without any sales signs.
The county has also addressed many neglected maintenance needs, replacing boilers and roofs, restoring the historic courthouse and repairing some roads and bridges, he said.
"We replaced the courthouse's electrical equipment from the 1950s Frankenstein Panel with a modern panel," Pedri told council. "Anyone who owns a home knows that an ounce of prevention avoids a pound of trouble, and we've been doing these things now so that act three can now come into play."
Pedri said he is now handing the ball over to the council and the next county manager. Chief lawyer Romilda Crocamo was appointed interim director on Tuesday, and the board plans to nominate volunteer citizen candidates to a new director search committee next week.
He also congratulated the division heads and the rest of the workforce. The council members he has worked with, critical or not, "are trying to do the best thing for the county," he said.
"We didn't agree on everything, but we agreed on a lot, and we've accomplished so much here," Pedri told the board.
More thinking
In a follow-up interview on Wednesday, Pedri said he did not view his relationship with the board as controversial and said most of his proposals were approved.
"We have accomplished a lot in the past five years, and we couldn't do it without cooperation," said the 41-year-old resident of Butler Township. "For the most part, they supported me. Of course, I won't get everything I want. What executive does he do? "
He began considering a career change after the board rejected increases for non-union workers in the 2021 budget, saying he believed increases were deserved for workers who "committed suicide" by supporting operations throughout the coronavirus pandemic. That said, he stressed that he understands that council members rightly have their own views and priorities on what is best for the county.
The hardest part of the job, he said, is keeping tabs on 1,400 employees in more than 30 departments who provide services to more than 320,000 residents in a county the size of Rhode Island.
"It's a small company," Pedri said.
Initially, he loaded as a manager, with the intention of quickly making changes "everywhere". Over time, he said he learned to surround himself with "a few great managers" and rely more on their feedback and specialist expertise to make incremental improvements that often lead to bigger ones. He recommends this approach to his successor, but cautioned that it is still crucial to maintain a "full understanding" of finances and operations and to be prepared to explain them to the public.
One regret is not being there to help decide how the county will use its $ 112.89 million US bailout allocation, he said.
This funding will be an opportunity for the next manager and board to "have a huge impact on our community," said Pedri. More detailed federal rules on how the money can be spent are expected in August, and the administration plans to present a proposal to council for consideration and approval. Some board members have also started to come up with their own ideas on how the money should be spent.
Pedri also wants the county to have fewer electoral problems, although he believes the county is on the path to continuous improvement in this area.
Looking back, Pedri said he will always remember the awe he saw on faces when the restored courthouse rotunda was unveiled, with thousands of residents enjoying the 'Rockin' the concert series. River "he co-founded at River Common and positive reactions from citizens. on the direction of the county.
"I think my job was to leave the county better than I found it and to start planning for the future of success," said Pedri. "I just feel like Luzerne County is about to be in a position where it's really going to be successful."
Andrew Wiggins can't play Warriors home games if he's not vaccinated, says SF
To 'maintain self-reliance' and increase Greenwich's affordable housing stock, the RTM OK Housing Trust Fund
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It is my favorite part of one of my favorite rides at Disneyland--the moment in Peter Pan's Flight when the flying galleon leaps over the rooftops and out onto London Town. I remember being absolutely entranced by this scene as a child. It didn't hurt that I had always wanted the ability to fly when I was younger (who doesn't?), and being able to soar above the city was sweetly wonderful. That was the magic of Disneyland, where I really could fly, with the help of some magic pixie dust. And so, even now, as a fully grown adult, whenever I ride Peter Pan's Flight, my heartstrings are wholly tugged when I come over this part of the classic and beloved attraction.


text:
The Uncommon Life of Alfred Warner in Six Days by Juliet Conlin
Published by Black and White Publishing on February 23, 2017
Sometimes you can be drawn to a book without entirely knowing why. The Uncommon Life of Alfred Warner in Six Days was one of those books for me. Although the blurb sounds completely fascinating, this is not the kind of book I normally read and the length of the book is a bit longer than someone like me with a million books to read would then choose to read. But like I say, there was really just something about this book that drew me in and I knew I had to read this book. I knew Alfred Warner was a character I had to meet, and honestly I am so glad I picked up this book because Alfred is one of the most exceptional characters I have ever read about.
Alfred is nearly eighty years of age, and the voices in his head are telling him he only has six days left to live. Which is why, when his granddaughter doesn't show up on the day he is supposed to be meeting her for the first time, he knows he has to find another way of telling her the story of his life, which contains a secret that will impact on her life forever. It took me about half a page maximum before I knew Alfred was a character who was going to keep me on my toes throughout eager to learn more about him and the life he has lived. There are interesting characters and then there is Alfred, a person who has lived one heck of a life which had me absolutely captivated, practically glued to the pages, as we unfold the layers of his character and the amazing stories he has to tell.
This is a book that captured my heart. It had me feeling every emotion under the sun. I laughed and I shed tears and I found parts to be heart-breaking and other parts to be thoroughly heart-warming. I smiled at the happy moments of his life and sympathised with him at others. Alfred has lived such a life that, despite me not normally having an attention span that will keep me reading a book longer then about 400 pages, made me want a book about double that size because I believed he had lived a life with so many more stories to tell.
Juliet Conlin has really written such a wonderful character in Alfred, but there were two other characters in this book that I enjoyed getting to know as well. The first one was Julia, a woman who lets Alfred into her home and befriends him through her concern. She isn't the most pivotal character in this book but I really warmed to her straight away as her caring nature was really appealing. She listened to Alfred and cared for his stories, and I found her a really genuine, likeable character. Then there is Brynja, Alfred's granddaughter, who I found to be really engaging too. I was fascinated by her character from the moment she is first supposed to meet Alfred but doesn't. Though this is Alfred's story to tell, the other characters helped to make this book what it is, an incredibly touching and unmissable treat of a novel.
The structure of this book is an unusual one which I thought I would find difficult to get into and yet I found it the complete opposite. The format of the six days of the story sees the book switch in perspective between Alfred, Julia and Brynja, whilst the chapters switch between the various and utterly absorbing experiences that Alfred has lived. Ranging from tales of the Holocaust to tales of his marriage and family life, the storytelling in this book covers multiple locations and multiple decades and there was not one single part of Alfred Warner's life that did not move me. The Uncommon Life of Alfred Warner in Six Days is absolutely fantastic, and truly a book you simply must read.
Gone Without A Trace by Mary Torjussen
The Housekeeper by Suellen Dainty
Where The Wild Cherries Grow by Laura Madeleine
The Jungle by Pooja Puri
Puzzle Girl by Rachael Featherstone
Quieter Than Killing by Sarah Hilary
The Song of the Stork by Stephan Collishaw
An Unsuitable Marriage by Colette Dartford
The Little Theatre by the Sea by Rosanna Ley
Cursed by Thomas Enger
All Our Wrong Todays by Elan Mastai
Lie in Wait by G.J. Minett
All Is Not Forgotten by Wendy Walker
The Silent Fountain by Victoria Fox
The Uncommon Life of Alfred Warner in Six Days by ...
The Gingerbread House by Kate Beaufoy
Stasi Wolf by David Young


The JV structure is the same approach these three Newspaper Co's already use to share ownership and operate CareerBuilder, ShopLocal, and Classified Ventures, which includes Cars.com, Apartments.com and Homescape.
Topix is already part of the local search ecosystem of folks like AOL, CitySearch, Ask Jeeves, The NY Times, My Yahoo… all folks I'm sure the News Co's would love to build/extend relationships with.
P.S. A shout-out to Elle for being one of the first to build something useful with the WordPress Structured Blogging plug-in!
CHICAGO, McLEAN, VA. and SAN JOSE, CALIF, March 23, 2005 .Topix.net announced today that Gannett Co., Inc. (NYSE: GCI), Knight-Ridder, Inc. (NYSE: KRI) and Tribune Company (NYSE: TRB) jointly have acquired a 75 percent stake in Topix.net. Ownership is split evenly, with each media company owning 25 percent and the Topix.net founders retaining a 25 percent stake. Financial terms of the transaction were not disclosed.
Topix.net continuously monitors breaking news from over 10,000 online sources and categorizes daily news content into over 300,000 topics, 24 hours a day. Topix.net will operate as an editorially independent, stand-alone company managed by Topix.net.s executive team. Topix.net will use content and funding from Gannett, Knight Ridder and Tribune to expand and refine its NewsRank. technology, services and operational infrastructure.
"This transaction provides us with the resources to be the best place to connect to a very wide range of news on the Web," said Rich Skrenta, CEO and co-founder of Topix.net. .We all serve the news enthusiast, so it.s a natural partnership. Gannett, Knight Ridder and Tribune generate valuable online content and Topix.net is dedicated to making that content easily found. By believing in the potential of our ideas and technology, they have made an investment in the future, and we.re thrilled to be a part of it..
Topix.net features links to news and information from more than 10,000 news sources including newspapers, magazines, radio, television stations and Web logs. Topix.net distributes news through its Web site, RSS feeds and email news alerts. It syndicates its content through partnerships and tens of thousands of Web sites that host news feeds from Topix.net.
Topix.net uses its NewsRank. technology to deliver highly relevant news, using concept, instead of simple keyword searches. It delivers published news stories organized by ZIP code, industry and hundreds of thousands of topics.
.People want the news that.s relevant to them . where they live, the business that they are in, the topics they care about. Topix.net.s technology will leverage the media assets and broad reach of our three companies in entirely new ways,. said Jack Williams,senior vice president diversified business and development at Gannett. .We are extremely pleased to partner with Topix.net to grow and develop its service and help audiences find the news they want online..
Collectively, Gannett, Knight Ridder and Tribune operate more than 140 newspaper Web sites with nearly 30 million unique visitors monthly. The companies have partnered together in other joint ventures such as ShopLocal.com and CareerBuilder.com.
Founded in 2002 with the specific mission of providing users with the ability to quickly and easily find targeted news on the Internet, Topix.net is a news aggregator with unique technology to find and categorize news into 300,000 topics, from Autos to ZIP code level local news. Topix.net distributes content via partnerships with Citysearch, Ask Jeeves, My Yahoo!, Bloglines, Metro Newspapers, Newsgator, Findory, Wondir, Infospace, Link Silicon Valley and HelloMetro.com.
Gannett Co., Inc. is a leading international news and information company that publishes 101 daily newspapers in the USA, including USA TODAY, the nation.s largest-selling daily newspaper. The company also owns in excess of 750 non-daily publications in the USA and USA WEEKEND, a weekly newspaper magazine. Gannett subsidiary Newsquest is the United Kingdom.s second largest regional newspaper company. Newsquest publishes more than 300 titles, including 17 daily newspapers, and a network of prize-winning Web sites. Gannett also operates 21 television stations in the United States and is an Internet leader with sites sponsored by its TV stations and newspapers including USATODAY.com, one of the most popular news sites on the Web.
Knight Ridder (NYSE: KRI) is the nation.s second-largest newspaper publisher, with products in print and online. The company publishes 31 daily newspapers in 28 U.S. markets, with a readership of 9.0 million daily and 12.7 million Sunday. Knight Ridder also has investments in a variety of Internet and technology companies and two newsprint companies. The company.s Internet operation, Knight Ridder Digital, develops and manages the company.s online properties. It is the founder and operator of Real Cities (http://www.blogger.com/www.RealCities.com), the largest national network of city and regional Web sites in more than 105 U.S. markets. Knight Ridder and Knight Ridder Digital are headquartered in San Jose, Calif.
Tribune (NYSE: TRB) is one of the country.s premier media companies, operating businesses in broadcasting and publishing. It reaches more than 80 percent of U.S. households and is the only media organization with television stations, newspapers and websites in the nation.s top three markets. In publishing, Tribune operates 14 leading daily newspapers including the Los Angeles Times, Chicago Tribune, Newsday and Spanish-language Hoy, plus a wide range of targeted publications. The company.s broadcasting group operates 26 television stations; Superstation WGN on national cable; WGN-AM in Chicago; and the Chicago Cubs baseball team. Popular news and information websites complement Tribune.s print and broadcast properties and extend the company.s nationwide audience.
‹ RUMOR: Knight Ridder buys Topix.net!


Since 2005 the unemployment rate in Hoffman-Estates, Illinois has ranged from 2.3% in December 1999 to 9.6% in January 2010. The current unemployment rate for Hoffman-Estates is 3.0% in February 2019. See how other local areas compare by using our Unemployment Compare tool.