text:
\section{Introduction}\label{sec1}

It has been observed \cite{N1} that binary charge-exchange reactions of
hadrons at high energies go through the charge-exchange scattering of
fast quarks located in the beginning of quantum fluctuations, the
splitting and recombination of partons inside hadrons. This fact has
far-reaching effects. First, the mentioned fast quark scattering is
accompanied by high virtualities \cite{N1}. Therefore the appropriate
subprocesses are hard and may be described in the parton model. Second,
the mentioned scattering interrupts the fluctuations, which results in
the formation of a cloud of uncorrelated partons. In the exclusive
reaction they are to be captured by the flying away clusters, and this
induces destruction of the coherence. So one can expect destruction of
the coherence of intermediate contributions. Due to increasing duration
of the interaction with increasing energy of the collisions
\cite{Gribov,Low,Petrov1,Petrov2}, the latter effect should be
increasing with increasing the energy.

Based on the above considerations \cite{N1} proposed a model for the
description of $\pi^- p \to M^0 n$ and $K^- p \to M^0 \Lambda$, $M^0 =
\eta, \eta',\pi^0$. The model also used the idea that contributions of
soft interactions that follow the hard scattering, obey the Regge
behavior \cite{Regge,Irving}. However, the mode of summations of
elementary contributions may be coherent or non-coherent. In the former
case the model reproduced conventional Regge approach. In the case of
non-coherent summation the model gave non-trivial predictions. Since the
mode of summation should change with increasing the energy, one of
non-trivial predictions was the emergence of the energy dependence in
the vertex functions, prohibited in the Regge approach. Such a
dependence is really observed at comparing at different energies the
dependence on the transfer of the ratio of yields of $\eta'$ and $\eta$
in the $\pi^-$ beams \cite{N1,Stanton}. However, the most striking
effect was  disappearance of a dip near $|t| =$ 0.4 (GeV$\!$/c)$^2$ with
increasing the energy in the differential cross-section $K^- p \to
\eta\Lambda$. In reality, the dip is observed at the $K^-$ momentum 3--8
GeV$\!$/c in the laboratory frame \cite{Moscoso,Mason,Marzano,Harran}
and disappears at the momentum 32.5 GeV$\!$/c \cite{GAMS}. In the Regge
approach the dip is explained as a consequence of a dominance of the
vector-exchange trajectory and simultaneously its zeroing by the
signature factor in the region $\alpha_{V}(t) \approx 0$, where
$\alpha_{V}(t)$ is the vector trajectory \cite{Martin}. However, this
mechanism is independent of the energy, and it is not clear why it
ceases to operate with the increasing energy. Ref.~\cite{N1} explained
the effect by the mode change of summation of intermediate
contributions. But the data fit was not carried out in \cite{N1} and
therefore the explanation was a qualitative.

In this paper we carry out a full fit of available data at the
relatively low and high energies, and on this basis we confirm the above
explanation. In addition, we solve the problem of the description at
intermediate energies where contributions of both modes are possible.
In effect, we propose an algorithm for measuring a relative value of the
coherent and non-coherent contributions. As a by-product we get an
independent estimate of the $\eta$--$\eta'$ mixing. A possible admixture
of the glueball is considered, as well.

We carry out the analysis in a modified version of the model \cite{N1}.
The modification concerns mainly the version of the Regge phenomenology.
First of all, we defreeze the parameters responsible for the quark
symmetry breaking. This modification is particularly important for
determining the $\eta$--$\eta'$ mixing. Further, following \cite{Irving}
we change a parameterization of the vertex functions in the spirit of
Veneziano model. At last, we consider the trajectories non-degenerate
and we take into account the effect of interference between their
contributions.

The paper is organized as follows. In the next section we formulate
modifications in the model. The data fit is carried out in
sect.~\ref{sec3}. A generalization of the model to a simultaneous
consideration of the coherent and non-coherent contributions is
discussed in sect.~\ref{sec4}. In sect.~\ref{sec5} we discuss the
results and make conclusions. In appendix A we transform data
\cite{GAMS} from the form of numbers of pairs of gamma-quanta to the
form of differential cross-sections.

\section{The model}\label{sec2}

As noted in the introduction, the high-energy charge-exchange reactions
occur via the inelastic scattering of fast quarks. The appropriate
subprocesses are hard and can be described in the QCD pertubation
theory. The calculations in the cases of $\pi^- p \to (\eta,\eta',\pi^0)
\, n$ and $K^- p \to (\eta,\eta',\pi^0)\Lambda$ \cite{N1} show that at
low transfers the relevant contributions to the amplitude are equal in
absolute value and independent of the flavors and energies of the
colliding quarks. On the background of hard subprocesses the
contributions of soft processes are further formed. We assume that they
are independent of the flavors in the hard subprocesses. In this case
the contributions of hard subprocesses actually are factorized, and
their non-trivial result is a certain mode of summation of soft
contributions. For a quantitative description of the soft contributions
we apply the Regge phenomenology. In doing so, we consider contributions
associated with the valence quarks in the final state as elementary
ones. They are formed in different ways and they are summed differently
in the different modes. As a result, in the mode of coherent summation
with a certain choice of the signs the formulas of the conventional
Regge approach are reproduced. In the non-coherent mode the elementary
contributions are summed in the cross-section. In general, this leads to
non-trivial consequences.

Basically, the above discussion defines the model up to the definition
of Regge amplitudes. Turning to the latter issue, we recall that binary
processes in the leading approximation at large $s$ and small $t$ are
described as a sum of contributions of the leading trajectories
\cite{Regge,Irving},
\begin{equation}\label{F1}
A_{ab}(s,t) \; = \; \sum_{i} \beta_{aib}^{\pm}(t) \;
\frac{1 \pm e^{-i\pi\alpha_{i}^{\pm}(t)}}
     {\sin \left(\pi\alpha_{i}^{\pm}(t)\right)} \;
\left( s/s_0 \right)^{\alpha_{i}^{\pm}(t)} .
\end{equation}
Here $a$ and $b$ mean initial and final states, $\alpha_{i}^{\pm}(t)$
are the trajectories of particular parity, $\beta_{aib}^{\pm}(t)$ are
the vertex functions, $s_0$ is a scale parameter. The numerator in
(\ref{F1}) represents the signature factor. The zeros in the denominator
give Regge poles in the region of bound states ($t>0$). In the
scattering region ($t<0$) the poles must be compensated by zeros in
$\beta_{aib}^{\pm}(t)$. The latter property is explicitly realized in
the parameterization in the spirit of the Veneziano model, which
includes the gamma function instead of the sine in the denominator
\cite{Irving}:
\begin{equation}\label{F2}
A_{ab}(s,t) \; = \; \sum_{i} \beta_{aib}^{\pm}(t) \;
\Gamma\left(1-\alpha_{i}^{\pm}(t)\right)
\left[1 \pm e^{-i\pi\alpha_{i}^{\pm}(t)}\right]
\left( s/s_0 \right)^{\alpha_{i}^{\pm}(t)} .
\end{equation}
Here vertex functions $\beta_{aib}^{\pm}(t)$ differ from those in
(\ref{F1}) and have smoother behavior. Phenomenologically they are well
described by an exponential function, possibly with a modification by
polynomial factors describing spin-flip contributions.

In fact, the presence of exponential factors in the vertex functions is
set by formula (\ref{F2}) itself. Really, the r.h.s in (2) does not
change with simultaneous  substitutions $s_0 \to \tilde{s}_0$,
$\beta_{aib}(t) \to \tilde{\beta}_{aib}(t)$, where
\begin{equation}\label{F3}
\tilde{\beta}_{aib}(t) = \beta_{aib}(t)
\left( \tilde{s}_0/s_0 \right)^{\alpha_{i}(t)}.
\end{equation}
Hereinafter we omit the sign of the signature ($\pm$) assuming it is
included in the index $i$. In linear approximation for the trajectory,
$\alpha_{i}(t) = \alpha_{i}(0) + \alpha'_{i}(0)\,t$, we have
\begin{equation}\label{F4}
\tilde{\beta}_{aib}(t) =
\left( \tilde{s}_0/s_0 \right)^{\alpha_{i}(0)} \beta_{aib}(t) \times
e^{\alpha'_{i}(0)\,\ln(\tilde{s}_0/s_0) \, t } \,.
\end{equation}
So, if at some $s_0$ the vertex functions do not include an exponential
factor, then this factor does appear at the transition to another scale
parameter. For example, in the Veneziano model the vertex
factor initially appears as a constant at the scale parameter $s_0 =
1/\alpha'$, where $\alpha'$ is a slope of the trajectory. However,
without changing the amplitude, we can choose another scale parameter
with simultaneous change of the vertex function in accordance with
(\ref{F4}). In particular, at the transition to the scale $\tilde{s}_0$
the vertex factor in the Veneziano model is converted to an exponential
function with the slope $\alpha' \ln(\tilde{s}_0 \alpha')$.

Below we assume that the vertex functions have purely exponential
behavior (up to the spin-flip factors) and we consider their
normalizations and slopes as free parameters. Simultaneously we put $s_0
= s$ in formula (\ref{F2}). This condition unifies the normalizations
and collects exponential $t$-dependence completely in the vertex
functions. At the same time, at the transition to different energies the
results can be recalculated by means of (\ref{F4}).

Further, we assume that the slopes in the vertex functions are
determined by the trajectories, not by real particles in the final
state. So at small $t$, we put
\begin{equation}\label{F5}
\beta_{aib}(t) = \beta_{aib}(0) \exp( c_i t )\,.
\end{equation}
The dependence on real particles is included in the norma\-lization
$\beta_{aib}(0)$, which is manifested in the overall strength of the
coupling, in the group factors and the mixing parameters. Violation of
the flavor symmetry is described by additional exponent-like factors,
\begin{equation}\label{F6}
\xi_{i} = \xi_{0i} \; \exp(\xi'_{i} t)\,.
\end{equation}
Here $\xi_{0i}$ and $\xi'_{i}$ are determined by the trajectory and the
valence quarks in the final state. (Indices for the valence quarks are
omitted to avoid bulkiness.) Actually (\ref{F6}) defines the splitting
of the contributions of trajectories due to violation of the flavor
symmetry.\footnote{Here we proceed from the provision, that the
splitting accumulates during formation of the trajectories and is weakly
related to the formation of real particles in the final state. In
support we note that the characteristic time of the former process is
much greater than that of the latter one. Really, the formation time of
a relativistic particle is of order ${\cal E}/\mu^2$, where ${\cal E}$
and $\mu$ are its energy and mass. In the case of trajectories the
characteristic time is determined by the formation time of fast partons
(at the recombination of the fluctuations). Since the masses of the
partons are much smaller than the hadron masses and their momenta are
comparable, the above ratio occurs.}

Let us consider for definiteness the charge-exchange processes in the
$\pi^-$ beams with yields of $\eta$ and $\eta'$. In the leading
approximation they are determined by the $a_2$-trajectory \cite{Regge}
and the corresponding formulas are independent of the mode of summation
of intermediate contributions \cite{N1}. Taking into account above
comments, we proceed directly to the differential cross-sections,
\begin{eqnarray}\label{F7}
 \frac{\mbox{d}\sigma}{\mbox{d} t} (\pi^{-} p \to \eta  n)\, &=&
g_{a_2}^2 (t) \,  (1 - r t) \,
 \cos^2 \!\frac{\pi \alpha_{a_2}}{2}
 \left(\cos\theta - \xi_{a_2}\sqrt{2}\sin\theta\right)^2 \,,
\\ [0.2\baselineskip] \label{F8}
 \frac{\mbox{d}\sigma}{\mbox{d} t} (\pi^{-} p \to \eta' n) &=&
g_{a_2}^2 (t) \,  (1 - r' t)
 \cos^2 \!\frac{\pi \alpha_{a_2}}{2}
 \left(\sin\theta + \xi_{a_2}\sqrt{2}\cos\theta\right)^2 \,.
\end{eqnarray}
Here in both formulas the first factor is a flavor-indepen\-dent
contribution to the vertex function. In the case of arbitrary trajectory
``$i\,$'', in accordance with (\ref{F2}) and (\ref{F5}), it is defined
as
\begin{equation}\label{F9}
g_{i} (t)  =
g_{0 i} \; \Gamma\left(1-\alpha_{i}(t)\right) \; \exp(c_{i} t)\,,
\end{equation}
where $g_{0 i}$ and $c_{i}$ are phenomenological parameters. (Remember,
in (\ref{F7}) and (\ref{F8}) $i=a_2$). The second factor in (\ref{F7}),
(\ref{F8}) is the spin-flip contribution. The third is the signature
factor. The last factors stand for flavor-dependent contributions to the
vertex functions. They are determined by the angle of $\eta$--$\eta'$
mixing, by the group factors, and by the nonet-symmetry-breaking factor
$\xi_{a_2}$ introduced in (\ref{F6}). Here, we consider the mixing in
the simplest scheme \cite{PDG},
\begin{eqnarray}\label{F10}
|\eta \rangle\, &=&
 \cos\theta \; |\eta^8 \rangle \, - \,\sin\theta \; |\eta^0 \rangle ,
 \nonumber\\ [0.4\baselineskip]
 |\eta' \rangle &=& \,
 \sin\theta \; |\eta^8 \rangle \, + \,\cos\theta \; |\eta^0 \rangle .
\qquad
\end{eqnarray}
It is worth mentioning that in the case of exact nonet symmetry
($\xi_{a_2} \! = \! 1$) the sine and cosine in (\ref{F7}), (\ref{F8})
define the non-strange component in the wave functions of $\eta$,
$\eta'$.

By this means formulas (\ref{F7}), (\ref{F8}) describe the reactions
with the aid of seven parameters. Six of them, $g_{0 a_2}$, $c_{a_2}$,
$r'$, $r$, $\xi_{0a_2}$, $\xi'_{a_2}$, are specific. Parameter $\theta$
is universal.

An important characteristic in the $\pi^-$ beams is the ratio of the
differential cross-sections. After simple transformations, we get
\begin{equation}\label{F11}
 R_{\pi}^{\,\eta'/\eta}(t)    \equiv
\frac{{\rm d} \sigma / {\rm d} t (\pi^- p \to \eta' n)}
 {{\rm d} \sigma / {\rm d} t (\pi^- p \to \eta  n)}  =
 \frac{1 \! - \! r' t}{1 \! - \! r t} \tan^2 (\theta + \theta_{id} -
\delta)\,.
\end{equation}
Here $\theta_{id} = \arctan\!\sqrt{2}$ \ ($\theta_{id} \approx
54.7^0$) and
\begin{equation}\label{F12}
\delta = \arctan \frac{\sqrt{2}(1-\xi_{a_2})}{1+2\xi_{a_2}}\,.
\end{equation}
It is seen from (\ref{F11}) that at zero transfer $R_{\pi}^{\eta'/\eta}$
is determined by the difference $\theta - \delta_0$, where $\delta_0 =
\delta(0)$. Traditionally $R_{\pi}^{\eta'/\eta}(0)$ is used for
determining the mixing angle $\theta$. However, we see that on this
basis only the difference $\theta - \delta_0$ can be determined, not
$\theta$ alone.

In the case of charge-exchange reactions in the $K^-$ beam with yields
of $\eta$ and $\eta'$, there are two leading trajectories, $K^{*}$ and
$K^{*}_2$ \cite{Regge}. They have different signatures and different
symmetry properties, which complicates the description. Moreover, the
trajectories can be non-degenerate. In this case an interference therm
should appear.

In the coherent mode the model leads to the following formulas for
differential cross-sections (the same formulas arise in the Regge
approach):
\begin{eqnarray}\label{F13}
& \displaystyle \frac{\mbox{d}\sigma}{\mbox{d} t} (K^{-} p \to \eta \Lambda) \;=\;
\qquad &
\nonumber
\\ [0.2\baselineskip]
& \displaystyle 3 \, g_{_{V}}^2 (t) \, \sin^2 \!\frac{\pi
\alpha_{_{V}}}{2}
\cos^2 \theta \,+\,
\frac{1}{3} \, g_{_{T}}^2 (t) \, \cos^2 \frac{\pi \alpha_{_{T}}}{2}
\left(\cos\theta + 2\sqrt{2}\,\xi\,\sin\theta \right)^2 \qquad &
\nonumber
\\ [0.5\baselineskip]
& \displaystyle  - \; 2 \, g_{_{V}} (t) \, g_{_{T}} (t) \,
\cos \! \frac{\pi \alpha_{_{T}}}{2} \,
\sin \! \frac{\pi \alpha_{_{V}}}{2}   \,
\sin \! \frac{\pi (\alpha_{_{V}}\!-\!\alpha_{_{T}})}{2} \;
\cos\theta \left(\cos\theta + 2\sqrt{2}\,\xi\,\sin\theta \right)\,,
\qquad  &
\end{eqnarray}
\begin{eqnarray}\label{F14}
& \displaystyle \frac{\mbox{d}\sigma}{\mbox{d} t} (K^{-} p \to \eta' \Lambda) \;=\;
\qquad &
\nonumber
\\ [0.2\baselineskip]
& \displaystyle 3 \, g_{_{V}}^2 (t) \, \sin^2 \!\frac{\pi
\alpha_{_{V}}}{2}
\sin^2 \theta \,+\,
\frac{1}{3} \, g_{_{T}}^2 (t) \, \cos^2 \frac{\pi \alpha_{_{T}}}{2}
\left(\sin\theta - 2\sqrt{2}\,\xi\,\cos\theta \right)^2 \qquad &
\nonumber
\\ [0.5\baselineskip]
& \displaystyle  - \; 2 \, g_{_{V}} (t) \, g_{_{T}} (t) \,
\cos \! \frac{\pi \alpha_{_{T}}}{2} \,
\sin \! \frac{\pi \alpha_{_{V}}}{2}   \,
\sin \! \frac{\pi (\alpha_{_{V}}\!-\!\alpha_{_{T}})}{2} \;
\sin\theta \left(\sin\theta - 2\sqrt{2}\,\xi\,\cos\theta \right)\,.
\qquad &
\end{eqnarray}
Hereinafter we introduce indices {\small $V$} and {\small $T$} instead
of $K^{*}$ and $K^{*}_2$, respectively, and we omit $T$ in $\xi_{_{T}}$.
The spin-flip factors are not included since the data do not need this
\cite{Moscoso,Mason,Marzano,Harran}. So, above formulas include seven
parameters: $\theta$, $g_{_{V}}$, $c_{_{V}}$, $g_{_{T}}$, $c_{_{T}}$,
$\xi_{0}$, $\xi'$, where $\xi_{0}$ and $\xi'$ are involved in $\xi$,
{\it cf.}~(\ref{F6}). Notice that the vertex functions with purely
singlet final states, in conformity with antisymmetric properties, are
zero in the case of vector trajectories. Accordingly, $\xi$ does not
appear in the vector channel.

It is helpful noting that expressions in the large round brackets in
(\ref{F13}), (\ref{F14}) are reduced to cosine and sine of $\theta +
\widetilde\theta_{id} - \widetilde\delta$, where $\widetilde\theta_{id}
= -\arctan(2\sqrt{2})$ and
\begin{equation}\label{F15}
\widetilde\delta = -\arctan \frac{2\sqrt{2}(1-\xi)}{1+8\xi}\,.
\end{equation}
With the aid of this property it is easy to understand the reason of the
appearance of a dip in $K^{-} p \to \eta \Lambda$. The point is that
$(\cos\theta \! + \! 2\sqrt{2}\,\xi\,\sin\theta)$ in the tensor
contributions in (\ref{F13}) is proportional to $\cos(\theta +
\widetilde\theta_{id} - \widetilde\delta)$, which in view of
$\widetilde\theta_{id} \approx -70.5^0$ is approximately zero at $\theta
\approx -20^{\mbox{\scriptsize o}}$, $\widetilde\delta \approx 0$.
Consequently the contributions of the tensor trajectory in (\ref{F13})
are strongly suppressed in the region where $\alpha_{K^{*}}(t) \approx
0$, {\it i.e.} at $t \approx - 0.4$ (GeV$\!$/c)$^2$ \cite{Martin}.

In the non-coherent mode the formulas for the differential
cross-sections are significantly different. Recall that in this case
the elementary contributions, associated with the valence quark, are
summed in the cross-section. In doing so, the strange and non-strange
quark-antiquark valence pairs appear in the mesonic final states in the
equal parts  (as in the coherent mode, as well) \cite{N1}. This gives
\begin{eqnarray}\label{F16}
& \displaystyle
\frac{\mbox{d}\sigma}{\mbox{d} t} (K^{-} p \to \eta \Lambda) \;=\;
\frac{5}{3}\; g_{_{V}}^2(t) \, \sin^2 \!\frac{\pi\alpha_{_{V}}}{2}
                             \, \cos^2 \theta \qquad & \nonumber
\\ [0.2\baselineskip]
& \displaystyle
+ \; \frac{1}{3} \; g_{_{T}}^2(t) \, \cos^2 \frac{\pi
\alpha_{_{T}}}{2}\left[
\left( \cos\theta - \sqrt{2}\,\xi\,\sin\theta \right)^2 +
\left(2\cos\theta + \sqrt{2}\,\xi\,\sin\theta \right)^2 \right] \qquad
& \nonumber
\\ [0.2\baselineskip]
& \displaystyle
+ \; \frac{2}{3} \, g_{_{V}}(t) \, g_{_{T}}(t) \;
\cos \! \frac{\pi \alpha_{_{T}}}{2} \,
\sin \! \frac{\pi \alpha_{_{V}}}{2}   \,
\sin\frac{\pi (\alpha_{_{V}}\!-\!\alpha_{_{T}})}{2} \;
\cos\theta
\left(5\cos\theta + \sqrt{2}\,\xi\,\sin\theta \right), \quad &
\end{eqnarray}
\begin{eqnarray}\label{F17}
& \displaystyle
\frac{\mbox{d}\sigma}{\mbox{d} t} (K^{-} p \to \eta'\Lambda) \;=\;
\frac{5}{3}\; g_{_{V}}^2(t) \, \sin^2 \!\frac{\pi\alpha_{_{V}}}{2}
                             \, \sin^2 \theta \qquad & \nonumber
\\ [0.2\baselineskip]
& \displaystyle
+ \; \frac{1}{3} \; g_{_{T}}^2(t) \, \cos^2 \frac{\pi
\alpha_{_{T}}}{2} \left[
\left( \sin\theta + \sqrt{2}\,\xi\,\cos\theta \right)^2 +
\left(2\sin\theta - \sqrt{2}\,\xi\,\cos\theta \right)^2 \right] \qquad
& \nonumber
\\ [0.2\baselineskip]
& \displaystyle
+ \; \frac{2}{3} \, g_{_{V}}(t) \, g_{_{T}}(t) \;
\cos \! \frac{\pi \alpha_{_{T}}}{2} \,
\sin \! \frac{\pi \alpha_{_{V}}}{2}   \,
\sin\frac{\pi (\alpha_{_{V}}\!-\!\alpha_{_{T}})}{2} \;
\sin\theta
\left( 5\sin\theta  - \sqrt{2}\,\xi\,\cos\theta \right). \quad &
\end{eqnarray}
These formulas include the same parameters as (\ref{F13}), (\ref{F14}),
but the parameters values may be different.

Finally, we note that there is a third charge-exchange reaction
in the $K^-$ beams, the $K^{-} p \to \pi^0 \Lambda$, which is determined
by the same trajectories. In this case the formula for the differential
cros-section is independent of mode of summation of elementary
contributions \cite{N1}. Owing to the absence of mesonic singlets in the
final state, it does not contain the mixing parameter $\theta$ and the
singlet-channel splitting $\xi$. However, a similar splitting can occur
because of another isotopic spin in the final state. So we have
\begin{eqnarray}\label{F18}
& \displaystyle
\frac{\mbox{d}\sigma}{\mbox{d} t} (K^{-} p \to \pi^0 \Lambda) \;=\;
\zeta^2 \, \biggl\{
g_{_{V}}^2 (t)  \, \sin^2 \!\frac{\pi \alpha_{_{V}}}{2}  +
g_{_{T}}^2 (t) \, \cos^2 \frac{\pi \alpha_{_{T}}}{2} \biggr. & \nonumber
\\ [0.5\baselineskip]
&  \biggl. \displaystyle + \; 2 \, g_{_{V}} (t) \, g_{_{T}} (t) \;
\sin \!\frac{\pi \alpha_{_{V}}}{2} \,
\cos \frac{\pi \alpha_{_{T}}}{2} \,
\sin\frac{\pi (\alpha_{_{V}}\!-\!\alpha_{_{T}})}{2} \biggr\}\,. &
\end{eqnarray}
Here
\begin{equation}\label{F19}
\zeta = \zeta_{0} \; \exp(\zeta' t),
\end{equation}
and are $\zeta_{0}$, $\zeta'$ are parameters. So, the process with
$\eta$,
$\eta'$, $\pi^0$ are described in total by nine parameters.

\section{The fit}\label{sec3}

To determine the mode of summation of intermediate contributions we
proceed to the fit of data. Recall that in accordance with previous
qualitative analysis \cite{N1} at relatively low and high energies the
coherent and non-coherent mode is realized, respectively. Our task
is to confirm or deny this result on the basis of the fit of data.
Correspondingly, we do the fit of data at relatively low and high
energies independently. In order to eliminate false solutions, we
introduce restrictions on the parameters. Namely, we assume that a
solution is physical if $\theta$, $\xi_0$, $\zeta_0$  belong to
intervals $-35^{\mbox{\scriptsize o}} < \theta < -5^{\mbox{\scriptsize
o}}$, \ $0.5 < \xi_0 <1.5$, $0.5 < \zeta_0 <1.5$. The first condition
cuts off solutions clearly inconsistent with the results of other
studies \cite{PDG}. The second and third conditions mean that the
violation of the flavor symmetry should not be too large. A similar
condition for the slope parameters implies that $\xi'$ and $\zeta'$ in
absolute value should not exceed $c_{_{T}}$ and $c_{_{V}}$, and within
each trajectory the slopes should not split significantly. We demand
also a positivity of the resulting slopes in the vertex functions, which
means decreasing of contributions of the trajectories with increasing
$-t$. Lastly, we consider linear trajectories and we define them based
on the spectroscopy~data.

\subsection{{\boldmath $\pi^-$} beams}\label{sec3.1}

\begin{figure*}
\hspace*{0.08\textwidth}
\includegraphics[width=0.31\textwidth]{F1a.eps}
 \hfill
 \includegraphics[width=0.31\textwidth]{F1b.eps}
\hspace*{0.1\textwidth}
\caption{\small Differential cross-sections $\pi^{-} p \to (\eta, \eta')
n$ [$\mu$b/(GeV$\!$/c)$^2$] at 32.5 GeV$\!$/c \cite{GAMS}.
Theoretical curves correspond to (\ref{F7}), (\ref{F8}) with parameter
values specified in the text.}
\label{Fig1}
\end{figure*}

In the first place we consider data in the $\pi^{-}$ beams. Recall that
appropriate cross-sections are given in (\ref{F7}), (\ref{F8}) and they
are independent of the mode of summation of intermediate contributions.

Based on the spectroscopy data \cite{PDG} the $a_2$-trajectory in the
linear approximation is
\begin{equation}\label{F20}
 \alpha_{a_2}(t) = 0.45 + 0.89\,t \,,
\end{equation}
where $t$ is given in (GeV$\!$/c)$^2$. The errors in (\ref{F20}) are
of order of percents, which is insignificant for subsequent results.
Notice that earlier determinations of $\alpha_{a_2}(t)$ proceeding
directly from the scattering data, gave $\alpha_{a_2}(t) = 0.4 + 0.7\,t$
\cite{apel} and $\alpha_{a_2}(t) = 0.37 + 0.79\,t + 0.03\,t^2$
\cite{Gahl}. The discrepancy with our result (\ref{F20}) is explained by
the differences in the formulas for the differential cross-sections used
in the mentioned references.

The charge-exchange reactions in the $\pi^{-}$ beams with yields of
$\eta$, $\eta'$ have been studied in detail at 8.45 GeV$\!$/c
\cite{Stanton} and at higher momenta \cite{GAMS,apelp,apel}.
Unfortunately, data \cite{Stanton} and \cite{apelp} are not available,
and data \cite{apel} without \cite{apelp} are not complete. For this
reason we to consider only data \cite{GAMS} at 32.5 GeV$\!$/c. (Of
course, this narrows our capabilities, but we will be able to work out
important details and verify the accuracy of the model itself.) The fit
with these data reveals a series of solutions with close $\chi^2$
distributed in the ($\theta$,$\,\xi_0$)-plane near a curve defined by
condition $\theta - \delta_0 = -18.5^{\mbox{\scriptsize o}}$. When doing
the fit with different fixed ($\theta,\,\xi_0$), the surface of minima
of $\chi^2$, which appears over the ($\theta,\,\xi_0$)-plane, has a
trough of almost constant depth located along the mentioned curve. The
boundaries of the curve are determined by the boundaries of the physical
region for the parameters:
($-5^{\mbox{\scriptsize o}}$, $0.62$) and ($-28.6^{\mbox{\scriptsize
o}}$, $1.5$). In these points $\chi^2/\mbox{d.o.f.}$ takes values
30.4/35 and 30.0/35, respectively, and along the curve $\chi^2$ remains
within the mentioned limits. (The other parameters vary typically within
10--20\%.) So, all the points of the curve determine practically
equivalent solutions.

For an illustration we point out one of the solutions that appear when
all the parameters are free: $\theta \! = \!\! -21^{\mbox{\scriptsize
o}} \! \pm 8^{\mbox{\scriptsize o}}$, $g_{0 a_2} \! =  1.7 \pm 0.4$,
$c_{a_2} \! = 3.37 \pm 0.08$, $\xi_{0 a_2} \! =  1.1 \pm 0.4$,
$\xi'_{a_2} \! = 0.40 \pm 0.08$, $r \! = \! 19.7 \pm 0.3$, $r' \! = 18.4
\pm 1.7$, $\chi^2/\mbox{d.o.f.} \! = \! 31.0/37$. Hereinafter the
normalization constant(s) are determined in $\mu$b$^{1/2}$/(GeV$\!$/c)
and the slopes in (GeV$\!$/c)$^{-2}$ (see footnote\footnote{We have
recalculated data \cite{GAMS}, obtained initially in the form of numbers
of pairs of gamma-quanta arising from $\eta$ and $\eta'$, into the units
of differential cross-sections, see appendix A. The error of the
conversion factor is not included in $g_{0 a_2}$.}). In fig.~\ref{Fig1}
the corresponding differential cross-sections are presented together
with the data used in the fit.

It is worth noting that in all the solutions the errors of $\theta$ and
$\xi_{0 a_2}$ are significant. However, both these parameters are highly
correlated and their difference is determined quite accurately. In
particular, for the solution given above $\theta - \delta_0 = (-18.5
\pm 0.6)^{\mbox{\scriptsize o}}$, and this result is kept for all the
solutions along the curve. From here and (\ref{F11}) we restore the
ratio of the cross sections $R_{\pi}^{\eta'/\eta}(0) = 0.53 \pm 0.02$
with $R_{\pi}^{\eta'/\eta}(0) = 0.54 \pm 0.04$ in \cite{GAMS}. In fact,
$R_{\pi}^{\eta'/\eta}(0)$ is practically independent of energy. Really,
\cite{Stanton} obtained $R_{\pi}^{\eta'/\eta}(0) = 0.500 \pm 0.092$ at
8.45 GeV$\!$/c, and \cite{apelp} obtained $R_{\pi}^{\eta'/\eta}(0) =
0.55 \pm 0.06$ at 25 GeV$\!$/c and 40 GeV$\!$/c. So, regardless
$\theta$, the value of $\xi_{0 a_2}$ remains constant (within errors)
with changing the energy. This means that $\xi_{0 a_2}$ is independent
of (weakly dependent on) the mode of summation of intermediate
contributions. However, $\xi'_{a_2}$ depends significantly which follows
from the energy dependence of the slope of $R_{\pi}^{\,\eta'/\eta}(t)$,
see (\ref{F11}), (\ref{F12}) and discussion in \cite{N1}.

Concluding, we note that high reliability of the solutions along the
curve means that our model describes data well and it is quite suitable
for the analysis of the charge-exchange reactions. In particular, our
ansatz about the presence of the gamma function in the amplitude and
the parameterization of the vertex functions is confirmed. If one
removes the gamma function but preserves the signature factor, the
quality of the description falls catastrophically: in the physical
region the minimum of $\chi^2/\mbox{d.o.f.}$ constitutes $126/35$.

\subsection{{\boldmath $K^-$} beams}\label{sec3.2}

{$\!$\it A priori}, we do not know whether the leading $K^{*}$
and~$K^{*}_2$ trajectories are non-degenerate. An independent fit of
spectroscopy data \cite{PDG} clearly indicates a preference of their
non-degeneracy.\footnote{The hypothesis of the degeneracy leads to
$\chi^2/\mbox{d.o.f.} = 26.6$ against $\chi^2/\mbox{d.o.f.} = 1.7$ in
the case of non-degeneracy.} Assuming linearity of the trajectories, we
obtain
\begin{equation}\label{F21}
\alpha_{K^{*}}   = 0.33 + 0.84\,t \,,
\end{equation}
\begin{equation}\label{F22}
\alpha_{K^{*}_2} = 0.11 + 0.93\,t \,. \vspace*{0.3 \baselineskip}
\end{equation}
The errors in the coefficients in (\ref{F21}), (\ref{F22}) do
not exceed 2\%, which is insignificant for our purposes.

Further, at relatively low energies we use formulas (\ref{F13}),
(\ref{F14}) and (\ref{F18}). The differential cross-sections in this
energy range were measured by several groups
\cite{Moscoso,Mason,Marzano}, but only data \cite{Marzano} at 4.2
GeV$\!$/c are suitable for the fit. Unfortunately, we can not use the
totality of these data as they cover a large area of the transfer while
our formulas with one-reggeon exchanges are valid at small $t$ only.
Since we do not {\it a priori} know how small $t$ should be, we do a
series of the fits gradually expanding the area of $t$. We continue this
procedure until the quality of the description worsens sharply or
physical solutions disappear. In this manner we define the limiting
values $-t =$ 1.4, 1.0, 0.35 (GeV$\!$/c)$^2$ with the numbers of
experimental points 8, 10, 10 in the cases with $\eta$, $\eta'$,
$\pi^0$, respectively. In the mentioned areas there is only one
solution. We present it in table~\ref{T1} as Fit1. (The contributions to
$\chi^2$ associated with $\eta$, $\eta'$, $\pi^0$ are 4.5, 3.8, 3.2,
respectively.) In fig.~\ref{Fig2} we show this solution by solid (green)
thick curves. The solid~thin curves continue the solution beyond the
region of the fit. The discrepancy with the data in this region means
that some unaccounted contributions dominate. Typically, they are
reggeon-pomeron or multi-reggeon contributions characterized by a lower
slope and hence prevailing at large $-t$ \cite{Regge}.

As we can see, the mentioned solution describes data well, but most
parameters are determined with large errors. In particular, the
mixing angle $\theta$ in essence is undetermined. So we can assign a
certain value to $\theta$ and then with this value do the fit again. We
take $\theta = -20.8^{^{\mbox{\scriptsize o}}}$, the value obtained
below. In this case again there is only one solution. We present it in
table~\ref{T1} as Fit2. (The contributions to $\chi^2$ associated with
$\eta$, $\eta'$, $\pi^0$ are 4.4, 4.0, 3.7, respectively.) In
fig.~\ref{Fig2} we show it by dashed (red) curves. We see that Fit1 and
Fit2 coincide within the errors, and the curves almost coincide in the
region of the fit. This means that in terms of conformity to the data
both solutions are equivalent.

\begin{table*
\caption{Solutions of the fit of data \cite{Marzano} (Fit1, Fit2) and
\cite{GAMS} (Fit3--Fit6). Fit1--Fit3 and Fit4--Fit6 match the coherent
and non-coherent mode, respectively. Parameters $g_{_{T}}$, $g_{_{V}}$
are given in $\mu$b$^{1/2}$/(GeV$\!$/c), the slopes in
(GeV$\!$/c)$^{-2}$.
}\label{T1}
\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}lllllll@{}}
\hline\noalign{\medskip}
            & Fit1    & Fit2    & Fit3   & Fit4    & Fit5    & Fit6
\\[1mm]
\hline\noalign{\medskip}
{\footnotesize  $\chi^2/\mbox{d.o.f.}$ } $\!\!\!\!\!\!\!$
            & 11.4/19 & 12.1/20 & 36.4/37 & 37.4/37 & 35.0/37 & 37.5/39
\\[1mm]
{\footnotesize  $\theta$ [$\, ^{^{\mbox{\scriptsize o}}} \,$] }
                 & {\footnotesize $\!\!\!\!\!-12   \pm 12$  }
                 & {\footnotesize $\!\!\!\!\!-20.8$ [input] }
                 & {\footnotesize $\!\!\!\!\!-25.9 \pm 2.32$}
                 & {\footnotesize $\!\!\!\!\!-21.8 \pm 5.0$ }
                 & {\footnotesize $\!\!\!\!\!-22.2 \pm 4.3$ }
                 & {\footnotesize $\!\!\!\!\!-20.8 \pm 4.9$ }
\\[1mm]
{\footnotesize  $g_{_{T}}$ }
                 & {\footnotesize $8.1  \pm 5.3$ }
                 & {\footnotesize $12.2 \pm 4.4$ }
                 & {\footnotesize $1.7  \pm 1.3$ }
                 & {\footnotesize $2.0  \pm 0.4$ }
                 & {\footnotesize $2.1  \pm 0.1$ }
                 & {\footnotesize $2.07 \pm 0.09$}
\\[1mm]
{\footnotesize  $c_{_{T}}$  }
                 & {\footnotesize $1.2  \pm 0.8$ }
                 & {\footnotesize $1.5  \pm 0.8$ }
                 & {\footnotesize $12.1 \pm 6.4$ }
                 & {\footnotesize $4.1  \pm 0.4$ }
                 & {\footnotesize $4.3  \pm 0.3$ }
                 & {\footnotesize $4.1  \pm 0.2$ }
\\[1mm]
{\footnotesize  $g_{_{V}}$  }
                 & {\footnotesize $12.2 \pm 0.6$ }
                 & {\footnotesize $12.8 \pm 0.6$   }
                 & {\footnotesize $2.2  \pm 0.4$ }
                 & {\footnotesize $0.2  \pm 1.4$  }
                 & {\footnotesize $\!\!\!\!\!-2.2 \pm 1.2$ }
                 & {\footnotesize $0$ [input]    }
\\[1mm]
{\footnotesize  $c_{_{V}}$  }
                 & {\footnotesize $1.7 \pm 0.1$ }
                 & {\footnotesize $1.7 \pm 0.1$ }
                 & {\footnotesize $3.0  \pm 0.9$ }
                 & {\footnotesize $0.7  \pm 13.7$}
                 & {\footnotesize $110  \pm 70$  }
                 & ---
\\[1mm]
{\footnotesize  $\xi_0$ }
                 & {\footnotesize $1.13 \pm 0.77$ }
                 & {\footnotesize $0.72 \pm 0.31$ }
                 & {\footnotesize $1.02 \pm 0.86$ }
                 & {\footnotesize $0.96 \pm 0.18$ }
                 & {\footnotesize $0.91 \pm 0.08$ }
                 & {\footnotesize $0.95 \pm 0.08$ }
\\[1mm]
{\footnotesize  $\xi'$  }
                 & {\footnotesize $1.0 \pm 0.8$ }
                 & {\footnotesize $0.8 \pm 0.9$ }
                 & {\footnotesize $\!\!\!\!\!-10.0 \pm 6.5$ }
                 & {\footnotesize $\!\!\!\!\!-2.3 \pm 0.6$ }
                 & {\footnotesize $\!\!\!\!\!-2.4 \pm 0.4$ }
                 & {\footnotesize $\!\!\!\!\!-2.2 \pm 0.4$ }
\\[1mm]
{\footnotesize  $\zeta_0$ }
                 & {\footnotesize $1.19 \pm 0.07$ }
                 & {\footnotesize $0.91 \pm 0.24$   }
                 & --- & --- & --- & ---
\\[1mm]
{\footnotesize  $\zeta'$  }
                 & {\footnotesize $\!\!\!\!\!-0.6 \pm 0.5$ }
                 & {\footnotesize $\!\!\!\!\!-0.5 \pm 0.4$ }
                 & --- & --- & --- & ---
\\[1mm]
\noalign{\smallskip}\hline
\vspace*{\baselineskip}
\end{tabular*}
\end{table*}

We can clarify the reason why angle $\theta$ in Fit1 is poorly defined.
The point is that formula (\ref{F18}) for the cross-section with $\pi^0$
does not contain $\theta$, while dependence on $\theta$ in formulas
(\ref{F13}), (\ref{F14}) in the tensor channel appears in the
combination $\theta - \widetilde\delta$. So when considering the two
latter reactions, we get a situation which is similar to that in the
case in the $\pi^-$ beams, although with a correction for the presence
of two trajectories. Namely, two series of solutions with close $\chi^2$
appear, which differ by a sign of the interference between contributions
of the trajectories. All these solutions are distributed in the
($\theta,\,\xi_0$)-plane along the curve $\theta - \widetilde\delta_0 =
\mbox{constant}$, where $\widetilde\delta_0 = \widetilde\delta (0)$.
Simultaneously, they all describe the dip in $K^{-} p \to \eta \Lambda$
and monotonic behavior of $K^{-} p \to \eta' \Lambda$. The inclusion of
data with $\pi^0$ excludes one of the series with ``wrong''
interference, and fixes $\theta$, $\xi_0$ via the correlation with other
parameters. However, fixing via the correlation implies poor definition,
as it happens.

\begin{figure*}
 \includegraphics[width=0.31\textwidth]{F2a.eps}
 \hfill
 \includegraphics[width=0.31\textwidth]{F2b.eps}
 \hfill
 \includegraphics[width=0.31\textwidth]{F2c.eps}
\caption{\small Differential cross-sections $K^{-} p \to (\eta, \eta',
\pi^{0}) \Lambda$ [$\mu$b/(GeV$\!$/c)$^2$] at 4.2 GeV$\!$/c
\cite{Marzano}. Theoretical curves in (a), (b), (c) correspond to
(\ref{F13}), (\ref{F14}), (\ref{F18}), respectively. Solid (green) and
dashed (red) curves represent solutions Fit1 and Fit2, respectively. In
the regions of the fit the data and curves are shown by thick lines.}
\label{Fig2} \vspace*{\baselineskip}
\end{figure*}

\begin{figure*}
 \includegraphics[width=0.31\textwidth]{F3a.eps}
 \hfill
 \includegraphics[width=0.31\textwidth]{F3b.eps}
 \hfill
 \includegraphics[width=0.31\textwidth]{F3c.eps}
 \hfill

\caption{\small Differential cross-sections $K^{-} p \to (\eta, \eta',
\pi^{0}) \Lambda$ [$\mu$b/(GeV$\!$/c)$^2$] at 32.5 GeV$\!$/c. The data
in (a) and (b) are taken from \cite{GAMS}. Solid (blue) curves represent
Fit6. Solution Fit2 in recalculation to the energy \cite{GAMS} is shown
by (red) dashed curves. Solution Fit3 is shown by (orange) points. In
the frame (c) the curves for Fit3 and Fit6 are shown at $\zeta =1$ in
(\ref{F18})} \label{Fig3}
\end{figure*}

Now we turn to relatively high energies. The available data in this
range are \cite{GAMS}, obtained at the $K^-$ momentum 32.5 GeV$\!$/c.
Unfortunately, these data cover reactions with $\eta$ and $\eta'$ only.
However, the fit in this case leads to more precise outcomes.

First of all, we check whether there is a solution in the coherent mode.
It turns out that such a solution exists, see Fit3 in table~\ref{T1}.
Its essential features are the negative sign of $\xi'$ and the
approximate equality in absolute value of $\xi'$ and $c_{_{T}}$. The
negative sign of $\xi'$ means exponential growth of $\xi(t)$ with
increasing $-t$. So at $t = - 0.4$ (GeV$\!$/c)$^2$ we have $\xi \approx
55$. Such a great value destroys the mechanism of appearance of a dip in
$K^{-} p \to \eta \Lambda$. However, both features are unacceptable to a
physical solution since they mean large splitting between the slope
parameters in the vertex function. Really, the slope in the pure singlet
component in this case is $c_{_{T}} + \xi' = 2.1 \pm 0.3$ (strong
correlation is taken into account). This is almost 6 times smaller than
the slope $c_{_{T}}$ in the octet component. The difference is too great
for the flavor symmetry violation. So Fit3 must be considered as
unphysical solution. However that is not all. Besides the slopes in Fit3
are incompatible with Regge behavior. Really, $\xi'$ should be unchanged
with changing the energy while $c_{_{T}}$ and $c_{_{V}}$ should evolve
in accordance with (\ref{F4}). However, $\xi'$ in Fit3 changes the sign
and in absolute value exceeds tenfold $\xi'$ in Fit1. Moreover,
proceeding from Fit2, $c_{_{T}}$ and $c_{_{V}}$ both must equal 3.3 at
the energy of Fit3. However, $c_{_{T}}$ in Fit3 is significantly larger.
This means violation of Regge behavior. The mentioned inconsistencies
are clearly visible in fig.~\ref{Fig3}(a,b), where solution Fit2
recalculated to the energy of Fit3 is shown by dashed (red) curves, and
solution Fit3 is shown by (orange) points. In fig.~\ref{Fig3}(c) the
analogous curves have meaning of predictions for $K^{-} p \to \pi^0
\Lambda$ at 32.5 GeV$\!$/c.

On the basis of the above discussion, we conclude that the high-energy
data are incompatible with predictions of the Regge approach. So in this
energy range a modified approach is required. For this purpose we take
advantage of our model in the non-coherent mode.

The corresponding fit of data \cite{GAMS} with formulas (\ref{F16}),
(\ref{F17}) leads to two solutions, shown in table~\ref{T1} as Fit4 and
Fit5. They almost coincide, except $g_{_{V}}$ and $c_{_{V}}$. Namely,
$g_{_{V}}$ in Fit4 is compatible with zero, whereas $g_{_{V}}$ in Fit5
is not. Besides, $c_{_{V}}$ in Fit5 is anomalously large. However, on
close examination the mentioned differences are illusory. Really, due to
the large $c_{_{V}}$ the vector contribution in Fit5 is nonzero at $t
\approx 0$ only. So, both solutions are compatible with zero
contribution of the vector trajectory. Assuming initially
$g_{_{V}}\!=\!0$, we come to solution Fit6. In view of above discussion,
Fit6 is practically equivalent to Fit4 and Fit5. In fig.~\ref{Fig3} we
show Fit6 by solid (blue) curves.

An essential feature of the solutions Fit4-Fit6 as compared to Fit1, are
essentially smaller errors in $\theta$ and $\xi_0$. This is a
consequence of the fact that both these parameters are determined based
on the data in the reactions with $\eta$ and $\eta'$ only, without
involving the data with $\pi^0$ (see above discussion). As the final
outcome for $\theta$ we take its value in Fit6,
\begin{equation}\label{F23}
  \theta = -(20.8 \pm 4.9)^{\mbox{\scriptsize o}}.
\end{equation}

Our main conclusion in this subsection is that  in the framework of
conventional Regge approach the data in the $K^{-}$ beams are well
described at relatively low energies, but are no longer described at the
transition to higher energies. At the same time, at the higher energies
the data are well described in the mode with non-coherent summation of
intermediate contributions. The data fit at low energies does not
determine the $\eta$--$\eta'$ mixing angle. The fit at higher energies
leads to estimate (\ref{F23}).

\subsection{Gluonium admixture}\label{sec3.3}

The above analysis was carried out in the particular scheme (\ref{F10})
of the $\eta$--$\eta'$ mixing. However analysis can easily be
generalized to any scheme. Here we consider a generalization which
includes a gluonium admixture in the $\eta'$. In this scheme
\begin{eqnarray}\label{F24}
|\eta \rangle \; &=&
 \cos\theta \, |\eta^8 \rangle -  \sin\theta \, |\eta^0 \rangle\,,
 \nonumber\\[0.4\baselineskip]
 |\eta' \rangle &=&
 \cos\theta_G \Bigl(\sin\theta \, |\eta^8 \rangle +
 \cos\theta \, |\eta^0 \rangle \Bigr)\! +
 \sin\theta_G \, |\eta^G \rangle. \qquad\quad
\end{eqnarray}
where $|\eta^G \rangle$ is a gluonium state and $\theta_G$ is an
additional mixing angle. This scheme was first proposed as a solution to
the axial Ward identities for the relevant composite interpolating
fields on the condition of the renormalization-group invariance of the
pattern of the mixing \cite{N2}. Afterwards this scheme was repeatedly
considered on purely phenomenological basis, see bibliography in
\cite{N1}. In scheme (\ref{F24}) the above formulas do not change in the
case of $\eta$ and only slightly change in the case of $\eta'$. Namely,
a factor $\cos^2 \theta_G$ appears in the r.h.s. in (\ref{F8}),
(\ref{F11}), (\ref{F14}), (\ref{F17}). We complement the allowable
range of parameters by condition $0.5 < \cos^2 \theta_G \le 1$, where
the lower bound means that $\eta'$ is predominantly a quark state, not a
glueball.

The analysis of data \cite{GAMS} in the $\pi^-$ beams, similar to that
in sect.~\ref{sec3.1}, leads to a series of solutions with close
$\chi^2$ grouped in the ($\theta,\xi_0,\theta_G$)-space near a surface
specified by condition
\begin{equation}\label{F25}
R_{\pi}^{\,\eta'/\eta}(0) \> = \>
\cos^2 \theta_G \; \tan^2 (\theta + \theta_{id} - \delta_0) \,.
\end{equation}
Among these solutions $\cos^2 \theta_G$ varies from $0.68 \pm 0.14$ to
$0.92 \pm 0.20$, in all cases with $\chi^2/\mbox{d.o.f.} \approx 30/34$.
Thereby, the data in the $\pi^-$ beams do not provide solid information
about the gluonium admixture in $\eta'$.

The fit of data \cite{Marzano} and \cite{GAMS} in the $K^-$ beams with
all the parameters free, do not reveal any physical solutions.
Nevertheless, at various fixed $\cos^2 \theta_G$ the solutions appear,
and we can trace their evolution. Recall that at $\cos^2 \theta_G = 1$
all the solutions are presented in table~\ref{T1}. With decreasing
$\cos^2 \theta_G$ from 1, the $\chi^2$ is typically increasing in the
appropriate branches of the solutions with the parameters are smoothly
evolving. In so doing, the solutions in the branches Fit1 and Fit3
continue to be incompatible with each other, and the solutions in Fit3
are characterized by abnormal splitting of the slope parameters as was
discussed in sect.~\ref{sec3.2}.

Within $0.6 \le \cos^2 \theta_G \le 1$ the solutions exist in all cases
with free $\theta$. In particular, at $\cos^2 \theta_G = 0.6$ the
$\chi^2/\mbox{d.o.f.}$ in branches Fit1 and Fit6 takes values 11.7/19
and 38.0/39, respectively, and in both cases $\theta$ remains within the
errors of table~\ref{T1}. In addition, in the branch Fit2 with $\theta =
-20.8^{\mbox{\scriptsize o}}$ and free $\cos^2 \theta_G$, there is only
one solution:
$\cos^2 \theta_G = 0.68 \pm 0.67$, $g_{_{T}} = 14.3 \pm 7.9$, $c_{_{T}}
= 1.6 \pm 0.8$, $g_{_{V}} = 12.8 \pm 0.6$, $c_{_{V}} = 1.7 \pm 0.1$,
$\xi_0 = 0.74 \pm 0.30$, $\xi' = 0.7 \pm 1.0$, $\zeta_0 = 0.82 \pm
0.33$, $\zeta' = - 0.5 \pm 0.5$, $\chi^2/\mbox{d.o.f.} = 12.0/19$.
Notice that $\cos^2 \theta_G$ is compatible with 1.

So the data on the charge-exchange reactions are rather indifferent
to a possible gluonium admixture in the $\eta'$, and on this basis we
can not draw a conclusion about the presence or absence of this
admixture. However the conclusion about the mode change of summation of
intermediate contributions remains in force in the presence of the
gluonium admixture. In the case of small gluonium admixture the estimate
for the angle $\theta$ undergoes insignificant changes.

\section{Solution at intermediate energies}\label{sec4}

We have seen above that the charge-exchange reactions in the $K^{-}$
beams at relatively low and high energies are well described by the
formulas obtained in the coherent and non-coherent modes, respectively.
Now we consider the problem of the description in the intermediate
energy region where contributions of both modes are possible.

We offer a solution in the spirit of the density matrix. Specifically,
we assume that each mode can be implemented with a certain probability.
Since there is no third, the sum of the two probabilities must be equal
to one. So, we consider the differential cross-section in the~form
\begin{equation}\label{F26}
\frac{{\rm d} \sigma}{{\rm d} t} (s,t) \;=\;
w \; \frac{{\rm d} \sigma_{\mbox{\scriptsize c}}}{{\rm d} t} (s,t) \;+\;
(1-w) \; \frac{{\rm d} \sigma_{\mbox{\scriptsize nc}}}{{\rm d} t}
(s,t)\,.
\end{equation}
Here $w$ is the probability to find the system in the coherent mode,
indices ``c'' and ``nc'' mean the modes of coherent and non-coherent
summation. In the most general case $w$ is a function of $s$ and $t$.
Asymptotically at low and high $s$, we expect $w \to 1$ and $w \to 0$,
respectively. Our task is to find an algorithm of experimental
measurement of $w$.

Below, we do this under the assumption that at small transfer one can
neglect in $w$ the dependence on $t$, so that $w = w(s)$. In fact,
reliable measurements are needed at least at two energies. At $s=s_1$ we
have
\begin{equation}\label{F27}
\frac{{\rm d} \sigma}{{\rm d} t} (s_1,t) = \left[
w_1  \frac{{\rm d} \sigma_{\mbox{\scriptsize c}}}{{\rm d} t}
(s_1,t)\right] +
\left[ (1 \! - \! w_1 ) \frac{{\rm d}
\sigma_{\mbox{\scriptsize nc}}}{{\rm d} t} (s_1,t)\right] \! . \;
\end{equation}
Here $w_1 = w(s_1)$, ${\rm d} \sigma_{\mbox{\scriptsize c}} / {\rm d} t$
and ${\rm d} \sigma_{\mbox{\scriptsize nc}} / {\rm d} t$ are described
by appropriate formulas in the coherent and non-coherent modes. The
large square brackets mean that factors $w_1$ and $(1-w_1 )$ are not
considered as independent parameters, but at the fit are considered
absorbed by normalization constants in the cross-sections.
At $s = s_2$ an analogous formula includes $w_2 = w(s_2)$ as a hidden
parameter. However, we write this formula in a modified form. Namely,
we factor out numbers $x$ and $y$, defined as
\begin{equation}\label{F28}
x = \frac{w_2}{w_1}\,, \qquad \quad
y = \frac{1-w_2}{1-w_1}\,.
\end{equation}
So at $s=s_2$, we have
\begin{equation}\label{F29}
\displaystyle
\frac{{\rm d} \sigma}{{\rm d} t} (s_2,t) = x \left[
w_1 \frac{{\rm d} \sigma_{\mbox{\scriptsize c}}}{{\rm d} t} (s_2,t)
\right]
 + y \left[(1 \! -\! w_1 ) \frac{{\rm d}
\sigma_{\mbox{\scriptsize nc}}}{{\rm d} t} (s_2,t) \right] \! . \quad
\end{equation}

Further, we notice that all the parameters in the square brackets in
(\ref{F29}) are actually determined by the fit at $s=s_1$ with
(\ref{F27}). Really, some parameters are unchanged at changing the
energy, and the remaining ones are changed by means of (\ref{F4}) with
$\tilde{s}_0 = s_2$, $s_0 = s_1$. So in (\ref{F29}) only numerical
factors $x$ and $y$ are unknown. They may be determined based on the fit
at $s=s_2$. Then, solving system (\ref{F28}), we obtain the
probabilities:
\begin{equation}\label{F30}
w_1  =      \frac{y-1}{y-x}\,, \qquad \quad
w_2 = x \, \frac{y-1}{y-x}\,.
\end{equation}

In this way, we have presented an algorithm to measure $w(s)$. Of
course, in practice it is expedient to do a joint fit of data
simultaneously at different energies. On this basis we can determine
$w(s)$ in a wide range of energy. We expect that $w(s)$ is monotonically
decreasing with increasing the energy from asymptotic value 1 at
relatively low energies up to asymptotic value 0 at higher energies.

Unfortunately, the currently available data are not sufficient to carry
out such an analysis because data \cite{Marzano} contain too few
measured points and \cite{GAMS} cover too small region of $t$. Besides,
in the case of the common use the data \cite{GAMS} from the very
beginning must be recalculated to units of the differential
cross-sections, which leads to an additional 12\%-error, see appendix A.
This further reduces the quality of the fit. As for the data
\cite{Harran} at 8.25 GeV$\!$/c, which fall into the intermediate
region, their quality is not sufficient to do the fit with them because,
in particular, they contain too few measured points.

\begin{figure*}
\hspace*{0.08\textwidth}
\includegraphics[width=0.31\textwidth]{F4a.eps}
 \hfill
 \includegraphics[width=0.31\textwidth]{F4b.eps}
\hspace*{0.1\textwidth}

\caption{\small Differential cross-sections $K^{-} p \to (\eta, \eta')
\Lambda$ [$\mu$b/(GeV$\!$/c)$^2$] at 8.25 GeV$\!$/c \cite{Harran}. Solid
(blue) and dashed (red) curves represent solutions Fit6 and Fit2,
respectively, recalculated to energy \cite{Harran}. Dash-dot (violet)
curves represent solution built by (\ref{F26}) at $w=0.95$.}\label{Fig4}
\end{figure*}

However, all the mentioned data may be used for illustrative purposes.
Namely, let us suppose that at the energies of \cite{Marzano} and
\cite{GAMS} the probability $w$ takes values 1 and 0, respectively.
Then, recalculating solutions Fit2 and Fit6 to the intermediate energy
\cite{Harran} and substituting the result into (\ref{F26}), we obtain a
prediction for the differential cross-section at 8.25 GeV$\!/c$. In
fig.~\ref{Fig4} we show data \cite{Harran} and we present by dashed
(red) and solid (blue) curves solutions Fit2 and Fit6, respectively,
recalculated to this energy. The dash-dot curve gives the result defined
by (\ref{F26}) at $w=0.95$, where 0.95 is chosen arbitrarily. We see
that the latter curve lies between the former two and on the whole
better matches the data. One should bear in mind, however, that the
positions of the curves in fig.~\ref{Fig4} are shown without taking into
account the appropriate errors.

\section{Discussion and conclusion}\label{sec5}

The analysis in this work confirms the earlier qualitative conclusion
\cite{N1} about the mode change with increasing the energy of summation
of intermediate contributions in the charge-exchange reactions. In
support of this conclusion, we put forward the following results
obtained based on the fit of data. First, we found a solution which
describes in the Regge approach the charge-exchange reactions $K^{-} p
\to (\eta, \eta', \pi^{0}) \Lambda$ at relatively low energies. In our
model this solution implies the mode of coherent summation of
intermediate contributions. Then, we showed the absence of a similar
solution at relatively high energies. Instead, a solution appears that
corresponds to the mode of non-coherent summation. Finally, we compared
the obtained solutions by recalculating to a common energy and
established fundamental difference between them.

We emphasize that the above solutions are statistically well-founded.
Moreover, the data on the charge-exchange reactions  in the $\pi^-$
beams are fitted very well.\footnote{In the case of $\pi^-$ beams the
mode change is independently confirmed by effect of the energy
dependence of the slope of $R_{\pi}^{\,\eta'/\eta}(t)$, see discussion
in sect.~\ref{sec3.1} and in \cite{N1}.} This indicates that our model
describes the charge-exchange reactions well, and results obtained on
its basis are reliable.

At the same time, the range of applicability of the model is limited by
small transfers. This is a consequence of the fact that the model takes
into account the contributions of leading trajectories only,\footnote{We
do not take into consideration the daughter trajectories as they have
close slope but lower intercept. Therefore their contributions are
suppressed.} and ignores the contributions of the Regge cuts that manifest
themselves at higher $-t$. In our approach the region where the
contributions of the cuts may be ignored is determined empirically by
gradually increasing the upper limit of $|t|$ until the data are well
described. So, in principle, the range of applicability of the model can
be expanded by including in analysis the reggeon-pomeron or
multi-reggeon contributions. We do not consider such an extension since
this implies a substantial increasing in the number of parameters in
conditions of
limited data.

In our analysis we proceeded from the assumption that the angle of the
$\eta$--$\eta'$ mixing is unknown, and we considered it as a parameter
which is to be determined by the fit. This approach is valuable in
itself in view of the lack of well-established estimate of the mixing
on the basis of the charge-exchange reactions \cite{Gilman,Scadron}.
Furthermore, unlike \cite{N1} we considered the parameter of violation
of the nonet symmetry as unknown, which must be determined by the fit,
as well. Unfortunately, this leads to the impossibility of determining
the mixing angle on the basis of data in the $\pi^-$ beams, since
corresponding cross-sections are determined by the difference of the
mixing angle and the parameter of the nonet symmetry breaking. For this
reason, we determined the mixing by basing on the data in the $K^-$
beams. Unluckily, the appropriate estimate includes large error and
therefore in practical terms is little informative. However, we show
that the gluonium admixture in the $\eta'$ has little effect on the $\eta
$--$ \eta'$ mixing. Moreover, the gluonium admixture does not
affect the conclusion about the mode change with increasing the energy.

Finally, we found a solution to the problem of description of the
charge-exchange reactions at the intermediate energy range where both
modes are possible, the coherent and non-coherent. The solution is based
on an idea that each mode occurs with a certain probability, and the
probability is dependent on the energy. By this means the differential
cross-section is determined as a sum of contributions with certain
modes, weighted with the probabilities of finding the system in the
particular mode. We propose an algorithm for measuring the
probabilities. A study of the charge-exchange reactions with taking into
account this solution might be a subject of future research provided
that sufficiently detailed data become available.

In summary, with high reliability we confirmed the effect of the mode
change with increasing the energy of summation of intermediate
contributions in the differential cross-sections of charge-exchange
reactions. The analysis in this paper more thoroughly specifies the
model for further systematic investigation of this phenomenon. In
particular, one can study the energy dependence of the coherent and
non-coherent contributions to the cross-sections of the mentioned
reactions.

\bigskip\noindent
The author is grateful for useful discussions to V.V.Ezhela, V.A.Petrov,
R.N.Rogalyov, and S.R.Slabospitsky. Special thanks to V.D.Samoylenko for
providung the GAMS-4$\pi$ data and for pointing out on a possibility of
converting them to the form of differential cross-sections using the
NICE results.

\medskip
\begin{flushleft}
{\bf\large Appendix A}
\end{flushleft}

The data \cite{GAMS} are presented in the form of numbers of pairs of
gamma-quanta arising due to the decays of $\eta$ and $\eta'$. Our task
is to determine the proportionality factor between the numbers of the
registered pairs of gamma-quanta and the corresponding differential
cross-sections.

In the case of $\pi^-$ beams we take advantage of formula (12) in
\cite{apel},
\begin{equation}\label{A1}
\left.\frac{\mbox{d}\sigma}{\mbox{d} t} (\pi^{-} p \to \eta  n \to \gamma\gamma n)
\right|_{t=0}\;=\;
(37.1 \pm 3.6) \left(\frac{s}{s_0}\right)^{-1.26 \pm 0.04} \times
\mu b \,(\mbox{GeV\!/c})^{-2} \,.
\end{equation}
Here $s_0 = 10\,(\mbox{GeV\!/c})^2$. At $s=62.1\,(\mbox{GeV\!/c})^2$,
which means $p_{\mbox{\scriptsize LAB}} = 32.5\,\mbox{GeV\!/c}$, the
r.h.s. in (\ref{A1}) is equal to $(3.71 \pm 0.45) \times \mu b
\,(\mbox{GeV\!/c})^{-2}$. On the other hand, at $p_{\mbox{\scriptsize
LAB}} = 32.5$ $\mbox{GeV\!/c}$ \cite{GAMS} obtained
\begin{equation}\label{A2}
\left.\frac{\mbox{d} N}{\mbox{d} t} (\pi^{-} p \to \eta  n \to \gamma\gamma n)
\right|_{t=0}\;=\;
(2.21 \pm 0.02) \times 10^{6} \times (\mbox{GeV\!/c})^{-2} \,,
\end{equation}
where $N$ is the number of pairs of gamma-quanta. The sought-for factor
is a ratio of (\ref{A1}) to (\ref{A2}). Numerically it is $r_{\pi^-} =
(1.68 \pm 0.20) \times 10^{-6} \mu b$. In the case of yield of $\eta'$
the appropriate factor numerically is the same, because the gamma-quanta
in \cite{GAMS} were registered in the same experiment. Moreover, this
factor determines also the ratio of ${\rm d} \sigma (\pi^{-} p \to
\eta^{[\prime]} n) / {\rm d} t$ to $\Delta N (\pi^-,\eta^{[\prime]}) /
\Delta t$, where $\Delta N (\pi^-,\eta^{[\prime]})$ is the number of
$\eta^{[\prime]}$ formed in the $\pi^-$ beam in the course of experiment
\cite{GAMS} in the bin $t$ of the width~$\Delta t$.

In the case of $K^-$ beams, we take advantage of the fact that the beam
of negative particles in \cite{GAMS} consisted of 98\% $\pi^-$ and 2\%
$K^-$, and the admixture of other particles was negligible. On this
basis, we get $r_{K^-} = (0.98/0.02) r_{\pi^-} = (8.23 \pm 0.98)
\times 10^{-5} \mu b$. Here $r_{K^-}$ has the meaning of the ratio of
${\rm d} \sigma (K^{-} p \to \eta^{[\prime]} n) / {\rm d} t$ to $\Delta
N(K^-,\eta^{[\prime]}) / \Delta t$, where $\Delta
N(K^-,\eta^{[\prime]})$ is the number of  $\eta^{[\prime]}$ formed in
the $K^-$ beam in the corresponding bin $t$ of the width $\Delta t$.

Given the branchings of $\eta^{[\prime]} \to \gamma\gamma$, the obtained
factors determine different conversion factors depending on the choice
of the final state.

redpajama_set_name: RedPajamaArXiv


text:
Three things we learned from Newcastle – Southampton
By Nicholas MendolaFeb 6, 2021, 12:07 PM EST
Newcastle – Southampton: Allan Saint-Maximin set up for two goals and Miguel Almiron bagged a brace as 9-man Newcastle United rode their luck in a 3-2 win over Southampton at St. James' Park on Saturday.
Newcastle finished with nine men after an early second-half sending off of Jeff Hendrick was joined by a bad injury to Fabian Schar after Steve Bruce had made all three substitutions in part due to injuries to Callum Wilson and Javier Manquillo.
[ MORE: How to watch PL in the USA ]
Arsenal loanee Joe Willock scored on debut for Newcastle and Liverpool's Takumi Minamino did the same for Southampton and James Ward-Prowse scored a typically-wonderful free kick in the loss.
Newcastle moves 11 points clear of 18th-place Fulham in avenging an early-season loss at St. Mary's. The Magpies are four points behind 12th-place Southampton.
STREAM NEWCASTLE – SOUTHAMPTON FULL MATCH REPLAY
1. Chaos reigns at St. James' Park: If you wanted to introduce a new fan to the "anything is possible" nature to the Premier League, slap this one on DVD or some other mode of safekeeping. The Magpies took a 2-0 lead via free-flowing football — a welcome respite from recent performances regardless of result — and Saints struck for their two goals in highlight-reel style. Jeff Hendrick's ridiculous second yellow card looked like he had no idea about the first, but set up a wild finish. That was made even crazier by a bad injury to Schar, who was stretchered off with what looked like a long-term problem after being kicked squarely below the knee, as Newcastle finished with nine men.
2. Almiron's finest hour: Ex-Atlanta United star Miguel Almiron had arguably the best game of his Newcastle career in his 78th appearance for the club, producing two goals and showing not just his threat and industry — which is usually apparent — but control of his talent and cool in the final moment. He upped his NUFC goals from 11 to 13, and added three tackles while winning 6-of-9 duels when forced to deploy his best defensive stance.
3. Southampton woes continue: Saints' fifth-straight loss showed that there were still new ways for them to drop points. Deserved 2-0 losers at Leicester and home to Arsenal, Southampton was vastly superior to Villa in 1-0 loss at St. Mary's and then finished with nine men in a 9-0 midweek loss at Manchester United. The side's bad run of injuries combined with the overall chaos of Saturday to highlight that sometimes things just aren't going to go your way in the Premier League.
Man of the Match: Allan Saint-Maximin
Only went 67 minutes during his first start in ages, but the Frenchman was electric in setting up the first two goals and completing seven of nine dribble attempts while winning 12-of-15 duels and drawing three fouls. He's their no-doubt best talent and it's hard to imagine someone won't be waving big money at Newcastle for his services soon.
Newcastle – Southampton recap
Willock, 21, feasted off the spoils of Allan Saint-Maximin playmaking to open the scoring, as the assisting Frenchman made his first start in months.
The youngster was on the right side when Saint-Max took a long ball out of the air with his chest and dusted off his mark to sent a pass through the 18 for Willock to slot home.
Almiron had it 2-0 when he took a pass from Saint-Maximin and hit a cross/shot that turned off Bednarek's heel to fool Alex McCarthy.
That's when Minamino got on the score sheet with a vicious finish to pull Saints back within one, Ryan Bertrand getting the assist.
But Almiron was back amongst the goals on a 3v2 break before halftime, restoring Newcastle's two-goal lead from inside the 18.
Shelvey came close to scoring Newcastle's fourth just after the break and would've been ruing that miss when Ward-Prowse whipped out his latest set-piece marvel in the 48th minute.
[ VIDEO: Premier League highlights ]
No sooner had that happened that Jeff Hendrick took one of the worst (read: least necessary) second yellow cards you'll ever see to put Newcastle down a man but up a goal for the final 40 minutes.
Saints were so nearly level when Danny Ings struck the far post and the ball headed back into traffic but straight into Karl Darlow's hands.
Che Adams then had the ball in the goal with a 65th-minute goal but Ings was offside in the build-up.
But Southampton found no answer even after Schar was kicked hard on the shin and could not move his right leg, leaving Newcastle with nine men and no subs for the final 15 minutes.
More PL action
Liverpool vs Watford final score: Reds go top
Three things we learned from Norwich City – Liverpool; video highlights
Three things we learned from Sheffield United – Burnley

redpajama_set_name: RedPajamaCommonCrawl


A lot of valuable experiences related to success and dedicated production was shared between participants. The importance of dedicated and loyal employees was stressed with focus on how a company can create a workplace with this spirit. In the speed dating seminar, many meetings were conducted and new relations created across the network.


text:
Our International Alliance
Our Saleroom
Operating since 1826, Lyon & Turnbull are one of the longest established fine art and antiques auction houses in the United Kingdom. In the 21st century, we combine dedicated personal advice and attention with innovative, thoughtful marketing and a deep understanding of the world of art and antiques to offer an unparalleled auctioneer service.
We hold approximately 30 auctions per annum, including specialist categories such as Scottish paintings, British & European paintings, Asian works of art, jewellery, silver, antiques, books, furniture and contemporary art and design. For details of the categories of art and antiques we sell, please visit the departments page.
In addition to our specialist auctions, our raison d'etre are single-owner collection sales. Each one carefully crafted to accentuate the quality and uniqueness of the particular collection, generating un-missable events for collectors, and high (often record) prices for our clients.
Lyon & Turnbull operates internationally, providing valuation and auction services worldwide, and ensuring our client's works of art and antiques are seen, desired, and purchased by the world's largest art markets in Europe, America and Asia.
"You did everything you said you would in respect to marketing the collection and I think the enormous interest generated was testament to that. Creative, passionate and enthusiastic - exactly what you promised us!"
Jonathan Brown, Drambuie Liqueur Company
We maintain offices and galleries in London and Glasgow, and our neoclassical headquarters and saleroom in Edinburgh is, we believe, the most beautiful in the UK. We also maintain an alliance with the oldest auction house in the USA, Freeman's, to take advantage of their Philadelphia salerooms and well developed client contacts.
However, our international reach and commercial success are only part of our appeal. The real reason why our clients choose us over our competitors is the knowledge and approachability of our experts, and the efforts we make to understand and fulfil our clients' tastes and needs.
Expanding its reach into the international market, Lyon & Turnbull formed a marketing alliance with Freeman's of Philadelphia, in 2000.
Freeman's holds an esteemed place in America's history as its oldest auction house, and as one of the country's first family owned businesses. For seven generations, since 1805, Freeman's has made up an integral part of the country's auction culture, taking part in countless, often historically significant sales of jewelry, fine art, furniture, and antiques on behalf of private collections, estates, and museums.
The company began by the efforts of a printseller who came to America from London, Tristram Bamfylde Freeman. After an order from Pennsylvania Governor Thomas McKean, Freeman was appointed to the office of auctioneer in Philadelphia, where he subsequently began the company. Over the years Freeman's has grown both nationally, opening regional offices throughout the US, and internationally, expanding into the European market after forming the alliance with Lyon & Turnbull. The current Chairman of Freeman's is Samuel M "Beau" Freeman II, the sixth generation of the Freeman family to run the auction house.
Lyon & Turnbull Directors, John Mackie and Paul Roberts, present a silver gavel to Samuel M. "Beau" Freeman, as part of Freeman's bi-centennial celebrations in 2005.
This alliance boosts sales potential in the US market, while maintaining one of the core values of both companies: one-to-one client service.
Our joint magazine, International View is produced twice a year – highlighting both past and upcoming sales for both houses as well as editorial pieces about other areas of the art world.
Visit Freeman's of Philadelphia
"Having recently commenced business as auctioneer and appraiser, I take the liberty most respectfully to solicit your patronage..." Jonathan Lyon, 1826
Established in Edinburgh in 1826, Lyon & Turnbull are Scotland's oldest firm of auctioneers. In the nineteenth century the firm conducted regular auctions in Edinburgh, handling a bewildering variety of art and antiques including such highlights as furniture from the Queen's residence at Balmoral. In the early 1840's Lyon & Turnbull moved premises to George Street, and continued selling fine art and antiques at this address (changing ownership in 1864) for over 130 years until 1999. The legacy of Lyon & Turnbull's George Street premises can still be seen, as some of the old signage is still visible on Thistle Street Lane North.
In 1999 Lyon & Turnbull was acquired by a group of auctioneers previously part of the Phillips team, an auction house that was the third largest in the world during the 1990s. They were joined in the enterprise by Sir Angus Grossart, Chairman of one of Scotland's leading merchant banks Noble Grossart and a past Chairman of The Trustees of the National Galleries of Scotland. The company's stated aim on regeneration was: "To rescue a national institution for Scotland and to establish a high quality auction house with an international footprint from a base in Edinburgh."
Since re-launching, Lyon & Turnbull has become the largest and most successful auction house in Scotland, and has become respected internationally as a diligent and trusted firm of auctioneers.
We are very proud of our stunning saleroom in the centre of Edinburgh.
This building is an important example of the work of Archibald Elliot (1760-1823), one of Scotland's leading architects in the early 19th century.
Designed and built as a church for the great and good of Edinburgh's fashionable New Town, the design is somewhat unusual due to Elliot's adoption of the Greek Revival style for an ecclesiastical commission. The church construction began in May 1820 and it opened for its first service in May 1821. A plan and elevation signed by Elliot shows that he originally intended the building to have a tower and spire, but, perhaps for financial reasons, these were never built. The ceiling rose in the auditorium originally held a large octagonal gasolier, which was removed in 1853.
Broughton Place Church featured in the 1981 multi-Oscar winning film Chariots of Fire. The Broughton McDonald congregation left in 1991 and most of the interior furnishings were dispersed; the organ and organ screen being relocated to a church in Italy. Malcolm Fraser Architects won the Edinburgh Architectural Association Regeneration/Conservation Award and Roses Design Award in 2006, having completed the conversion of the church to an auction house in 2003.
Today the auction house is a bright and airy space, combining the grandeur, materials, and high quality workmanship of the past with a distinctly modern design aesthetic. As such, the Broughton Place saleroom is perfectly suited to Lyon & Turnbull's style and we are proud to be the guardians of the grade-A listed property for future generations of architecture, art and antique lovers.
Lyon & Turnbull Ltd | 33 Broughton Place, Edinburgh, EH1 3RR | Registered in Scotland No: 191166


text:
iniva iniva
Stuart Hall Library
Library & Archive Collection
Access & Visiting
Emotional Learning Cards
Iniva Team & Board
Annotating Art's Histories
© Iniva 2020. All rights reserved. Site by Fiasco Design.
Welcome to Iniva's new website. We are in the process of updating content throughout. We welcome your feedback at info@iniva.org
All Events News Projects Shop
Scat: Sound and Collaboration Sonia Boyce
05 Jun-27 Jul 2013
Scat: Sound and Collaboration is an exhibition by Sonia Boyce which explores the significance of sound in art. Presented by Iniva across all the public spaces of Rivington Place, Scat brings together two immersive video works for the first time along with artefacts from the Devotional Collection, Boyce's archive of CDs cassettes, vinyl records and other ephemera.
Sonia Boyce
Still taken from Oh Adelaide (2010) ©Sonia Boyce, courtesy the artist
Sonia Boyce's new exhibition Scat, presented by Iniva, brings together two immersive video works for the first time with The Devotional Collection, Boyce's archive and collective memorialisation of black British women in the music industry. As a result, the exhibition places a spotlight on her interest in the archive as arts practice. 'Just the very act of putting something in an archive suggests its future use is beyond the control of the past. But we don't have to settle for the past as it is presented. The past is not fixed'.
This question of playing with history is a recurring thematic in Boyce's work. Combined with the title's emphasis on dispersal, the works presented in Scat have a visceral effect and impact on the visitor's sense of time and place, and will encourage them to reflect on the significance of sound in different settings.
The Devotional Collection, is an archive of CDs, cassettes, vinyl records and other ephemera relating to black British women in the music industry that Boyce has been developing, through the involvement of a wide range of participants, since 1999. It includes The Devotional Wallpaper (2008 -), a work in which Boyce sets out a roll-call of 200 female luminaries, memorialised as a large-scale printed wallpaper. As she says, 'Many of the named performers would probably hate being collected under that rubric. The act of collecting is not on their behalf, it's not to represent them. It's really about an unplanned way that a diverse range of public listeners have built a collective memory.'
Oh Adelaide (2010), a collaborative work by Sonia Boyce and sound artist Ain Bailey, incorporates found film footage of the jazz singer and entertainer, Adelaide Hall (1901-1993). This video work is also symptomatic of the collective memorialisation in The Devotional Collection. As Boyce says, 'Oh Adelaide is a digital mash up where vision and sound sit awkwardly side-by-side. I decided to treat this digital footage as something elastic. Light and dazzling whiteness becomes the material presence that reveals and threatens to obliterate everything in its path, which Adelaide Hall and her accompanying pianist emerge and disappear within. As the audience, we're urged to fight to keep track of her – to capture her.'
In For you, only you (2007), Boyce orchestrates an unlikely meeting between an Early Music consort, Alamire, and a contemporary sound artist, Mikhail Karikis. Boyce's video captures the performance of Karikis's sound-work, which in turn imagines an encounter between his fractured vocalisations and a deconstructed Renaissance masterpiece- Tu solus qui facis mirabilia/'You alone can do wonders' by the Franco-Flemish composer Josquin Desprez. While challenging the boundaries between noise and music, For you, only you interrogates notions of harmony and dissonance, conformity and difference.
Sonia Boyce has had a long-standing relationship with Iniva. Indeed, an artist-residency sponsored by Iniva in 1997, kick-started the particular focus on sound that we see in her work today. Initially an image-maker and painter, more recently Boyce has become increasingly involved in collaborative practice. For you, only you was commissioned by the Ruskin School of Drawing & Fine Art in partnership with the De La Warr Pavilion, Locus+, Milton Keynes Gallery and Model Arts and Niland Gallery and with the support of Arts Council England.
Sonia Boyce is a British Afro-Caribbean artist, living and working in London. Her early work addresses issues of race, gender and contemporary urban experience expressed in large pastel drawings and photographic collages. Since the 1990s her work has shifted materially and conceptually to incorporate a variety of media that combines photographs, collages, films, prints, drawings, installation and sound. Increasingly, she works with other people to produce 'improvisational collaborations', with recent work bringing the audience into sharper focus as an integral part of the artwork. By doing this Boyce demonstrates how cultural differences might be articulated, mediated, enjoyed.
About Iniva
Iniva (Institute of International Visual Arts) works at the intersection of society and politics. It engages with new ideas and emerging debates in the contemporary visual arts, reflecting in particular the diversity of contemporary society. We work with artists, curators, creative producers, writers and the public to explore the vitality of visual culture. Iniva is supported by Arts Council England.
About Rivington Place
Opened in 2007, Rivington Place is home to Iniva and Autograph ABP. Designed by architect David Adjaye OBE, this award winning building is dedicated to the display, debate and reflection of global diversity issues in the contemporary visual arts. An ongoing programme of exhibitions and events is presented by Iniva in the 2 project spaces and Iniva's Learning Space. It is also home to the Stuart Hall Library, Iniva's unique research library with specialist resources and collections about contemporary international visual art.
Exhibition listings Information
Exhibition: Scat – Sonia Boyce: Sound and Collaboration
Dates: 5 June – 27 July 2013
Venue: Rivington Place, London, EC2A 3BA
Rivington Place public opening hours:
Tuesday, Wednesday, Friday: 11am – 6pm
Late Thursdays: 11am – 9pm (last admission 8.30pm)
Saturday: 12noon – 6pm
info@rivingtonplace.org
www.rivingtonplace.org
Tubes: Old Street/Liverpool Street/Shoreditch High St
Rivington Place is fully accessible, for parking & wheelchair facilities call +44 (0)20 7749 1240
For high resolution images please contact:
Giulia Crossley, Communications
gcrossley@iniva.org or call 020 7749 1246
Good Morning Freedom
Scat: Sound and Collaboration Press Release
Introduction to Scat: Sound and Collaboration
Read the Scat Exhibition Guide
Scat Gallery
Sonia Boyce Still taken from Oh Adelaide 2010 ©Sonia Boyce, courtesy the artist
Sonia Boyce Devotional Wallpaper ©Sonia Boyce, courtesy the artist
Sonia Boyce Installation view of Scat, Iniva at Rivington Place 2013
Artist Talk 13 Jun 2013
Nonsense: working with voices
Panel Discussion 27 Jun 2013
Talking about artists making records*
Artist's Presentation 04 Jul 2013
Ode To No. 6 | The Pitch Sisters Live!
Curatorial Exhibition Tour 20 Jul 2013
Scat: Curator's Tour
Designed and crafted by


text:
Parents want police who pointed guns over 4yo daughter's shoplifting fired
By 9Honey with AP| 7 months ago
A US couple has called for police who pointed guns at them after their four-year-old daughter took a doll from a store without their knowledge fired.
Video emerged this week of officers confronting Dravon Ames and his pregnant fiancée Iesha Harper in Phoenix, Arizona after they visited a Family Dollar store on May 29.
The couple say they had pulled into an apartment complex to drop their daughter off at a babysitter after shopping when police began banging against their window and yelling threats.
"Our hands are up, we're just trying not to get shot, trying to stay calm," Ames recalled to reporters.
"He had a gun drawn."
Now, the couple, who have two children, say they don't accept the apologies of the city's police chief and mayor and have not received a face-to-face apology since outcry erupted over the video.
"It was very frightening for me and my children," said Harper, who is six months pregnant.
The couple's four- and one-year-old daughters witnessed the encounter.
Cops pulled guns on parents after daughter, 4, took doll from shop, they claim
"It was very frightening for me and my children," Iesha Harper says of the incident. (PA AAP)
Once taught to trust law enforcement, Harper said the couple's older daughter "is now terrified of the police, wets the bed, wakes up crying."
The 12-minute video released on Friday shows officers aiming guns and yelling profane commands at a man and a pregnant woman holding a baby.
One officer can be seen handcuffing Ames on the ground and a second time against a police car, kicking him in the process.
He is also heard yelling: "When I tell you to do something, you f------ do it."
Another officer appears to draw a gun on the passenger side of the couple's vehicle before Harper exits holding a small child, with a second next to her.
Neighbours are seen gathering in the parking lot, with one women witnessing the confrontation stepping in and taking the children to get them out of harm's way.
"I really thought he was gonna shoot me in front of the kids," Harper tells CNN.
The race of the officers investigating the shoplifting report last month is not known.
Ames and Harper have filed a US$10 million claim against the city alleging civil rights violations by officers.
They allege there were no sirens or lights to indicate they were being pulled over, and that the officer who pointed a gun at them did not identify himself as police straight away.
The couple are calling for the officers who pulled guns on their family to be fired. (PA AAP)
The Phoenix Law Enforcement Association is urging calm, saying in a statement the police union will not form an opinion until an official investigation is complete.
Mayor Kate Gallego posted an apology to the family on Twitter on Saturday.
Police Chief Jeri Williams, a black woman, spoke out Friday on Twitter about how the incident was handled and then apologised to the family, community and public during a television interview Sunday.
"I don't accept the apology," Harper said Monday, with Ames saying, "It's not sincere."
The police chief has said an internal investigation into the officers is underway.
"It's absolutely absurd you can talk about apologising and seek to move beyond something without actually disciplining and firing these officers," said the Rev. Jarrett Maupin, a local civil rights advocate who organised the news conference.
The city has organised a community meeting about the encounter for Tuesday.


Thank you to everyone who came out to donate blood! The event was a success! Because of you all, we will be donating 5 pair of glasses to 5 different children in need within the Lincoln County School System. Thank you, again!


You are Rowdy... Love America... and Support the 2nd Amendment. We made this shirt for you!! Introducing the "Rowdy Patch" which will be present on all of our sleeves moving forward.
This Item is in stock and ships within 1-2 business days.


text:
\section{The Eisenhart-Duval Lift}

The Eisenhart-Duval lift \cite{Eisenhart,Duval:1984cj,Duval:1990hj}
 has proved itself a powerful tool
for ``geometrising'',  and thus elucidating,
the symmetries of, Lagrangian    dynamical systems
\cite{Cariglia:2014ysa,Cariglia:2016oft}.
In the simplest case one starts from
a Lagrangian of the form
\begin{eqnarray}
L= \frac{1}{2} \dot {\bf x} ^2 - V({\bf x} ,t) \,, \label{system}
\end{eqnarray}
where ${\bf x}$ is a position vector in $\mathbb{E}^n$
with components $x^i$,
and regards  solutions of the Euler-Lagrange equations
of (\ref{system}) as corresponding to the null geodesics
of an $n+2$ Lorentzian spacetime with coordinates
$X^\mu = (x^i,t,s)\,,i=1,2,\dots,n$
admitting an $n+2$ dimensional  Bargmann structure,
that is   covariantly constant null killing vector field
$\frac{\partial}{\partial s}$.

The Lorentzian metric takes the form
\begin{eqnarray}
g_{\mu \nu} dX^\mu dX^\nu = 2Ldt^2 + 2 dt ds =
d {\bf x}^2 + 2 dt ds - 2V({\bf x},t) dt ^2
\label{Elift}
\end{eqnarray}
which may be recognized as a Brinkmann
or pp-wave \cite{Brinkmann:1925fr,Ehlers:1962zz}.
In particular, if for every  $t$,   $V({\bf x},t)$  is a harmonic function,
then the metric $g_{\mu \nu}$ is Ricci flat. That is,
a plane fronted gravitational wave with parallel rays  \cite{Ehlers:1962zz}.
If $V({\bf x},t)$ is both harmonic and quadratic in ${\bf x}$
we have plane gravitational waves with  partially  broken Carroll
symmetry \cite{Duval:2017els} which may be used to model
the gravitational waves currently being detected by LIGO
\cite{Zhang:2017rno,Zhang:2017geq}. The coordinates $t$ and $s$ may be thought
of as Newtonian and Carrollian time respectively \cite{Duval:2014uoa}.

In addition to elucidating the symmetries, of and
equivalences between, dynamical systems
the Eisenhart-Duval construction offers the prospect of applying
global techniques
in  Lorentzian geometry and causal theory  to the  dynamical systems
which are their  shadows  \cite{Minguzzi:2006gq,Minguzzi:2012cx}.

\section{Isometric Embedding of the  Eisenhart-Duval Lift
into a flat space with two times}

An immediate  generalisation of a result valid for $n=3$
\cite{Collinson} is that the $n+2$  dimensional Lorentzian
spacetime with metric (\ref{Elift})
 may be isometrically embedded into flat space
$\mathbb{E} ^{n+2,2}$  equipped with the flat metric
\begin{eqnarray}
 \eta_{AB}dZ^A dZ^B =
d Z^i dZ ^i   - (dZ^{n+1})^2  +(dZ^{n+2}) ^2 + (dZ^{n+3})^2
  -(dZ^{n+4} )^2 \label{flat}
\end{eqnarray}
 $A=1,2,\dots ,n+4$, whose signature is $n+2,2$
as a co-dimension 2  Lorentzian submanifold, i.e. as an
$9n+1)$-brane \footnote{The
 word ``p-brane'' in its most general sense refers
to an object of $p$-dimensional spatial extent.}
with world volume coordinates $( {\bf x},s,t )$,
whose  induced metric coincides with
(\ref{Elift}).  We need  not  assume any special property of
$V({\bf x},t)$.

The embedding is given explicitly by
$Z^A=Z^A ( {\bf x}, t,s) $, $A=1,2,\dots , n,n+1,n+2,n+3,n+3 $ where
\begin{eqnarray}
Z^i &=& x^i\,,\qquad i=1,2,\dots, n  \nn\\
Z^{n+1}&=& \frac{1}{\sqrt{2}} (tV({\bf x},t) +s + t)\,, \nn\\
Z^{n+2}&=& \frac{1}{\sqrt{2}} (tV({\bf x},t)  +s - t )\,,\nn \\
Z^{n+3}&=& \frac{1}{\sqrt{2}}  (V({\bf x},t) + \frac{1}{2} t^2) \,,\nn\\
Z^{n+4}&=& \frac{1}{\sqrt{2}} (V({\bf x},t) -  \frac{1}{2} t^2) \,. \label{embed}
\end{eqnarray}
On may check that if one substitutes (\ref{embed}) into the flat metric
(\ref{flat}) one obtains the Eisenhart-Duval (\ref{Elift})  lift of the
dynamical system (\ref{system}).
\section{Equation of motion}

We now enquire whether our  $(n+1)$-brane satisfies the
Euler-Lagrange equations arising from extremizing the world-volume.
It is traditional in much of the geometry literature to refer such surfaces
as ``mininal'' without prejudice as to whether the extremum does actually
minimize  the world-volume, hence its use in my title. It certainly
seems to me better than ``on-shell'' for example.
Bearing that proviso in mind, we vary the action functional
\begin{eqnarray}
S[Z^A(x^\mu)]= \int \sqrt{-g} \,d^{n+2} \, x
\end{eqnarray}
with respect to the $n+4$ world-volume scalar fields $Z^A(x)$.
\begin{eqnarray}
\delta S&=& \frac{1}{2} \int \sqrt{-g} g^{\mu \nu}
\delta (\eta _{AB} \partial_\mu Z^A \partial _\nu Z^B )\, d^{n+2}\,  x \\
&=&  \int \sqrt{-g} g^{\mu \nu} \eta _{AB}
\partial_\mu Z^A \partial _\nu \delta Z^B \, d^{n+2}\, x \\
&=-& \int \partial_\nu ( \sqrt{-g} g^{\mu \nu} \eta _{AB}
\partial_\mu Z^A)  \delta Z^B \, d^{n+2}\, x \qquad
        +{\quad boundary \quad term}
\end{eqnarray}
Thus the Euler-Lagrange equations are
\begin{eqnarray}
\partial_\nu ( \sqrt{-g} g^{\mu \nu} \eta _{AB}
\partial_\mu Z^A) =0 \,,
\end{eqnarray}
which is equivalent that the embedding coordinates $Z^A$ are harmonic
functions with respect to the world-volume metric.
In out case $\sqrt{-g}=-1 $,  therefore we  require
\begin{eqnarray}
0&=&  \partial _\mu \bigl(  g^{\mu \nu}  \partial_\nu Z^A \bigr )  \\
&=& \Bigl \{  \frac{\partial^2}{\partial {\bf x} \cdot  {\bf x} } -
2\frac{\partial ^2 }{\partial t \partial s }
+ 2V({\bf x},t) \frac{\partial ^2}{\partial s ^2} \Bigr \}  Z^A =0 \,.
\end{eqnarray}

Thus in order to satisfy the equations of motion of an $(n+1)$-brane
it suffices that $V({\bf x},t)$ is harmonic in the $n$ vector ${\bf x}$.
That is, if (\ref{Elift}) is  a Ricci flat pp-wave.

An especially appealing example is the case of
the Newtonian equations of motion of  $N$ point particles in $\mathbb{E}^3$
for which $n=3N$.

\section{Newtonian Cosmology}

Although often thought of as a quintessentially
general relativistic effect, the Slipher-Lemaitre-Hubble
expansion may be described (at least for distance small
compared with the Hubble radius, i.e. for or small redshifts)
using Newtonian point particle mechanics (see  \cite{Gibbons:2013msa}
and references therein).
For $N$ point self-gravitating particles with position vectors
${\bf r}_a$, $a=1,2,\dots, N$,
and positive masses $m_a$, a suitable Lagrangian is
\begin{eqnarray}
L_{\rm Newton} = \frac{1}{2} \sum_{1\le a \le N}  m_a {\dot {\bf r}_a}  ^2 +
\sum_{1\le  a< b \le N} \frac{Gm_am_b }{|{\bf r}_b - {\bf r}_a   | }  \,. \label{Lag} \end{eqnarray}
To convert (\ref{Lag}) to the form (\ref{system}) we set
\begin{eqnarray}
 {\bf x}= ( \sqrt{m_1}{\bf r}_1,  \sqrt{m_2}{\bf r}_2, \dots,
        \sqrt{m_N}{\bf r}_N )
\end{eqnarray}
Then
\begin{eqnarray}
\frac{\partial^2 V({\bf x},t)  }{\partial {\bf x}  \cdot  \partial {\bf x}}
= \sum _{ 1\le a \le N  }
-\frac{1}{m_a}  \frac{\partial^2}{\partial {\bf r}_a  \cdot
  \partial {\bf r}_a }   \Bigl( \sum_{1\le  a< b \le N} \frac{Gm_am_b }{|{\bf r}_b - {\bf r}_a   | }      \Bigr ) =0 \,.
\end{eqnarray}

One may also treat perturbations around a background
flat Friedmann-Lemaitre model with scale factor $S(t)$
using the Dmitriev-Zeldovich equations \cite{Ellis:2014sla}
For which a suitable Lagrangian is
\begin{eqnarray}
L_1= \frac{1}{2} \sum_{1\le a \le N}  m_a  \big( {\dot {\bf r}_a}  ^2  + \frac{\ddot S}{S} {\bf r}_a^2 \bigr )    +
\sum_{1\le  a< b \le N} \frac{Gm_am_b }{|{\bf r}_b - {\bf r}_a   | }  \,. \label{Lag} \end{eqnarray}
In this case the Eisenhart-Duval lift is no longer  Ricci flat.

In fact one may define a comoving coordinates ${\bf y}_a$ by
\begin{eqnarray}
{\bf r}_a= S(t) {\bf y}_a \,.
\end{eqnarray}
The equations in terms of the comoving coordinates  ${\bf y}_a$
may be obtained from a Lagrangian which differs from (\ref{Lag})
by a total time derivative
\begin{eqnarray}
L_2= \frac{1}{2} S(t)^2 \sum_{1\le a \le N}  m_a {\dot {\bf y}_a}  ^2 +
\frac{1}{S(t)} \sum_{1\le  a< b \le N} \frac{Gm_am_b }{|{\bf y}_b - {\bf y}_a   | }
\,. \label{DZ}
\end{eqnarray}

One might wonder whether $L_1$ or $L_2$ give rise to
an Eisenhart-Duval lift which lifts to a brane in higher dimensions.
We leave that question for future study. Another intriguing possible
connection is with \cite{Cariglia:2015dha}.

\section{Curved Configuration spaces}

It is possible to replace Euclidean $\mathbb{E}^n$  space by
a curved configuration space $\{Q,g_{ij}\}$ with time independent
metric $g_{ij}(x^k)$ and the discussion
goes through almost unchanged. The embedding
equations (\ref{embed}) provide an embedding into the metric product
$Q \times \mathbb{E}^{2,2}$. Of course it may be that
the configuration space admits an isometric embedding
into some higher dimensional flat space $\mathbb{E}^{u,v}$
in which case we have a brane in $\mathbb{E}^{u+2,v+2}$.

A further extension is obtained by the addition of a velocity dependent
term to the Lagrangian so that
\begin{eqnarray}
L= \frac{1}{2} g_{ij}(x^k)  \dot x^i \dot x^j + A_i(x^k) \dot x^i -V(x^k,t)
\end{eqnarray}
and the lift becomes
\begin{eqnarray}
g_{\mu \nu} dX^\mu dX^\nu = - \bigl(2 V + g^{rs}A_rA_s  \bigr )
  dt ^2 + 2 dt ds + g_{ij} (dx^i + g^{ir} A_r dt)(dx^j + g^{js} A_s dt )\,.
\end{eqnarray}
If we define
\begin{eqnarray}
\tilde V= V + \frac{1}{2} + g^{rs}A_rA_s
\end{eqnarray}
we have define
\begin{eqnarray}
\tilde Z^{n+1}&=& \frac{1}{\sqrt{2}} (t \tilde V({\bf x},t) +s + t)\,, \nn\\
\tilde Z^{n+2}&=& \frac{1}{\sqrt{2}} (t \tilde V({\bf x},t)  +s - t )\,,\nn \\
\tilde Z^{n+3}&=& \frac{1}{\sqrt{2}}  (\tilde V({\bf x},t) + \frac{1}{2} t^2) \,,\nn\\
\tilde Z^{n+4}&=& \frac{1}{\sqrt{2}} (\tilde V({\bf x},t) -  \frac{1}{2} t^2) \,. \label{embed}
\end{eqnarray}
and lift the Eisenhart-Duval lift  to $Q \times \mathbb{E}^{2,2}$
with a twisted metric product
\begin{equation}
g_{ij} (dx^i + g^{ir} A_r dt)(dx^j + g^{is} A_s dt ) -
 (dZ^{n+1})^2  +(dZ^{n+2}) ^2 + (dZ^{n+3})^2
 -(dZ^{n+4} )^2 \,.
\end{equation}

\section{Discussion}

Using the fact that  one may regard the motion of any
natural mechanical system defined on an $n$-dimensional,
configuration space as   the shadows of    null geodesics moving
in an $(n+2)$ dimensional  Lorentzian spacetime, we have shown that
in the case that the the configurations space
is $n$-dimensional Euclidean space, the Eisenhart-Duval lifted spacetime
may be isometrically embedded in  an $(n +4)$ dimensional
flat space  with two time coordinates.
The  Eisenhart-Duval  lift is then the history of an $(n+1)$-
brane moving in an ambient flat space
with two time coordinates. In the particular case case that
the Eisenhart-Duval  lift is
Ricci flat, the $n+1$ is a ``minimal''  surface: it  extremises
the spacetime volume of the $(n+1)$ brane. In other words it satisfies
the equations of motion arising from the generalisation to
$(n+1)$ dimensions of the Nambu-Goto action
of string theory.

That the embedding requires two time coordinates
may seem worrying from a physical point of view
but, as pointed out by Penrose \cite{Penrose:1965rx}  many years  ago, is mathematically inevitable
because the Eisenhart lift is a Bargmann manifold
which  may be regarded  as  gravitational wave.
If they   satisfy   the standard energy conditions, such spacetimes cannot be globally-hyperbolic
and hence cannot admit a time function.
However if the embedding were into a  higher dimensional
Minkowski spacetime with just one time, the intersections with parallel  spacelike hyperplanes would
provide a time function.

Embeddings into flat spaces with two times   are in fact not unusual. The Ur-example is of course
$(n+1)$-dimensional anti-de-Sitter spacetime. This isometrically embeds
as  a quadric in
$\mathbb{E}^{n,2}$. It has closed timelike curves
(CTC's) and  is not
globally-hyperbolic.  Even if one eliminates the CTC's by passing  to its
universal covering spacetime,
it remains non-globally-hyperbolic

In fact it is possible to isometrically embed more
complicated supergravity solutions representing self-gravitating branes rather  than the test or probe branes
considered in this paper,
into  flat spacetimes with two times. In particular BPS branes.
In this case the embedding  covers the horizon and hence embeds a  non-globally-hyperbolic spacetime
\cite{Gibbons:1998th,Andrianopoli:1999kx, Andrianopoli:1999fn}.
As pointed out in \cite{Tipler:1986qu} the intuitively appealing picture of
strings, or
branes splitting and joining, changing  their
topology as they do so is incompatible with an everywhere non-singular
Lorentzian world  sheet or world volume metric . At the level of classical
solutions, examples show that the resolution of this paradox is that the
world sheet or world volume can remain a smooth submanifold of
Minkowski spacetime, but allow the induced metric to change signature
\cite{Gibbons:2004dz}. That does not happen in the examples exhibited above,
but changes of signature of various types can certainly be expected for
branes moving in a spacetime with more than one time.

At the quantum level, S-matrix amplitudes are typically calculated in
practice by considering Riemann surfaces embedded in Euclidean
space as a function of external Euclidean momenta  and then analytically
continued to timelike   Lorentzian momentum. This is fine
for S-matrix elements but one then faces the problem
of what to do in the context of  cosmology.

For many years Itzhak Bars has been advocating
an approach to the symmetries
of string theory and supergravity based on two  time physics
\cite{Bars:2000qm,Bars:2010zz,Araya:2013bca} .
It would be surprising if there were not some relation of these considerations
to the present paper.  Eisenhart-Duval lifts with  two times
have already been shown
to arise for higher derivative systems \cite{Galajinsky:2016zer}.
If they can be isometrically embedded in higher dimensions even more
time dimensions are likely to be required.

Finally let me remark that very recently  the authors of
\cite{Bergshoeff:2020xhv} found, using branes,  a  relation  between
the Carroll and Galilei groups which is differs  from that
in \cite{Duval:2014uoa}. An interesting question is how this fits in
with the construction of the present paper.

\section{Acknowledgements}

The author thanks Marco Cariglia, Peter Horvathy and Paul Townsend
for helpful comments on the manuscript.