\section{Introduction}
\label{sect.01}

Meson spectroscopy continues to be an important issue both for
experimentalists and theoreticians.  More precise experimental data
have appeared in the last years \cite{1,2,3,4,5,6,7,8,9} and a large
number of theoretical works are devoted to light-meson properties
\cite{10,11,12,13,14,15,16,17,18,19,20}. The important idea that
mesons have universal properties, from light-light to heavy quarkonia,
is supported in many studies \cite{21,22,23,24,25,26,27,28} and a
kind of universal  $q\bar q$ potential was used for all mesons in
different models \cite{9,10}, \cite{12,13,14,15,16}, \cite{22,23,24}.
The detailed analysis of meson spectra was done in the relativized
potential model (RPM), introducing a phenomenological (universal)
potential \cite{9}. A convenient systematics of radial excitations
was suggested in Ref.~\cite{11}, where it was assumed that the slope
of the radial Regge trajectories (RTs) has a universal value (with
a good accuracy) for all mesons. However, up to now the discussions
continue about the true value of the slopes of the radial RTs
\cite{24,25,26,27}, and even the linearity of the radial RTs is
disputed \cite{28}. However, the physical effects which are responsible
for the observed universality remain unclear up to now.

Here we use the relativistic string Hamiltonian (RSH) \cite{12,13},
derived in the framework of the field correlator method \cite{29,30},
which allows for expressing the meson properties via two fundamental
parameters: the string tension and the QCD constant $\Lambda$. In
principle, the RSH contains both perturbative and nonperturbative
dynamics, and yields also the spin-dependent interactions, so that,
in general, all possible dynamical regimes in the $q\bar q$ systems
can be addressed. It is the main purpose of our  work to start a
general analysis of the light meson dynamics both in radial and
orbital excitations. However, in the present paper we confine
ourselves to the case of the radial excitations of the vector mesons
$\rho(nS)$ and the ground states with $l=0,1,2$, where the physical
picture is more simple and transparent. Thus, our analysis can be
considered as the first step towards the overall picture, which may
be more complicated.

We show that in light mesons the dynamics is more complicated than
in heavy quarkonia, which manifests itself in two effects: the
so-called screening of the gluon-exchange (GE) interaction  and the
flattening of the linear confining potential, which is especially
important for high excitations. These phenomena occur for extended
objects owing to $q\bar q$ holes (loops), which are created inside
the film subtended by the Wilson loop. These two effects can be
described by the RSH and will be the main subject of our analysis.
As was shown in Ref.~\cite{12}, the RSH defines two regimes: the
string regime, valid for the states with large $l$, $l\geq 3$, and
the potential-like regime, taking place for low-lying states. For
the ground states (with large $l$)  the mass formula:  $M^2(l,n_r=0)
= 2\pi\sigma\sqrt{l(l+1)}$ was derived, which explicitly shows that
the slope of the leading RT is equal to $2\pi\sigma=(1.13\pm
0.02)$~GeV$^2$  with great accuracy (e.g. for $l=3$ the accuracy
is $0.8\%$). To derive this expression it was assumed that in RSH
the centrifugal term (the rotation of the string) gives a large
contribution, while the corrections to the mass from the GE, the
spin-dependent potentials, and the self-energy term are considered
to be small and may be neglected.  However, for low-lying states
these terms in the $q\bar q$ potential are not small \cite{22,23}
and therefore the question arises how to match the mass relations,
valid for the states with large $l$, and those with $l=0,1,2$.
First, we discuss the well-established features of the universal
$q\bar q$ interaction.

In the RSH approach the $q\bar q$  potential is defined in a
gauge-invariant way via the Wilson loop \cite{29,30} and the confining
potential is shown to be scalar and linear (if no quark loops occur):
$V_{\rm C}(r)=\sigma_0 r$, with the string tension
$\sigma_0=0.18(2)$~GeV$^2$ fixed by the slope of the leading RT
(with $j=l+s$) \cite{31,32}. In the static potential, the confining
and GE potentials enter as a sum to satisfy the Casimir scaling,
observed on the lattice with very good accuracy \cite{33,34}. A
very important point is that the parameters of the GE potential
cannot be taken arbitrarily but have to be determined in full
correspondence with the existing information from pQCD \cite{35}.
From high energy experiments the QCD constant
$\Lambda_{\overline{MS}}(n_f=5)$ is now well established, while the
QCD constants for $n_f=3,4$ are defined by matching the coupling
at the quark mass thresholds \cite{35,36,37}; it gives the value
$\Lambda_{\overline{MS}}(n_f=3)=(339\pm 10)$~MeV, if
$\alpha_s(M_Z)=0.1184(7)$ is used  \cite{35}, or a bit smaller
$\Lambda_{\overline{MS}}(n_f=3)=(327\pm 12)$~MeV is obtained for
the new world-average $\alpha_s(M_Z)=0.1177(13)$ \cite{37}. Knowledge
of $\Lambda_{\overline{MS}}(n_f=3)$ is very important, because its
value determines the ``vector" constant $\Lambda_V(n_f=3)$, entering
the vector coupling in the GE potential: $\Lambda_{\rm
V}(n_f=3)=1.4753~\Lambda_{\overline{MS}}(n_f=3)= (485\pm 25)$~MeV
\cite{36}. Besides, as shown recently in Ref.~\cite {38}, the
infrared regulator (IR) $M_{\rm B}$ is not an extra parameter, but
can be expressed via the string tension:
$M_B^2=2\pi\sigma=1.13(11)$~GeV$^2$ (the accuracy of calculations
is estimated to be $\sim 10\%$). Then taking the central values,
$\Lambda_V=0.485$~GeV and $M_B=1.13$~GeV, one defines the two-loop
freezing coupling (called {\em critical}), $\alpha_{\rm crit}=\alpha_{\rm
V}(q^2=0)=0.6065$.  Surprisingly, this value of the two-loop
$\alpha_{\rm crit}$ with a large $\Lambda_{\rm V}\sim 480$~MeV
coincides  with the one-loop phenomenological $\alpha_{\rm crit}$
from Ref.~\cite{9}, where a very small (unrealistic)
$\Lambda(n_f=3)=200$~MeV is used. Knowledge of this freezing constant
$\alpha_{\rm crit} = 0.60\pm 0.04$ is crucially important for heavy
quarkonia, where the GE interaction remains important up to high
excitations. Unfortunately, the role of the GE interaction in light
mesons is not fully understood and here we pay special attention
to the correct definition of the universal potential to distinguish
between true dynamical effects and artifacts coming from  different
fitting parameters, including the constituent masses.  Many features
of the light-meson dynamics become more transparent if one studies
the $S-$wave isovector mesons, which are more simple from the
theoretical point of view, since they are not subject to chiral
effects (with exception of the $\pi$-meson) and for them there does
not exist a complicate centrifugal term in the RSH.

We now pay special attention to the radial RTs with the systematics,
suggested in Ref.~\cite{11}, assuming that the radial RTs are linear in
the ($n_r,M^2$)-plane ($J^{PC}$ is fixed),
\begin{equation}
 M^2(n_r) = M^2(n_r=0) + n_r\mu^2.
\label{eq.1}
\end{equation}
Here  $M(0)$ is the mass of the lowest-lying meson on the RT and
$\mu^2$ is the slope parameter. According to Refs.~\cite{8,11} the
slope $\mu^2$ is approximately the same for all radial RT trajectories,
$\mu^2=1.25\pm 0.15$~GeV$^2$. This strong statement cannot be checked
in many cases, since no sufficient experimental information is
available about high radial excitations, with exception of the
$\rho$ family, and in the literature there are also other predictions
for $\mu^2$  \cite{18,25}, and even the linear behavior is disputed
\cite{28}. The question is whether this slope is universal or not.
Notice, that in different RPMs \cite{9,16,23} a much larger mass
difference $\mu_1^2= M^2(\rho(1450)) - M^2(\rho(775)) = (1.54\pm
0.04)$~GeV$^2$ is obtained and this value agrees with experimental
$\mu_1^2({\rm exp.})$, if the central values of the mass, $M(\rho(775))$
and $M(\rho'(1450))$ from the Partical Data Group (PDG) \cite{1}
are taken. An even larger value $\mu_1^2({\rm exp.)}=1.63(2)$~GeV$^2$
corresponds to the recent BaBar data for $M(\rho')=(1493\pm 15)$~MeV
\cite{5}. In the present paper we will show that this large mass
difference is not accidental and occurs because the $\rho(775)$
mass is ``too small'' due to a large GE and self-energy contributions.
For that reason (in contrast to other RTs) the linear behavior of
the radial $\rho$-trajectory starts with the first excitation
$n_r=1$.

Our calculations here are done in closed-channel approximation,
neglecting the widths and hadronic shifts, while in a strict sense
light mesons have to be studied as many-channel systems, taking
into account a contribution of every channel to the meson wave
function (w.f.). But such many-channel calculations form a very
difficult task, which needs individual consideration of every meson
and a complete theory of meson decays, which does not exist now.
Therefore, calculations in closed-channel approximation continue
to be very important: they allow for a separation of the conventional
$q\bar q$ mesons from multi-quark systems of a different nature
\cite{7}. Moreover, the influence of open channels can be effectively
taken into account, introducing the string tension $\sigma(r)$
depending on the separation $r$ \cite{23}. This effect occurs owing
to the creation of virtual quark loops in the Wilson loop, causing
the string tension to decrease and depend on $r$, and this effect
is very important for higher radial excitations,  while the ground
states are not affected by this {\em flattening} effect, since they
have relatively small sizes.

An important point is that one can introduce the critical value
of the string tension, $\sigma_{\rm crit}$, when the breaking of
the $q\bar q$ string takes place. If the string tension is taken
as in Ref.~\cite{23}: $\sigma(r) = \sigma_0 (1 - \gamma f(r))$ (with
$\sigma_0=0.18(2)$~GeV$^2$), then at not too large distances, $r\leq
1.2$~fm the string tension is almost constant, $\sigma(r)\approx
\sigma_0$, while at larger distances the function $f(r)\rightarrow
1$ and the critical value is
\begin{equation}
 \sigma_{\rm lim} = \sigma_0 (1 - \gamma).
\label{eq.2}
\end{equation}
The calculations show that a good description of the radial excitations
is reached if the parameter $\gamma=0.43\pm 0.03$ is used as a
fitting parameter. Moreover, the value of $\gamma$ strongly affects
the slope of the radial RT and therefore it can be extracted from
this slope, if there are good experimental data for the mass of the
excitations with $n_r\geq 2$. In particular, more precise data on
the masses of $\rho(1900)$ and $\rho(2150)$ could allow for
distinguishing between the value $\mu^2=1.43(13)$~GeV$^2$, suggested
in Ref.~\cite{18}, and $\mu^2=1.365(108)$ obtained in Refs.~\cite{8,11}
from the analysis of the Crystal Barrel data \cite{2}.

Here we also calculate the leptonic widths of the $\rho(n\,{}^3S_1)$
and $\rho(n\,{}^3D_1)$ states. However, the accuracy of these calculations is
limited by the fact that they are done in closed-channel approximation,
where the norm of the $q\bar q$ component of the w.f. at the origin
remains undetermined, e.g. for the states with $J^{PC}=1^{--}$ the
w.f. can be schematically written as
\begin{eqnarray}
 \psi_{\rm S}(r) & = & C_{q\bar q}(\cos\theta~ \psi_{\rm S}(r)
 -  \sin\theta~ \psi_{\rm D}(r)) + C_{\rm cont}(S) \psi_{\rm CS},
\nonumber \\
 \psi_{\rm D}(r) & = & C_{q\bar q}(\cos\theta~ \psi_{\rm D}(r)+
 \sin\theta ~\psi_{\rm S}(r))  + C_{\rm cont}(D) \psi_{\rm CD},
\label{eq.3}
\end{eqnarray}
assuming that the $q\bar q$ components of the $S$- and $D$-wave
w.f.s have equal (or close) values and allowing for $S-D$ mixing.
Fortunately, knowledge of the continuum component is not important
for the leptonic widths, since a multi-quark component of the w.f.,
even if it is large, gives a small contribution to the w.f. at the
origin \cite{39}. Thus the weight $C_{q\bar q}^2$ remains as the
relevant unknown parameter in the closed-channel approximation which
produces an uncertainty in the theoretical predictions of the
leptonic widths.  Here, in our calculations of the leptonic widths
of $\rho(nS), \rho(nD)$ with $J^{PC}=1^{--}$ we take $C_{q\bar
q}=1$.

\section{The string regime}
\label{sect.02}
In the RSH of light mesons, the quark mass $m_q=0$ and all spin-dependent
potentials are considered as a perturbation; then  the RSH is given
by the expression \cite{12,22}:
\begin{equation}
 H(\omega, \nu) = \omega + \frac{p_r^2}{\omega} +
 \frac{l(l+1)}{\omega + \int^1_0{\rm d}\beta \nu(\beta)
 \left(1 -\frac{\beta^2}{2} \right)^2} +
 \frac{\sigma^2 r^2}{2}\int^1_0{\rm d}\beta
 \frac{1}{\nu(\beta)} + \frac{1}{2} \int^1_0 {\rm d}\beta\nu(\beta).
\label{eq.4}
\end{equation}
This Hamiltonian contains two variables $\omega, \nu(\beta)$, which
are defined from the extremum conditions. The variable $\nu(\beta)$
is shown to be different for the states with large $l\geq 3$
(in the so-called the string regime) and for small $l\leq 2$ (the
potential-like regime) in order to provide the minimal value of the
mass  \cite{12,13}. In the string regime the centrifugal term  and
the term proportional to $\sigma^2 r^2$ dominate and thus the
ground state masses $M_{\rm str}(l,n_r=0)$ were obtained neglecting
the contributions from the GE and the fine structure potentials.
In that approximation the masses of all members of the multiplet
are equal to the centroid mass, which for the ground state ($n_r=0$)
with large $l$ is
\begin{equation}
 M_{\rm str}^2(l,n_r=0) = 2\pi\sigma\sqrt{l(l+1)}.
\label{eq.5}
\end{equation}
From this formula one can see that the mass difference, $\beta_l=
M_{\rm str}^2(l+1) - M_{\rm str}^2(l)$, is practically equal to
$2\pi\sigma=1.13(1)$~GeV$^2$~ ($\sigma=0.18(2)$~GeV$^2$) with high
accuracy, e.g. for $l=3$ the accuracy is $0.8\%$. The values of
$M_{\rm str}(l,n_r=0)$ are given in Table~\ref{tab.01} together
with experimental masses with $j=l+s$ and the centroid masses $M_{\rm
cog}(l,n_r=0)$ for $l=1,2$.
\begin{table}[!htb]
\caption{The masses $ M_{\rm str}(l,n_r=0)$ (in MeV) in the string regime
Eq.~(\ref{eq.4}) \label{tab.01}}
 \begin{center}
\begin{tabular}{|c|c|c|c|c|c| }
\hline
  $l$            &    1     &     2   &   3    &     4    &   5   \\
\hline
 $M_{\rm str}(l) $&  1265  &    1664    &  1979   &   2249    & 2489  \\
 $M({\rm exp.},j=l+s)  $ &   1318(1) & 1689(2) &  1982(14) &  2330(35)  & 2450(130)\\
\hline
\end{tabular}
\end{center}
\end{table}

Table~\ref{tab.01} shows the good agreement between the masses calculated
according to Eq.~(\ref{eq.5}), and the experimental masses for
$a_4(2040),~\rho_5(2350), ~a_6(2450)$ \cite{1}.  Surprisingly, even
for $a_2$ and $\rho_3$ with $l=1,2$ ,the centroid masses have reasonable
values, although the low-lying states have to be studied in the
potential-like regime and for them all kinds of the interactions:
the confining, the GE, the centrifugal term, are important. In the
potential-like regime the RSH can be rewritten in a more convenient
form, $H=H_0 + \Delta({\rm str})$, where the unperturbed part $H_0$ has
the form of the Hamiltonian occurring in the spinless Salpeter
equation (SSE) ($m_q=0$) \cite{22,23}:
\begin{equation}
 H_0 = 2\sqrt{\bm p^2+ m_q^2} + V_0(r),
\label{eq.6}
\end{equation}
and the operator $p_r^2$ is replaced by $\bm p^2$, while the remaining
part of the centrifugal term, the so-called string correction,
\begin{equation}
 \Delta({\rm str}) = - \frac{l(l+1)\sigma
 \langle r^{-1}\rangle_{nl}}{8 \omega^2(nl)},
\label{eq.7}
\end{equation}
is considered as a perturbation. This correction is not very large,
$\sim 50-100$~MeV for $l=1,2$, still it cannot be neglected.  In
Eq.~(\ref{eq.7}) the variable $\omega(nl)$ is the kinetic energy of a
light quark, defined by the solutions of the SSE:
\begin{equation}
(2\sqrt{\bm p^2} + V_0(r))\psi_{nl}(r) =M_0(nl)\psi_{nl}(r).
\label{eq.8}
\end{equation}

To define the solutions of the unperturbed Hamiltonian $H_0$ with
$l=0,1,2$, it is important to use  the universal quark-antiquark
potential, which has no fitting parameters and therefore allows to
separate physical effects from the artifacts introduced by fitting
parameters. This potential has the form of  linear plus GE terms
(observed on the lattice \cite{33} and derived in the field correlator
method \cite{34}), and successfully describes heavy quarkonia spectra
\cite{40},
\begin{equation}
 V_0(r)= \sigma_0 r + V_{\rm GE}(r),
\label{eq.9}
\end{equation}
with $\sigma_0=0.18(2)$~GeV$^2$, fixed by the slope of the leading
RT. In the  GE potential the vector coupling in coordinate space,
\begin{equation}
 V_{\rm GE}(r)= - \frac{4}{3} \frac{\alpha_{\rm V}(r)}{r},
label{eq.10}
\end{equation}
is taken in two-loop approximation, where it does not depend on the
renormalization scheme, and defined via the vector coupling in
momentum space:
\begin{equation}
 \alpha_{\rm V}(r)=\frac{2}{\pi} \int\limits_0^\infty{\rm d}q
 \frac{\sin(qr)}{q}\alpha_{\rm V}(q^2).
\label{eq.11}
\end{equation}
Here
\begin{equation}
 \alpha_{\rm V}(q^2) = \frac{4\pi}{\beta_0 t}
 \left(1 - \frac{\beta_1}{\beta_0^2} \frac{\ln t}{t}\right),
\label{eq.12}
\end{equation}
where for $n_f=3,~\beta_0=9,~\beta_1=64$ and in the logarithm $t(q^2)
= \ln \left(\frac{q^2 + M_{\rm B}^2}{\Lambda_{\rm V}^2}\right)$,
the vector constant $\Lambda_{\rm V}(n_f=3)=(480\pm 20)$~MeV
corresponds   to $\Lambda_{\overline{MS}}(n_f=3) = (327\pm 15)$~MeV
from pQCD, while the IR regulator $M_{\rm
B}=\sqrt{2\pi\sigma}=1.13(11)$~GeV$^2$ was defined in Ref.~\cite{38}
(see the discussion in the Introduction). At $q^2=0$ the logarithm
\begin{equation}
 t_0 = t(q^2=0) =\ln \frac{M_{\rm B}^2}{\Lambda_{\rm V}^2},
\label{eq.13}
\end{equation}
defines the freezing constant, $\alpha_{\rm crit}(q^2=0)=\alpha_{\rm
V}(r\rightarrow \infty)=0.60\pm 0.04$, which is rather large (for the
admissible values, $\Lambda_{\rm V}=480\pm 20$~MeV and $ M_{\rm
B}=1.1-1.15$~GeV ). In bottomonium, this strong GE interaction remains
important up to high excitations and  gives a good description of
the charmonium  and bottomonium spectra \cite{40}.

However, for the light mesons this universal potential appears to
be too strong, giving  smaller masses for the $1S$, $1P$, and $1D$
ground states (see Table~\ref{tab.02}). This result does not change
if the parameters of the vector coupling vary within the admissible
range. It also shows that the dynamics in light mesons, which all
lie above open hadronic thresholds, is  more complicated due to
their large spatial extensions and the appearance of virtual $q\bar
q$ loops in the Wison loop of large size, and hence modifying the
gluon exchange propagator.  As we shall discuss later in
Section~\ref{sect.05}, the gluon effectively acquires the screening
mass due to these loops as obstacles and the color-magnetic confinement
\cite{41}. For that reason we consider also a screened potential.

The creation of  the virtual quark loops (scalars in the ${}^3 P_0$
mechanism) decreases the string tension, making it dependent on the
separation $r$.  Due to this flattening effect the masses of excited
states decrease, e.g. the mass  $M(4S)$ becomes by $\sim 300-350$~MeV
smaller than for a purely linear potential  $\sigma_0 r$. However,
it is not so for the ground states with $l=0,1,2$, which have
relatively small sizes ($\langle r \rangle \leq 1.2$~fm) and are
not affected by the flattening effect.  Below, we shall find out
the direct connection between the parameter responsible for the
flattening of the potential, and the slope of the radial RT.

There is one more difference between  light meson masses and those
of heavy quarkonia, where the centroid masses just coincide with
the eigenvalue (e.v.) of the SSE. For a light meson its centroid
mass $M_{\rm cog}(nl)$ also includes a negative self-energy
contribution $\Delta({\rm SE})$ \cite{22,23,42} and negative string
correction $\Delta({\rm str})~(l=1,2)$, which do not introduce extra
parameters. The self-energy term is very important for the mass
value, since it gives contribution to the intercept of RT. In the
case  $l=0$,
\begin{eqnarray}
 M_{\rm cog}(nS) & =  & M_0(nS) +\Delta({\rm SE}),
\nonumber \\
 \Delta({\rm SE}) & = & - \frac{3\sigma}{\pi \omega(nl)}.
\label{eq.14}
\end{eqnarray}

In heavy quarkonia $\Delta({\rm SE})\sim (1-5$)~MeV is very small
and can be neglected, while for light-light,  $K$,~ and $\phi$
mesons,  $\Delta(SE)$ is rather large due to the small value of the
kinetic energy  $\omega(nl)\sim (400-500)$~MeV in the denominator.
Because of this term, the squared mass $M^2(nl)$ does not contain
a term linear in $M(nl)$)  and provides the linear behaviour of the
RT \cite{10,23}. Notice, that in the RPM a negative subtractive
constant (a fitting parameter), usually added to the potential (or
the mass) \cite{9}, violates the linearity of the orbital and radial
RT (see also the discussion in Ref.~\cite{15}).

In the mass of the $n\,{}^3S_1$ states, $M(n\,{}^3S_1)= M_0(nS) +
\Delta({\rm SE}) + \frac{1}{4} \Delta({\rm HF})$, the hyperfine
correction is defined as in Ref.~\cite{43},
\begin{equation}
 \Delta({\rm HF}) =
 \frac{32\pi \alpha_s(\mu_{\rm hf}) |\psi_{nS}(0)|^2}{9\omega^2(nS)},
\label{eq.15}
\end{equation}
where the kinetic energy $\omega(nS)$ enters in the denominator.
Here it is important to underline that in Eq.~(\ref{eq.15}) the
coupling $\alpha_s(\mu_{\rm hf})$ is not an arbitrary parameter.
As shown in Ref.~\cite{43}, this coupling is defined at the universal
scale (for all mesons, light and heavy)  $\mu_{\rm hf}\simeq
T_g^{-1}$, where $T_g\simeq 0.12$~fm is the vacuum correlation
length. Since the scale $\mu_{\rm hf}$ is close to the mass of the
$\tau$-lepton, the value of $\alpha_s(\mu_{\rm hf})$ must be close
to  $\alpha_s(M_{\tau})= 0.33(2)$ \cite{1}. We take here
$\alpha_s(\mu_{\rm hf})=0.31$, as it  was used in Ref.~\cite{44}
in the analysis of the hyperfine splitting of  the B mesons and
bottomonium.

In Table~\ref{tab.02} the calculated $\rho(nS)$ masses are  given in the typical
case, when the universal potential has no screening in the GE term
and the freezing constant $\alpha_{\rm
crit}=0.608,~~\sigma_0=0.18$~GeV$^2$.

\begin{table}[!htb]
\caption{{The masses of the $n\,{}^3S_1$ light mesons (in MeV) for the
universal potential with $\sigma=0.18$~GeV$^2$ and $\alpha_{\rm crit}=0.6086$.
Experimental data are taken from  Refs.~\cite{1,7}\label{tab.02}}}
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
 $n=n_r+1 $      &    1   &    2      &   3     &   4      \\
\hline
 $M(n\,{}^3S_1)$  &   693    &   1478    &  2046   &    2510  \\
 exper. [1]  &    775    &   1465 (25)  &  1909(42)  & 2150(90)  \\
 data   [7]  &    775     &  1493(15)   &  1861 (17)  & 2254(22) \\
\hline
\end{tabular}
\end{center}
\end{table}

From Table ~\ref{tab.02} one can see (i) that the strong (universal)
GE potential gives the $\rho(775)$ mass, as well as the masses of
the $M(a_2(1318)) =1.240$~GeV, $M(\rho_3)=1.59$~GeV, smaller by
$\sim 80$~MeV than their experimental values.  (ii) On the contrary,
for high excitations, where the influence of the GE potential is
small, the masses $M(3\,{}^3S_1)$ and $M(4\,{}^3S_1)$ are by $\sim
150$~MeV and $\sim 300$  MeV larger than in experiment, irrespective
of the strength of the GE potential, if the linear $\sigma_0 r$
potential is used. Therefore one needs to look for another effect
(reason), responsible for the strong decrease of the $nS$ masses
observed in experiments.

\section{The flattening effect}
\label{sect.03}
In Section~\ref{sect.05}  we present the physical picture explaining
the flattening phenomena, observed on the lattice \cite{45} and
studied in Ref.~\cite{23}, while here we give concrete results of
our calculations with the confining potential, where the string
tension depends on the quark-antiquark separation $r$,
\begin{equation}
 \sigma(r) = \sigma_0 f(r),\quad
 \lim_{r \to \infty} \sigma(r) = \sigma_0 (1 - \gamma).
\label{eq.16}
\end{equation}
Here $\sigma_0=0.18(2)$~GeV$^2$ and the function $f(r)= 1 -
\gamma\frac{\exp(\sigma_0 (r - R_0))} {B + \exp(\sigma_0 (r -
R_0))}$, contains three parameters; two of them,  $B\cong 15-20$
and $R_0\cong (1.2-1.4)$~fm, are chosen in  such  way that the
flattening slowly starts at rather large distances, $\sim 1.2$~fm,
while the lowest lying states with  $l=0,1,2$ are not affected by
the flattening effect. At large distances, the string tension goes
to the limiting value, $\sigma_{\rm lim}=\sigma_0 (1 - \gamma)$.
Direct calculations show that the decrease of the $\rho(nS)~(n\geq
2)$ masses occur  mostly due to the flattening effect and the mass
shifts are very sensitive to the value of the parameter $\gamma$
in $\sigma(r)$. To reach agreement with experiment, the fitting
parameter $\gamma$ is to be taken in the narrow range, $\gamma=(0.43\pm
0.03)$, however, the values of $\gamma=0.40$ and 0.45 give rise to
different slopes of the $\rho$-trajectory.  If $\gamma=0.40$ is
taken, then a contribution from the screened GE potential is more
important than  for $\gamma=0.45$  (see Table~\ref{tab.03}), but
both variants have common features:
\begin{itemize}
\item[1. ]The linear behavior of the $\rho$-trajectory  starts with
$n_r=1$, although these RTs have slightly different slopes: for
$\gamma=0.40~(0.45)$ the slope  $\mu^2(\rho)=1.40~(1.35)$~GeV$^2$.

\item[2.] At the same time the mass difference, $\mu_1^2=M^2(\rho(2S)) -
M^2(\rho(1S))= 1.52(4)$~GeV$^2$ remains relatively large (in both
cases) and agrees  with $\mu_1^2({\rm exp.})=M^2(\rho(1465)) -
M^2(\rho(775))=(1.55\pm 0.07)$~GeV$^2$, if the central values of
the experimental mass are taken. The reason why $\mu_1^2$ is large,
is discussed below, in Section~\ref{sect.04}.

\item[3.] The choice of $\gamma$ directly determines the slope of the radial
RT and therefore it could be extracted from  the experimental masses
$M(\rho(3S))$ and $M(\rho(4S))$, if they would be  measured with better
accuracy.
\end{itemize}
In Table~\ref{tab.03} we give the masses in three cases (in all cases
$\sigma_0=0.182$~GeV$^2$, $m_q=0$): in case A there is no
screening of the GE potential, i.e., $\delta=0$; in the cases B and C the
exponential form of the screening, $V_{\rm scr}=V_{\rm GE} \exp(-\delta
r)$ with the screening parameter $\delta=0.20$~GeV, is taken.  In
the cases A and  B the other parameters coincide,
\begin{eqnarray}
 & & \Lambda_{\rm V}(n_f=3)=465~{\rm MeV},~ M_B=1.15~{\rm GeV},~
 \alpha_{\rm crit}=0.5712, ~
\nonumber \\
  & & \sigma_0=0.182~{\rm GeV}^2,~ \gamma=0.40,~ B=20,~R_0=6.0~{\rm GeV}.
\label{eq.17a}
\end{eqnarray}
In case C the stronger GE potential, with $\alpha_{\rm crit}=0.635$,
is taken, while $\gamma=0.45$ in the flattening potential is larger
than in Eq.~(\ref{eq.17a}). The other parameters in case C are as follows:
\begin{eqnarray}
 & & \Lambda_{\rm V}=500~{\rm MeV}, \quad M_B=1.15~{\rm GeV},
\nonumber \\
 & &\sigma_0=0.182~{\rm GeV}^2, \quad B= 15, \quad R_0=6.0~{\rm GeV}^2.
\label{eq.17b}
\end{eqnarray}
For all $nS$-states  the hyperfine correction to the masses is calculated with
$\alpha_s(\mu_{\rm hf})=0.31$.

As seen from Table~\ref{tab.03}, without screening ($\delta=0$) the
ground state masses of $\rho(775)$, $a_2(1318)$, and $\rho_3(1690)$
appear to be $50 - 100$ MeV  smaller than in experiment and variations
of the parameters within reasonable ranges do not change this result.
On the contrary, in the case B for the screened GE potential
($\delta=0.20$~GeV,  the other parameters remaining the same, as
in case A) a reasonable agreement with experiment is reached. The
choice of $\delta=0.30$~GeV, i.e., stronger  suppression of the GE
potential, gives rise to large  masses of  $\rho(775)$ and $\rho(1465)$
and was neglected.

The best agreement with experimental data takes place in the case C
Eq.~(\ref{eq.17b}), when the screening parameter $\delta=0.20$~GeV is the
same, but  $\Lambda_V=500$~MeV and $\gamma=0.45$ are larger than
in the cases A and B.
\begin{table}[!htb]
\caption{The masses of the lowest lying states $(l=0,1,2)$ and excited
$n\,{}^3S_1$ states (in MeV) for the flattening potential:  case
A with $\gamma=0.40,~ \delta=0$~GeV;  case B with
$\gamma=0.40,~\delta=0.20$~GeV;  case C with
$\gamma=0.45,~\delta=0.20$~GeV, $\Lambda=500$~MeV. \label{tab.03}}
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
     &$\delta=0$ & $\delta=0.20$ &  $\delta=0.20$ &      \\
\hline
 state              &  Case A     &  Case B     &   Case C  &  exp. \\
 \hline
  $1\,{}^3S_1 $ &  $ 698^{a)} $   &  790    &  774  &  775   [1] \\
  $2\,{}^3S_1 $  &  1430      &   1474      & 1468 &  1465 [1] \\
               &            &         &             &   1493(15) [7]\\
  $3\,{}^3S_1 $ &  1876    &    1920    & 1880  & 1909(42) [1] \\
              &           &           &        &  1861(17)  [7] \\
 $ 4\,{}^3S_1$  &  2172    &   2239    &  2170   &  2150(90) [1] \\
              &           &          &          &   2254(22) [7] \\
  $1\,{}^3P_2  $ &  1240     &  1312    &   1309   &   1318(1) \\
 $ 1\,{}^3D_3$  & 1590     &   1696    &   1690  &    1689(2)  \\
\hline
\end{tabular}

$^{a)}$ Here the hyperfine contribution $\sim 65$~MeV is taken into account.
\end{center}
\end{table}

Our conclusion is that screening of the universal GE potential is
necessary to obtain correct values of the masses of the lowest lying
states with $l=0,1,2$, otherwise they are  $\sim 80$~ MeV smaller
than in experiment. For the higher $nS$ excitations the contribution
from the GE potential cannot be neglected and agreement with
experiment is reached both in the cases B and C. The masses of the
$nP$ and $nD~(n_r\geq 1)$ states weakly depend on the screened GE
potential  and will be discussed in the next Section.

We give here also the radii (r.m.s.) $R_s(nS)= \langle
\sqrt{r^2}\rangle_{nS}$ of $\rho(1S)$ and $\rho(2S)$, which weakly
change in all three cases: $R_s(1S)=0.71(0.72)$~fm in the cases B
(C) and a bit smaller, 0.68~fm in the case A, where there is no
screening effect. The r.m.s. of $\rho(2S)$ is significantly larger,
$R_s(2S)=1.0(1)$~fm in all cases.

\section{Radial Regge Trajectories}
\label{sect.04}
We have shown that the GE potential gives a small contribution to the
masses of the high radial excitations ($n_r\geq 1$) and therefore, in
first approximation, the GE potential with  screening can be
neglected. This allows to reveal more explicitly the role of the
flattening effect for formation of the radial RT. The most important
contribution to the light meson masses comes from the e.v. of the SSE,
Eq.~(\ref{eq.8}), the unperturbed part of the RSH,  which in the case of the
linear $\sigma_0 r$ potential ($\sigma_0$ is a  constant) is well
known. Namely, the e.v. of the SSE ($m_q=0$) can be approximated
with great accuracy (for $n_r\geq 1$) by the expression
\cite{16,23},
\begin{equation}
 M_0^2(nl)= \sigma_0 ( 8 l + 4\pi n_r  +3\pi).
\label{eq.18}
\end{equation}
This formula explicitly shows that the e.v.s $M_0^2(n_r=0,l)$ for
the ground states for given $l$, lie on the orbital  RT with the
slope $\beta_0=8\sigma_0=1.44$~GeV$^2$, which is $\approx 27\%$
larger than $\beta({\rm exp.})= 2\pi\sigma_0=1.13(1)$~GeV$^2$,
observed in experiment.  For the radial excitations, the difference
between the slope in Eq.~(\ref{eq.18}) and the one found  in
experiment is very large: $\mu_0^2= 4\pi\sigma_0 = 2.26$~GeV$^2$
is $1.6-2.0$ times larger than $\mu^2({\rm exp.})=(1.25\pm
0.15)$~GeV$^2$ \cite{8,11}. The question is why such a large
difference occurs.

First of all, we look at the contribution to the centroid mass from
the self-energy correction, Eq.~(\ref{eq.14}), for which we use the
relation, $M_0(nS) = 4\omega_0(nS)$ (valid for $\sigma=\rm const.$)
and rewrite $\Delta({\rm SE})= -\frac{12\sigma_0}{\pi M_0}$; then
\begin{equation}
 M_{\rm cog}(nS) = M_0(nS)- \frac{3.82\sigma_0}{ M_0(nS)}.
\label{eq.19}
\end{equation}
In the squared mass we neglect the small squared self-energy term
(although it is not small for the $1S$ state)  and obtain
\begin{equation}
 M_{\rm cog}^2(nS) = M_0^2(nS) - 7.6 \sigma_0 =
 \sigma_0 (4\pi n_r - 7.6 +3\pi)= (0.33 +2.26 n_r)~{\rm GeV}^2.
\label{eq.20}
\end{equation}
From here one can see that owing to $\Delta(SE)$ the value of $M_{\rm
cog}^2$ is smaller than $M_0^2$ given in Eq.~(\ref{eq.18}), while
the slope $\mu_0^2=4\pi\sigma_0=2.26(2)$~GeV$^2$ does not change.
Thus we have confirmed the well-known result that the purely linear
potential with $\sigma=\rm const.$ produces always a large slope
of the radial RT.

The situation strongly changes, if the flattening potential $V_{\rm
C}(r)= \sigma(r) r$ is considered, for which the representation
Eq.~(\ref{eq.18}) is not valid anymore (in this case the e.v. of the SSE will
be denoted as $\tilde{M}_0(nS)$). Our calculations show that
\begin{itemize}
\item[1.] The linear behavior of the $\rho$ RT starts with $n_r=1$, because
for the flattening potential (with $\gamma=0.40$ or $0.45$) the mass
difference $\mu_1^2 = \tilde{M}^2(2S) - \tilde{M}^2(1S)$ remains
large, $\mu_1^2\sim 1.87(5)$~GeV$^2$, being still  20\% smaller
than $\mu_1^2=4\pi\sigma_0$ in Eq.~(\ref{eq.18}).

\item[2.] For the $nP$ and $nD$ states the linear behaviour starts with $n_r=0$.

\item[3.] The slope $\mu^2(l)$ strongly depends on the parameter $\gamma$
in $\sigma(r)$ , Eq.~(\ref{eq.16}), which characterizes the weakening of the
confining potential.
\end{itemize}

The squared e.v., $\tilde{M}_0^2(nS)$ (in GeV$^2$) $(n_r\geq 1$),
with $\gamma=0.40, 0.45, 0.50$ can be approximated as
\begin{eqnarray}
 \tilde{M}_0^2 & = & (2.42 +1.40\,n_r)~{\rm GeV}^2,~{\rm for}~\gamma=0.40,
\nonumber \\
\tilde{M}_0^2  & =  & (2.31 + 1.27\, n_r)~{\rm GeV}^2,~{\rm for}~\gamma=0.45,
\nonumber \\
\tilde{M}_0^2  & =  & (2.25 + 1.15\, n_r)~{\rm GeV}^2,~{\rm for}~ \gamma=0.50.
\label{eq.21a}
\end{eqnarray}
The accuracy of these expressions is $\sim 1\%$.

From Eq.~(\ref{eq.21a}) the important result follows that for the
flattening potential the squared e.v.s of the SSE have a much smaller
slope (two times smaller for $\gamma=0.50$), than in the case of
the purely linear potential Eq.~(\ref{eq.18}), which decreases for
larger values of $\gamma$, i.e., a stronger flattening effect. For
the centroid mass the intercept is changed, while the value of the
slope is the same, so that the $\rho$ trajectory ($n_r\geq 1$) is,
\begin{eqnarray}
 M_{\rm cog}^2(\rho) & = & (0.77 +1.40\, n_r)~{\rm GeV}^2,~{\rm for}~
\gamma=0.40,
\nonumber \\
 M_{\rm cog}^2 & = & (0.80+ 1.27\, n_r)~{\rm GeV}^2,~{\rm for}~\gamma=0.45,
\nonumber \\
 M_{\rm cog}^2 & = & (0.90 + 1.15\, n_r)~{\rm GeV}^2,~{\rm for}~\gamma=0.50.
\label{eq.21b}
\end{eqnarray}
In all cases $M_{\rm cog}(\rho(1450))=(1.44-1.47)$~GeV. However, if the
exponential screening of $V_{\rm GE}$ and the hyperfine interaction are
taken into account, then the slope increases while the intercept does
practically not change. In the cases A and C (see the parameters of the
GE potential in Eqs.~(\ref{eq.17a},\ref{eq.17b})) and  for $n_r \geq 1$
we have
\begin{eqnarray}
 M^2(n\,{}^3S_1) & = & (0.78 + 1.40(2)\, n_r)~{\rm GeV}^2~ (\gamma=0.40),
\nonumber \\
 M^2(n\,{}^3S_1) & = & (0.81 + 1.34(1)\, n_r)~{\rm GeV}^2~ (\gamma=0.45).
\label{eq.22}
\end{eqnarray}
Thus for $\gamma=0.45$ the calculated $\rho$-trajectory has $\mu^2=
1.34(1)$~GeV$^2$, in agreement with the results in Refs.~\cite{11,26},
where $\mu^2=1.365(108)$~GeV$^2$ was obtained from the analysis of
the Crystal Barrel data \cite{2}. On the contrary, the larger
$\mu^2=1.40(2)$~GeV$^2$ for $\gamma=0.40$ agrees with  the slope,
$\mu^2=1.43(13)$~GeV$^2$, predicted in Ref.~ \cite{27}. Notice, that for
$\gamma=0.45$ a better agreement is obtained for the $\rho(1450)$
mass (see Table~\ref{tab.03}).

In the same way, the radial RTs for the $nP$ and $nD$ states were
considered; it appears that for $l=1,2$ the linear behavior of the
radial RT starts with $n_r=0$ and the squared e.v. of SSE $M_0^2(nP)$
can be approximated as
\begin{equation}
 \tilde{M}_0^2(nP)= (3.11 + 1.25\, n_r)~{\rm GeV}^2~ (\gamma=0.45).
\label{eq.23a}
\end{equation}
Then, taking into account the self-energy and string corrections we
obtain for the centroid masses,
\begin{equation}
 M_{\rm cog}^2(nP)=(1.64(2) + 1.25\, n_r)~{\rm GeV}^2.
\label{eq.23b}
\end{equation}
From this expression one can obtain the $a_{\rm j}$ radial RT,
taking into account the fine-structure splitting, which does
practically not change the slope, but introduces a fitting parameter. For
that reason we restrict ourselves to the  RTs for the centroid masses.
Notice that $\mu^2(nP)=1.25$~GeV$^2$ practically coincides with
the slope for the centroid masses of the $nS$ states, if $\gamma=0.45$.

For the $nD$ trajectory  the e.v.s of the SSE have a smaller slope
($n_r\geq 0$),
\begin{equation}
 \tilde{M}_0^2(nD)=(4.36 + 1.11(5)\, n_r)~{\rm GeV}^2 ~(\gamma=0.45).
\label{eq.24a}
\end{equation}
and
\begin{equation}
 M_{\rm cog}^2(nD) = (2.8(1) + 1.11(5)\, n_r)~{\rm GeV}^2 (\gamma=0.45).
\label{eq.24b}
\end{equation}
Notice, that the slope $\mu^2(l)$ decreases for increasing
angular-momentum $l$.  At this point it is important to stress that
for physical $nD$ states the GE contribution is much smaller than
that for the $nS$ states, and therefore for the $\rho_3$, $\rho_2$,
and $\rho(n\,{}^3D_1)$ trajectories the slopes have to be close to
the one given in  Eq.~(\ref{eq.24b}), where $\mu^2(D)=1.11(5)$~GeV$^2$,
if the fine-structure effects are neglected. Our result is in
agreement with  $\mu^2(a_2)=1.00(6)$~GeV$^2$ and $\mu^2(a_1)=(1.084\pm
0.63)$~GeV$^2$, predicted for the $a_1$ and $a_2$ RTs  from the
analysis of experimental data in Ref.~\cite{26}.

Now we briefly discuss the reasons why the mass difference
$\mu_1^2=M^2(2S)- M^2(1S)$, is large. The first reason is that this
factor is very large for the flattening potential (without GE
interaction), where  $\mu_1^2=1.87(5)$~GeV$^2$ for the squared e.v.
of SSE. This result does not change, if a reasonable choice of the
parameters in $\sigma(r)$ is made.  Secondly, if the GE interaction
is taken into account, then for the $1S$ state, localized at rather
small distances, the self-energy and hyperfine corrections decrease
the mass difference $\mu_1^2$ but its value remains  rather large,
$\mu_1^2=1.56(6)$~GeV$^2$.  This number appears to be very close
to what is observed in experiment, $\mu_1^2(\rm exp.)=1.55(7)$~GeV$^2$
\cite{1}, if the central values of the $\rho(775)$ and $\rho(1465)$
masses are used. Just for that reason, the linear behaviour of the
$\rho$ RT begins with $n_r=1$, while other radial RTs start with
$n_r=0$.

\section{Flattening phenomenon - the physical picture}
\label{sect.05}
The dynamics of light mesons is more complicated than that in heavy
quarkonia, since light mesons, as rather extended objects, are sensitive
to detailed properties of the confinement mechanism, which also
affects the gluon exchanges. Our approach is based  on the background
perturbation theory (BPT) \cite{46}, which takes into account the
non-perturbative background with confinement and does not contain
unphysical singularities (the Landau ghost poles and IR renormalons),
present in standard perturbation theory. Below, we illustrate
how the BPT predicts three effects, which are observed in experiment
and especially important for light mesons:
\begin{itemize}
\item[1.] Stabilisation of the coupling $\alpha_s(q^2)$ at
$q^2\rightarrow 0,~\alpha(0)\equiv \alpha_{\rm crit}$;

\item[2.] Screening of $\alpha_s$ at large distances;

\item[3.] Flattening of the string tension at large distances,
$\sigma\rightarrow \sigma(r)$.
\end{itemize}
Item 1. The basic feature of BPT is the gauge-invariant treatment
of confinement and gluon-exchanges, when both phenomena occur owing
to the Wilson loop, where confinement creates the minimal-area
surface (the so-called confining film) and the gluon-exchange
trajectories are necessarily present inside this surface. As a
result, the gluon loops, appearing on these trajectories and
responsible for asymptotic freedom, create open loops in the confining
film, and this effect strengthens with increasing $\alpha_s$, leading
finally to the saturation of $\alpha_s(q^2)$ which in two-loop
approximation is given by $\alpha_{\rm crit}=\frac{4\pi}{\beta_0
t_0} \left( 1 - \frac{\beta_1}{\beta_0^2} \frac{\ln t_0}{t_0}\right)$,
where $t_0=\ln(\frac{M_{\rm B}^2}{\Lambda^2})$ and with $10\%$
accuracy $M_{\rm B}^2=2\pi\sigma$ \cite{41}.

Item 2. For the same reason, the scalar $q\bar q$ loops, appearing
in the film (of large size), lead to the screening of the GE
interaction, since any gluon trajectory, propagating inside the
confining film, is interrupted by the scalar loops and those create
an effective mass of the gluon. In addition, there is a difference
between the free propagation (free Green's function of the gluon)
and the gluon propagation inside the surface with confinement. This
complicated phenomenon was studied in Ref.~\cite{47} for zero
temperature and in Ref.~\cite{48} for the deconfined phase, where
it occurs due to the color-magnetic confinement.  This effect at
zero temperature, when both color-electric and color-magnetic
confinement collaborate, is not yet finally settled and therefore
in our paper we exploit the effective screening parameter $\delta$
for the screening mass. Our analysis has shown that for the exponential
form of screening $\delta=0.20$~GeV is the preferable value, while
suppression of the GE potential is too strong for $\delta=0.30
(0.10)$~GeV.

Item 3. For excited light mesons, confinement occurs in a highly
excited string, when the Wilson loop has a free boundary and several
typical features, partly discussed above. Namely, there exist

(i)  a finite density of the $q\bar q$ loops in the confining film,
which leads to the dependence of the string tension on $r$,
$\sigma\rightarrow \sigma(r)$;

(ii)  a possibility to decay, virtually or really, into a pair (or
several) mesons, so that if the distance $r$ in the confining
potential $\sigma r$ exceeds the separation, $R_{\rm f}\sim 2
r_{\pi}\simeq 1.2$~fm, then the flattening of the potential is
expected.

The first feature can also be seen in the  $T$-dependence of
$\sigma(r)$: when the density of the $q\bar q$ loops grows with
increasing temperature $T$,  then the potential $V(r,T)$ becomes
more and more flat, as it was observed on the lattice \cite{49}.
Another manifestation of the flattening phenomenon was recently
studied in Ref.~\cite{50}:  while applying a magnetic field parallel
to the confining film, it was observed that the density of the
$q\bar q$ loops increases and the string tension $\sigma(r)$ flattens,
in agreement with the lattice data \cite{51}.

Both features, flattening due to a finite $q\bar q$ density and the
existence of the critical length $R_{\rm f}\sim 2 r_{\pi}$ are
embodied in the form of the string tension, Eq.~(\ref{eq.16}), which
is used in our paper.

In conclusion we give  the r.m.s of the $\rho(nS)$ mesons, $r_s=\langle
\sqrt{r^2}\rangle_{nS}$, calculated for the sets of the parameters
Eqs.~(\ref{eq.17a}, \ref{eq.17b}): for $\rho(775)$,
$r_s(1S)=(0.71-0.73)$~fm and for  $\rho(1450)$, $r_s=(0.9-1.0)$~fm.

\section{The leptonic widths of $\rho(n\,{}^3S_1)~l=0,2$}
\label{sect.06}
The decay constants $f_{\rm V}$ and leptonic widths of the
$\rho(n\,{}^3S_1)$ mesons are calculated here,  considering them
as purely $q\bar q$ states, i.e., taking  $C_{q\bar q}=1.0$ in the
w.f. given in Eq.~(\ref{eq.3}). For the decay constant in the vector
channel $f_{\rm V}$ we use the expression from Ref.~\cite{52}, where
the correlator of the currents (in different channels) is derived
using the functional integral representation and on the final stage
expanding this correlator in the complete set of  eigenfunctions
of the RSH $H_0$, Eq.~(\ref{eq.6}). This  gives
\begin{equation}
 f_{\rm V}^2 = 12\bar{e}_q^2 \frac{|\psi_n(0)|^2 \xi_{\rm V}}{M_{\rm V}(nS)}=
 \frac{3 \bar{e}_q^2|R_n(0)|^2 \xi_{\rm V}}{\pi M_{\rm V}(nS)},
\label{eq.25}
\end{equation}
and
\begin{equation}
 \Gamma_{\rm ee}(n\,{}^3S_1) =
 \frac{4\pi\alpha^2 f_{\rm V}^2 \beta_{\rm QCD}}{3 M_{\rm V}}.
\label{eq.26}
\end{equation}
Here, for a light meson with $m_q=0$ the relativistic factor
$\xi_{\rm V}(nS)$  is
\begin{equation}
 \xi_{\rm V} = \frac{\omega_n^2 + \frac{1}{3} \bm p^2}{2 \omega_n^2},
\label{eq.27}
\end{equation}
which for the ground and excited states are almost equal,
$\xi(1S)=0.70(1)$ and $\xi(nS)=0.72(1),~ (n=2,3,4)$, if the static
potential with the parameters Eqs.~(\ref{eq.17a},\ref{eq.17b}) is
used (for the $\rho$-mesons the average $\bar{e}_q^2 =1/2$).  The
factor $\beta_{\rm QCD}=1-\frac{16}{3\pi} \alpha_s =0.40$ takes
into account the radiative corrections \cite{53} and here we use
for all ($n\,{}^3S_1$)-states the same coupling $\alpha_s(\mu_s)=0.353$
(at the scale  $\mu_s\sim 1.0$~GeV).  If the confining potential
flattens at large distances, then the w.f.s at the origin $R_{\rm
nS}(0)\sim (0.36-0.33)$~GeV$^{3/2}~(n=2-4)$ have close values ,
while for the ground state  the w.f. $R_{\rm 1S}(0)=(0.376\pm
0.008)$~GeV$^{3/2}$ is larger, and for the $\rho(775)$  the decay
constant and leptonic width are
\begin{equation}
 f_{\rm V}=(245\pm 6)~{\rm MeV},
\label{eq.28a}
\end{equation}
where the uncertainty comes from that in the w.f. at the origin, and
\begin{equation}
 \Gamma_{\rm ee}(\rho(775))=( 7.0\pm 0.3)~{\rm keV}.
\label{eq.28b}
\end{equation}

To calculate the leptonic widths of the higher $\rho(nS)$, it is
convenient to use the ratio of the leptonic widths,
$\Gamma_{\rm ee}(n\,{}^3S_1)/\Gamma_{\rm ee}(\rho(775))$, where the
factors $\xi(nS)$ and $\beta_{\rm QCD}$ drop out. This gives
\begin{eqnarray}
 \Gamma_{\rm ee}(2\,{}^3S_1) & = & 0.24\;\;
 \Gamma_{\rm ee}(\rho(775))= 1.7(1)~{\rm keV},
\nonumber \\
 \Gamma_{\rm ee}(3\,{}^3S_1) & = & 0.14\;\;
 \Gamma_{\rm ee}(\rho(775))=1.0(1)~{\rm keV},
\nonumber \\
 \Gamma_{\rm ee}(4\,{}^3S_1) & = & 0.096\;
 \Gamma_{\rm ee}(\rho(775))=0.7(1)~{\rm keV},
\label{eq.29}
\end{eqnarray}
where the uncertainties come from the experimental errors in the
$\rho(nS)$ masses and the w.f.s at the origin. Notice that in a
realistic situation the leptonic widths of the excited $\rho(nS)$
mesons may be smaller, if the $q\bar q$ component $C_{q\bar q}$ in
their w.f.s is less than 1.0.

The leptonic widths of the ($n\,{}^3D_1$)-states is calculated
defining their w.f.s at the origin via the second derivative, according
to the prescription  from Ref.~\cite{54}: $R_{nD}(0)=\frac{5
R''(0)}{2\sqrt{2}\omega_{nD}^2}$, where $R_{1D}''(0)=0.026(1)$~GeV$^{7/2}$
and $\omega(1D)=0.536$~GeV. It gives $R_{1D}(0)=0.163(3)$~GeV$^{3/2}$,
which is not very small due to the small value of the kinetic energy
$\omega_{nD}\sim 0.5$~GeV. The other parameters are $\xi(1D)=0.69,~
\beta_{\rm QCD}=0.40,~ M(1\,{}^3D_1)=1.72(2)$~GeV, so that the
leptonic width,
\begin{equation}
 \Gamma_{\rm ee}(1\,{}^3D_1)=0.26(5)~{\rm keV}
\label{eq.30}
\end{equation}
is rather small. However, its value may increase owing to the $2S-1D$
mixing, and for a mixing angle $\theta=21^\circ$ the leptonic widths
of  $\rho(1450)$ and $\rho(1700)$ become almost equal:
\begin{equation}
 \Gamma_{\rm ee}(\rho(1450))=1.0(1)~{\rm keV},\quad
 \Gamma_{\rm ee}(\rho(1700))=0.75(6)~{\rm keV}~(\theta=21^\circ).
\label{eq.31}
\end{equation}
Here it was assumed that in the w.f.s of $\rho(1450)$ and $\rho(1700)$
the $q\bar q$ components are equal.

\section{Conclusions}
\label{sect.07}
We have studied  the light meson properties with the use of the
RSH, which allows us to investigate the light-meson dynamics without
introducing fitting parameters.  It appears that the universal
static potential, successfully applied to heavy quarkonia, gives
rise to small masses of the lowest states with $l=0,1,2$ and at the
same time large masses of the excited states. To explain the physical
spectrum, two effects: the screening of the GE interaction and the
flattening of the confining potential, which appear owing to
quark-loop creation, are to be taken into account. We have demonstrated
the following properties.

\begin{itemize}
\item[1.] The screening of the GE potential, taken as an exponential
function  with the screening parameter $\delta=0.20$~GeV, gives the
masses of the lowest lying states for each $l$ in agreement with
experiment.

\item[2.] The slope of the radial RT is very sensitive to the value
of the parameter $\gamma$, which determines the flattening of the
string tension $\sigma(r)$:  at large distances  $\sigma(r)\rightarrow
\sigma_0(1 - \gamma)$. The parameter $\gamma$ could be extracted
from the experimental masses of $\rho(1900)$ and $\rho(2150)$, if
these were measured with better accuracy, while now it is taken
from the range $\gamma=0.43\pm 0.03$.

\item[3.] From our calculations two values for the  slope of the
$\rho$-trajectory are obtained, $\mu^2(\rho)=1.40(2)$~GeV$^2$ for
$\gamma=0.40$ and $\mu^2(\rho)=1.34(1)$~GeV$^2$ for $\gamma=0.45$,
neither result contradicts the existing experimental data.

\item[4.] The linear behaviour of the radial RT starts with $n_r=0$
for the $nP$ and $nD$ trajectories, while the linear behaviour of
the $\rho$ trajectory begins with the first excitation, $n_r=1$,
since the large value of the mass difference,
$M^2(\rho(1450))-M^2(\rho(775))=1.56(6)$GeV$^2$ (or the relatively
small value of the $\rho(775)$ mass) is a dynamical property of the
$1S$ ground state.

\item[5.] The leptonic widths $\Gamma_{\rm ee}(\rho(775))=7.0(3)$~keV,
$\Gamma_{\rm ee}(\rho(1450))=1.7(1)$~keV, $\Gamma_{\rm
ee}(\rho(1900))=1.0(1)$~keV, $\Gamma_{\rm ee}(2150))=0.7(1)$~keV,
and $\Gamma_{\rm ee}(\rho(1700)) =0.26(5)$~keV are calculated
considering them as purely $q\bar q$ states. If $2S-1D$ mixing is
possible, then for the mixing angle $\theta=21^\circ$ comparable
values of the leptonic widths $\Gamma_{\rm ee}(\rho(1450))=1.0(1)$~keV
and $\Gamma(\rho(1700)=0.75(6)$ ~keV are obtained.

\end{itemize}

\begin{acknowledgments}
The financial support of the grant RFBR 1402-00395 is gratefully
acknowledged by A.~M.~Badalian.
\end{acknowledgments}


\section{Introduction}

Ultracold neutral plasmas studied in the laboratory offer access to a regime of plasma physics that scales to describe thermodynamic aspects of important high-energy-density systems, including strongly coupled astrophysical plasmas \cite{VanHorn,Burrows}, as well as terrestrial sources of neutrons \cite{Hinton,Ichimaru_fusion,Atzeni,Boozer} and x-ray radiation \cite{Rousse,Esarey}.  Yet, under certain conditions, low-temperature laboratory plasmas evolve with dynamics that are governed by the quantum mechanical properties of their constituent particles, and in some cases by coherence with an external electromagnetic field.

The relevance of ultracold plasmas to such a broad scope of problems in classical and quantum many-body physics has given rise to a great deal of experimental and theoretical research on these systems since their discovery in the late 90s.  A series of reviews affords a good overview of progress in the last twenty years \cite{Gallagher,Killian_Science,PhysRept,Lyon}.  Here, we focus on the subset of ultracold neutral plasmas that form via kinetic rate processes from state-selected Rydberg gases, and emphasize in particular the distinctive dynamics found in the evolution of molecular ultracold plasmas.

While molecular beam investigations of threshold photoionization spectroscopy had uncovered relevant effects a few years earlier \cite{Scherzer,Alt}, the field of ultracold plasma physics began in earnest with the 1999 experiment of Rolston and coworkers on metastable xenon atoms cooled in a magneto optical trap (MOT) \cite{Killian}.

This work and many subsequent efforts tuned the photoionization energy as a means to form a plasma of very low electron temperature built on a strongly coupled cloud of ultracold ions.  Experiment and theory soon established that fast processes associated with disorder-induced heating and longer-time electron-ion collisional rate processes act to elevate the ion temperatures to around one degree Kelvin, and constrain the effective initial electron temperature to a range above 30 K \cite{Kuzmin,Hanson,Laha}.

This apparent limit on the thermal energy of the electrons can be more universally expressed for an expanding plasma by saying that the electron correlation parameter, $\Gamma_e$, does not exceed 0.25, where,
\begin{equation}
\Gamma_e = \frac{e^2}{4\pi \epsilon_0 a_{ws}}\frac{1}{k_B T_e}
\label{eqn:gamma_e}
\end{equation}
defines the ratio of the average unscreened electron-electron potential energy to the electron kinetic energy.  $a_{ws}$ is the Wigner-Seitz radius, related to the electron density by, $\rho_e = 1/(\frac{4}{3} \pi a_{ws}^3)$.  These plasmas of weakly coupled electrons and strongly coupled ions have provided an important testing ground for ion transport theory and the study of electron-ion collision physics \cite{Strickler}.

Soon after the initial reports of ultracold plasmas formed by direct photoionization, a parallel effort began with emphasis on the plasma that forms spontaneously by Penning ionization and electron-impact avalanche in a dense ultracold Rydberg gas \cite{Mourachko}.  This process affords less apparent control of the initial electron temperature.  But, pulsed field-ionization measurements soon established that the photoionized plasma and that formed by the avalanche of a Rydberg gas both evolve to quasi-equilibria of electrons, ions and high-Rydberg neutrals \cite{Rolston_expand,Gallagher}.

Early efforts to understand plasmas formed by Rydberg gas avalanche paid particular attention to the process of initiation.  Evolution to plasma in effusive atomic beams was long known for high-Rydberg gases of caesium and well explained by coupled rate equations \cite{Vitrant}.  But, low densities and ultracold velocity distributions were thought to exclude Rydberg-Rydberg collisional mechanisms in a MOT.

In work on ultracold Rydberg gases of Rb and Cs, Gallagher, Pillet and coworkers describe the initial growth of electron signal by a model that includes ionization by blackbody radiation and collisions with a background of uncooled Rydberg atoms \cite{Mourachko,Gallagher,Li,Comparat,Tanner}. This picture was subsequently refined to include many-body excitation and autoionization, as well as attractive dipole-dipole interactions \cite{Viteau,Pillet}, later confirmed by experiments at Rice \cite{Mcquillen}.

The Orsay group also studied the effect of adding Rydberg atoms to an established ultracold plasma.  They found that electron collisions in this environment completely ionize added atoms, even when selected to have deep binding energies \cite{Vanhaecke}.  They concluded from estimates of electron trapping efficiency that the addition of Rydberg atoms does not significantly alter the electron temperature of the plasma.

Tuning pair distributions by varying the wavelength of the excitation laser, Weidem\"uller and coworkers confirmed the mechanical effects of van der Waals interactions on the rates of Penning ionization in ultracold $^{87}$Rb Rydberg gases \cite{Amthor_mech}.  They recognized blackbody radiation as a possible means of final-state redistribution, and extended this mechanical picture to include long-range repulsive interactions \cite{Amthor_model}.  This group later studied the effects of spatial correlations in the spontaneous avalanche of Rydberg gases in a regime of strong blockade, suggesting a persistence of initial spatial correlations \cite{RobertdeSaintVincent}.

Robicheaux and coworkers have recently investigated the question of prompt many-body ionization from the point of view of Monte Carlo classical trajectory calculations \cite{Goforth}.  For atoms on a regular or random grid driven classically by an electromagnetic field, they find that many-body excitation enhances prompt ionization by about twenty percent for densities greater than $5.6 \times 10^{-3}/(n_0^2 a_0)^3$, where $n_0$ is the principal quantum number of the Rydberg gas and $a_0$ is the Bohr radius.  They observed that density fluctuations (sampled from the distribution of nearest neighbour distances) have a greater effect, and point to the possible additional influence of secondary electron-Rydberg collisions and the Penning production of fast atoms not considered by the model, but already observed by Raithel and coworkers \cite{Knuffman}.

The Raithel group also found direct evidence for electron collisional $\ell$-mixing in a Rb MOT \cite{Dutta}, and used selective field ionization to monitor evolution to plasma on a microsecond timescale in ultracold $^{85}$Rb $65d$ Rydberg gases with densities as low as $10^8$ cm$^{-3}$ \cite{WalzFlannigan}.  Research by our group at UBC has observed very much the same dynamics in the relaxation of Xe Rydberg gases of similar density prepared in a molecular beam \cite{Hung2014}.  In both cases, the time evolution to avalanche is well-described by coupled rate equations (see below), assuming an initializing density of Penning electrons determined by Robicheaux's criterion \cite{Robicheaux05}, applied to an Erlang distribution of Rydberg-Rydberg nearest neighbours.

Theoretical investigations of ultracold plasma physics have focused for the most part on the long- and short-time dynamics of plasmas formed by direct photoionization \cite{PhysRept,Lyon}.  In addition to studies mentioned above, key insights on the evolution dynamics of Rydberg gases have been provided by studies of Pohl and coworkers exploring the effects of ion correlations and recombination-reionization on the hydrodynamics of plasma expansion \cite{Pohl:2003,PPR}.  Further research has drawn upon molecular dynamics (MD) simulations to reformulate rate coefficients for the transitions driven by electron impact between highly excited Rydberg states \cite{PVS}, and describe an effect of strong coupling as it suppresses three-body recombination \cite{Bannasch:2011}.  MD simulations confirm the accuracy of coupled rate equation descriptions for systems with $\Gamma$ as large as 0.3.  Newer calculations suggest a strong connection between the order created by dipole blockade in Rydberg gases and the most favourable correlated distribution of ions in a corresponding strongly coupled ultracold plasma \cite{Bannasch:2013}.

Tate and coworkers have studied ultracold plasma avalanche and expansion theoretically as well as experimentally.  Modelling observed expansion rates, they recently found that $^{85}$Rb atoms in a MOT form plasmas with effective initial electron temperatures determined by initial Rydberg density and the selected initial binding energy, to the extent that these parameters determine the fraction of the excited atoms that ionize by electron impact in the avalanche to plasma \cite{Forest}.  This group also returned to the question of added Rydberg atoms, and managed to identify a crossover in $n_0$, depending on the initial electron temperature, that determines whether added Rydberg atoms of a particular initial binding energy act to heat or cool the electron temperature \cite{Crockett}.

Our group has focused on the plasma that evolves from a Rydberg gas under the low-temperature conditions of a skimmed, seeded supersonic molecular beam.  In work on nitric oxide starting in 2008 \cite{Morrison2008,Plasma_expan,Morrison_shock,PCCP}, we established an initial kinetics of electron impact avalanche ionization that conforms with coupled rate equation models \cite{Saquet2011,Saquet2012,Scaling,haenelCP} and agrees at early times with the properties of ultracold plasmas that evolve from ultracold atoms in a MOT.  We have also observed unique properties of the NO ultracold plasma owing to the fact that its Rydberg states dissociate \cite{Haenel2017}, and identified relaxation pathways that may give rise to quantum effects \cite{SousMBL,SousNJP}.  The remainder of this review focuses on the nitric oxide ultracold plasma and the unique characteristics conferred by its evolution from a Rydberg gas in a laser-crossed molecular beam.

\section{Avalanche to strong coupling in a molecular Rydberg gas}

\subsection{The molecular beam ultracold plasma compared with a MOT}

When formed with sufficient density, a Rydberg gas of principal quantum number $n_0>30$ undergoes a spontaneous avalanche to form an ultracold plasma \cite{Li,Morrison2008,RobertdeSaintVincent}.  Collisional rate processes combine with ambipolar hydrodynamics to govern the properties of the evolving plasma.  For a molecular Rydberg gas, neutral fragmentation, occurs in concert with electron-impact ionization, three-body recombination and electron-Rydberg inelastic scattering.  Neutral dissociation combined with radial expansion in a shaped distribution of charged particles, can give rise to striking effects of self-assembly and spatial correlation \cite{Schulz-Weiling2016,Haenel2017}.

The formation of a molecular ultracold plasma requires the conditions of local temperature and density afforded by a high mach-number skimmed supersonic molecular beam.  Such a beam propagates at high velocity in the laboratory, with exceedingly well-defined hydrodynamic properties, including a propagation-distance-dependent density and sub-Kelvin temperature in the moving frame \cite{MSW_tutorial}.  The low-temperature gas in a supersonic molecular beam differs in three important ways from the atomic gas laser-cooled in a magneto-optical trap (MOT).

The milli-Kelvin temperature of the gas of ground-state NO molecules entrained in a beam substantially exceeds the sub-100 micro-Kelvin temperature of laser-cooled atoms in a MOT.  However, the evolution to plasma tends to erase this distinction, and the two further characteristics that distinguish a beam offer important advantages for ultracold plasma physics:  Charged-particle densities in a molecular beam can exceed those attainable in a MOT by orders of magnitude.  A great many different chemical substances can be seeded in a free-jet expansion, and the possibility this affords to form other molecular ultracold plasmas, introduces interesting and potentially important new degrees of freedom governing the dynamics of their evolution.

\subsection{Supersonic molecular beam temperature and particle density}

Seeded in a skimmed supersonic molecular beam, nitric oxide forms different phase-space distributions in the longitudinal (propagation) and transverse coordinate dimensions.  As it propagates in $z$, the NO molecules reach a terminal laboratory velocity, $u_{\parallel}$, of about 1400 ${\rm ms^{-1}}$, which varies with the precise seeding ratio.

The distribution of $v_{\parallel}$, narrows to define a local temperature, $T_{\parallel}$, of approximately 0.5 K.  The beam forms a Gaussian spatial distribution in the transverse coordinates, $x$ and $y$.  In this plane, the local velocity, $v_{\perp}(r)$ is defined for any radial distance almost entirely by the divergence velocity of the beam, $u_{\perp}(r)$.  Phase-space sorting cools the temperature in the transverse coordinates, $T_{\perp}$ to a value as low as $\sim 5$ mK \cite{MSW_tutorial}.

The stagnation pressure and seeding ratio determine the local density distribution as a function of $z$.  For example, expanding from a stagnation pressure of 500 kPa with a 1:10 seeding ratio, a molecular beam propagates 2.5 cm to a skimmer and then 7.5 cm to a point of laser interaction, where it contains NO at a peak density of $1.6 \times 10^{14}$ cm$^{-3}$.

Here, crossing the molecular beam with a laser beam tuned to the transition sequence, ${\rm X} ~^2 \Pi_{1/2} ~N'' = 1 \xrightarrow{\omega_1} {\rm A} ~^2\Sigma^+ ~N'=0  \xrightarrow{\omega_2} n_0 f(2)$ forms a Gaussian ellipsoidal volume of Rydberg gas in a single selected principal quantum number, $n_0$, orbital angular momentum, $\ell = 3$, NO$^+$ core rotational quantum number, $N^+ = 2$ and total angular momentum neglecting spin, $N=1$.

A typical $\omega_1$ pulse energy of 2 $\mu$J and a Gaussian width of 0.2 mm serves to drive the first step of this sequence in a regime of linear absorption.  Overlapping this volume by an $\omega_2$ pulse with sufficient fluence to saturate the second step forms a Rydberg gas ellipsoid with a nominal peak density of $5 \times 10^{12}$ cm$^{-3}$  \cite{Morrison2008,MSW_tutorial}.  Fluctuations in the pulse energy and longitudinal mode of $\omega_1$ cause the real density to vary.  For certain experiments, we find it convenient to saturate the $\omega_1$ transition, and vary the density of Rydberg gas by delaying $\omega_2$.  An $\omega_1$-$\omega_2$ delay, $\Delta t$, reduces the Rydberg gas density by a precise factor, $e^{-\Delta t/\tau}$, where $\tau$ is the 200 ns radiative lifetime of NO ${\rm A} ~^2\Sigma^+ ~N'=0$ \cite{Carter,Hancock}.

\subsection{Penning ionization}

The density distribution of a Rydberg gas defines a local mean nearest neighbour distance, or Wigner-Seitz radius of $ a_{ws} =  \left(3/4 \pi \rho \right)^{1/3} $, where $\rho$ refers to the local Rydberg gas density.  For example, a Rydberg gas with a density of $ \rho_0=0.5 \times 10^{12}$ cm$^{-3} $ forms an Erlang distribution \cite{Torquato.1990} of nearest neighbour separations with a mean value of $ 2 a_{ws}=1.6$  $\mu$m.

A semi-classical model \cite{Robicheaux05} suggests that 90 percent of Rydberg molecule pairs separated by a critical distance, $ r_c = 1.8 \cdot 2 n_0^2 a_0 $ or less undergo Penning ionization within 800 Rydberg periods.  We can integrate the Erlang distribution from $ r=0 $ to the critical distance $r = r_c$ for a Rydberg gas of given $n_0$, to define the local density of Penning electrons ($ \rho_e$ at $t=0$) produced by this prompt interaction, for any given initial local density, $\rho_0$ by the expression:
\begin{equation}
\rho_e(\rho_0,n_0) = \frac{0.9}{2} \cdot 4 \pi \rho_0 ^2\int_0^{r_{c}} r^2 \mathrm{e}^{-\frac{4\pi}{3}\rho_0 r^3}\mathrm{d}r \quad.
\label{eqn:Erlang}
\end{equation}

Evaluating this definite integral yields an equation in closed form that predicts the Penning electron density for any particular initial Rydberg density and principal quantum number.
\begin{equation}
\rho_e(\rho_0,n_0) =\frac{0.9 \rho_0}{2}(1-\mathrm{e}^{-\frac{4\pi}{3}\rho_0 r_c^3}) \quad.
\label{Eq:PenDens}
\end{equation}
\begin{figure}[h!]
\centering
\includegraphics[scale=0.33]{Penning_Latice.pdf}
\caption{Distributions of ion-ion nearest neighbours following Penning ionization and electron-impact avalanche simulated for a predissociating molecular Rydberg gas of initial principal quantum number, $n_0$, from 30 to 80, and density of 10$^{12}$ cm$^{-3}$.  Dashed lines mark corresponding values of $a_{ws}$. Calculated by counting ion distances after relaxation to plasma in 10$^6$-particle stochastic simulations. Integrated areas proportional to populations surviving neutral dissociation.}
\label{fig:PL}
\end{figure}

Prompt Penning ionization acts on the portion of the initial nearest-neighbour distribution in the Rydberg gas that lies within $r_c$.  When a molecule ionizes, its collision partner relaxes to a lower principal quantum number, $n'<n_0/\sqrt{2}$.  This close-coupled interaction disrupts the separability of Rydberg orbital configurations in the Penning partner.  This causes mixing with core penetrating states that are strongly dissociative.  Penning partners are thus very likely to dissociate, leaving a spatially isolated distribution of ions.  We refer to the spatial correlation that results as a Penning lattice \cite{Sadeghi:2014}.  The extent of this effect varies depending on the local density and the selected initial principal quantum number.  Figure \ref{fig:PL} shows the degree to which Rydberg gases with initial principal quantum numbers from 30 to 80 form a Penning lattice for an initial density of $1 \times 10^{12} ~{\rm cm}^{-3}$.

\subsection{Spontaneous electron-impact avalanche}

The electrons produced by prompt Penning ionization start an electron impact avalanche.  The kinetics of this process are well described by a set of coupled rate equations that account for state-to-state electron-Rydberg inelastic scattering, electron-impact ionization and three-body ion-electron recombination \cite{PPR,Saquet2011,Saquet2012,Scaling} using detailed rate coefficients,  $k_{ij}$, $k_{i,ion}$ and $k_{i,tbr}$ validated by MD simulations \cite{PVS}.
\begin{eqnarray}
-\frac{d\rho_i}{dt}&=&\sum_{j}{k_{ij}\rho_e\rho_i}-\sum_{j}{k_{ji}\rho_e\rho_j} \nonumber\\
&& +k_{i,ion}\rho_e\rho_i-k_{i,tbr}\rho^3_e
  \label{level_i}
\end{eqnarray}
\noindent and,
\begin{equation}
\frac{d\rho_e}{dt}=\sum_{i}{k_{i,ion}\rho_e^2}-\sum_{i}{k_{i,tbr}\rho^3_e}
  \label{electron}
\end{equation}

The relaxation of Rydberg molecules balances with collisional ionization to determine an evolving temperature of avalanche electrons to conserve total energy per unit volume.
\begin{equation}
E_{tot}=\frac{3}{2}k_BT_e(t)\rho_e(t)-R\sum_i{\frac{\rho_i(t)}{n_i^2}},
  \label{energy}
\end{equation}
Here, for simplicity, we neglect the longer-time effects of Rydberg predissociation and electron-ion dissociative recombination \cite{Saquet2012}.

Such calculations show that the conversion from Rydberg gas to plasma occurs on a timescale determined largely by the local Penning electron density, or Penning fraction, $P_f = \rho_e/\rho_0$, which depends on the local density of Rydberg molecules and their initial principal quantum number.

Avalanche times predicted by coupled rate equation calculations range widely.  For example, in a model developed for experiments on xenon, simulations predict that a Rydberg gas with $n_0 = 42$ at a density of $8.8 \times 10^8 ~{\rm cm}^{-3}$ ($P_f = 6 \times 10^{-5}$) avalanches with a half time of  40 $\mu$s \cite{Hung2014}.  At an opposite extreme, rate equations estimate that a Rydberg gas of NO with $n_0=60$ at a density of $1 \times 10^{12} ~{\rm cm}^{-3}$ ($P_f = 0.3$) rises to plasma in about 2 ns \cite{Saquet2012}.

\begin{figure}[h!]
\centering
\includegraphics[width= .49 \textwidth]{SFI_n=49.pdf}
   \caption{Contour plots showing SFI signal as a function the applied field for an $nf(2)$ Rydberg gas with an initial principal quantum number, $n_0=49$.  Each frame represents 4,000 SFI traces, sorted by initial Rydberg gas density.  Ramp field beginning at 0 and 150 ns (top, left to right), and  300 and 450 ns (bottom) after the $\omega_2$ laser pulse.  The two bars of signal most evident at early ramp field delay times represent the field ionization of the $49f(2)$ Rydberg state respectively to NO$^+$ X $^1\Sigma^+$ cation rotational states, $N^+=0$ and 2.  The signal waveform extracted near zero applied field represents the growing population of plasma electrons.
   }
\label{fig:SFI}
\end{figure}

Selective field ionization (SFI) probes the spectrum of binding energies in a Rydberg gas.  Applied as a function of time after photoexcitation, SFI maps the evolution from a state of selected initial principal quantum number, $n_0$, to plasma \cite{Haenel2017}.  Figure \ref{fig:SFI} shows SFI spectra taken at a sequence of delays after the formation of $49f(2)$ Rydberg gases of varying density.

Here, we can see that a $49f(2)$ Rydberg gas with an estimated initial density $\rho_0 = 3 \times 10^{11} ~{\rm cm}^{-3}$ relaxes to plasma on a timescale of about 500 ns.  Observations such as these agree well with the predictions of coupled rate-equation calculations.  We can understand this variation in relaxation dynamics with $\rho_0$ and $n_0$ quite simply in terms of the corresponding density of prompt Penning electrons these conditions afford to initiate the avalanche to plasma.

Figure \ref{fig:scaled_rise} illustrates this, showing how rise times predicted by coupled rate-equation simulations for a large range of initial densities and principal quantum number match when plotted as a function of time scaled by the ultimate plasma frequency and fraction of prompt Penning electrons.  The dashed line gives an approximate account of the scaled rate of avalanche under all conditions of Rydberg gas density and initial principal quantum number in terms of the simple sigmoidal function:

\begin{equation}
\frac{\rho_e}{\rho_0} = \frac{a}{b+e^{-c\tau}},
  \label{scaledEq1}
\end{equation}
where,
\begin{equation}
\tau = t \omega_e P_f^{3/4},
  \label{scaledEq2}
\end{equation}
in which $\omega_e$ is the plasma frequency after avalanche, $P_f$ is the fraction of prompt Penning electrons, and $a = 0.00062$,  $b =   0.00082$ and $c =     0.075$ are empirical coefficients.

\begin{figure}[h!]
\centering
\includegraphics[width= .4 \textwidth]{sim_analytical_density.pdf}
   \caption{Rise in fractional electron density as a function of time scaled by the plasma frequency, $\omega_e$ and fraction, $\rho_e(t=0)/\rho_0 = P_f$, of prompt Penning electrons.  Simulation results shown for $n_0 = 30$, 50 and 70 with initial densities, $\rho_0 = 10^9,~10^{10},~10^{11},~{\rm and}~10^{12}~{\rm cm}^{-3}$.
   }
\label{fig:scaled_rise}
\end{figure}

\subsection{Evolution to plasma in a Rydberg gas Gaussian ellipsoid}

As outlined above, the local density and principal quantum number together determine the rate at which a Rydberg gas avalanches to plasma.  Our experiment crosses a 2 mm wide cylindrically Gaussian molecular beam with a 1 mm diameter TEM$_{00}$ $\omega_1$ laser beam to produce a Gaussian ellipsoidal distribution of molecules excited to the A $^2\Sigma^+$ $v=0, ~N'=0$ intermediate state.  A larger diameter $\omega_2$ pulse then drives a second step that forms a Rydberg gas in a single $n_0f(2)$ state with the spatial distribution of the intermediate state.

We model this shaped Rydberg gas as a system of 100 concentric ellipsoidal shells of varying density \cite{haenelCP}.  Coupled rate equations within each shell describe the avalanche to plasma.  This rate process proceeds from shell to shell with successively longer induction periods, determined by the local density as detailed above.  The rising conversion of Rydberg molecules to ions plus neutral dissociation products conserves the particle number in each shell.  We assume that local space charge confines electrons to shells, conserving quasi-neutrality.  Electrons exchange kinetic energy at the boundaries of each shell, which determines a single plasma electron temperature.

\begin{figure}[h!]
\centering
\includegraphics[width= .5 \textwidth]{shell_model_100}
   \caption{(top frame) Cross-sectional contour diagram in the $x,y$ plane for $z=0$ describing the distribution of ion plus electron density over 100 shells of Gaussian ellipsoid with initial dimensions, $\sigma_x= 0.75$ mm and $\sigma_y= \sigma_z = 0.42$ mm and an initial $n_0 = 50$ Rydberg gas density, $\rho_0 = 2 \times 10^{11}$ cm$^{3}$ after an evolution time of 100 ns.  (bottom frame) Curves describing the (dashed) ascending ion and (solid) descending Rydberg gas densities of each shell as functions of evolution time, for $t=20$, 40, 60, 80 and 100 ns.
   }
\label{fig:shell}
\end{figure}

The upper frame of Figure \ref{fig:shell} shows contours of NO$^+$ ion density after 100 ns obtained from a shell-model coupled rate-equation simulation of the avalanche of a Gaussian ellipsoidal Rydberg gas of nitric oxide with a selected initial state, $50f(2)$ and a density of $2 \times 10^{11}$ cm$^{-3}$.  Here, we simulate a relaxation that includes channels of predissociation at every Rydberg level and redistributes the energy released to electrons, which determines a uniform rising electron temperature for all shells.

For comparison, the lower frame plots curves describing the ion density of each shell as a function of time from 20 to 100 ns, as determined by applying Eq \ref{scaledEq1} for the local conditions of initial Rydberg gas density.  This numerical approximation contains no provision for predissociation.  Coupled rate-equation simulations for uniform volumes show that predissociation depresses yield to some degree, but has less effect on the avalanche kinetics \cite{Saquet2012}.  Therefore, we can expect sets of numerically estimated shell densities, scaled to agree with the simulated ion density at the elapsed time of 100 ns to provide a reasonable account of the earlier NO$^+$ density profiles as a function of time.

\begin{figure}[h!]
\centering
\includegraphics[width= .4 \textwidth]{Shell_pop}
   \caption{Global population fractions of particles as they evolve in the avalanche of a shell-model ellipsoidal Rydberg gas with the initial principal quantum number and density distribution of Figure \ref{fig:shell}
   }
\label{fig:shell_yields}
\end{figure}

For each time step, the difference, $\rho_0 - \rho_e$ defines the neutral population of each shell.  We assign a fraction of this population to surviving Rydberg molecules, such that the total population of NO$^*$ as a function of time agrees with the prediction of the shell-model simulation, as shown in Figure \ref{fig:shell_yields}.  We consider the balance of this neutral population to reflect NO$^*$ molecules that have dissociated to form N($^4$S) $+$ O($^3$P).  Figure \ref{fig:shell} plots these surviving Rydberg densities as a function of radial distance for each evolution time.  At the initial density of this simulation, note at each time step that a higher density of Rydberg molecules encloses the tail of the ion density distribution in $x$.

\subsection{Plasma expansion and NO$^+$ - NO$^*$ charge exchange as an avenue of quench}

We regard the ions as initially stationary.  The release of electrons creates a radial electric potential gradient, which gives rise to a force, $-e\nabla \phi_{k,j}(t)$, that accelerates the ions in shell $j$ in direction $k$ according to \cite{Sadeghi.2012}:

\begin{align}
\frac{-e}{m'}\nabla \phi_{k,j}(t) = & \frac{\partial u_{k,j}(t)}{\partial t} \notag  \\
= & \frac{k_BT_e(t)}{m'\rho_j(t)} \frac{\rho_{j+1}(t) - \rho_j(t)}{r_{k,j+1}(t) - r_{k,j}(t)},
  \label{dr_dt}
\end{align}

\noindent where $\rho_j(t)$ represents the density of ions in shell $j$.

The instantaneous velocity, $u_{k,j}(t)$ determines the change in the radial coordinates of each shell, $r_{k,j}(t)$,
\begin{equation}
\frac{\partial r_{k,j}(t)}{\partial t}=u_{k,j}(t) = \gamma_{k,j}(t) r_{k,j}(t),
  \label{dr_dt}
\end{equation}
\noindent which in turn determines shell volume and thus its density, $ \rho_j(t)$.
The electron temperature supplies the thermal energy that drives this ambipolar expansion.  Ions accelerate and $T_e$ falls according to:

\begin{equation}
\frac{3k_B}{2}\frac{\partial T_e(t)}{\partial t}= -\frac{m'}{\sum_{j}{N_j}}\sum_{k,j}{N_j u_{k,j}(t)\frac{\partial u_{k,j}(t)}{\partial t}},
  \label{dr_dt}
\end{equation}
\noindent where we define an effective ion mass, $m'$, that recognizes the redistribution of the electron expansion force over all the NO$^+$ charge centres by resonant ion-Rydberg charge exchange, which occurs with a very large cross section \cite{PPR}.
\begin{equation}
m' =\left (1+ \frac{\rho^*_{j}(t)}{ \rho_j(t)}\right) m ,
  \label{dr_dt}
\end{equation}
\noindent in which $\rho^*_{j}(t)$ represents the instantaneous Rydberg density in shell $j$.

The initial avalanche in the high-density core of the ellipsoid leaves few Rydberg molecules, so this term has little initial effect.  Rydberg molecules predominate in the lower-density wings.  There, momentum sharing by charge exchange assumes a greater importance.

We see this most directly in the $\omega_2$ absorption spectrum of transitions to states in the $n_0 f(2)$ Rydberg series, detected as the long-lived signal that survives a flight time of 400 $\mu$s to reach the imaging detector.  The balance between the rising density of ions and the falling density of Rydberg molecules depends on the initial density of electrons produced by prompt Penning ionization.  As clear from Eq \ref{Eq:PenDens}, this Penning fraction depends sensitively on the principal quantum number, and for all principal quantum numbers, on the initial Rydberg gas density.

\begin{figure}[h!]
\centering
\includegraphics[width= .4 \textwidth]{w2_spectra}
   \caption{Double-resonant spectra of nitric oxide Rydberg states in the $nf$ series converging to NO$^+$ $v=0$, $N^+=2$ (designated, $nf(2)$), derived from the late-peak signal obtained after a flight time of 400 $\mu$s by scanning $\omega_2$ for $\omega_1$ tuned to NO A $^2\Sigma^+$ $v=0$, $N'=0$ for initial $nf(2)$ densities from top to bottom of 0.07, 0.10, 0.13, 0.19, 0.27, 0.30, 0.32 and $3 \times 10^{12}$ cm$^{3}$.
  }
\label{fig:w2_spectra}
\end{figure}

Figure \ref{fig:w2_spectra} shows a series of $\omega_2$ late-signal excitation spectra for a set of initial densities.  Here, we see a clear consequence of the higher-order dependence of Penning fraction - and thus the NO$^+$ ion - NO$^*$ Rydberg molecule balance - on $n_0$, the $\omega_2$-selected Rydberg gas initial principal quantum number.  This Penning-regulated NO$^+$ ion - NO$^*$ Rydberg molecule balance appears necessary as a critical factor in achieving the long ultracold plasma lifetime required to produce this signal.  We are progressing in theoretical work that explains the stability apparently conferred by this balance.

\subsection{Bifurcation and arrested relaxation}

Ambipolar expansion quenches electron kinetic energy as the initially formed plasma expands.  Core ions follow electrons into the wings of the Rydberg gas.  There, recurring charge exchange between NO$^+$ ions and NO$^*$ Rydberg molecules redistributes the ambipolar force of the expanding electron gas, equalizing ion and Rydberg velocities.  This momentum matching effectively channels electron energy through ion motion into the overall $\pm x$ motion of gas volumes in the laboratory.  The internal kinetic energy of the plasma, which at this point is defined almost entirely by the ion-Rydberg relative motion, falls.  Spatial correlation develops, and over a period of 500 ns, the system forms the plasma/high-Rydberg quasi-equilibrium dramatically evidenced by the SFI results in Figure \ref{fig:SFI}.

\begin{figure}[h!]
\centering
\includegraphics[width= .4 \textwidth]{Bifurcation.pdf}
   \caption{$x,y$ detector images of ultracold plasma volumes produced by 2:1 aspect ratio ellipsoidal Rydberg gases with selected initial state, $40f(2)$ after a flight time of 402 $\mu$s over a distance of 575 mm.  Lower frame displays the distribution in $x$ of the charge integrated in $y$ and $z$.  Both images represent the unadjusted raw signal acquired in each case after 250 shots.
   }
\label{fig:bifurcation}
\end{figure}

In the wings, momentum redistribution owing to cycles of ion-Rydberg charge transfer retards radial expansion \cite{Pohl2003,PPR}.  By redirecting electron energy from ambipolar acceleration to $\pm x$ plasma motion, NO$^+$ to NO$^*$ charge exchange dissipates electron thermal energy.  This redistribution of energy released in the avalanche of the Rydberg gas to plasma, causes the ellipsoidal Rydberg gas to bifurcate \cite{Schulz-Weiling2016,Haenel2017}, forming very long-lived, separating charged-particle distributions.  We capture the electron signal from these recoiling volumes on an imaging detector as pictured in Figure \ref{fig:bifurcation}.  Here, momentum matching preserves density and enables ions and Rydberg molecules to relax to positions that minimize potential energy, building spatial correlation.

The semi-classical description of avalanche and relaxation outlined above forms an important point of reference from which to interpret our experimental observations.  The laser crossed molecular beam illumination geometry creates a Rydberg gas with a distinctively shaped high-density spatial distribution.  This initial condition has an evident effect on the evolution dynamics.  We have developed semi-classical models that explicitly consider the coupled rate and hydrodynamic processes governing the evolution from Rydberg gas to plasma using a realistic, ellipsoidal representation of the ion/electron and Rydberg densities \cite{haenelCP}.  No combination of initial conditions can produce a simulation that conforms classically with the state of arrested relaxation we observe experimentally.

\subsection{A molecular ultracold plasma state of arrested relaxation}

Thus, we find that spontaneous avalanche to plasma splits the core of an ellipsoidal Rydberg gas of nitric oxide. As ambipolar expansion quenches the electron temperature of this core plasma, long-range, resonant charge transfer from ballistic ions to frozen Rydberg molecules in the wings of the ellipsoid quenches the ion-Rydberg molecule relative velocity distribution. This sequence of steps gives rise to a remarkable mechanics of self-assembly, in which the kinetic energy of initially formed hot electrons and ions drives an observed separation of plasma volumes. These dynamics redistribute ion momentum, efficiently channeling electron energy into a reservoir of mass-transport. This starts a process that evidently anneals separating volumes to a state of cold, correlated ions, electrons and Rydberg molecules.

We have devised a three-dimensional spin model to describe this arrested state of the ultracold plasma in terms of two, three and four-level dipole-dipole energy transfer interactions (spin flip-flops), together with Ising interactions that arise from the concerted pairwise coupling of resonant pairs of dipoles \cite{SousMBL,SousNJP}.

The Hamiltonian includes the effects of onsite disorder owing to the broad spectrum of states populated in the ensemble and the unique electrostatic environment of every dipole.  Extending ideas developed for simpler systems \cite{Burin1,Sondhi}, one can make a case for slow dynamics, including an arrest in the relaxation of NO Rydberg molecules to predissociating states of lower principal quantum number.

Systems of higher dimension ought to thermalize by energy transfer that spreads from rare but inevitable ergodic volumes (Griffiths regions) \cite{Sarang2, Roeck_griffith, RareRegions_rev, Thermal_inclusions}.  However, a feature in the self-assembly of the molecular ultracold plasma may preclude destabilization by rare thermal domains:  Whenever the quenched plasma develops a delocalizing Griffiths region, the local predissociation of relaxing NO molecules promptly proceeds to deplete that region to a void of no consequence.

In summary, the classical dynamics of avalanche and bifurcation appear to create a quenched condition of low temperature and high disorder in which dipole-dipole interactions drive self-assembly to a localized state purified by the predissociation of thermal regions.  We suggest that this state of the quenched ultracold plasma offers an experimental platform for studying quantum many-body physics of disordered systems. \\

\section{Acknowledgments}
This work was supported by the US Air Force Office of Scientific Research (Grant No. FA9550-17-1-0343), together with the Natural Sciences and Engineering research Council of Canada (NSERC), the Canada Foundation for Innovation (CFI) and the British Columbia Knowledge Development Fund (BCKDF).