text:
\section*{Introduction}

Hyperfine splitting of the ground state in light H-like systems,
such as hydrogen, deuterium, tritium, and helium-3 ion, has long
been known experimentally with extremely high precision. The
present-day theory of the ground-state hyperfine structure ({\em
hfs}) is still far behind the experiment, due to a relatively
large contribution of the nuclear-structure effects, which cannot
be accurately calculated at present. One of the possibilities to
overcome this difficulty \cite{sternheim:63} is to study the
normalized difference $\Delta_{21} = 8\, \nu_{2s}-\nu_{1s}$, where
$\nu_{1s}$ and $\nu_{2s}$ are the $1s$ and the $2s$ {\em hfs } interval,
respectively. A large class of corrections to $\nu_{1s}$ and
$\nu_{2s}$ (among them, all lowest-order nuclear effects) are
proportional to the nonrelativistic electron density at the
position of the nucleus ($r=0$) and, therefore, do not contribute
to the difference $\Delta_{21}$. Consequently, the theoretical
study of this difference can be performed up to a much higher
accuracy than that of $\nu_{1s}$ and $\nu_{2s}$ separately.

The experimental value of the difference $\Delta_{21}$ is obtained
by combining results of two independent measurements of $\nu_{1s}$
and $\nu_{2s}$ and is known less precisely than $\nu_{1s}$. The
best accuracy is obtained for the helium-3 ion in a combination of
two relatively old results \cite{schluessler:69,prior:77},
\begin{equation}\label{1}
    \Delta_{21}(^3{\rm He}^+) = 1\,189.979\,(71)\,\,{\rm kHz}\,.
\end{equation}
Recent progress was achieved in the measurement of $\nu_{2s}$
in hydrogen \cite{kolachevsky:04:prl} and
deuterium \cite{kolachevsky:04:pra}, which significantly improved the
corresponding experimental values for the difference $\Delta_{21}$.

Theory of {\em hfs } and, specifically, of the difference $\Delta_{21}$
in light H-like atoms has recently been examined in detail in
Ref.~\cite{karshenboim:02:epjd}. It is demonstrated that one of
the major uncertainties in the theoretical prediction of
$\Delta_{21}(^3{\rm He}^+)$ stems from the one-loop self-energy
correction. The self-energy correction is also responsible for a
significant part of the theoretical uncertainty for the
ground-state hyperfine splitting in muonium \cite{mohr:05:rmp}.

The goal of the present investigation is to improve the numerical
accuracy of the one-loop QED correction for the $1s$ and $2s$
states in light H-like atoms. Our consideration will be carried
out to all orders in the parameter ${Z \alpha}$ ($Z$ is the nuclear
charge number and $\alpha$ is the fine-structure constant).
All-order calculations of the self-energy {\em hfs } correction in
H-like ions have been previously performed by numerous authors
\cite{persson:96:hfs,yerokhin:96:pisma,blundell:97:pra,yerokhin:97:eprint,%
blundell:97:prl,sunnergren:98:pra,yerokhin:01:hfs,sapirstein:01:hfs}.
Different evaluations are generally in good agreement (except for
the first two calculations, see a discussion in
Ref.~\cite{yerokhin:01:hfs}). For high- and middle-$Z$ ions,
results for this correction can be presently considered as well
established at the level of the experimental interest. In the
low-$Z$ region, however, the experimental accuracy is much higher
and technical problems encountered in all-order calculations are
more demanding than for higher-$Z$ ions. It would be clearly
preferable to perform a direct all-order calculation for $Z=1$ and
2 with an accuracy significantly higher than the one obtained from
the ${Z \alpha}$ expansion, as it was done for the Lamb shift
\cite{jentschura:99:prl}. However, such a project has not been
realized yet. Blundell {\em et al.} \cite{blundell:97:prl}
obtained the higher-order (in ${Z \alpha}$) contribution to the $1s$
self-energy correction for $Z=1$ by extrapolating their numerical
results for higher $Z$. Similar procedure was employed in the
investigation by two of us \cite{yerokhin:01:hfs} for the
self-energy correction to the $1s$ and $2s$ {\em hfs } intervals for
$Z=1$ and 2.

The vacuum-polarization {\em hfs } correction was evaluated to all
orders in ${Z \alpha}$ in
Refs.~\cite{persson:96:hfs,shabaev:97:pra,shabaev:98:hfs} (without
the magnetic-loop Wichmann-Kroll correction) and in
Refs.~\cite{sunnergren:98:pra,artemyev:01} (complete
calculations). However, the above studies were mainly concerned
with high- and middle-$Z$ ions, so that little information was
provided about the behavior of the Wichmann-Kroll part of the
vacuum-polarization correction in the low-$Z$ region.

In the present work, we perform a calculation of the one-loop
self-energy and vacuum-polarization corrections to the $1s$ and
$2s$ {\em hfs } intervals in H-like ions. The paper is organized as
follows. In Sec.~\ref{sec:se} we evaluate the self-energy
correction by employing the additional-subtraction scheme
\cite{yerokhin:05:se} that improves the convergence properties of
the resulting partial-wave expansion. In this way, we
significantly increase the accuracy of the numerical results for
$Z \ge 5$ as compared to the previous evaluations. The
vacuum-polarization correction is evaluated in Sec.~\ref{sec:vp}.
The higher-order one-loop QED contribution is inferred from our
all-order results  in Sec.~\ref{sec:ho} by subtracting the known
terms of the ${Z \alpha}$ expansion. Finally, we discuss the experimental
consequences of our calculation.

Relativistic units ($\hbar = c = m = 1$) are used throughout the paper.

\section{Self-energy correction}
\label{sec:se}

In this section we describe the evaluation of the self-energy {\em hfs } correction
without any expansion in ${Z \alpha}$. We start with general
formulas for the self-energy correction in the presence of an
additional perturbing potential $\delta V$. To the first order in
$\delta V$, the self-energy correction is given by the sum of the
{\em irreducible}, the {\em reducible}, and the {\em vertex}
correction \cite{shabaev:02:rep},
\begin{equation}\label{4a}
    \Delta E_{\rm SE} = \Delta E_{\rm ir}+\Delta E_{\rm red}+\Delta E_{\rm
    ver}\,.
\end{equation}
The irreducible part arises through a perturbation of the wave
function,
\begin{equation}\label{5}
\Delta E_{\rm ir} = \langle \delta a|\gamma^0
    \widetilde{\Sigma}(\varepsilon_a) |a\rangle
              + \langle a|\gamma^0 \widetilde{\Sigma}(\varepsilon_a) |\delta a\rangle\,,
\end{equation}
where $\widetilde{\Sigma} = \Sigma -\delta m$, $\delta m$ is the
one-loop mass counterterm, $\Sigma$ is the one-loop self-energy
function,
\begin{eqnarray} \label{6}
 \Sigma(\varepsilon,{\bm{x}}_1,{\bm{x}}_2) &=&
2\,{\rm i}\alpha\,\gamma^0 \int^{\infty}_{-\infty} d\omega\,
      \alpha_{\mu}\,
 \nonumber \\ && \times
         G(\varepsilon-\omega,{\bm{x}}_1,{\bm{x}}_2)\, \alpha_{\nu}\,
    D^{\mu\nu}(\omega,{\bm{x}}_{12})\,
         \,,
\end{eqnarray}
$G$ is the Dirac Coulomb Green function $G(\varepsilon) = [\varepsilon-{\cal
H}(1-{\rm i} 0)]^{-1}$, $\cal H$ is the Dirac Coulomb Hamiltonian,
$D^{\mu\nu}$ is the photon propagator, ${\alpha}^{\mu} = (1,
\bm{\alpha})$, and ${\bm{x}}_{12} = {\bm{x}}_1-{\bm{x}}_2$. The perturbed wave
function is given by
\begin{equation}\label{7}
|\delta a\rangle = \sum_n^{\varepsilon_n \ne \varepsilon_a} \frac{|n\rangle
        \langle n|\delta V|a\rangle}{\varepsilon_a-\varepsilon_n} \,.
\end{equation}
The reducible part can be considered as a correction due to the
first-order perturbation of the binding energy,
\begin{equation}\label{8}
    \Delta E_{\rm red} = \delta \varepsilon_a\, \langle a| \gamma^0 \left.
    \frac{\partial}{\partial \varepsilon} \widetilde{\Sigma}(\varepsilon) \right|_{\varepsilon = \varepsilon_a} |
            a\rangle\,,
\end{equation}
where $\delta \varepsilon_a = \langle a|\delta V|a\rangle$. The vertex part is
given by
\begin{align} \label{9}
&\Delta E_{\rm ver} = \ \frac{{\rm i}}{2\pi}\int^{\infty}_{-\infty} d\omega\,
  \nonumber \\
& \times   \sum_{n_1n_2}
    \frac{\langle n_1|\delta V|n_2\rangle\,
            \langle an_2| I(\omega)|n_1a\rangle}
                   {[\varepsilon_a-\omega-\varepsilon_{n_1}(1-{\rm i} 0)]
                   [\varepsilon_a-\omega-\varepsilon_{n_2}(1-{\rm i} 0)]}\,,
\end{align}
where $I(\omega) = e^2\,
\alpha_{\mu}\alpha_{\nu}\,D^{\mu\nu}(\omega)$.

The self-energy correction to {\em hfs } is given by the above formulas,
in which we should assume the perturbing potential to have the
form of the Fermi-Breit interaction (the nuclear magnetic moment
is denoted by $\bm{\mu}$),
\begin{equation}\label{10a}
    \delta V \to V_{hfs}({\bm{r}}) =
    \frac{|e|}{4\pi}\,\frac{\bm{\alpha}\cdot[\bm{\mu}\times{\bm{r}}]}{r^3}\,,
\end{equation}
and the initial-state wave function $|a\rangle$ to be the wave
function of the coupled system (electron$+$nucleus),
\begin{equation}\label{10}
    |a \rangle \to  |FM_FIj\rangle =  \sum_{M_Im_a} C^{FM_F}_{IM_I j_a
        m_a}\, |IM_I\rangle \,|j_a m_a\rangle\,,
\end{equation}
where $|IM_I\rangle$ denotes the nuclear wave function, $|j_a
m_a\rangle$ is the electron wave function, $F$ is the total momentum
of the atom, and $M_F$ is its projection. Radial integrations over
the nuclear coordinates can easily be performed already in the
general expressions. One can show that formulas
(\ref{5})-(\ref{9}) yield corrections to {\em hfs } if we employ the
perturbing interaction in the form
\begin{equation}\label{11}
    \delta V({\bm{r}}) = \frac{E_F}{4/3({Z \alpha})^3}\, \frac{[{\bm{r}} \times
            \bm{\alpha}]_z}{r^3}\,,
\end{equation}
where $E_F$ is the nonrelativistic Fermi energy, and consider the
initial-state wave function to be the electron wave function with
the moment projection $m_a = 1/2$,
\begin{equation}\label{12}
    |a\rangle = |j_a\,1/2\rangle\,.
\end{equation}

\subsection{Irreducible part}

As follows from Eq.~(\ref{5}), evaluation of the irreducible part
of the self-energy {\em hfs } correction implies a calculation of a
non-diagonal matrix element of the self-energy function and,
therefore, is very similar to the evaluation of the first-order
self-energy correction to the Lamb shift. Since our present
approach to this problem is somewhat different from the standard
potential-expansion method, we now give a short description of the
scheme used for the evaluation of a self-energy matrix element.

Ultraviolet divergencies in the self-energy function (\ref{6}) are
traditionally isolated by separating the first two terms in the
expansion of the bound-electron propagator $G$ in terms of the
binding potential $V$,
\begin{eqnarray} \label{13}
G(E,{\bm{x}}_1,{\bm{x}}_2) &=&
G^{(0)}(E,{\bm{x}}_1,{\bm{x}}_2)+G^{(1)}(E,{\bm{x}}_1,{\bm{x}}_2)
  \nonumber \\ &&
                         +G^{(2+)}(E,{\bm{x}}_1,{\bm{x}}_2)\,,
\end{eqnarray}
where $G^{(0)} = [\omega-{\cal H}_0(1-{\rm i} 0)]^{-1}$ is the free
Dirac Green function, $G^{(1)}$ is the first-order expansion term
\begin{equation} \label{14}
G^{(1)}(E,{\bm{x}}_1,{\bm{x}}_2) = \int d{\bm{z}}\, G^{(0)}(E,{\bm{x}}_1,{\bm{z}})\,
             V(z)\, G^{(0)}(E,{\bm{z}},{\bm{x}}_2)\,,
\end{equation} and $G^{(2+)}$ is the remainder. Representing $G$
in the form (\ref{13}) leads to splitting the matrix element of
the self-energy function into the zero-potential, one-potential,
and many-potential parts (see Refs.
\cite{snyderman:91,blundell:92,yerokhin:99:pra} for details),
\begin{equation}\label{15}
\langle a |\gamma^0 \widetilde{\Sigma}(\varepsilon_a)|a\rangle
 = \Delta E_{\rm zero}+
    \Delta E_{\rm one}+
     \Delta E_{\rm \, many}\,,
\end{equation}
with the mass-counterterm part naturally ascribed to the
zero-potential term.

Modifications of the standard potential-expansion approach
introduced in our previous investigation \cite{yerokhin:05:se}
concern the many-potential term, which is given by
\begin{align} \label{16}
\Delta E_{\rm \, many} =&\  2\,{\rm i}\alpha \int_{C_{LH}} d\omega
\int d{\bm{x}}_1\,
        d {\bm{x}}_2\,
        D^{\mu \nu}(\omega, {\bm{x}}_{12})
  \nonumber \\ & \times
        \psi^{\dag}_a({\bm{x}}_1)\, \alpha_{\mu}\,
        G^{(2+)}(\varepsilon_a-\omega,{\bm{x}}_1,{\bm{x}}_2)\, \alpha_{\nu}\,
        \psi_a({\bm{x}}_2)\,
        \,.
\end{align}
The integration contour $C_{LH}$ consists of two parts, the
low-energy ($C_L$) and the high-energy ($C_H$) one. The low-energy
part extends from $\varepsilon_0-{\rm i} 0$ to $-{\rm i} 0$ on the lower bank of
the branch cut of the photon propagator and from ${\rm i} 0$ to
$\varepsilon_0+{\rm i} 0$ on the upper bank of the cut. In order to avoid
the appearance of poles of the electron propagator near the
integration contour, each part of $C_L$ is bent into the complex
plane if the calculation is performed for an excited state. The
high-energy part of the contour is $C_H =
(\varepsilon_0-{\rm i}\infty,\varepsilon_0-{\rm i} 0]+[\varepsilon_0+{\rm i}
0,\varepsilon_0+{\rm i}\infty)$. The parameter $\varepsilon_0$ of the contour is
chosen arbitrary from the interval $\varepsilon_0 \in
(\varepsilon_a-\varepsilon_{1s},1+\varepsilon_a)$, where $\varepsilon_{1s}$ is the
ground-state energy.

The function $G^{(2+)} = G-G^{(0)}-G^{(1)}$ that enters
Eq.~(\ref{16}) is not known in its closed form at present and,
consequently, an evaluation of the many-potential term has to be
performed by expanding $G$ (and, therefore, $G^{(2+)}$) into
eigenfunctions of the Dirac angular momentum with the eigenvalue
$\kappa$. This expansion will be referred to as the partial-wave
expansion in the following. The convergence properties of this
expansion are of crucial importance for the numerical evaluation
of the self-energy correction. In our previous investigation
\cite{yerokhin:05:se}, it was demonstrated that the convergence
rate of the partial-wave expansion could be significantly enhanced
by separating from $\Delta E_{\rm \, many}$ a part that is
calculated in a closed form.

It was shown that in the region ${\bm{x}}_1 \approx {\bm{x}}_2$ the
function $G^{(2+)}$ could be approximated by a simpler function
$G_a^{(2+)}$,
\begin{align}\label{17}
    G_a^{(2+)}(E,{\bm{x}}_1,&\,{\bm{x}}_2)  =
    G^{(0)}(E+\Omega,{\bm{x}}_1,{\bm{x}}_2)
  \nonumber \\ &
   - G^{(0)}(E,{\bm{x}}_1,{\bm{x}}_2)
   -         \Omega\, \frac{\partial}{\partial E}\,
             G^{(0)}(E,{\bm{x}}_1,{\bm{x}}_2)\,,
\end{align}
where
\begin{equation}\label{18}
    \Omega  = \frac{2{Z \alpha}}{x_1+x_2}\,.
\end{equation}
The above approximation can be obtained from the exact expression
for $G^{(2+)}$ by neglecting the commutators $[V,G^{(0)}]$ to all
orders. The function $G^{(2+)}_a$ is expressed in terms of the
free Green function and can be easily evaluated in a closed form.
We thus write $\Delta E_{\rm \, many}$ as a sum of two terms,
\begin{equation}\label{19}
    \Delta E_{\rm \, many} = \Delta E_{\rm \, many}^{\rm
\, sub}+ \Delta E_{\rm \, many}^{\rm \, remd} \,.
\end{equation}
The subtraction term $\Delta E_{\rm \, many}^{\rm \, sub}$ is
obtained from the high-energy part of Eq.~(\ref{16}) by the
substitution $G^{(2+)}\to G^{(2+)}_a$. The second term $\Delta
E_{\rm \, many}^{\rm \, remd}$ is the remainder. The subtraction
term is evaluated numerically in its closed form ({\em i.e.,}
without any partial-wave expansion), whereas the remainder yields
a rapidly converging partial-wave expansion; for the details of
the evaluation see Ref.~\cite{yerokhin:05:se}.

\subsection{Reducible part}

The reducible part is defined by Eq.~(\ref{8}). Using the
definition of the self-energy function and employing the contour
$C_{LH}$ for the integration over $\omega$, we write the
expression in the form
\begin{align} \label{20}
\Delta E_{\rm red} =&\ 2\,{\rm i}\alpha\, \delta \varepsilon_a
\int_{C_{LH}} d\omega\,\int d{\bm{x}}_1\,d{\bm{x}}_2\,
    D^{\mu\nu}(\omega,{\bm{x}}_{12})\,
 \nonumber \\ & \times
  \psi_a^{\dag}({\bm{x}}_1)
      \alpha_{\mu}\,
      \left. \frac{\partial}{\partial \varepsilon}\,
         G(\varepsilon-\omega,{\bm{x}}_1,{\bm{x}}_2)
    \right|_{\varepsilon = \varepsilon_a}\!\!\!\!
         \alpha_{\nu}\,
         \psi_a({\bm{x}}_2)\,.
         \nonumber \\
\end{align}
Ultraviolet (UV) and infrared (IR) divergences present in this
expression can be conveniently isolated by separating the Green
function $G$ into 3 parts,
\begin{equation}\label{21}
    G(E) = G^{(0)}(E)+ G^{(a)}(E)+ \bigl[G(E) - G^{(0)}(E)- G^{(a)}(E)\bigr]\,,
\end{equation}
where $G^{(a)}$ incorporates the part of the spectral
decomposition of the bound-electron propagator with $\varepsilon_n =
\varepsilon_a$,
\begin{equation}\label{22}
    G^{(a)}(E,{\bm{x}}_1,{\bm{x}}_2) = \sum_{\mu_a}
        \frac{\psi_a({\bm{x}}_1)\,\psi_a^{\dag}({\bm{x}}_2)}{E-\varepsilon_a(1-{\rm i} 0)}\,,
\end{equation}
and $\mu_a$ denotes the momentum projection of the states $\psi_a$
in this expression. The terms on the right-hand-side of
Eq.~(\ref{21}) substituted in Eq.~(\ref{20}) give rise to the
splitting of $\Delta E_{\rm ir}$, respectively, into 3 parts:
\begin{equation}\label{23}
    \Delta E_{\rm red} = \Delta E_{\rm red}^{(0)}+ \Delta E_{\rm red}^{(a)}
       + \Delta E_{\rm red}^{\rm \, many}\,.
\end{equation}
In this sum, the term $\Delta E_{\rm red}^{(0)}$ contains all UV
divergences. It is calculated in momentum space in a way similar
to that for the zero-potential part of the first-order self-energy
correction. UV-divergent terms are covariantly isolated; they
disappear when combined with the free-propagator contribution in
the vertex part. The term $\Delta E_{\rm red}^{(a)}$ contains all
IR divergences present in the reducible part. (IR divergences of
this type are sometimes also termed as the {\em reference-state}
singularities). These divergences are regularized by employing the
photon propagator with a finite photon mass $\mu$. The limit
$\mu\to 0$ can be taken when $\Delta E_{\rm red}^{(a)}$ is
combined with the corresponding contribution from the vertex part.

The term $\Delta E_{\rm red}^{\rm \, many}$ is finite.
Contributions of such type are frequently encountered in all-order
QED calculations. Usually, they are evaluated in coordinate space
after expanding into an infinite partial-wave series. In the
present investigation, we modify the standard scheme in order to
achieve a better convergence of the partial-wave expansion,
analogously to that for  the irreducible part. We thus separate
$\Delta E_{\rm red}^{\rm \, many}$ into two parts,
\begin{equation}\label{24}
    \Delta E_{\rm red}^{\rm \, many} = \Delta E_{\rm red}^{\rm \, sub}
       + \Delta E_{\rm red}^{\rm \, remd}\,.
\end{equation}
The remainder term $\Delta E_{\rm red}^{\rm \, remd}$ is obtained
from $\Delta E_{\rm red}^{\rm \, many}$ by replacing the standard
subtraction $\bigl[G(E) - G^{(0)}(E)- G^{(a)}(E)\bigr]$ by
$\bigl[G(E) - G^{(0)}(E+\Omega)- G^{(a)}(E)\bigr]$ in the
high-energy part of the expression. The remaining difference
$\bigl[G^{(0)}(E+\Omega)- G^{(0)}(E)\bigr]$ gives rise to the
subtraction term $\Delta E_{\rm red}^{\rm \, sub}$. More
explicitly, the subtraction term is written as
\begin{eqnarray} \label{25}
\Delta E_{\rm red}^{\rm \, sub} &=&2\,{\rm i}\alpha\, \delta \varepsilon_a
\int_{C_{H}} d\omega\,\int d{\bm{x}}_1\,d{\bm{x}}_2\,
    D^{\mu\nu}(\omega,{\bm{x}}_{12})\,
 \nonumber \\ && \times
\psi_a^{\dag}({\bm{x}}_1)\,
      \alpha_{\mu}\,
   \frac{\partial}{\partial \varepsilon}\,
         \Bigl[ G^{(0)}(\varepsilon-\omega+\Omega,{\bm{x}}_1,{\bm{x}}_2)
 \nonumber \\ &&
    \left.
           - G^{(0)}(\varepsilon-\omega,{\bm{x}}_1,{\bm{x}}_2) \Bigr]
    \right|_{\varepsilon = \varepsilon_a}\!\!\!\!
         \alpha_{\nu}\,
         \psi_a({\bm{x}}_2)\,,
\end{eqnarray}
where $\Omega$ is given by Eq.~(\ref{18}). This expression is
calculated in its closed form in coordinate space. The calculational
formulas are immediately obtained from the corresponding
expressions for the subtraction term for the first-order
self-energy correction \cite{yerokhin:05:se}.
The remainder term is calculated by a partial-wave expansion. Due to
the additional subtraction in the high-energy part, the convergence
properties of this partial-wave expansion are much better than in
the standard approach.

\subsection{Vertex part}

Rewriting expression~(\ref{9}) for the vertex part of the
self-energy {\em hfs } correction in terms of the bound-electron
propagators, we obtain
\begin{eqnarray}   \label{26}
\Delta E_{\rm ver} &=& 2\,{\rm i}\alpha \int_{C_{LH}} d\omega\,
           \int d\bm{x}_1\,d\bm{x}_2\,d\bm{x}_3\,
          \psi_a^{\dag}(\bm{x}_1)\, \alpha_{\mu}\,
  \nonumber \\ && \times
         G(\varepsilon_a-\omega,\bm{x}_1,\bm{x}_2)\,
       \delta V(\bm{x}_2)\,
         G(\varepsilon_a-\omega,\bm{x}_2,\bm{x}_3)\,
  \nonumber \\ && \times
         \alpha_{\nu}\,
          \psi_a(\bm{x}_3)\, D^{\mu\nu}(\omega,\bm{x}_{13})\,.
\end{eqnarray}
UV and IR divergences present in this expression
can be conveniently isolated by the following separation
\begin{eqnarray}    \label{27}
G\,\delta V\,G &=& G^{(0)}\,\delta V\,G^{(0)} +
              G^{(a)}\,\delta V\,G^{(a)}
\nonumber \\ &&
              +  \left[ G\,\delta V\,G - G^{(0)}\,\delta V\,G^{(0)} - G^{(a)}\,\delta V\,G^{(a)}\right]\,.
\nonumber \\
\end{eqnarray}
This separation, being substituted into Eq.~(\ref{26}), gives rise
to the following three parts of $\Delta E_{\rm ver}$,
respectively,
\begin{equation}\label{28}
    \Delta E_{\rm ver} = \Delta E_{\rm ver}^{(0)}+ \Delta E_{\rm ver}^{(a)}
       + \Delta E_{\rm ver}^{\rm \, many}\,.
\end{equation}
Only the first term in this sum is UV divergent. UV divergences in
$\Delta E_{\rm ver}^{(0)}$ are covariantly isolated by employing a
momentum-space representation; they disappear when combined with
the corresponding contribution from the reducible part, see, {\em
e.g.}, Ref.~\cite{blundell:97:pra}. The second term $\Delta E_{\rm
ver}^{(a)}$ is IR divergent. In order to retain its finite part,
we consider it together with the corresponding contribution from
the reducible part,
\begin{eqnarray}   \label{29}
\Delta E_{\rm ver}^{(a)} + \Delta E_{\rm red}^{(a)} &=&
    \frac{{\rm i}}{2\pi} \int_{C_{LH}}d\omega\,
      \frac{1}{(\omega-{\rm i} 0)^2}\,
  \nonumber \\ && \times
         \sum_{\mu_{a'}\mu_{a''}}\,
           \biggl[ \langle a'|\delta V|a''\rangle\,
                        \langle aa''|I(\omega)|a'a\rangle
  \nonumber \\ &&
   - \langle a|\delta V|a\rangle\,
                        \langle aa'|I(\omega)|a'a\rangle
       \biggr]\,,
\end{eqnarray}
where $a'$ and $a''$ denote the intermediate states with
$\varepsilon_n=\varepsilon_a$ and with the momentum projection $\mu_{a'}$ and
$\mu_{a''}$, respectively. The integration over $\omega$ can be
carried out analytically, which leads to an explicitly finite
result. Sometimes it is more convenient to calculate this
contribution directly according to Eq.~(\ref{29}) (as long as the
contour $C_{LH}$ is employed for the integration over $\omega$,
this expression is suitable for the numerical evaluation).

The third term $\Delta E_{\rm ver}^{\rm \, many}$ does not contain
any divergences and is calculated in coordinate space after
expanding into a partial-wave series. We note that the additional
subtraction, similar to the one introduced for the reducible part,
does not improve the convergence properties of the partial-wave
expansion in this case. This is due to the fact that a significant
contribution to the partial-wave expansion terms originates from
the first-order commutator $[V,G^{(0)}]$. In order to achieve a
significant improvement, one needs to separate the complete
contribution of the vertex with one magnetic and one Coulomb
interaction. Such contribution was evaluated in a closed form for
the self-energy correction to the $g$-factor
\cite{yerokhin:02:prl,yerokhin:04:pra} using the explicit form of
the interaction with the constant magnetic field. In the case of
the self-energy {\em hfs } correction, we are presently unable to obtain
a closed representation for this term. Nevertheless, the
partial-wave expansion for $\Delta E_{\rm ver}^{\rm \, many}$ is
converging significantly faster than that for $\Delta E_{\rm
ir}^{\rm \, many}$ and $\Delta E_{\rm red}^{\rm \, many}$, so that
the enhanced convergence achieved for the irreducible and
reducible parts results finally in a significant improvement of
the total accuracy of the calculation.

\subsection{Numerical results}
\label{sec:se:res}

The self-energy {\em hfs } correction is conveniently represented in
terms of the dimensionless function $D_n^{\rm \, SE}$ defined as
\begin{equation}\label{2}
    \Delta E_{\rm SE} = \frac{E_F}{n^3}\, \frac{\alpha}{\pi}\, D_n^{\rm \, SE}({Z \alpha})\,,
\end{equation}
where $n$ is the principal quantum number. The results of our
numerical calculation for the individual contributions of this
correction for the $1s$ and $2s$ states and $Z=10$ are presented
in Table~\ref{tab:breakdown}. The calculation was performed for
the point nuclear model and in the Feynman gauge. In
Table~\ref{tab:total} we list the final results for the
self-energy {\em hfs } correction for H-like ions with $Z$ varying from
5 to 30. A comparison of the results of different theoretical
evaluations for this correction in the low-$Z$ region is given in
Table~\ref{tab:compar}.

\begingroup
\squeezetable
\begin{table*}
\begin{center}
\begin{minipage}{16.0cm}
\caption{ Individual contributions to the self-energy {\em hfs }
correction for $Z=10$, in units of the function $D_n$ defined by
Eq.~(\ref{3}). \label{tab:breakdown} }
\begin{ruledtabular}
\begin{tabular}{ccccccc}
     &  $\Delta E_{\rm ir}$ &  $\Delta E_{\rm red}^{(0)}+\Delta E_{\rm ver}^{(0)}$
                                   & $\Delta E_{\rm red}^{(a)}+\Delta E_{\rm ver}^{(a)}$
                                           & $\Delta E_{\rm red}^{\rm \, many}$
                                                             & $\Delta E_{\rm ver}^{\rm \, many}$
                                                                 & Total
\\[0.5pt]
\hline
1$s$ & $-$0.263902 &  1.896440  &$-$0.002385 & $-$1.607827 & $-$0.185180& $-$0.162853(1) \\
2$s$ & $-$0.223649 &  3.433554  &$\ \ \ $0.176267 & $-$1.878883 & $-$1.649848& $-$0.142559(3) \\
\end{tabular}
\end{ruledtabular}
\end{minipage}
\end{center}
\end{table*}
\endgroup

\begin{table}
\caption{The self-energy {\em hfs } correction for the $n=1$ and $n=2$
states of light H-like ions. \label{tab:total}}
\begin{ruledtabular}
\begin{tabular}{r..}
 $Z$ & \multicolumn{1}{c}{$D_1({Z \alpha})$}& \multicolumn{1}{c}{$D_2({Z \alpha})$}\\
\hline
5    &    0.174x\,026\,(2)   &     0.181x\,940\,(2) \\
6    &    0.106x\,815\,(2)   &     0.117x\,124\,(2) \\
7    &    0.039x\,476\,(2)   &     0.052x\,265\,(2) \\
8    &   -0.027x\,933\,(1)   &    -0.012x\,626\,(2) \\
9    &   -0.095x\,379\,(1)   &    -0.077x\,559\,(3) \\
10   &   -0.162x\,853\,(1)   &    -0.142x\,559\,(3) \\
12   &   -0.297x\,905\,(1)   &    -0.272x\,913\,(1) \\
15   &   -0.501x\,056\,(1)   &    -0.470x\,078\,(1) \\
20   &   -0.843x\,572\,(1)   &    -0.807x\,153\,(1) \\
25   &   -1.196x\,242\,(2)   &    -1.162x\,717\,(2) \\
30   &   -1.566x\,491\,(3)   &    -1.547x\,535\,(2) \\
\end{tabular}
\end{ruledtabular}
\end{table}

\begin{table}
\caption{ Comparison of the results of different calculations of
the self-energy {\em hfs } correction for the $1s$ and $2s$ states in
light H-like ions, in units of $D_n({Z \alpha})$. \label{tab:compar} }
\begin{ruledtabular}
\begin{tabular}{c...c}
           &  \multicolumn{1}{c}{$Z=5$}&   \multicolumn{1}{c}{$Z=10$}& \multicolumn{1}{c}{$Z=20$}  & Ref.
\\ \hline
$1s$       &  0.174x\,026\,(2)         &   -0.162x\,853\,(1)     &  -0.843x\,572\,(1)   &     \\
           &  0.174x\,028\,(20)        &   -0.162x\,860\,(20)    &  -0.843x\,588\,(15)  & \cite{yerokhin:01:hfs}\\
           &  0.174x\,05\,(1)          &   -0.162x\,83\,(1)      &  -0.843x\,56\,(1)    & \cite{blundell:97:prl}\\
           &  0.174x\,1(1)             &   -0.162x\,8(1)         &                      & \cite{sunnergren:98:pra}\\
\hline
$2s$       &  0.181x\,940\,(2)         &   -0.142x\,559\,(3)     &  -0.807x\,153\,(1)   &    \\
           &  0.181x\,96\,(10)         &   -0.142x\,51\,(10)     &  -0.807x\,16\,(6)    & \cite{yerokhin:01:hfs}\\
\end{tabular}
\end{ruledtabular}
\end{table}

As can be seen from Table~\ref{tab:compar}, the present
calculation improves the numerical accuracy of the self-energy
{\em hfs } correction by about an order of magnitude for the $1s$ state
and even more for the $2s$ state, as compared to our previous
calculation \cite{yerokhin:01:hfs}. This progress is due to the
additional subtraction scheme employed in the present work for the
evaluation of the irreducible and reducible parts of the
correction. In order to illustrate the improvement in the
convergence properties of the partial-wave expansion introduced by
this scheme, in Figs.~\ref{fig1} and \ref{fig2} we plot (in the
decimal logarithmic scale) the dependence of the absolute value of
the individual terms of the partial-wave series on the expansion
parameter $|\kappa|$ within the standard potential-expansion
approach and within the new subtraction scheme. (The parameter
$\kappa$ is the Dirac angular-momentum eigenvalue of one of the
electron propagators in the vertex function.) Figs.~\ref{fig1} and
\ref{fig2} represent this comparison for the irreducible and the
reducible part, respectively.

\begin{figure}
\includegraphics[width=0.95\columnwidth]{irconv.eps}
\caption{\label{fig1} The absolute magnitude of the individual
terms of the partial-wave expansion for the irreducible part of
the self-energy {\em hfs }  correction for $Z=5$, within the standard
potential-expansion scheme (filled dots) and with the additional
subtraction employed in the present work (open dots). Plotted are
the contributions to the function $D_n({Z \alpha})$ as a function of the
absolute value of the relativistic angular momentum parameter
$\kappa$. A discontinuity of the curve on the upper graph around
$|\kappa|=19$ is due to the change of the sign of the
contributions to $D_1({Z \alpha})$. }
\end{figure}

\begin{figure}
\includegraphics[width=0.95\columnwidth]{redconv.eps}
\caption{\label{fig2} The same as in Fig.~\ref{fig1}, but for the
reducible part of the self-energy {\em hfs } correction. }
\end{figure}

As a result of the improvement achieved, we were able to eliminate
completely the uncertainty arising from termination of the
partial-wave expansion in the irreducible and reducible parts.
Still, there remains the partial-wave expansion of the
many-potential vertex term $\Delta E_{\rm ver}^{\rm \, many}$,
which has to be terminated and properly extrapolated to infinity.
(In actual calculations, the summation was terminated at
$|\kappa|=40$). The error due to this extrapolation yields one of
the main uncertainties of our numerical evaluation (the other
source of the uncertainty is the stability of numerical
integrations.) Fortunately, the partial-wave expansion of $\Delta
E_{\rm ver}^{\rm \, many}$ is monotonic and relatively
well-converging (not worse than $1/|\kappa|^3$ for all $Z\ge 5$),
and so the uncalculated tail of the expansion can be estimated
reasonably well.

\section{Vacuum-polarization correction}
\label{sec:vp}

The vacuum-polarization {\em hfs } correction can be conveniently split
into two parts, the so-called {\em electric-} and {\em
magnetic-loop} contributions.  The electric-loop part originates
from the diagrams with the {\em hfs } interaction attached to the
external electron line, whereas the magnetic-loop one comes from
the diagram with the {\em hfs } interaction attached to the
vacuum-polarization loop. These contributions are also
traditionally separated into the Uehling and Wichmann-Kroll (WK)
parts. The WK part is suppressed by a factor of $({Z \alpha})^2$ as
compared to the Uehling contribution and can often be regarded as
a small correction for low-$Z$ ions.

The ${Z \alpha}$ expansion of the one-loop vacuum-polarization {\em hfs }
correction reads
\begin{eqnarray}
  D_n^{\rm VP}({Z \alpha}) &=& ({Z \alpha})\,b_{10}+ ({Z \alpha})^2\Bigl[
     L\,b_{21}+b_{20}\Bigr]
   \nonumber \\ &&
     + ({Z \alpha})^3\Bigl[L\,b_{31}+ F_{n}^{\rm VP}({Z \alpha})\Bigr]\,,
\end{eqnarray}
where $L =\ln[(Z\alpha)^{-2}]$, the function $D_n^{\rm VP}$ is
related to the energy shift analogously to Eq.~(\ref{2}), and the
function $F_{n}^{\rm VP}$ incorporates all higher-order terms,
$F_{n}^{\rm VP}({Z \alpha}) = b_{30}+ {Z \alpha} \,(\ldots)\,$. The first
expansion coefficients up to $b_{31}$ stem from the Uehling part
of the vacuum-polarization correction; they are given by (see,
{\em e.g.}, \cite{karshenboim:02:epjd})
\begin{eqnarray}
  b_{10}(ns) &=&  \frac{3\,\pi}{4}\,, \\
  b_{21}(ns) &=& \frac{8}{30}\,,  \\
  b_{20}(1s) &=& \frac{34}{225}-\frac{8}{15}\,\ln 2\,,  \\
  b_{20}(2s) &=& -\frac{247}{450}\,, \\
  b_{31}(ns) &=& \frac{13\,\pi}{48}\,.
\end{eqnarray}
Higher-order coefficients starting with $b_{30}$ arise both from
the Uehling and WK contributions; only their Uehling part is
presently known \cite{karshenboim:01:jetp,karshenboim:02:epjd}.

The Uehling part of the one-loop vacuum-polarization {\em hfs }
correction can easily be calculated numerically; some results can
be found, {\em e.g.}, in
Refs.~\cite{sunnergren:98:pra,artemyev:01}. In the case of the
point nucleus, this contribution was evaluated also analytically
\cite{karshenboim:01:jetp,karshenboim:02:epjd}. For completeness,
we re-calculate it in the present work. The corresponding
contributions to the higher-order remainder $F_{n}^{\rm VP}({Z \alpha})$
for the point nuclear model are listed in the first and the second
columns of Table~~\ref{tab:vp} for the $1s$ and $2s$ states,
respectively. The results presented are in agreement with the
previous calculations of this correction.

\begin{table*}
\begin{center}
\begin{minipage}{16.0cm}
\caption{Individual contributions to $F^{\rm VP}_n({Z \alpha})$ for the
$1s$ and $2s$ states of light H-like ions. \label{tab:vp}}
\begin{ruledtabular}
\begin{tabular}{r......}
 $Z$ &  \multicolumn{2}{c}{Uehling}&
         \multicolumn{2}{c}{Electric-loop WK}& \multicolumn{2}{c}{Magnetic-loop WK}\\
&  \multicolumn{1}{c}{$1s$} & \multicolumn{1}{c}{$2s$}     &  \multicolumn{1}{c}{$1s$} & \multicolumn{1}{c}{$2s$} & \multicolumn{1}{c}{$1s$} & \multicolumn{1}{c}{$2s$} \\
\hline
1    &   7.x231  &   9.x546  &  -0.x117   &  -0.x117  &              &            \\
2    &   7.x337  &   9.x651  &  -0.x120   &  -0.x119  &              &            \\
5    &   7.x587  &   9.x901  &  -0.x128   &  -0.x125  &              &            \\
10   &   7.x947  &  10.x282  &  -0.x138   &  -0.x133  &  -0.x699(2)  &  -0.x706(2) \\
     &     x     &     x     &  -0.x139\,^a &           &  -0.x697\,^a   &            \\
12   &   8.x092  &  10.x441  &  -0.x142   &  -0.x136  &  -0.x701(2)  &  -0.x709(2) \\
14   &   8.x240  &  10.x609  &  -0.x145   &  -0.x138  &  -0.x703(2)  &  -0.x714(2) \\
16   &   8.x394  &  10.x788  &  -0.x149   &  -0.x141  &  -0.x705(2)  &  -0.x718(2) \\
18   &   8.x556  &  10.x978  &  -0.x153   &  -0.x145  &  -0.x707(2)  &  -0.x723(2) \\
     &     x     &     x     &  -0.x154\,^a &           &  -0.x706\,^a   &            \\
20   &   8.x725  &  11.x182  &  -0.x156   &  -0.x148  &  -0.x712(2)  &  -0.x732(2) \\
22   &   8.x904  &  11.x401  &  -0.x160   &  -0.x151  &  -0.x717(2)  &  -0.x740(2) \\
24   &   9.x093  &  11.x635  &  -0.x164   &  -0.x155  &  -0.x725(2)  &  -0.x752(2) \\
\end{tabular}

$^a$ Ref.~\cite{sunnergren:98:pra}\,.
\end{ruledtabular}
\end{minipage}
\end{center}
\end{table*}

The WK part of the vacuum-polarization {\em hfs } correction is more
difficult to calculate. Especially, this refers to the
magnetic-loop WK contribution. As outlined in
Ref.~\cite{artemyev:01}, this correction is divergent in the
point-dipole approximation for the nuclear magnetization
distribution. A finite result for this correction is obtained if
an extended nuclear magnetization distribution is employed. It
should be also taken into account that the magnetic-loop WK
interaction contributes to the measured value of the nuclear
magnetic moment \cite{milstein:89}. In order to prevent
double-counting, the corresponding contribution should be
subtracted from the magnetic-loop WK part of the
vacuum-polarization {\em hfs } correction. Practical calculations
\cite{sunnergren:98:pra,artemyev:01} show that, after such a
subtraction, the magnetic-loop WK correction depends weakly on
details of the nuclear magnetization distribution and has a finite
limit in the point-dipole approximation.

The results of our numerical evaluation of the electric- and
magnetic-loop WK contributions for the $1s$ and $2s$ states of
light H-like ions are listed in Table~\ref{tab:vp}, in terms of
the higher-order remainder $F_{n}^{\rm VP}({Z \alpha})$. The
electric-loop WK correction was calculated for the point nuclear
model by employing the analytical-approximation formulas for the
WK potential from Ref.~\cite{fainshtein:91}. The relative accuracy
of this approximation is considered by the authors to be not worse
than $10^{-4}$ for all $Z$ up to ${Z \alpha}=0.95$. As an independent
test of the accuracy of this approximation in the low-$Z$ region,
we checked that it reproduces well the first two terms of the
${Z \alpha}$ expansion of the WK correction to the Lamb shift. The
magnetic-loop WK correction was calculated for the point-dipole
nuclear magnetization model by using a code developed in our
previous investigation \cite{artemyev:01}. A comparison given in
Table~\ref{tab:vp} demonstrates good agreement of our numerical
values with the $1s$ results of Ref.~\cite{sunnergren:98:pra} for
$Z=10$ and 18.

\section{Higher-order one-loop QED correction}
\label{sec:ho}

One of the main goals of our investigation is to improve the
accuracy of the one-loop QED correction for $Z=1$ and 2, these
being the most interesting cases from the experimental point of
view. The present approach does not employ the ${Z \alpha}$ expansion
and, therefore, our numerical results do not suffer from omission
of the higher-order terms, as is the case with the perturbative
${Z \alpha}$-expansion approach. But on the other hand, technical
problems do not presently allow us to perform a direct numerical
evaluation for $Z=1$ and 2 with a sufficient accuracy. In the
present work, we employ an indirect method used previously in
Refs.~\cite{blundell:97:prl,yerokhin:01:hfs}. By subtracting the
known terms of the ${Z \alpha}$ expansion from the all-order results for
higher values of $Z$, we identify the higher-order remainder and
then extrapolate it to $Z=1$ and 2.

First we summarize the results obtained for the one-loop
self-energy {\em hfs } correction within the perturbative
${Z \alpha}$-expansion approach. The corresponding ${Z \alpha}$ expansion reads
\begin{align}\label{3}
  D_n^{\rm \, SE}({Z \alpha}) =&\ a_{00}+ ({Z \alpha})\,a_{10}+ ({Z \alpha})^2\Bigl[
     L^2 a_{22}+ L\,a_{21}+a_{20}\Bigr]
   \nonumber \\ &
     + ({Z \alpha})^3\Bigl[L\,a_{31}+ F_{n}^{\rm \, SE}({Z \alpha})\Bigr]\,,
\end{align}
where  $L =\ln[(Z\alpha)^{-2}]$  and $F_{n}^{\rm \, SE}$ is the
remainder containing all higher-order terms, $F_{n}^{\rm \,
SE}({Z \alpha}) = a_{30}+ {Z \alpha} \,(\ldots)\,$. The results presently
available for the expansion coefficients are (for the references
see, {\em e.g.}, \cite{karshenboim:02:epjd}):
\begin{eqnarray}\label{4}
  a_{00}(ns) &=& \frac12 \,, \\
  a_{10}(ns) &=& \left(\ln 2-\frac{13}{4}\right)\,\pi \,, \\
  a_{22}(ns) &=& -\frac23\,,\\
  a_{21}(1s) &=& -\frac{8}{3}\ln 2+\frac{37}{72}\,, \\
  a_{21}(2s) &=& a_{21}(1s)-\frac{8}{3}\ln 2+\frac{7}{2}\,, \\
  a_{20}(1s) &=& 17.122\,339\ldots \,, \\
  a_{20}(2s) &=& a_{20}(1s)-5.221\,233(3) \,,\\
  a_{31}(ns) &=& \left(\frac52 \ln 2-\frac{191}{32}\right)\, \pi\,.
\end{eqnarray}
For the $a_{30}$ term, there is a preliminary result \cite{nio:01}
for the $1s$ state, $a_{30}(1s) = -15.9(1.6)$, and a partial
result \cite{karshenboim:01:hfs,karshenboim:02:epjd} for the
difference $\Delta_{21}$, $a_{30}(2s)-a_{30}(1s) = 7.92$.

The higher-order self-energy remainder $F_n^{\rm \, SE}$ can be
inferred from our all-order numerical data. The corresponding
results for the function $F_1^{\rm \, SE}({Z \alpha})$ and the difference
$F_{21}^{\rm \, SE}({Z \alpha})\equiv F_2^{\rm \, SE}({Z \alpha})-F_1^{\rm \,
SE}({Z \alpha})$ are plotted on the upper graphs of Figs.~\ref{fig:fho1s}
and \ref{fig:fho2s1s}, respectively. We note that both the
$F_1^{\rm \, SE}({Z \alpha})$ and $F_{21}^{\rm \, SE}({Z \alpha})$ functions
have a rapidly varying structure in the low-$Z$ region. In order
to demonstrate this more clearly, we subtract their ``linear''
part (obtained by a global linear fit), with the corresponding
plots presented on the middle graphs of Figs.~\ref{fig:fho1s} and
\ref{fig:fho2s1s}. The behavior observed apparently indicates that
the logarithmic term to order $\alpha({Z \alpha})^4 E_F$ enters with a
large coefficient, which complicates extrapolation considerably.

\begin{figure}
\includegraphics[width=\columnwidth]{fho1s.eps}
\caption{\label{fig:fho1s} The higher-order part of the $1s$
self-energy {\em hfs } correction $F_{1}^{\rm \, SE}$ as a function of
the nuclear charge number (the upper graph); $F_{1}^{\rm \, SE}$
with its linear part (obtained by a global linear fit) subtracted
(the middle graph); $F_{1}^{\rm \, SE}$ with its next-to-leading
contribution (obtained by a least-squares fit) subtracted (the
lower graph). The numerical values of the coefficients $c_1$ and
$c_2$ are: $c_1 = 14.83$ and $c_2 = 2.08$. }
\end{figure}

\begin{figure}
\includegraphics[width=\columnwidth]{fho2s1s.eps}
\caption{\label{fig:fho2s1s} The same as in Fig.~\ref{fig:fho1s}
but for the difference $F_{21}^{\rm \, SE} = F_{2}^{\rm \,
SE}-F_{1}^{\rm \, SE}$. The numerical values of the coefficients
$c_1$ and $c_2$ are: $c_1 = 28.26$ and $c_2 = 36.85$. }
\end{figure}

Now we would like to extrapolate our results for the higher-order
remainder to the lower values of $Z$, namely $Z=0$, 1, and 2. For
this purpose we employ a procedure similar to the one recently
described in detail in Ref.~\cite{bigot:03}. The extrapolated
value of a function at $Z= z_0$ is obtained in two steps. First we
apply an (exact) linear  fit to each two consecutive points from
our data set and store the resulting value at $Z=z_0$ as a
function of the average abscissa of the points involved in the
fit. Then we perform a global parabolic least-squares fit to the
set of data obtained and take the fitted value at $Z=z_0$ as a
final result.

We tested this procedure for variation of the logarithmic
contribution to the next-to-leading order and found it rather
stable. However, in order to take into account the presence of
such contribution explicitly, we modify the procedure described
above as following. First, we approximate our numerical data by a
function
\begin{equation}
f(Z) = c_0+ ({Z \alpha})\,[c_1\, \ln ({Z \alpha})+c_2+({Z \alpha})\,c_3 ]\,
\end{equation}
with free coefficients $c_i$, which are determined by a
least-squares fit similar to the one described in
Ref.~\cite{ivanov:01}. Then, we use the values obtained for $c_1$
and $c_2$ in order to define a modified higher-order remainder
function as
\begin{equation}
\widetilde{F}_n^{\rm \, SE}({Z \alpha}) = F_n^{\rm \, SE}({Z \alpha}) -
({Z \alpha})\,[c_1\,\ln ({Z \alpha})+c_2]\,.
\end{equation}
The numerical results for this function are plotted on the lower
graphs of Figs.~\ref{fig:fho1s} and \ref{fig:fho2s1s}. The
function $\widetilde{F}_n^{\rm \, SE}$ is much flatter in the
low-$Z$ region than $F_n^{\rm \, SE}$ and, therefore, is more
suitable for the extrapolation. We obtain our final results
extrapolating the function $\widetilde{F}_n^{\rm \, SE}$ by means
of the procedure described above. The numerical values of the
higher-order self-energy remainder obtained in this way are given
in the first line of Table~\ref{tab:qedho}. In the next 3 lines of
the table, we present the results of the previous evaluations
\cite{blundell:97:prl,yerokhin:01:hfs,nio:01}. The numerical
values obtained for the function $F_{21}^{\rm \, SE}$ in this work
fall slightly outside the error bars ascribed to our previous
results \cite{yerokhin:01:hfs}, as a consequence of the
logarithmic contribution to the next-to-leading order being
apparently much larger than it was assumed in our former work. Our
present values for the function $F_1^{\rm \, SE}$ are in a
marginal agreement with the result by Blundell {\em et al.}
\cite{blundell:97:prl} and deviate by 1.5$\,\sigma$ from the
preliminary result by Nio \cite{nio:01}.

\begin{table*}
\begin{center}
\begin{minipage}{16.0cm}
\caption{One-loop higher-order QED correction. Acronyms are:
``SE'' denotes the self-energy contribution, ``Ue'' -- the Uehling
part, ``WK-EL'' -- the electric-loop WK part, ``WK-ML'' -- the
magnetic-loop WK part. \label{tab:qedho}}
\begin{ruledtabular}
\begin{tabular}{l......c}
        & \multicolumn{1}{c}{$F_1(0\alpha)$}
              & \multicolumn{1}{c}{$F_1(1\alpha)$}
                     & \multicolumn{1}{c}{$F_1(2\alpha)$}
                         & \multicolumn{1}{c}{$F_{21}(0\alpha)$}
                                & \multicolumn{1}{c}{$F_{21}(1\alpha)$}
                                           & \multicolumn{1}{c}{$F_{21}(2\alpha)$}
   &  Reference \\
\hline
  SE    & -13.x2(4)   &  -13.x8(3)   &  -14.x1(3)   &   8.x4(5)   &   7.x6(4)   &  7.x2(3)&   \\
        &             &  -12.x0(2.0) &              &             &             &         & \cite{blundell:97:prl}\\
        &             &  -14.x3(1.1) &  -14.x5(7)   &             &   6.x5(8)   &  6.x3(6)& \cite{yerokhin:01:hfs}\\
        & -15.x9(1.6) &              &              &             &             &         & \cite{nio:01}\\
  Ue    &   7.x06     &    7.x23     &    7.x34     &   2.x32     &   2.x32     &  2.x31  &   \\
  WK-EL &  -0.x11     &   -0.x12     &   -0.x12     &   0.x00     &   0.x00     &  0.x00  &   \\
  WK-ML &  -0.x69(15) &   -0.x69(12) &   -0.x69(7)  &   0.x00     &   0.x00     &  0.x00  &   \\
 Total  &  -6.x9(4)   &   -7.x4(3)   &   -7.x6(3)   &  10.x7(5)   &   9.x9(4)   &  9.x5(3)&   \\
\end{tabular}
\end{ruledtabular}
\end{minipage}
\end{center}
\end{table*}

The Uehling part of the vacuum-polarization {\em hfs } correction is
given in the next line of Table~\ref{tab:qedho}. The numerical
values are taken from Table~\ref{tab:vp} for $Z=1$ and 2 and from
Refs.~\cite{karshenboim:01:jetp,karshenboim:02:epjd} for $Z=0$.
The electric-loop WK correction for $Z=1$ and 2 was calculated
directly in Sec.~\ref{sec:vp}; the corresponding numerical value
for $Z=0$ was obtained by a simple extrapolation. Extrapolation
was also employed in order to obtain the results for the
magnetic-loop WK part of the vacuum-polarization correction
presented in the table. The error bars specified are obtained
under the assumption that the logarithmic contribution to the
next-to-leading order enters with a coefficient of about 2.

We now turn to the experimental consequences of our calculation.
As demonstrated in Ref.~\cite{karshenboim:02:epjd}, the
higher-order self-energy correction is one of the major sources of
uncertainty of the theoretical prediction for the normalized
difference of the {\em hfs } intervals $\Delta_{21} = 8\,
\nu_{2s}-\nu_{1s}$ for the $^3{\rm He}^+$ ion. Our present
calculation changes the theoretical value of this correction by
$-0.056$~kHz (as compared to our former result
\cite{yerokhin:01:hfs}) and improves its accuracy by a factor of
2. The resulting value of the one-loop QED contribution that
incorporates all orders in ${Z \alpha}$ starting with the constant term
to order $\alpha({Z \alpha})^3E_F$ for the $^3{\rm He}^+$ ion is given in
the first line of Table~\ref{tab:D21}. In the next lines of the
table, we give the total theoretical value for the difference
$\Delta_{21}(^3{\rm He}^+)$ taken from
Ref.~\cite{karshenboim:02:epjd}, this value modified by the
present calculation, and the corresponding experimental result. As
can be seen from the table, our calculation increases the
deviation of the theoretical prediction from the experimental
value from $0.9\,\sigma$ to $1.6\,\sigma$.

\begin{table}
\caption{Normalized difference of the {\em hfs } intervals $\Delta_{21}
= 8\, \nu_{2s}-\nu_{1s}$ for the $^3{\rm He}^+$ ion, in kHz.
\label{tab:D21}}
\begin{ruledtabular}
\begin{tabular}{l.c}
Higher-order QED correction  &       -0x.594\,(19)     &                \\
$\Delta_{21}$, old theory    &  -1\,190x.068\,(64)     &  \cite{karshenboim:02:epjd} \\
$\Delta_{21}$, new theory    &  -1\,190x.124\,(55)     &                \\
$\Delta_{21}$, experiment    &  -1\,189x.979\,(71)     &  \cite{schluessler:69}+\cite{prior:77} \\
\end{tabular}
\end{ruledtabular}
\end{table}

It should be noted that our numerical results for all corrections
at $Z=1$, except for the one for the magnetic-loop WK
contribution, can be directly applied to the hyperfine splitting
in muonium. Our calculation of the magnetic-loop WK correction
does not hold for muonium since it involves a regularization by an
extended magnetization distribution of the nucleus and a
subtraction of the related contribution to the measured value of
the nuclear magnetic moment. In the case of muonium, the nucleus
is substituted by a point-like muon and the regularization should
be performed by a finite mass of the muon rather than by a finite
size.

\section{Summary}

In the present investigation, we carried out an all-order (in
${Z \alpha}$) calculation of the one-loop QED correction to the hyperfine
splitting of the $1s$ and $2s$ states in light H-like ions. This
calculation significantly improved the accuracy of this
correction, as compared to the previous evaluations. By
subtracting the known terms of the ${Z \alpha}$ expansion and
extrapolating the remainder to lower values of $Z$, we obtained
the results for the higher-order remainder for $Z=0$, 1, and 2.
Our calculation shifts the theoretical value of the normalized
difference $\Delta_{21}$ of the $1s$ and $2s$ {\em hfs } intervals in
$^3{\rm He}^+$ by $0.056$~kHz and slightly improves its accuracy.

\section*{Acknowledgements}

This work was supported in part by RFBR (Grant No.~04-02-17574) and by
DFG.  A.N.A. and V.A.Y. acknowledge support from the ``Dynasty''
foundation and from INTAS (Grants No.~YS~03-55-960 and YS~03-55-1442).
G.P. acknowledges financial support from BMBF and GSI.

redpajama_set_name: RedPajamaArXiv


text:
1821 Token of Respect to Student Ann Caroline Potter in New York City from Teachers Hezekiah Gold Ufford and Lewis Bixby
Token of respect, dated July 24, 1821, given to student Miss Ann Potter in New York City by her teachers H. G. Ufford and L. Bixby. I believe these people were Ann Caroline Potter (1808-1882); Hezekiah Gold Ufford (1779-1863); and Lewis Bixby (1795-1866).
The token of respect came with a group of papers, including Family Record pages removed from a Bible, that pertain to the genealogy of several families:
Crone
Layman
As noted in the accompanying papers, Miss Ann Potter was Ann Caroline Potter, born in 1808 in New York City and future wife of New York City attorney Frederick Wait Burke, a native of Woodstock, Vermont. Ann died on August 23, 1882 at Brooklyn, New York; her death record indicates that her father was born in New Jersey and her mother on Long Island in New York. The Mary Potter living with the Burkes at the time of the 1850 and 1860s Censuses was presumably Ann's widowed mother.
H. G. Ufford was presumably Rev. Hezekiah Gold Ufford, born in 1779 and a graduate of Yale College. Below, his obituary in the Yale publication Obituary Record of Graduates of Yale College Deceased from July, 1859, to July, 1870, noting that he was born April 14, 1779 at Stratford, Connecticut, son of Samuel Ufford and Abigail (Gold) Ufford:
L. Bixby may have been Rev. Lewis Bixby, whose name appears with that of H. G. Ufford in a testimonial of the Story of Jack Halyard, the Sailor Boy. This Lewis Bixby was born in New Hampshire about 1795, graduated from Yale College and taught in several places in New York throughout his career.
Below, his mention on pages 563 and 564 in A Genealogy of the Descendants of Joseph Bixby, 1621-1701, of Ipswich and Boxford, Massachusetts..., compiled by Willard Goldthwaite Bixby and published in 1914:
If you have information on Ann Caroline Potter, Hezekiah Gold Ufford and/or Lewis Bixby, please leave a comment or contact me directly.
Calling card of Henry Burton Voter (1867-1944) of ...
1812 Document, possibly Freetown, Massachusetts ar...
Account of Sugar Refined in 1815 in the Sugarhouse...
1823 Document from Danvers, Massachusetts: Mrs. Ma...
1821 Token of Respect to Student Ann Caroline Pott...
1869 Document Certifying the Marriage of William H...
1743 Sampler Wrought by Mary Gisborne
Sampler with Family Register of the William Brown ...
1827 Sampler with Family Register, "Wrought by Han...
1863 Document: George F. Emery, Clerk of Maine Dis...
1836 Military Correspondence from Seward Dill, Com...
1812 List of Repairs to the Rollins House, presuma...
c1815 Document, Brunswick, Maine: David Stanwood; ...

redpajama_set_name: RedPajamaCommonCrawl


BUICED-ing: adding liquid vitamins to your drinks.
Ray Doustdar explains how his high cholesterol lead him to give up pill form supplements and start juicing and "buiced-ing" (pronounced "boosting").
Doustdar: Long story short for me, after thinking I was a pretty "fit" guy, at least externally, a blood test in Feb. 2012 revealed my cholesterol was high (234), as was my LDL (low-density lipoprotein—the bad cholesterol, 132).
My doctor wanted to put me on Lipitor [a brand of cholesterol-lowering drug], as high cholesterol does run in my family and currently my dad, mom, and brother are all on statins [a class of drugs used to lower cholesterol].
After watching several documentaries on plant-based nutrition and then juicing, I bought a juicer and started juicing fresh veggies. I was hooked immediately, and in 4 months, my cholesterol went from 234 to 168 (down 28 percent). Juicing [also] gave me portion control and helped curb my cravings.
Epoch Times: What misconceptions do people have about juicing? What should they know?
1. Do not use juice fasting as a long-term approach.
Why not? The goal of juicing is to supplement your day with bioavailable nutrients that will help you to be healthier and feel better, all so you can live better.
Fasting and cleansing are not things that can become sustainable in your daily routine. If you are eating clean and healthy [food], your body is cleansing itself every single day…. Juicing as a cleanse or as a fast treats juicing more as a "quick fix" instead of health solution. Just like anything in life, unless you sustain it, you will end up right back where you started.
There is also the misconception that it takes a lot of time and that it is messy. Well, if you know what you are doing and follow my instructions, you will find these things not to be true. An important point I want to get across is that to reap the optimal health benefits, I have found that you need to do it everyday.
What has been really interesting as well is that during this journey that I have been on to improve my health, I discovered that although the leafy greens are packed with phytonutrients, they lack many essential vitamins and minerals.
I tried taking vitamins in pill form after each juice, but they upset my stomach. (I would get nauseated when I took pills. With or without food, I could count on feeling nauseated for a good 20 minutes each time I took my pills. Also, I actually got a few of those horse pills stuck in my throat a few times, which was not pleasant.) Then I learned liquid vitamins are more readily available for easy absorption, 98 percent vs. 10-20 percent for pill forms, I decided to solve my own problem and created BUICED to boost my juice.
Now I BUICED my fresh veggie juice, my morning OJ or my water. And when I'm in a hurry, I just fire back a shot. BUICED is a great liquid multivitamin product that gives you 100 percent daily value of the entire spectrum of vitamins. By juicing and then BUICED-ing your juice, you get it all.
Epoch Times: One of the critiques of juicing is that because it removes the fiber from fruit and vegetables, the body absorbs the sugars from fruit more easily and this can upset blood sugar levels. What recommendations do you have to avoid this?
Doustdar: This is true, and this is why I recommend the 90/10 rule.
That is that you should juice 90 percent veggies and 10 percent fruits. Or another thing I tell people is to eat your fruits, juice your vegetables. That way you get both.


text:
$7 Billion Merger of Colgate and Centennial, the 2 Largest Permian Operators
Despite the circulating rumors concerning Colgate's attempt to launch an IPO, on May 19 the company decided to combine with Centennial Resource Development Inc. This merger of equals is estimated at $7 billion and will found the biggest pure-play E&P company in the Delaware Basin of the Permian. The transformative combination essentially enlarges companies' potential and hastens the growth across all financial and operating metrics. According to Centennial CEO Sean Smith, the combined company is anticipated to furnish shareholders with quickened capital return program due to a fixed dividend coupled with a share repurchase plan. Due to a recent report, the merger would increase production 7%, to 145,000 boe/d by the fourth quarter would further ratchet up next year. By third-quarter 2023, the company predicted 160,000 boe/d based on a drilling program of 140 wells per year. Colgate Energy was reported to be getting an IPO last December that sources said would value the company at approximately $4 billion. The combined company will have over 15-years of drilling inventory, assuming its current drilling pace, the companies will produce over $1 billion of free cash flow in 2023 at current strip prices.
Colgate Energy's owners are planning to go public
Colgate Energy is planning to float its shale oil producer in the Permian's Delaware Basin on the stock market. If successful, this IPO would be the first major U.S. oil producer offering since Jagged Peak Energy's IPO in January 2017. Looks like investors' confidence in the sector is returning as U.S. crude prices hit their highest in seven years late last year S&P energy index delivered roughly twice the return of the S&P 500 in 2021.


text:
Professor Dr Grant Jarvie (Honorary Doctorate)
University of Edinburgh Chair of Sport and Director (Academy of Sport) - Visiting Professor - University of Toronto
University of Edinburgh - Moray House School of Education and Sport
University of Edinburgh- Academy of Sport
University ot Toronto
Tel: +44 (0) 131 651 6546 - Direct
Academy of Sport Office : +44 (0) 131 651 6520
Email: grant.jarvie@ed.ac.uk
Academy of Sport (Founding Director )
East Renfrewshire Leisure and Culture Trust Board
Sportscotland Board 2009-2016
Scottish Football Association Advisory Board
Scottish Sports Hall of Fame Advisory Board
St Leonard's Land (Rm 3.18)
EH8 8AQ
Grant Jarvie was born in Bellshill and comes from an international sporting family. Having held senior leadership positions in both the university sector and the sports industry he has experience of and continues to build high performance boards and teams. He is currently Chair (third professorial chair) and Director of the Academy of Sport at the University of Edinburgh – one of the few dedicated think tanks supporting the sports industry to address local and global challenges.
Recent successes have included helping the University of Edinburgh secure significant partnerships with the University of Toronto and FC Barcelona while working with the 53 countries of the Commonwealth to advance education around governance, safeguarding and equalities. His review of the Scottish sporting landscape is about to be published by the Scottish Government.
Within the university sector Grant has held the offices of Acting University Principal, Vice-Principal, Head of Faculty (8 Schools) and Head of Department both within the UK and beyond. He has worked with Scottish Universities Principal's Executive and been an advisor to successive education and sport Ministers. Within the sports sector he has been Chair and Director with the National Sports Council, is currently Chair of East Renfrewshire Culture and Leisure Trust and an Independent Trustee with the Scottish Para Football Association. He has been chair of various Finance, Audit and Risk Committee's and been accountable for budgets in excess of £150million.
He has expertise in change management programmes, philanthropy and fund raising and has always strived to open up opportunities through both education and sport. In the development of Football More than a Game Grant he led the development of the world's first football MOOC and the subsequent opportunities which have helped the University of Edinburgh open up access to education in marginalized communities by delivering credit bearing football courses in local football stadiums. He held the university equalities portfolio at time of of the Equalities Act being introduced, helped to bring into being the National Para Sports Centre at Inverclyde, the delivery of the 2014 Commonwealth Games and the Winning Students Scholarship Scheme which has helped to open up access to education through sport across Scotland's Universities and Colleges.
Having benefited from leadership training including media training he is an experienced communicator beyond the university, comfortable in engaging with senior figures in government and industry and with high net worth individuals. This in part has been demonstrated through regularly appearing in committees, policy forums and as a practiced media performer, including radio, TV and international public audiences such as Celtic Connections.
He completed the Top Management Programme for future leaders taken by University Senior Management and Civil Servants and sits on the British Council Board for Erasmus and the CMB International Strategy Board. He has served as an advisor to UNESCO and the Prince Albert Foundation.
He is an associate member of the Iona Community, an Honorary Visiting Professor with the University of Toronto and has been awarded an Honorary Degree for services to international higher education.
Teaching & PhD supervision
Development of the first MOOC- Football More than a Game (Non-Credit bearing version, free access to education)
Global Impact of Sport Summer and Winter Schools
Football More than a Game- 5 Credits at level 7 - delivered in areas of multiple deprivation across Scotland
Postgraduate teaching
Supervision of Phd and Msc research students
Sport, Society and International Development MSc Core Course
Open to PhD supervision enquiries?
Areas of interest for supervision
It would be great to talk to potential PhD students from all over the world who want to make a difference in the world through sport and related activities. We have a good track record of success, introducing you to a world wide network, welcoming you into a critical mass and family of PhD students across the University working on sport, physical activity, physical education topics and much more.
Grant's work is concentrated in three areas education, sport and international relations through which he has been committed to social change, policies tackling inequality and interventions designed to alter life chances. He has consistently advocated for sport to be recognised as an effective national and international social and economic tool and championed the role of the university and sport in developing human capabilities, cities and opportunities. He has a wealth of experience of working in the real world of sport as well as the university.
Have a more detailed look at http://www.ed.ac.uk/education/institutes/spehs/academy-of-sport
Current PhD students supervised
Yujin Xu- An Experiential Approach to Cross Cultural Education - drawing on the case study of sail training
Mason Robbins- Football, Money and Sports Rankings
Matthew Holmes - Sport for Development in Zambia
Corissa Vella White - Small State Sport Systems
David Meir - Sport, International Development and Social Change
Tavis Smith - joint supervision with the University of Toronto
Christos Tomatzu- Education through Football
Past PhD students supervised
Grant has supervised more than 30 Phd's (Full-time; Part-Time and by Publication)
Many have gone on to secure posts including:
Irene Reid- Lecturer in Scotland
Fiona Grossert Sanders- Lecturer in Australia
Dong Jhy Hwang- Lecturer in Taiwan
Claire Parker- Lecturer in Australia
Susan Ahrens- Lecturer in Canada
Gilberto Galeazzi- Civil Servant in Italy
Comparative Analysis - Canada and Scotland
Sport, Cultural Relations and Diplomacy; Football- Social, Cultural, Economic Impacts; Cities; Canada and Scotland; Social Inequality and Social Change; Commonwealth Sport for Development - Report for Commonwealth Secretariat; The Scottish Sporting Landscape - Report for the Scottish Government; The Making of Sports Policy- Report for the Jimmy Reid Foundation;
Past research interests
Labour Migration of Kenyan and Ethiopian Athletes; Malaysian Sports Industry- Report for the Malaysian Government;
Affiliated research centres
Current sample of projects:
Sport and the Sustainable Development Goals
Welsh sport as a soft poweer asset
Scottish Government Review of sport
Widening access to education through sport
International Sports Diplomacy
Sport and social inequality in Scotland
Current project grants
Sport and the sustainable development goals (External Funding - Commonwealth and Australian Department of Foreign Affairs and Trade)
Welsh sport as a soft power asset (External Funding- British Council)
Widening access to education through football( University funded)
Enabling Gaelic sport (External funding)
Sports Policy in Scotland Report (External funding)
Past project grants
MOOC feasibility study (External funding)
Advancing Sports Heritage (External funding)
Handa Sports Fellowships (External funding)
Letter concerning the David Hume Tower Sep 2020
email correspondence › Other contribution (Published)
Sport, Physical Education, and Social Justice: Religious, Sociological, Psychological, and Capability Perspectives (206 pages) 20 Jul 2020
Research output: › Book (Accepted/In press)
Sports Observations: Sport, Protest and Black Lives Matter (10 pages) Jul 2020
Research output: › Other report (Published)
The Value of Scottish Football (23 pages) 2020
Sport, Physical Education, and Social Justice: Religious, Sociological, Psychological, and Capability Perspectives (120000 pages) 11 Dec 2019
Sport and Social Justice 9 Dec 2019
A Review of the Scottish Sporting Landscape: The Jarvie Report 2019 24 Sep 2019
Research output: › Commissioned report (Published)
Sport and social inequality (12 pages) May 2019
Research output: › Chapter (Published)
Edinburgh Social and Political Sports Research Update (3 pages) 24 Apr 2019
Sport, homelessness and capability: Voices from the street (14 pages) 3 Apr 2019 In: Quest, vol. 71, pp. 239-251
DOI: https://doi.org/10.1080/00336297.2018.1552160
Research output: Contribution to Journal › Article (Published)
Sport for social justice, capability and the common good: A position statement in honour of Tessa Jowell (23 pages) 3 Apr 2019 In: Quest, vol. 71, pp. 150-162
Editorial: Religious, Sociological and Capability Perspectives (8 pages) 7 Dec 2018 In: Quest, pp. 1-8
Research output: Contribution to Journal › Editorial (Published)
Arran, Countess of 1 Oct 2018
Research output: › Entry for encyclopedia/dictionary (Published)
Hamilton, Helen 1 Oct 2018
Newstead, Isabel 1 Oct 2018
View all 58 publications on Research Explorer
Professor Grant Jarvie's inaugural lecture "Sport: More Than Just a Game" (on You Tube)
This article was published on 17 Jan, 2021


Obesity rates tend to be higher in communities without access to fresh fruits and vegetables.
More Californians than ever are obese, according to a new report by the UCLA Center for Health Policy Research.
The study found that 24.8 percent of adults were obese in 2011–12, compared to 19.3 percent a decade earlier. Nearly 18 million California adults and adolescents are considered overweight or obese, and about 7.4 million of them can be classified as obese. Data for the research was drawn from California Health Interview Survey results starting in 2001. Adults with a body mass index of 25 or greater are considered overweight; those with a BMI of 30 or more are considered obese.
The authors also found that 16 percent of Californians ages 12 to 17 were overweight and 17 percent were obese — both figures were relatively unchanged from 2001. Nearly 1 million California adolescents were overweight or obese in 2011–12.
The study, funded by The California Endowment, found obesity rates had increased in every major region of the state since 2001. But current adult obesity rates varied widely by county: San Francisco County had the lowest rate (11.3 percent), while Imperial County had the highest (41.7 percent).
* Smaller counties were pooled to create a larger sample.
A sortable, county-by-county list is available on the UCLA Center for Health Policy Research website.
Lower-income Californians and those belonging to certain racial and ethnic groups had disproportionately high rates of obesity, the report found. Obesity rates among the poorest adults —those with household incomes below 200 percent of the federal poverty level — jumped during the past decade from 24.4 percent to 30.5 percent, a rate 10 points higher than that of higher-income Californians (those whose income was at least 400 percent of the poverty level).
Obesity rates also varied considerably by race, although the figures for all racial and ethnic groups increased during the decade. Groups with the highest obesity rates in 2011–12 were Pacific Islanders (37.1 percent), American Indians (36.2 percent), African-Americans (36.1 percent) and Latinos (32.6 percent) — all markedly higher than the rate for whites (21.9 percent). Asians had the lowest obesity rate, 9.7 percent, but even that figure represented a jump from 5.3 percent in 2001.
Rates within racial groups varied significantly, as well. Among Asian subgroups, for example, 2.1 percent of Koreans were obese versus 20 percent of Southeast Asians.
Access to fresh produce: 22 percent of adults who said fresh produce was always affordable in their neighborhood were obese, compared with 31 percent of those who said fresh produce was never affordable. Only three-quarters of Latinos and African-Americans said fresh produce was affordable, compared to 86 percent of whites.
Neighborhood safety: Among low-income adults, 28 percent felt their neighborhood was unsafe compared to 5 percent of higher-income adults. Twenty-two percent of Latinos and 18 percent of African-Americans reported their neighborhoods were unsafe compared to 7 percent of whites.
Park safety and exercise: Among adolescents, 27.4 percent of those who felt their neighborhood park was safe were obese; the obesity rate among adolescents who felt their park was unsafe was 42.3 percent.
Social connection: 26 percent of those who felt little connection with neighbors were obese, compared with 22 percent of those in high-connection neighborhoods.
The authors write that local governments should address these barriers by bringing more farmers' markets to underserved areas or by offering incentives to storefront markets to add fresh produce; improving safety at existing parks through better maintenance; developing crime prevention programs to encourage leisure walking; making neighborhoods more walkable and pedestrian-oriented; and encouraging development projects that include recreation space.


skin a dewy and smooth appearance with this revolutionary deep-cleansing, wash-off sheet mask.
1. Before use, please rub the surface of the pouch to mix the contents well.
2. Take out and unfold the mask.
Apply the mask over a dry face in alignment with the face shape.
3. Wear the mask for 10-15 minutes.
Remove the mask when the oxygen bubbles have risen sufficiently.
4. Rinse your face with lukewarm water.


text:
mostrapalazzofarnese » Electronic » New Order - Blue Monday
New Order - Blue Monday mp3 album
New_Order Electro Synth-pop 1983
Performer: New Order
Title: Blue Monday
Formats: AC3 VOC MMF MPC AIFF MIDI AA
Style: Electro, Synth-pop
It was released as a 12-inch single on 7 March 1983 through Factory Records and later as a 7-inch single through Tonpress in 1985. It appeared on certain cassette and CD versions of the band's second studio album, Power, Corruption & Lies (1983). Blue Monday" is a synth-pop and alternative dance song that drew inspirations from many works of other artists.
At nearly 7 and a half minutes in length, "Blue Monday" is one of the longest tracks ever to chart in the UK. It is recognised as the biggest selling 12" single of all time, but as Factory Records were not members of the British Phonographic Industry association, it was not eligible for an official gold disc.
Album Power, Corruption and Lies. The biggest selling 12 of all time was a pivotal moment in dance music. Mixing cold wave sentiment with the disco influence of Sylvester and Donna Summer, the result was a stunning explosion of drum machines, wibbly synths and Bernard Sumner's no-nonsense vocal style. More than 30 years later, it still sounds amazingly modern. The title of the song is a reference to an illustration by author Kurt Vonnegut. Blue Monday" Track Info.
The key enabling this to be deciphered was printed on the back sleeve of the album Power, Corruption And Lies. In Q Magazine's 1001 Best Songs Ever issue, Peter Hook said, "I go through stages of intense dislike for 'Blue Monday,' which I'm sure every group does when they get one song they're synonymous with, but the way it keeps getting reinvented is wonderful. It seems to be one of those tracks that's timeless, which is amazing. Later, Kraftwerk's ex-member Karl Bartos released an album called Communication. The release from there, "15 Minutes Of Fame" is EXTREMELY similar. How ironic! Martin from Groningen, NetherlandsRolling Stone 15/9/83 Interview by Debby Miller.
Modifier l'album Reporter une erreur. Membres possèdant cet album4. 1. Blue Monday 1988 (Single Mix). 2. Blue Monday 1988 (12" Mix). 3. Blue Monday 1988 (Dub Version).
Related to New Order - Blue Monday albums:
Tommy B. - Blue Monday download mp3
52nd Street - Cool As Ice / Twice As Nice download mp3
New Order - In Order download mp3
New Order - Confusion download mp3
New Order - Paradise Garage NY July 7th, 1983 download mp3
New Order - Ceremony download mp3
Coco Electrik / Zook - Tainted Love / Blue Monday download mp3
New Order - Procession / Everything's Gone Green download mp3
New Order - Sub-Culture download mp3
New Order - Touched by the hand of god 12 inch mix download mp3


text:
When Seeing (And Hearing) Isn't Believing: Deep Fakes & Digital Deception – Part II
There have already been instances of malicious actors creating deep fakes to deceive their adversaries, but it is important to note that knowledge of how they are made can help investigators and intelligence analysts tell the difference between a real piece of media and one that has been manipulated, or created from scratch.
There may be tell-tale signs in the metadata of a file (little tags of data that describe other data). This could help identify some digital fingerprints, such as the time of day the image or video was taken, with what device, and whether it was subsequently opened in an editor. This is not fool proof, though, since metadata itself can also be manipulated. Further, EXIF data attached to images is more routinely being removed from open sources – including as a matter of course when files are uploaded to social media or on messaging apps, where deep fakes are likely to be found. This presents further challenges for the investigator.
Alongside the most advanced deep fakes – the combining of voice and audio – the two component parts weaponised on their own can still have devastating consequences. We previously wrote about an FSB assassin falling victim to social engineering and revealing details of their unsuccessful attempt to poison Alexei Navalny on a phone call – with Navalny on the other end of the line posing as a senior FSB official. The weaponisation of synthetic voice cloning has further enhanced the scope of social engineering possibilities. In several fraud cases, fraudsters have spoofed voices to gain access to corporate systems and steal money. In one instance a bank manager, believing he was speaking to the director of a company (with whom he had spoken before) that the bank was preparing to lend money to for a potential acquisition, began transferring funds when told the acquisition was going ahead. Unfortunately he was actually duped by criminals using an audio deep fake and the transfers were received into those criminals' accounts.
It has sometimes been possible for analysts to verify the authenticity (or otherwise) of a suspected deep fake by measuring shadows and tracking the position of the sun to assess whether they tally with the time of day the deep fake was allegedly taken. Reverse image searching also remains a useful tool for analysts seeking to identify deep fakes. This may lead to the original source from which a fake has been generated but, due to the fact that the images which form part of the deep fake have themselves been manipulated, that original source may not be easily identifiable. Still, with advancements in the sophistication of image search algorithms, it is possible that individual components of the image may be identified, providing useful information for an analyst.
In some cases, the deep fake just looks distorted, and can be identified as having been computer generated with careful observation using the naked eye. There is sometimes inconsistent emotion between the face and voice track of a video, and breathing rates can vary from human norms. Beyond this, there is a growing arsenal of algorithmic detection tools being developed and constantly improved to automatically detect deep fakes. Some, for example, compare the forensic traces of two parts of an image or video to produce a measure of how similar or dissimilar they are. However, the effectiveness of these tools varies considerably and many are still unable to achieve a sufficiently high detection rate to efficiently combat the proliferation of deep fakes.
There is undeniably an ongoing arms-race between producers of deep fakes and investigators. Early on in the history of deep fakes, for instance, researchers noticed that the subjects did not blink, and developed a machine-learning algorithm to track eye movement and identify fake content. But only a week after publishing their findings deep fake creators had "fixed" this problem, and included blinking in their new fake videos. This arms-race will almost certainly continue into the future, as long as there remain incentives, be they financial, political or simply personal amusement, for those that wish to create and disseminate deep fakes.
©2021 KIRIS GROUP all rights reserved | London, UK


text:
This article contains a list of bands formed in New York City, New York.  Bands should be notable and linked to their articles which lists their New York origins in the lead.  References should be provided for any new entries on this list.  Bands may be temporarily red-linked (while an article is developed) as long as the reference establishes that the band is notable and from New York.

#
 108
 24-7 Spyz
 2 in a Room

A–F

 A Place To Bury Strangers
 A Tribe Called Quest
 Agnostic Front
 Alice Donut
 AJR
 American Classical Orchestra
 American Composers Orchestra
 The Amygdaloids
 The Antlers
 Anthrax
 Bad Ronald
 Band of Susans
 Barkmarket
 Battles
 Bayside
 Beach Fossils
 Beastie Boys
 Blondie
 Blood, Sweat & Tears
 Bloodsimple
 Bloody Panda
 The Blue Flames
 Blue Öyster Cult
 Blues Magoos
 The Books
 Boys Choir of Harlem
 Brand New
 The Brandos
 Breakfast Club
 Breaking Laces
 Brutal Truth
 Burn
 The Cake
 Cameo
 The Candles
 Castevet
 Certain General
 Chic
 The Chiffons
 Cibo Matto
 Circus Mort
 The City and Horses
 Class Actress
 The Coachmen
 Coheed and Cambria
 The Cold Crush Brothers
 Company Flow
 Contre
 The Cookies
 The Cramps
 The Cro-Mags
 The Crystals
 The Davenports
 Deee-Lite
 Defunkt
 De La Soul
 DeLeon
 The Devil Dogs
 The Dictators
 DIIV
 The Diplomats
 Dirty Projectors
 DJ Logic
 Dope
 Drowners
 The Druids of Stonehenge
 The Drums
 Elephant's Memory
 Envy On The Coast
 Errortype: Eleven
 Even Worse
 Fischerspooner
 Foreigner
 Fountains of Wayne
 Four to the Bar
 Freelance Whales
 Full Blown Chaos
 Fun
 Fun Lovin' Criminals

G–M

 Gato Loco
 The Ghost of a Saber Tooth Tiger
 Glassjaw
 The Golden Filter
 Goodbye Picasso
 Gorilla Biscuits
 Grizzly Bear
 Groove Theory
 The Groupies
 Growing
 Grupo Latin Vibe
 Gumball
 G-Unit
 The Heartbreakers
 Richard Hell and the Voidoids
 Helmet
 Hem
 Hercules and Love Affair
 Holy Ghost!
 The Hundred in the Hands
 Interpol
 The Indecent
 Ism
 Javelin
 Jawbreaker
 Jets to Brazil
 Kid Creole and the Coconuts
 Kinetics & One Love
 Kiss
 LCD Soundsystem
 Leftöver Crack
 The Little Orchestra Society
 Life Of Agony
 Living Colour
 Lord Tariq and Peter Gunz
 Lotion
 The Lounge Lizards
 The Lovin' Spoonful
 The Mamas & the Papas
 Method Man & Redman
 Michael Hill's Blues Mob
 Midtown
 Miller Miller Miller & Sloan
 Mindless Self Indulgence
 MisterWives
 MNDR
 Mod Fun
 The Mooney Suzuki
 Murphy's Law

N–S

 Nada Surf
 Naturally 7
 NBC Symphony Orchestra
 Neon Boys
 New York Bandura Ensemble
 New York Dolls
 New York Percussion Trio
 New York Philharmonic
 Ninja Sex Party
 Nuclear Assault
 One For All
 Orpheus Chamber Orchestra
 The Pains of Being Pure at Heart
 Paperdoll
 Park Avenue Chamber Symphony
 Parts & Labor
 Pretendo
 The Pretty Reckless
 Prong
 PS22 chorus
 Public Enemy
 Quicksand
 The Raelettes
 Ramones
 The Rapture
 Ratatat
 Riot
 The Ronettes
 The Ropes
 Run-DMC
 School of Seven Bells
 Scissor Sisters
 Seguida
 The Shangri-Las
 Sheer Terror
 Shelter
 The Shondes
 Shootyz Groove
 Shy Child
 Sick of It All
 Silos
 Simon & Garfunkel
 Skaters
 Sleigh Bells
 Sonic Youth
 Spacehog
 Spin Doctors
 Spread Eagle
 Steely Dan
 The Steinettes
 The Strangeloves
 The Strokes
 ST-X Ensemble
 Suffocation
 Swans
 Sylar (band)

T–Z

 Taking Back Sunday
 Talking Heads
 Teenage Jesus and the Jerks
 Television
 Terror Squad
 Texas Is the Reason
 They Might Be Giants
 Tiny Masters of Today
 Toilet Böys
 Tomandandy
 Trachtenburg Family Slideshow Players
 Trilogy
 TV On The Radio
 Type O Negative
 Ui
 The Unlovables
 Unsane
 Vampire Weekend
 The Van Pelt
 The Velvet Underground
 Richard Hell and the Voidoids
 Versus
 The Waldorf-Astoria Orchestra
 Warzone
 Weep
 White Zombie
 Wungabongas
 Wu-Tang Clan
 X27
 Yeah Yeah Yeahs

See also

 Lists of musicians
 List of people from New York City

References

New York City

Bands

redpajama_set_name: RedPajamaWikipedia


text:
OU football: TALE OF THE TAPE
Published: Tue, January 6, 2009 12:00 AM Updated: Tue, January 6, 2009 1:54 AM
First to
→Oklahoma: OU's football program took the field in 1895, 12 years before Oklahoma was granted statehood.
→Florida: UF's first recognized team in 1906 finished 5-3, splitting a pair of games with Rollins College (yes, that Rollins College).
→The edge: Oklahoma, because it's cool to say the Sooners debuted during the second Grover Cleveland administration.
Bowl appearances
→Oklahoma: The Sooners are 24-16-1 all time, 2-4 in BCS games.
→Florida: 16-19 all time, 2-1 in BCS.
→The edge: Oklahoma. You can't argue against the Ws.
Conference titles
→Oklahoma: 41 conference titles. The first came in 1915. Remember that year? Didn't think so.
→ Florida: Eight SEC titles. The first, in 1984, was forfeited because of NCAA infractions (more on that later).
→The edge: Oklahoma. The Sooners rule college football in tornado alley.
Pro hall-of-famers
→Oklahoma: Troy Aikman, Tommy McDonald, Lee Roy Selmon.
→Florida: Jack Youngblood.
→The edge: Oklahoma, although Aikman played only one season at Oklahoma before transferring to UCLA.
Players in the NFL
→Oklahoma: The NFL drafted 36 Sooners since Bob Stoops became the Sooners head coach, including 2007 Rookie of the Year Adrian Peterson ( Minnesota Vikings) and Davin Joseph (Tampa Bay Bucs).
→Florida: There were 38 Gators on NFL rosters this season, including Jacksonville Jaguars running back Fred Taylor and Dolphins linebacker Channing Crowder.
→The edge: Push. Both schools stockpile the Sunday league with plenty of talent.
Game-day traditions
→Oklahoma: The Ruf/Neks are some of the most vicious fans in college football. They shoot off pop guns in the end zone at Owen Field and mercilessly heckle the opposition. In fact, they so annoyed former Nebraska Coach Bill Callahan that he referred to them as a bunch of [expletive] hillbillies following a game there in 2005.
→Florida: George Edmondson Jr., aka Mr. Two-Bits, is hanging it up after leading cheers at the Swamp since 1949, but his "two bits, four bits, six bits, a dollar" cheer will undoubtedly live on. And the Gator chomp is something you won't see outside of Gainesville.
→The edge: Oklahoma, because not even Mr. Two-Bits can hold a candle to [expletive] hillbillies.
→Oklahoma: The Sooner Schooner, the school's official mascot, is a Conestoga wagon pulled by two white ponies. How adorable.
→Florida: Albert the Alligator used to be a real, live creature toted to and fro in a glass cage. Now he's just an oversized, but lovable, lug in a plush outfit.
→The edge: Florida, although neither mascot inspires greatness.
→Oklahoma: The Pride of Oklahoma marching band loves to toll the Rodgers and Hammerstein title tune from the hit 1943 musical Oklahoma! Indeed, the waving wheat can sure smell sweet.
→Florida: Florida's Fightin' Gator Marching Band also is known as Pride of the Sunshine.
→The edge: Push. They're bands, people. Unless a trombone player gets flattened in the end zone, do they really matter? Really?
Notorious coaches
→Oklahoma: Barry Switzer coached the Sooners to three national championships, but the NCAA twice put Oklahoma on probation during his head-coaching tenure, which lasted from 1973-1989. He was also known as a lax disciplinarian and some of his players became regulars on the Norman police blotter. Switzer later coached the Dallas Cowboys to a Super Bowl ring, but this part champion, part scoundrel couldn't survive long in the NFL.
→Florida: Three years after he became UF's head coach, Charley Pell's actions at his old school, Clemson, placed the Tigers on probation. But Pell got the Gators in hot water with the NCAA, too, so he was canned during the 1984 season. Pell was followed by Galen Hall, who stepped down in 1989 after the NCAA uncovered more wrongdoing at UF.
→The edge: Oklahoma. Pell and Hall weren't angels, but you somehow envision horns growing out of Switzer's head when he flashes that impish grin.
→Oklahoma: Five winners: Billy Vessels (1952), Steve Owens (1969), Billy Sims (1978), Jason White (2003), Sam Bradford (2008).
→Florida: Three winners: Steve Spurrier (1966), Danny Wuerffel (1996) and Tim Tebow (2007).
→The edge: Oklahoma, but since only a handful of you remember the likes of Billy Vessels and Steve Owens, the Sooners don't have a huge advantage.
National titles
→Oklahoma: Talk about a huge trophy case. The Sooners have seven national championships and this will be the fourth time they've played for a national title under Stoops.
→Florida: The Gators have two national championships to tout, including a win over Ohio State two years ago in the first BCS championship game.
→The edge: Oklahoma.
BY MATT HUMPHREY
Staff Writer for the Orlando Sentinel
Article: BCS National Championship: OU quarterback Sam Bradford inspiration to American Indians
Article: OU kickoff return unit lauded
Article: BCS National Championship: Florida's Brandon Spikes fires back at OU
Article: BCS National Championship: Manny Johnson quiet as leader for OU
Article: BCS National Championship: Juaquin Iglesias is tough, talented
Video: OU Show with John Helsley (2009-01-05)
Video: OU 13, Florida State 2 (2009-01-05)
Video: USC 55, OU 19 (2009-01-05)
LEE ROY SELMON, OU FOOTBALL: University of Oklahoma defensive tackle LeRoy Selmon.
CommentsOU football: TALE OF THE TAPE


text:
A sports comeback to behold
LEGEND: Roger Federer is proving to be one of the greatest athletes of our time. (Larry Smith, EPA)
Kyrgios heaps praise on Federer ahead of semi
Federer 'thinks young' to make Miami luck
Federer to battle Nadal for Miami crown
Silver Sibiya
Johannesburg - The man of the moment in tennis circles is none other than Roger Federer – one of the greatest tennis players in history and the record-breaking holder of 18 Grand Slam singles titles.
Federer has had an excellent run this year, claiming top honours at the ATP World Tour Masters 1000 at Indian Wells two weeks ago – in the process beating his biggest rival, Rafael Nadal.
Federer remains on course for the winner's podium at the Miami Open, which finishes on Sunday.
Second oldest man
In January, he won the Australian Open, also defeating his Spanish nemesis Nadal in the final.
The Swiss is yet to lose a match this year as he continues his impressive run.
The 35-year-old was thought to be finished and his name was fading in the tennis world, but after five years without a Grand Slam victory, as well as a season sidelined by injury and his world ranking down to number 17, he has risen like the mythical phoenix.
The five-time US Open champion did the unexpected by becoming a champion again.
Federer became the second oldest man to reach a major final – surpassed only by Australia's Ken Rosewall (39) at the 1974 US Open – and he is the oldest man to win a Grand Slam in more than 40 years.
He is now ranked sixth in the world.
The 2009 French Open Champion's victories at the Australian Open and Indian Wells have restored faith that he could recapture the world number one position, a goal that will be strengthened if he reaches and wins the Miami Open final on Sunday, where Andy Murray (ranked number one) and Novak Djokovic (number two) are notable absentees as they both have elbow injuries.
Federer has not topped the rankings since 2012, but says his focus is not rankings, but titles.
Defeated now retired
He told Sky Sports: "I understand the talk of world number one, so I will try to back it up with a good tournament. I think I'm definitely doing a few things better than I ever have. I do feel that I have improved. The game has evolved – I had to adjust – but, overall I do believe I'm probably a better player than I was 10 years ago."
Federer hasn't won the Miami Open in 11 years. He last won when he defeated now retired Croatian Ivan Ljubicic in 2006. With the way the five-time Australian Open champion is playing these days, that may change.
Is this Federer's year or will the injuries start creeping in again?
"I just have to be a bit more clever in terms of scheduling overall. Unfortunately, I can't do it all [play all games]," he told Tennis World Magazine. Only time will tell if Federer will be able to regain the number one spot at his age. He will play Nadal on Sunday.
Read more on: atp | rafael nadal | roger federer | miami | johannesburg | tennis
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When all is said and done, who of the 'Big 3' in men's tennis ends their career with the most Grand Slams?
Roger Federer - 20 to date
Rafael Nadal - 19 to date
Novak Djokovic - 16 to date
49% Roger Federer - 20 to date
14% Rafael Nadal - 19 to date
36% Novak Djokovic - 16 to date
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Looking around Amritapuri, the culture-transcending allure of Sri Krishna was abundantly evident. The ashram was full of people of all lands and walks of life celebrating Krishna's birth and life. And virtually everywhere one looked there were children of all nationalities dressed up as gopis, gopas, Radhas and Krishnas. Amritapuri has been transformed into a sea of colourful saris and swaying peacock feathers.
The children soon formed a procession, walking through the ashram and out into the surrounding villages—the jingle of their anklets adding to the joyous noise of their drums, horns and tambourines. Upon their return, the uriyadi [pot-striking] games began, and Amma and all the ashram residents came to watch as the children tried their best to smash open the clay pots for candy and coins.
At the end of the games, Amma burst into song, singing "Hari Bol Hari Bol!" When the song reached its climax, Amma cried out "Hari Bol!" and pointed her index fingers to skyward, a terrific smile spread across her face.
The actual time of Krishna's birth was marked with the recital of his birth story. The lines from the Malayalam Srimad Bhagavatam were sung with true pathos, piercing everyone's heart. Surrounding the lone voice was a pin-drop silence. Towards the end of the reading, puja was offered to a small idol of baby Krishna.
Amma also glorified the devotion of the gopis, saying how it is their love for Krishna that is remembered over anyone else's, even though they were with Krishna for a relatively short amount of time. To illustrate this point, Amma told a story wherein the gopis were able to remember the precise details of every single interaction they had ever had with Krishna, no matter how long before it had transpired.
The evening ended with Amma distributing sweet payasam for one and all. It was then that a devotee offered Amma with the opportunity to play some uriyadi of her own, presenting Amma with a stick and a couple clay pots of butter and curd. But he allowed Amma to break open the pots, he wrapped her head in an orange turban, transforming her into the cowherd Krishna of Vrindavan.