text:
\section{Introduction}

The polarizabilities of a composite system such as the nucleon are
elementary structure constants, just as its size and shape. They can
be studied by applying electromagnetic fields to the system. The
physical content of the nucleon polarizabilities can be visualized
best by effective multipole interactions for the coupling of the
electric $(\vec{E})$ and magnetic $(\vec{H})$ fields of a photon
with the internal structure of the nucleon. This structure can be
accessed experimentally by the Compton scattering process $\gamma +
N \to \gamma + N$ on the nucleon, see
Refs.~\cite{DPV_report,Schumacher} for recent reviews. When
expanding the Compton scattering amplitude in the energy of the
photon, the zeroth and first order terms follow from a low energy
theorem and can be expressed solely in terms of the charge, mass,
and anomalous magnetic moment of the nucleon. The second order terms
in the photon energy describe the response of the nucleon's internal
structure to an electric or magnetic dipole field, they are given by
the following effective interaction~:
\begin{eqnarray}
\label{h2order}
H_{\rm{eff}}^{(2)} = -4\pi \left[
{\textstyle\frac{1}{2}}\,\alpha_{E1}\,\vec{E}\,^2 +
{\textstyle\frac{1}{2}}\,\beta_{M1}\,\vec{H}\,^2\ \right],
\end{eqnarray}
where the proportionality coefficients are the electric
($\alpha_{E1})$ and magnetic ($\beta_{M1}$) dipole (scalar) polarizabilities,
respectively. These global structure coefficients are proportional
to the electric and magnetic dipole moments of the nucleon which are
induced when placing the nucleon in static electric and magnetic
fields. They have been measured extensively using unpolarized
Compton scattering. A fit to all modern low-energy Compton
scattering data yields the following results for the
proton~\cite{Olmos01}:
\begin{eqnarray}
\alpha_{E1}^p & = & \left[ 12.1  \pm
0.3(\rm{stat}) \mp 0.4(\rm{syst}) \pm 0.3(\rm{mod}) \right]
\times 10^{-4} \, \mbox{fm}^3 \ ,
\nonumber \\
\beta_{M1}^p & = & \left[ 1.6 \pm 0.4(\rm{stat}) \pm 0.4(\rm{syst}) \pm
0.4(\rm{mod}) \right] \times 10^{-4} \, \mbox{fm}^3 \ ,
\label{eq:exp}
\end{eqnarray}
with the statistical, systematical and model-dependent errors, respectively.
These values confirm, beyond any doubt, the dominance of the
electric polarizability $\alpha_{E1}$. The tiny value of the
magnetic polarizability, $\beta_{M1}$, comes about due to a
cancellation of the large paramagnetic contribution of the $N \to
\Delta$ spin-flip transition with a nearly equally large
diamagnetic contribution, partly due to pion loop effects.
\newline
\indent The internal spin structure of the nucleon appears at third
order in an expansion of the Compton scattering amplitude. It is
described by the effective interaction
\begin{eqnarray}
\label{h3order}
{H}_{\rm{eff}}^{(3)} &=&  - 4 \pi \left[
{\textstyle\frac{1}{2}}\gamma_{E1E1}
\,\vec{\sigma}\cdot(\vec{E}\times\dot{\vec{E}})
+ {\textstyle\frac{1}{2}}\gamma_{M1M1}\,
\vec{\sigma}\cdot(\vec{H}\times\dot{\vec{H}}) \right. \nonumber \\
&&\left. \hspace{.75cm}
- \gamma_{M1E2}\, E_{ij}\,\sigma_iH_j
+ \gamma_{E1M2}\, H_{ij}\,\sigma_iE_j\  \right],
\end{eqnarray}
which involves one derivative of the fields with regard to either
time or space~: $\dot{\vec{E}} = \partial_t\vec{E}$ and
$E_{ij}=\frac{1}{2}(\nabla_iE_j+\nabla_jE_i)$, respectively. The
four spin (or vector) polarizabilities $\gamma_{E1E1}$, $\gamma_{M1M1}$,
$\gamma_{M1E2}$, and $\gamma_{E1M2}$ describing the nucleon spin response
at third order, can be related to a multipole
expansion~\cite{BGM}, as is reflected in the subscript notation. For
example, $\gamma_{M1E2}$ corresponds to the excitation of the
nucleon by an electric quadrupole ($E2$) field and its de-excitation by a
magnetic dipole ($M1$) field. Expanding the Compton scattering amplitude to
higher orders in the energy, one obtains higher order
polarizabilities to the respective order, e.g., the quadrupole
polarizabilities to fourth order~\cite{BGM,HDPV}.
\newline
\indent The effective Hamiltonians of Eqs.~(\ref{h2order}) and
(\ref{h3order}) describe a shift in the nucleon energies at second
order in the electromagnetic fields. This has been implemented in
recent years as a tool to calculate nucleon polarizabilities in
lattice QCD. By calculating the mass shifts in a constant background
field, and isolating the quadratic response, it has thus been
possible to compute the electric polarizability of the neutron and
other neutral octet and decuplet baryons~\cite{Christensen:2004ca},
and the magnetic polarizability of the proton,  neutron, and all
other particles in the lowest baryon octet and decuplet
states~\cite{Lee:2005dq}. In Ref.~\cite{Lee:2005dq},
$\beta_{M1}^p$ has been calculated by use of the
Wilson action in the pion mass range
$0.5 \leq m_\pi \leq 1$~GeV and neglecting disconnected loop diagrams.
For the smallest calculated pion mass
of $m_\pi \simeq 500$~MeV, a value of $\beta_{M1}^p = ( 2.36 \pm
1.20 ) \times 10^{-4} \, \mbox{fm}^3$ was obtained. While it is
encouraging to see that the lattice result is in the right ballpark
when compared to the experimental value of Eq.~(\ref{eq:exp}), a
more precise comparison clearly requires a dynamical fermion
calculation, including disconnected loop diagrams and much smaller
pion masses. Such small pion masses would then allow one to
extrapolate safely to the physical pion mass within the framework of
chiral perturbation theory.
\newline
\indent To determine the spin polarizabilities, Eq.~(\ref{h3order})
can likewise be used to calculate energy shifts of a polarized
nucleon in an external field. As an example, consider a nucleon
polarized along the $z$-axis and apply a magnetic field rotating
with angular frequency $\omega$ in the $xy$ plane,
\begin{eqnarray}
\vec H = B_0 \left[ \cos (\omega t) \; \hat e_x + \sin (\omega t) \; \hat e_y
\right],
\end{eqnarray}
where $\hat e_i$ stands for the unit vector in the direction $i = x,
y$ and $B_0$ is the magnitude of the field. Such a field leads to
an energy shift $\Delta E = \mp 2 \pi \gamma_{M1 M1} \omega B_0^2$
if the nucleon spin is oriented along the positive ($-$) or negative
($+$) $z$-axis. The split between the two levels is then directly
proportional to the magnetic dipole spin polarizability
$\gamma_{M1M1}$.
\newline
\indent It has been shown in
Ref.~\cite{Detmold:2006vu} that allowing for
background fields with suitable variations in space and time,
lattice QCD should be able to calculate all six dipole
polarizabilities. In particular, calculations are in progress to
determine the electric polarizability of a charged particle such as
the proton as well as the four proton spin polarizabilities of
Eq.~(\ref{h3order})~\cite{Detmold:2006xh}.
\newline
\indent
A microscopic understanding of the nucleon's
polarizabilities requires to quantify the interplay between
resonance contributions, e.g., the $N \to \Delta$ transition, and
long range pion cloud effects. Such systematic studies of the pion
cloud effects became possible with the development of chiral
perturbation theory (ChPT), by systematically expanding the
lagrangian in the external momenta and the pion mass
(``$p$-expansion''). The first such calculation to the one-loop order,
at $\mathcal{O}(p^3)$, yielded the following leading terms for the
proton scalar polarizabilities~\cite{Bernard}:
\begin{eqnarray}
\alpha_{E1}^p = 10\beta_{M1}^p = \frac
{5\alpha_{em}g_A^2}{96\pi f_{\pi}^2m_{\pi}} = 12.2\ ,
\label{eq:lo}
\end{eqnarray}
where $\alpha_{em} = 1/137$, $g_A \simeq 1.27$, and $f_\pi =
92.4$~MeV. This result is in remarkable agreement with the
experimental result of Eq.~(\ref{eq:exp}). It also illustrates that
these quantities diverge in the chiral limit, which is a challenge
for the lattice QCD calculations. Conversely, if one is in the
``small $m_\pi$'' regime where the chiral expansion converges well,
ChPT can complement the lattice calculations by extrapolating to the
physical pion mass. The ChPT work was extended to $\mathcal{O}(p^4)$
within the heavy-baryon expansion~\cite{BKSM}. However, the
agreement with the experimental values of the scalar
polarizabilities became much more challenging when including the
$\Delta$ degree of freedom, which yields a large
contribution to $\beta_{M1}^p$~\cite{Hemmert,HHKK,Pascalutsa:2004wm}.
\newline
\indent The spin polarizabilities have been calculated within the
framework of ChPT at $\mathcal{O}(p^3)$~\cite{BKKM} and
$\mathcal{O}(p^4)$~\cite{Ji:1999sv,VijayaKumar:2000pv,Gellas:2000mx}.
However, much less is known about these observables on the
experimental side, except for the forward ($\gamma_0$) and backward
($\gamma_{\pi}$) spin polarizabilities of the proton, given
by the following linear combinations of the polarizabilities of
Eq.~(\ref{h3order}):
\begin{eqnarray}
\gamma_{0}&=&-\gamma_{E1E1}-\gamma_{M1M1}-\gamma_{E1M2}-\gamma_{M1E2}
\,,\label{eq:g0_mult}\\
\gamma_{\pi}&=&-\gamma_{E1E1}+\gamma_{M1M1}-\gamma_{E1M2}+\gamma_{M1E2}
\,.\label{eq:gpi_mult}
\end{eqnarray}
The forward spin polarizability has been determined from the
Gerasimov-Drell-Hearn sum rule experiments at MAMI and
ELSA~\cite{GDHmami,GDHelsa},
\begin{eqnarray}
\label{eq:go} \gamma_0=( -1.00\pm 0.08\pm 0.10)\times 10^{-4} \:
\mbox{fm}^4\,,
\end{eqnarray}
and the recent experimental value for the backward spin
polarizability has been obtained by a dispersive analysis of
backward angle Compton scattering~\cite{Schumacher},
\begin{eqnarray}
\gamma_\pi=(-38.7 \pm
1.8)\times 10^{-4} \: \mbox{fm}^4\;.\label{eq:gpi}
\end{eqnarray}
No data exist for the other two independent proton spin
polarizabilities. It is the aim of the present work to
show that polarized Compton scattering, both near pion threshold and
in the $\Delta(1232)$ resonance region, can be used to
determine the remaining two spin polarizabilities.
\newline
\indent The extraction of polarizabilities from Compton scattering
data has been performed by three techniques. The first one is a low
energy expansion of the Compton cross sections. Unfortunately this
procedure is only applicable at photon energies well below 100~MeV,
which makes a precise extraction a rather challenging task. The
sensitivity to the polarizabilities is increased by measuring
Compton scattering observables around pion threshold and into the
$\Delta(1232)$ resonance region.
A second formalism which has been successfully applied to Compton data
in this energy region makes use of dispersion relations.
This formalism has been worked out for both unsubtracted~\cite{lvov97} and
subtracted~\cite{DGPV} dispersion relations, and yield the values
of the scalar polarizabilities given in Eq.~(\ref{eq:exp}).
Recently, a third approach has been
developed within the framework of a chiral effective field
theory~\cite{McGovern,Pascalutsa,Hildebrandt_unp,Hildebrandt_pol}.
For energies below and around pion threshold
the full Compton scattering process has been calculated to fourth
order in the small momenta, allowing for an independent extraction
of the polarizabilities from Compton scattering data. The thus
obtained values for $\alpha_{E1}^p$ and $\beta_{M1}^p$
in the work of \cite{McGovern,Hildebrandt_unp} are nicely
compatible with the results given by Eq.~(\ref{eq:exp}).
\newline
\indent In this work we study the polarized proton Compton
scattering observables both below pion threshold and in the
$\Delta(1232)$ resonance region within the subtracted dispersion
relation formalism of Ref.~\cite{DGPV}. In particular we investigate
the sensitivity of different beam and beam-target polarization
observables to the extraction of the proton spin polarizabilities.
\newline
\indent
After a brief review of fixed-$t$ dispersion relations for the Compton
scattering process in Section 2, we discuss different single
and double polarization observables in Section 3. We present our results
for these different polarization observables in Section 4 and study the
sensitivity to the extraction of the proton spin polarizabilities.
Finally, we give our conclusions in Section 5.

\section{Fixed-t subtracted dispersion relations}
\label{disp}

In this section we review the essentials of the dispersion relation formalism
for real Compton scattering (RCS), a more detailed presentation can be found in
Refs.~\cite{DPV_report,DGPV}. Let us first define the kinematics of RCS on the
proton, the reaction \begin{equation} \gamma(q) + p(p) \rightarrow \gamma(q') + p(p'), \end{equation}
where  the variables in brackets denote the four-momenta of the participating
particles. The familiar Mandelstam variables are
\begin{equation}
s=(q+p)^2\ ,\ \ t=(q-q')^2\ ,\ \ u=(q-p')^2\ ,
\label{eq2.1}
\end{equation}
which are constrained by $s+t+u=2M^2$, where $M$ is the nucleon mass. The
crossing-symmetric variable $\nu$ is defined by
\begin{equation}
\nu=\frac{s-u}{4M}\, .
\label{eq2.3}
\end{equation}
The two Lorentz invariant variables $\nu$ and $t$ are related to the initial
($E_\gamma$) and final ($E'_\gamma$) photon lab energies and to the lab
scattering angle $\theta_{{\rm lab}}$  by
\begin{eqnarray}
\nu&=&E_\gamma+\frac{t}{4M}=\frac{1}{2}(E_\gamma+E'_\gamma),\nonumber\\
t&=&-4E_\gamma \, E'_\gamma \, \sin^2 (\theta_{\rm lab}/2)=
-2M(E_\gamma-E'_\gamma).\nonumber
\end{eqnarray}
The $T$ matrix of real Compton scattering can be expressed by 6 independent
structure functions $A_i(\nu, t)$, $i=1,...,6$, which were first introduced in
Ref.~\cite{lvov97}. These structure functions depend on $\nu$ and $t$, they are
free of kinematic singularities and constraints, and because of the crossing
symmetry they satisfy the relation $A_i(\nu, t)=A_i(-\nu, t)$. Assuming further
analyticity and an appropriate high-energy behavior, the amplitudes $A_i$
fulfill unsubtracted dispersion relations (DRs) at fixed $t$,
\begin{equation}
{\rm Re} A_i(\nu, t) \;=\; A_i^B(\nu, t) \;+\;
{2 \over \pi} \; {\mathcal P} \int_{\nu_{thr}}^{+ \infty} d\nu' \;
{{\nu' \; \mathrm{Im}_s A_i(\nu',t)} \over {\nu'^2 - \nu^2}}\;,
\label{eq:unsub}
\end{equation}
where $A_i^B$ are the nucleon pole contributions of the Born terms
describing the photon scattering off a point-like nucleon with
anomalous magnetic moment, as explicitly given in App.~A of
Ref.~\cite{lvov97}. Furthermore, $\mathrm{Im}_s A_i$ are the
discontinuities across the $s$-channel cut of the Compton process,
starting at pion production threshold, i.e.,  $\nu_{thr} = m_\pi +
(m_\pi^2 + t/2 )/(2 M)$, with $m_\pi$ the pion mass. However, as can
be deduced from the asymptotic behavior of the functions
$A_i(\nu,t)$  for $\nu \rightarrow \infty$ and fixed
$t$~\cite{DGPV}, such unsubtracted DRs do not converge for the
amplitudes $A_1$ and $A_2$. We therefore subtract the fixed-$t$ DRs of
Eq.~(\ref{eq:unsub}) at $\nu=0$, with the result
\begin{equation}
\mathrm{Re} A_i(\nu, t) \;=\; A_i^B(\nu, t) \;+\;
\left[ A_i(0, t) - A_i^B(0, t) \right]
\;+\;{2 \over \pi} \;\nu^2\; {\mathcal P} \int_{\nu_{thr}}^{+ \infty} d\nu' \;
{{\; \mathrm{Im}_s A_i(\nu',t)} \over {\nu' \; (\nu'^2 - \nu^2)}}\;.
\label{eq:sub}
\end{equation}
Because of the two additional powers of $\nu'$ in the denominator, these
subtracted DRs should now converge for all of the invariant amplitudes.
\newline
\indent The 6 subtraction functions $A_i(\nu=0, t)$ appearing in
Eq.~(\ref{eq:sub}) can be determined by once-subtracted DRs in the
variable $t$:
\begin{eqnarray}
A_i(0, t) \;-\; A_i^B(0, t) &=&
\left[ A_i(0, 0) \;-\; A_i^B(0, 0) \right]
\;+\;
\left[ A_i^{t-pole}(0, t) \;-\; A_i^{t-pole}(0, 0) \right] \nonumber\\
&+&\;{t \over \pi} \; \int_{4 \, m_\pi^2}^{+ \infty} dt' \; {{\mathrm{Im}_t
A_i(0,t')} \over {t' \; (t' - t)}} \;+\;{t \over \pi} \; \int_{-
\infty}^{-2m^2_\pi-4Mm_\pi} dt' \; {{\mathrm{Im}_t A_i(0,t')} \over {t' \; (t'
- t)}} \;, \label{eq:subt}
\end{eqnarray}
where $A_i^{t-pole}(0, t)$ represents the contribution of the poles in the $t$
channel, in particular of the $\pi^0$ pole in the case of $A_2$ as explicitly
evaluated in Ref.~\cite{DGPV}, and the subtraction constants $a_i = A_i(0, 0)
\;-\; A_i^B(0, 0)$ are related to the polarizabilities as explained below.
\newline
In order to evaluate the dispersion integrals in Eq.~(\ref{eq:sub}), the
imaginary parts in the $s-$channel are calculated from the unitarity relation,
taking into account the $\pi N$ intermediate states and the resonant
contributions of inelastic channels involving more pions in the intermediate
states. In particular, for the $\gamma N\rightarrow \pi N \rightarrow \gamma N$
contribution we use the multipole amplitudes from the analysis of Hanstein {\it
et al.}~\cite{HDT} for energies $\nu\leq 500 $ MeV, and at the higher energies
up to $\nu=1.5$ GeV we  take the solutions of the SAID analysis~\cite{said}.
The multi-pion intermediate states are approximated by the inelastic decay
channels of the $\pi N$ resonances as detailed in Ref.~\cite{DGPV}. This simple
approximation of the higher inelastic channels is quite sufficient, because
these channels are largely suppressed by the energy denominator
$\nu'(\nu'^2-\nu^2)$ in the subtracted DRs of Eq.~(\ref{eq:sub}).
\newline
The imaginary part in the $t$-channel integral from $4 m_\pi^2
\rightarrow + \infty$ in Eq.~(\ref{eq:subt}) is saturated by the
possible intermediate states for the $t$-channel process, which lead
to cuts along the positive $t$-axis. For values of $t$ below the $K
\bar K$ threshold, the $t$-channel discontinuity is dominated by the
two-pion intermediate states, $\gamma\gamma\rightarrow
\pi\pi\rightarrow N\bar N$. We calculate this contribution by
evaluating a unitarized amplitude for the $\gamma\gamma\rightarrow
\pi\pi$ subprocess, and then combine it with the $\pi \pi\rightarrow
N\bar N$ amplitudes as determined from dispersion theory by
analytical continuation of the $\pi N$ scattering
amplitudes~\cite{Hoehler83}. The second integral in
Eq.~(\ref{eq:subt}) extends from $-\infty$ to $ -2m^2_\pi
-4Mm_\pi\approx - 0.56$ GeV$^2.$ As long as we stay at small
(negative) values of $t$, this integral is strongly suppressed by
the denominator $t'(t'-t)$ in Eq.~(\ref{eq:subt}), and therefore it
can be approximated by the contributions of $\Delta$-resonance and
non-resonant $\pi N$ intermediate states. The latter contributions
are evaluated by first evaluating the imaginary parts of the Compton
amplitude in the physical $s$-channel region by unitarity, and then
extrapolating these results into the unphysical region at $\nu=0$
and negative $t$ by means of analytical continuation.
\newline
\indent The 6 subtraction constants $a_i$ in Eq.~(\ref{eq:subt}) are
related to the electric ($\alpha_{E1}$) and magnetic ($\beta_{M1}$)
scalar polarizabilities in the spin-independent sector,
\begin{eqnarray}
\label{eq:ai-spinind}
\alpha_{E1}=-\frac{1}{4\pi}(a_1+a_3+a_6)\,,\quad\quad\beta_{M1}=\frac{1}{4\pi}(a_1-a_3-a_6)\,,
\end{eqnarray}
and to the 4 spin-dependent or vector polarizabilities,
\begin{eqnarray}\label{eq:ai-spindep}
\gamma_{E1E1}&=&\frac{1}{8\pi M}(a_2-a_4+2 a_5+a_6), \qquad\qquad
\gamma_{M1M1}=-\frac{1}{8\pi M}(a_2+a_4+2a_5-a_6),
\nonumber\\
\gamma_{E1M2}&=&\frac{1}{8\pi M}(a_2-a_4-a_6),\qquad\qquad \qquad
\gamma_{M1E2}=-\frac{1}{8\pi M}(a_2+a_4+a_6)\,.
\end{eqnarray}

\indent Although in principle all 6 subtraction constants $a_1$ to $
a_6$ could be used as fit parameters, we restrict the fit to the
parameters $a_5$ and $a_6$, or equivalently to the two spin
polarizabilities $\gamma_{E1E1}$ and $\gamma_{M1M1}$. For fixed
values of the fit parameters $a_5$ and $a_6$, the subtraction
constants $a_1$ and $a_3$ are then determined by a recent global fit
of the scalar polarizabilities to the low-energy data~\cite{Olmos01},
\begin{eqnarray}\label{alpha-beta-exp}
\alpha_{E1}+\beta_{M1}&=&(13.8\pm0.4) \times 10^{-4} \, \mbox{fm}^3 \, ,
\nonumber\\
\alpha_{E1}-\beta_{M1}&=&(10.5\pm0.9\, (\mathrm{stat.+syst.})\pm0.7\,
(\mathrm{mod.})) \times 10^{-4} \, \mbox{fm}^3\,.
\end{eqnarray}
In the spin-dependent sector, we use the experimental values of the
forward ($\gamma_0$) and backward ($\gamma_\pi$) spin
polarizabilities  describing the Compton spin-flip amplitude at
$\theta=0^{\circ}$ and $180^{\circ}$, respectively. These
observables have been expressed in Eqs.~(\ref{eq:g0_mult}) and
(\ref{eq:gpi_mult}) by linear combinations of the polarizabilities
defined in Eq.~(\ref{eq:ai-spindep}), they are related to the
subtraction constants as follows:
\begin{eqnarray}
\gamma_{\pi}=-\frac{1}{2\pi M}(a_2+a_5) \,,\quad \quad \gamma_{0}=
\frac{1}{2\pi M}a_4. \label{eq:gammapi0}
\end{eqnarray}
In particular, the subtraction constant $a_2$ is fixed by the value
of $\gamma_\pi$ given by Eq.~(\ref{eq:gpi}). This definition of the
backward spin polarizability includes both the dispersive and the
large $\pi^0$-pole contribution. In the analysis of
Ref.~\cite{Schumacher}, the latter takes the value
$\gamma_\pi^{\pi^0-{\rm pole}}= -46.7 \times 10^{-4}$ fm$^4$, which
leads to $\gamma_\pi^{{\rm disp}}= (8.0\pm 1.8 )\times 10^{-4}$
fm$^4$. In fact, the pion pole contribution is not known to that
accuracy, and therefore the error in Eq.~(\ref{eq:gpi}) stems from
both the dispersive and the pole contributions to the backward spin
polarizability. The value of the forward spin polarizability, or
equivalently of $a_4$, is fixed by Eq.~(\ref{eq:go}). We conclude
that the choice of the subtraction constants $a_5$ and $a_6$ as fit
parameters is equivalent to varying the polarizabilities
$\gamma_{E1E1}$ and $\gamma_{M1M1}$. As can be seen from
Eqs.~(\ref{eq:ai-spindep}) and (\ref{eq:go}), another possibility
would be to fit $a_2$ and $a_6$ or $\gamma_{E1M2}$ and
$\gamma_{M1E2}$.

\section{Cross sections and asymmetries - formalism}
\label{formalism }

The general formalism for RCS with one or two polarized particles
has been originally derived in Ref.~\cite{BGM}. In the following we
only review some pertinent formulas necessary to define the
observables for polarized incident photons scattered by polarized
target nucleons.
\newline
\indent
We work in a reference frame with the $z$ axis along $\hat q$ (the direction of
the incoming photon), the $x$ axis in the scattering plane and in the
half-plane of the outgoing photon, and the $y$ axis perpendicular to the
scattering plane along the direction $(\hat q\times \hat q')$. The
photon-polarization  density matrix is defined by the Stokes parameters $\xi_i$
($i=1,2,3$) as follows~\cite{landau}:
\begin{eqnarray}\label{Stocks-def}
\langle \, \varepsilon_\alpha \varepsilon_\beta^* \, \rangle =
\frac12(1+\vec\sigma \cdot \vec\xi)_{\alpha\beta} =  \frac12 \pmatrix{ 1+\xi_3
& \xi_1 -i\xi_2 \cr \xi_1 +i\xi_2 & 1-\xi_3 }_{\alpha\beta} ~ ,
\end{eqnarray}
where $\varepsilon^\mu$ is the photon polarization vector chosen in
the radiation gauge, $ \vec \varepsilon\cdot\vec q=0$, and
$\alpha,\beta =1,2$ denote either of the two orthogonal directions
$x$ and $y.$ The total degree of photon polarization is given by
$\xi=\sqrt{\xi_1^2+\xi^2_2+\xi^2_3}$, and
$\xi_{\ell}=\sqrt{\xi^2_1+\xi_3^2}$ and $\xi_2$ describe the degrees of
linear and circular polarization, respectively. Furthermore,
$\xi_2=+1$ and $\xi_2=-1$ correspond to right- and left-handed
states, respectively, and in the case of linear polarization the
azimuthal angle $\phi$ between the electric field and the scattering
plane is defined by $\cos 2\phi=\xi_3/\xi_{\ell}$ and $\sin
2\phi=\xi_1/\xi_{\ell}$. With these definitions, the Stokes parameters
take the same value in the c.m. and lab frames.
\newline
\indent The nucleon polarization density matrix is described by a
polarization four-vector $S^\mu$ that is orthogonal to the nucleon
four-momentum \cite{landau},
\begin{eqnarray}
\langle \, u(p)\bar u(p) \, \rangle=\frac12\Big( \gamma \cdot p + M\Big)
\Big(1+\gamma_5\gamma \cdot S \Big),
\end{eqnarray}
where $u(p)$ is a nucleon Dirac spinor normalized as $\bar u(p)\,u(p)=2M$.
\newline
\indent
The differential cross section is related to the $T$-matrix by
\begin{eqnarray}\label{cs-generic}
\frac{d\sigma}{d\Omega}= \Phi^2\left|T\right|^2, \qquad \mbox{with} \qquad \Phi
= \cases{  \displaystyle \frac {1}{8 \pi M}\frac{E'_\gamma}{E_\gamma}  & (lab frame) \cr
\displaystyle \frac {1}{8 \pi \sqrt{s}}  & (c.m. frame). \rule{0ex}{4ex}}
\end{eqnarray}
The $T$-matrix for polarized photons and polarized targets can be decomposed in
8 independent functions $W_{ij}$,
\begin{eqnarray}\label{TWgN}
\left|T(\roarrow\gamma \roarrow N \to \gamma  N)\right|^2 &=& W_{00} +
W_{03}\xi_3  + N \cdot S\,(W_{30} + W_{33}^+\xi_3) \nonumber\\&+& K\cdot S\,(
W_{11}^+\xi_1 + W_{12}^+\xi_2) +  Q \cdot S\,( W_{21}^+\xi_1 + W_{22}^+\xi_2),
\end{eqnarray}
with the orthogonal four-vectors $K$, $N$, and $Q$ defined as
\begin{eqnarray} K_{\mu}=\frac{1}{2}(q'+q)_{\mu}, \qquad
N_{\mu}=\epsilon_{\mu\alpha\beta\gamma}P'^\alpha Q^\beta K^\gamma, \qquad
Q_{\mu}=\frac12(p-p')_{\mu}=\frac12(q'-q)_{\mu},
\end{eqnarray}
where $\epsilon_{0123}=1$. The photon asymmetry $\Sigma$ follows if the nucleon
polarization vector is set to zero, and the unpolarized case $\left|T\right|^2 = W_{00}$ is
recovered for vanishing photon and nucleon polarization vectors.
\newline \indent
In terms of the invariant amplitudes $A_i$, the functions $W_{ij}$
read~\cite{frolov60, BGM}:
\begin{eqnarray}\label{Wij}
   W_{00}
   &=&  \frac14 (4 M^2 -t) \Big( t^2 |A_1|^2 +
   \eta^2 |A_3|^2\Big) - \frac14\Big( t^3 |A_2|^2 -
   \eta^3 |A_4|^2\Big)  \nonumber\\ && {}
   -\nu^2 t\,(t + 8 \nu^2)|A_5|^2
   +\frac12 \eta\, (t^2 + 2 M^2 \eta)|A_6|^2 \nonumber\\ && {}
   +{\rm Re}\Big\{2 \nu^2 t^2(A_1 + A_2)A_5^* +
   \frac12\eta^2 (4 M^2 A_3 + t A_4)A_6^*\Big\}, \\[2ex]
   W_{03}
   &=&  \frac{\eta t}{2} {\rm Re}\Big\{ \Big( (4 M^2 - t) A_1
    + 4 \nu^2 A_5 \Big) A_3^* \,+\, 4 M^2 A_1 A_6^* \Big\}, \\[2ex]
   W_{30}
   &=& - 8 \nu{\rm Im}(t A_1 A_5^* + \eta A_3 A_6^*), \\[2ex]
   W_{33}^{\,\pm}
   &=& {\rm Im} \Big\{ {-} 8\nu\, \Big[ \Big(
     t A_1 - (t+4\nu^2) A_5 \Big) A_6^*  + \eta A_3 A_5^* \Big]
  \nonumber\\ && {} \qquad
     \pm \frac2m (t A_2 - 4\nu^2 A_5)(\eta A_4^* + t A_6^*)\Big\}, \\[2ex]
   W_{11}^{\,\pm}
   &=& {\rm Im}\Big\{ \frac{t}{2M}
   \Big( (4M^2-t)A_1 + 4\nu^2 A_5\Big) (\eta A_4^* + t A_6^*)
  \nonumber\\ && \qquad {}
   \pm 2\nu t \, (t A_2 - 4\nu^2 A_5) A_6^* \Big\}, \\[2ex]
   W_{12}^{\,\pm}
         &=&  {\rm Re} \Big\{ {-} \frac{\eta}{2M}
     \Big( (4M^2-t) A_3 + 4M^2 A_6 \Big) (\eta A_4^* + t A_6^*)
  \nonumber\\ && \qquad {}
   \pm 2\nu t\, (t A_2 - 4\nu^2 A_5) A_5^*  \Big\}, \\[2ex]
   W_{21}^{\,\pm}
         &=& 2 {\rm Im}\Big\{   {-}M\,(t A_2 - 4\nu^2 A_5)
     \Big( \eta A_3^* + (t+4\nu^2) A_6^* \Big)  \nonumber\\ && \hspace{8em} {}
      \pm \nu \Big( t A_1 - (t+4\nu^2) A_5 \Big)
     (\eta A_4^* + t A_6^*) \Big\}, \\[2ex]
   W_{22}^{\,\pm}
           &=&  2 {\rm Re}\Big\{ {-}M t\,(t A_2 - 4 \nu^2 A_5) A_1^*
   \mp \nu \eta A_3\,(\eta A_4^* + t A_6^*) \Big\},
\end{eqnarray}
where we have introduced the invariant variable $\eta= 4\nu^2+t-t^2/(4M^2).$
Below the pion photoproduction threshold the functions $A_i$ are real, and
therefore only the 6 structures $W_{00}, W_{03}, W_{12}^\pm$, and $W_{22}^\pm$
contribute below threshold.
\newline \indent
In the following, we focus on the asymmetries that can be obtained by varying
the photon and target polarizations in Eq.~(\ref{TWgN}):
\begin{itemize}
\item{}
circular photon polarization ($\xi_2=\pm 1$) and
target spin aligned in the $\pm z$ direction,
\begin{eqnarray}
\Sigma_{2z}
= \frac{\sigma^R_{+z}-\sigma^L_{+z}}{\sigma^R_{+z}+\sigma^L_{+z}}
= \frac{\sigma^R_{+z}-\sigma^R_{-z}}{\sigma^R_{+z}+\sigma^R_{-z}}
=\frac{C^{K}_{z}W_{12}^+ + C^{Q}_{z}W_{22}^+}{W_{00}} \, ,
\end{eqnarray}
where the coefficients $C^{K,Q}_{z}$ can be written in terms of lab or
invariant variables as
\begin{eqnarray}\label{eq:cqkz}
C^{K}_z&=&-\frac12\,(E_\gamma + E'_\gamma\cos\theta_{{\rm lab}})
= -\frac{s-M^2}{2M} - \frac{t\,(s+M^2)}{4M\,(s-M^2)}, \nonumber \\
C^{Q}_z&=& \frac12\,(E_\gamma - E'_\gamma\cos\theta_{{\rm lab}})
= - \frac{t\,(s+M^2)}{4M\,(s-M^2)}\, ,
\end{eqnarray}
\item{}
circular photon polarization ($\xi_2=\pm 1$) and
target spin aligned in the $\pm x$ directions,
\begin{eqnarray}\label{Sigxz}
\Sigma_{2x}
= \frac{\sigma^R_{+x}-\sigma^L_{+x}}{\sigma^R_{+x}+\sigma^L_{+x}}
= \frac{\sigma^R_{+x}-\sigma^R_{-x}}{\sigma^R_{+x}+\sigma^R_{-x}}
= \frac{C^{K}_{x}W_{12}^+ + C^{Q}_{x}W_{22}^+}{W_{00}}   \, ,
\end{eqnarray}
with \begin{eqnarray}\label{eq:cqkx} C^{K}_x=
C^{Q}_x=-\frac{1}{2}\,E'_\gamma\sin\theta_{{\rm lab}}
= - \frac{M\sqrt{-\eta t}} {2(s-M^2)} \,,
\end{eqnarray}
\item{}
linearly polarized photons, either parallel or
perpendicular to the scattering plane $(\xi_3 = \pm 1)$, and target nucleon
polarized perpendicularly to the scattering plane,
\begin{eqnarray}\label{eq:cny}
\Sigma_{3y}
&=& \frac{(\sigma^\parallel - \sigma^\perp)_{y}
-(\sigma^\parallel - \sigma^\perp)_{-y}}
{(\sigma^\parallel + \sigma^\perp)_{y}
+(\sigma^\parallel + \sigma^\perp)_{-y}}
=C^{N}_{y}\frac{W_{33}^+}{W_{00}}\,,
\end{eqnarray}
with
\begin{eqnarray}
C^N_y=\frac{M}{2}E_\gamma E'_\gamma\sin\theta_{{\rm
lab}}=\frac{M}{4}\sqrt{-\eta t}\, ,
\end{eqnarray}
\item{}
linearly polarized photons, either parallel or
perpendicular to the scattering plane $(\xi_3 = \pm 1)$, and
unpolarized target nucleons,
\begin{eqnarray}\label{eq:Sigma}
\Sigma_3
&=& \frac{\sigma^\parallel - \sigma^\perp}
{\sigma^\parallel + \sigma^\perp}
=\frac{W_{03}}{W_{00}}\,,
\end{eqnarray}
\item{}
photons linearly polarized at $\varphi = \pm\pi/4$ with respect to the
scattering plane ($\xi_1 = \pm 1)$ and the
nucleon target polarized in the scattering plane in the $\pm z$ direction,
\begin{eqnarray}\label{Sig1xz}
\Sigma_{1z}
= \frac{\sigma^{\pi/4}_{+z}-\sigma^{-\pi/4}_{+z}}
{\sigma^{\pi/4}_z+\sigma^{-\pi/4}_z}
= \frac{\sigma^{\pi/4}_{+z}-\sigma^{\pi/4}_{-z}}
{\sigma^{\pi/4}_{+z}+\sigma^{\pi/4}_{-z}}
= \frac{C^{K}_{z}W_{11}^+ + C^{Q}_{z}W_{21}^+}{W_{00}}\, ,
\end{eqnarray}
with coefficients $C^{K,Q}_z$ as defined in Eqs.~(\ref{eq:cqkz}),
\item{}
photons linearly polarized at $\varphi =
\pm\pi/4$ with respect to the scattering plane ($\xi_1 = \pm 1)$ and nucleon
targets polarized in the $\pm x$ direction,
\begin{eqnarray}
\Sigma_{1x}
= \frac{\sigma^{\pi/4}_{+x}-\sigma^{-\pi/4}_{+x}}
{\sigma^{\pi/4}_{+x}+\sigma^{-\pi/4}_{+x}}
= \frac{\sigma^{\pi/4}_{+x}-\sigma^{\pi/4}_{-x}}
{\sigma^{\pi/4}_{+x}+\sigma^{\pi/4}_{-x}}
= \frac{C^{K}_{x}W_{11}^+ + C^{Q}_{x}W_{21}^+}{W_{00}}\; ,
\end{eqnarray}
with $C^{K,Q}_x$ given in Eq.~(\ref{eq:cqkx}).
\end{itemize}

\section{Results and discussion}
\label{results}

In this section we present our predictions for the polarization
observables. Our results are based on subtracted DRs evaluated with
the pion photoproduction multipoles of Refs.~\cite{HDT,said} as
input. Because of the subtraction, two-pion and heavier intermediate
states will generally yield only small corrections. Four of the
subtraction constants are determined by the experimental values for
the polarizabilities $\alpha_{E1}, \beta_{M1}, \gamma_{0},$ and
$\gamma_{\pi}$. The remaining two constants are obtained by fixing
the spin polarizabilities $\gamma_{E1E1}$ and $\gamma_{M1M1}$. In
the following figures, we start from the predictions of fixed-$t$
DRs for the dispersive part of the spin
polarizabilities~\cite{HDPV,DPV_report},
\begin{eqnarray}
\label{eq:gammaEEMM}
\gamma_{E1E1}=-4.3 \times 10^{-4}~{\rm {fm^4}}
\,,\quad\quad \gamma_{M1M1}=2.9 \times 10^{-4}~{\rm {fm^4}} \,,
\end{eqnarray}
which are then varied by $\pm$~2 units. We note that here and in the
following discussion including the figures, the values of the spin
polarizabilities refer only to their dispersive parts, that is, the
pion pole contribution,
$\gamma_{E1E1}^{\pi0-{\rm pole}}=-\gamma_{M1M1}^{\pi0-{\rm pole}}=
11.68\times 10^{-4}$ fm$^4$, has been subtracted.
Because of the
relatively large error bar for $\gamma_{\pi}$, we also vary this
polarizability within its error band while keeping $\gamma_{E1E1}$
and $\gamma_{M1M1}$ fixed at their central values.
\newline
\indent Figure~\ref{fig1} shows the asymmetry $\Sigma_{2z}$, with
circular photon polarization and target aligned parallel to the
incoming photon. We observe asymmetries up to 90~$\%$ and a strong
dependence on both angle and energy, with distinct structures near
the threshold for pion photoproduction. Although the asymmetry near
$\theta_{{\rm lab}}=90^{{\rm o }}$ is smaller than for the forward
and backward directions, it is rather sensitive to a variation of
$\gamma_{M1M1}$ both near threshold and in the $\Delta~(1232)$
resonance region. Within the range of the variation, $\Sigma_{2z}$
changes by 15-20~$\%$, which provides a promising signal to
determine the spin polarizability $\gamma_{M1M1}$. The following
Fig.~\ref{fig2} displays the same information for the asymmetry
$\Sigma_{2x}$, with circular photon polarization and target aligned
sideways to the incident photon. Contrary to the previous figure,
the maximum asymmetry is now reached at scattering angles
$\theta_{{\rm lab}}\approx 90^{{\rm o }}$, and the maximum
sensitivity occurs by changing $\gamma_{E1E1}$. Within the range of
variation, this observable changes by 15~$\%$ near threshold and
40~$\%$ in the $\Delta$ region. The right panels of Figs.~\ref{fig1}
and~\ref{fig2} show that these observables are hardly changed by a
variation of $\gamma_{\pi}$. As a result, the observables
$\Sigma_{2z}$ and $\Sigma_{2x}$ sample conclusive and complementary
information on the nucleon's spin structure. Furthermore, through
the input of the DRs, they are related to the physics of the
observables E and F of pion photoproduction.
\newline
\indent The full angular distribution for $\Sigma_{2z}$ and
$\Sigma_{2x}$ are shown in Fig.~\ref{fig3} for a photon beam of
240~MeV. It is seen that the discussed sensitivity extends over a
large angular range between $30^{{\rm o }}$ and $150^{{\rm o }}$. In
view of the error bars of the backward scalar and vector
polarizabilities, $\alpha_{E1}-\beta_{M1}$ and $\gamma_{\pi}$,
respectively, the cleanest information should however come from
angles around $\theta_{{\rm lab}}\approx 90^{{\rm o }}$.
\newline
\indent Figure~\ref{fig4} displays the results for the beam
asymmetry $\Sigma_{3}$, for linear photon polarization, which is
obtained by averaging the double-polarization observable
$\Sigma_{3y}$ over the target polarizations.
This observable rises from large negative values at low energies to
positive values in the resonance region. Moreover, there appear interesting
cusp effects, escpecially at forward angles. The figure shows strong
sensitivity to changes of $\gamma_{M1M1}$.
Within the range of our variation, the asymmetry $\Sigma_3$ changes
by about 0.15 near threshold and nearly 0.30 in the resonance region.
The LEGS data~\cite{Blanpied01} taken at $65^{{\rm o }}$ and $90^{{\rm o }}$
are compatible with our central value for $\gamma_{M1M1}$ (full
lines), they scatter about this value at $135^{{\rm o }}$. As in the
previous examples, the variation of $\gamma_{\pi}$ leads to much
smaller effects. The following Fig.~\ref{fig5} shows the full
angular distribution for $\Sigma_{3}$ and the double-polarization
observable $\Sigma_{3y}$ in the resonance region. Large asymmetries
are predicted for both observables over a wide angular range.
Moreover, the sensitivity to the variation is large, and while
$\Sigma_{3}$ is particularly sensitive to $\gamma_{M1M1}$, the
related double-polarization asymmetry $\Sigma_{3y}$ changes strongly with
$\gamma_{E1E1}$. We conclude that these two observables are
particularly useful to disentangle the unknown spin
polarizabilities.
\newline
\indent
The physics behind the asymmetry $\Sigma_3$ is addressed in
Fig.~\ref{fig5b} as function of the photon energy. The Born terms,
subtraction constants, and $t$-channel contributions provide a negative
background whose height at the resonance depends strongly
on the value of $\gamma_{M1M1}$. The S-wave multipoles in the
dispersion integral provide the interesting cusp effect near threshold
through the opening of the imaginary part, whereas the P-waves are
responsible for the further increase to large positive values in the
$\Delta(1232)$ resonance region. In conclusion, the large effect is
mainly given by the $M_{1+}^{(3/2)}$ multipole of pion photoproduction.
\newline
\indent
Let us now address the question of the model dependence of the
dispersive approach. The general answer is that the errors in
fixed-$t$ dispersion relations increase with
both the beam energy and the scattering angle.
If the energy gets larger, the unknown contributions of the higher resonances
and backgrounds become more and more important for the evaluation of the
$s$-channel dispersion integrals. Large angles and energies, on the other
hand, require the knowledge of the subtraction functions at large (negative)
$t$-values. Of course, the error also depends on the specific variable under
discussion. We have studied these error sources for all the shown variables
and found them to yield only small corrections in all the interesting cases,
that is for large asymmetries and large sensitivity to the variation.
Specifically for $\Sigma_3$ and $\Sigma_{3y}$ in Fig.~\ref{fig5},
the neglect of the two-pion production changes these asymmetries by less
than 0.01 at all energies and angles. It is therefore reasonable to assume
that also the neglected higher resonance and background contributions above
$\nu = 1.5$~GeV are irrelevant for our discussion. Concerning the error from
the $t$-channel integral, we find that the contribution from the negative
$t$-channel cut is negligible for $E_\gamma \leq 230$~MeV. At the
energy $E_\gamma = 275$~MeV shown in Fig.~\ref{fig5}, the asymmetry
$\Sigma_3$ increases by less than 0.02 if we completely neglect that
contribution, while $\Sigma_{3y}$ is even less affected by the
$t$-channel integral. In conclusion, even a very conservative estimate yields
errors of a few percent at most, which are almost negligible in view of the
large range of variation predicted for $\Sigma_3$ and $\Sigma_{3y}$.
\newline
\indent
Let us finally discuss the beam-target asymmetries with
photons linearly polarized at azimuthal angles $\phi= \pm 45^{{\rm o
}}$ with respect to the scattering plane. These asymmetries are
related to the observables G and H of pion photoproduction. They
vanish below the one-pion threshold, because they thrive on the
imaginary part of the scattering amplitudes. For the same reason,
Fig.~\ref{fig6} does not show a peak near the $\Delta$ resonance,
because all $\Delta$ multipoles carry the same phase. Instead, these
observables are strongly enhanced by interference of non-resonant
S-wave and resonant P-wave pion production. Since the pion
photoproduction multipoles are quite well known in the $\Delta$
region, we expect a rather model-independent information. The
asymmetry $\Sigma_{1z}$ turns out to be small at all angles and
energies. Figure~\ref{fig6} shows the most promising kinematics at
angles near $\theta_{{\rm lab}}=90^{{\rm o }}$, where the asymmetry
changes up to about 8~$\%$ over the range of variation. The
asymmetry $\Sigma_{1x}$, on the other hand, has its maximum
sensitivity at the backward angles. However, the variations of
$\gamma_{E1E1}$ and $\gamma_{M1M1}$ yield similar effects, such that
the observed change is qualitatively proportional to the difference
of the two polarizabilities. In conclusion, these asymmetries
$\Sigma_{1x}$ and $\Sigma_{1z}$ would
have to be measured with an accuracy of a few percent in order to
get useful information on the spin polarizabilities.

\section{Summary and conclusions}
\label{conclusions} Compton scattering probes the response of the
nucleon to an external electromagnetic field. At low energies, this
response is described by 2 scalar and 4 vector polarizabilities,
which contain global information on the excitation spectrum. Whereas
the scalar polarizabilities are now known with relatively small
error bars, our knowledge on the spin-dependent sector is as yet
incomplete. Only two combinations of the vector polarizabilities
have been measured. The forward spin polarizability has been
determined by forward dispersion relations as an energy-weighted
integral over the helicity-dependent total cross sections.
Furthermore, the backward spin polarizability has recently been
measured both below and in the $\Delta(1232)$ resonance region,
however with a larger model dependence. It has been known for some
time that it will take a full-fledged experimental program including
several polarization observables in order to achieve a complete
separation of the spin polarizabilities. Considerable theoretical
work has been dedicated to finding the appropriate observables and
kinematics for this purpose.
\newline
\indent In our present work we study polarized Compton scattering
within the framework of dispersion relations at $t$=const.
Subtraction of these relations speeds up the convergence of the
dispersion integrals, and the subtraction constants of the 6
relativistic amplitudes turn out to be linear combinations of the 6
polarizabilities. In most cases also the unsubtracted integrals
converge and hence a prediction for the polarizabilities is given.
However, we choose to subtract all 6 integrals in order to reduce
the model dependence on the high-energy spectrum as much as
possible. Four of the subtraction constants are then fixed by the
experimental values for the electric ($\alpha_{E1}$) and magnetic
($\beta_{M1}$) scalar polarizabilities as well as the forward
($\gamma_{0}$) and backward ($\gamma_{\pi}$) spin polarizabilities.
The remaining two independent polarizabilities are taken to be
$\gamma_{E1E1}$ and $\gamma_{M1M1}$. They are varied about a central
value as predicted by dispersion relations, within a range which
comprises several predictions from effective field theories and
dispersive approaches.
\newline
\indent Whereas the scalar polarizabilities contribute to the
differential cross section already at second order in the photon
beam energy, the vector polarizabilities appear only at third or
fourth order, depending on the observable. It is therefore not
surprising that it takes at least 100~MeV to get visible effects of
the vector polarizabilities. Even at energies near the pion
production threshold, the asymmetries change at most by $10~\%$ over
the range of our variation. Of course, the experiments could
eventually yield polarizabilities far outside the range of the
existing predictions. However we choose to stay on the conservative
side regarding our predictions. Much larger effects are obtained in
the first resonance region, in which both the cross sections and the
asymmetries increase considerably. Specifically, we find a good
chance to solve the open questions by the following two independent
setups:
\begin{itemize}
\item{}
Circularly polarized photons and targets aligned in the beam
direction ($\Sigma_{2z}$) or transverse to the beam in the
scattering plane ($\Sigma_{2x}$). Both asymmetries should yield a
reasonably large sensitivity to the spin polarizabilities in the
threshold region and increasingly large effects closer to the
resonance region. Moreover, the response to the variation shows
interesting cusp effects at threshold and a distinctly different
sensitivity of these observables with regard to $\gamma_{E1E1}$ and
$\gamma_{M1M1}$.
\item{}
Linearly polarized photons, parallel or perpendicular to the
scattering plane, and with unpolarized targets ($\Sigma_{3}$) or
with targets polarized perpendicular to the scattering plane
($\Sigma_{3y}$). The sensitivity of these two observables is
relatively small near threshold but very large in the resonance
region. Moreover, the variations of the spin polarizabilities affect
the observables in a completely different way,
with $\Sigma_3$ being mainly sensitive to $\gamma_{M1M1}$ whereas
$\Sigma_{3y}$ being mainly sensitive to $\gamma_{E1E1}$.

\end{itemize}

We further looked at the observables $\Sigma_{1z}$ and
$\Sigma_{1x}$, obtained with photons linearly polarized at an angle
of $45^{\rm o }$ against the scattering plane and target alignment
perpendicular to this plane. However, these asymmetries turn out to
be small in general, and even though the relative effect of the
variations may be large, such experiments will require asymmetry
measurements at the few percent level.
\newline
\indent With the advent of new experimental tools as polarized
targets and photon beams it will be possible to study the full spin
structure of Compton scattering. The new HIGS project~\cite{HIGS} of
a high-intensity beam with circularly polarized photons in an energy
range up to 140-160~MeV is ideally suited to perform the discussed
experiments in the threshold region. Complementary investigations
should be performed at energies closer to the first resonance
region, in which we expect much larger cross sections and
sensitivities to the spin polarizabilities. Such experiments in the first
resonance region are planned
using the Crystal Ball detector at MAMI~\cite{CBall}.
We strongly believe that
only a combination of such experimental projects will provide the
"sharp knife" to extract the spin polarizabilities in an unambiguous
way. The spin polarizabilities of the nucleon are fundamental
structure constants of the nucleon, just as shape and size of this
strongly interacting many-body system, which strongly justifies the
experimental effort. Such activities are both important and timely,
they will provide stringent precision tests for the existing
predictions of effective field theories and the new results expected
from the lattice gauge community for the polarizability of the
nucleon.

\section*{Acknowledgments}

The authors are grateful to J. Ahrens, R. Miskimen, and
V. Pascalutsa for useful discussions.
This research was supported by the Deutsche Forschungsgemeinschaft (SFB443),
by the DOE grant DE-FG02-04ER41302 and DOE contract DE-AC05-06OR23177 under
which Jefferson Science Associates operates the Jefferson Laboratory,
and the EU Integrated Infrastructure Initiative Hadron Physics
Project under contract number RII3-CT-2004-506078.

redpajama_set_name: RedPajamaArXiv


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Assistant/Associate Professor Acute Care Surgeon
Advertising Copy The Department of Surgery at the University of Vermont College of Medicine and The University of Vermont Medical Center are seeking to fill one surgeon position for the Acute Care Surgery Division. Successful candidates will be...
Details Posted: 13-Jan-22 Location: Burlington, Vermont Type: Full Time Categories: Academic / Research Internal Number: F2092PO HEMATOPATHOLOGY FACULTY UNIVERSITY OF VERMONT The Department of Pathology and Laboratory Medicine of the Robert Larner,...
Part Time Faculty - Agriculture - AGR 1030, Randolph Center
Job Details Description The Agriculture Department at Vermont Technical College invites applications for Spring 2022 (January 18, 2022 to May 09, 2022). The course to be taught is AGR 1030 Animal Reproduction and Genetics This course focuses on the...
Employer:Lyndon State College
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Part Time Faculty - CIS 3030-TH1
Job Details Description The Computer & Information Systems Department at Vermont Technical College invites applications for Spring 2022 (January 18, 2022 - May 13, 2022 The course to be taught is CIS 3030 Programming Languages This course covers...
Location:Williston, VT
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Part Time Faculty -MEC 3121
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Company Description: Founded in 1819, Norwich University is the oldest private military college in the country and thebirthplace of the Reserve Officers' Training Corps (ROTC). Today we are a diversified academic institution that offers...
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Well Mothers Day is over. The card isle at your local store will have all the Mothers Day cards pulled and replaced with Fathers Day cards. Flowers and candy will be less conspicuous and sadly just as all the Mothers Day paraphernalia will disappear so will a lot of the attention and affection we all endeavored to do our best to show. It is amazing how we as human beings move on to the next thing and all too often back to a world that revolves primarily around ourselves and our own wants, needs and desires.
Today my daughters, my wife Annette and my sister Julie drove up to Seattle today and had lunch at the infamous Red Mill burgers. We had a great time driving up and talking, laughing and telling stories but the thing that struck me the most (besides the burgers) was the incredible variety and diversity of people, clothing, styles and interests that were all gathered around this little burger joint. Everybody there was ordering some variety of burgers with some combination of fries, onion ringers, milk shakes and soda or all of the above. We seem to go out of the way to try and highlight how different we are but at that burger joint we were all hungry people looking to enjoy some great greasy food with people we know and love. What is so unique and different about that? I don't know, I think we work to hard at seeking out our differences and exceptions as opposed to recognizing our similarities and common interests.
So all in all it was a great day together, however I am grateful the Red Mill is all the way up in Seattle or I would have a serious waistline issue and if you feel I already do please keep it to yourself.


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Work on strategic roads and flagship road schemes underway
Date published: 12 June 2020
Road improvement schemes
Infrastructure Minister Nichola Mallon has announced her intention to progress a number of strategic infrastructure schemes and flagship projects as part of her plan to aid economic recovery and community transformation
Within the Minister's budget statement to the Assembly on Wednesday, she included the funding for the continued development of strategic roads schemes and flagship projects. The Minister has now asked officials to push forward on the development of these important schemes such as the A1 Junctions Phase 2, A24 Ballynahinch By Pass, A4 Enniskillen Southern By Pass, Newry Southern Relief Road, A29 Cookstown By Pass, A2 Buncrana Road and A32 ( Omagh to Enniskillen) at Cornamuck. The Minister has now set the wheels in motion and has asked officials to work at pace to progress these schemes. The Minister intends to provide further announcements later to keep the public updated on progress.
The Minister has also reaffirmed her commitment to the A5 and A6 schemes. While Covid-19 has impacted construction work on the A6, this is continuing and it is hoped that the Toome to Castledawson section could be completed in Spring 2021.
In relation to the A5, following a public inquiry and the successful completion of the statutory processes, construction could commence in mid 2021.
Minister Mallon said:
"These strategic and flagship road improvement schemes span Northern Ireland and will deliver for many communities and help to address regional imbalance.
"Many communities across the North have waited a long time for progress to be made and while Covid-19 has left considerable challenges, I have been determined to get the wheels in motion and get these schemes back on track.
"Each scheme is at a different stage and I have asked my officials to bring forward plans to press ahead in the short and longer term.
"I have also asked officials to bring forward proposals to progress Narrow Water Bridge, a commitment in New Decade New Approach, which would fully explore the environmental and economic benefits this project would deliver for the people of South Down.
"While these are uncertain and difficult times, with a focus on green recovery, I also recognise that our road infrastructure needs updated to relieve traffic in towns, explore options for active travel and reshape our places to deliver towns and cities that work for our communities and aid economic growth.
"I hope to be in a position in the coming weeks to provide a further update on the progress being made."
All media queries should be directed to the Department for Infrastructure Press Office at: press.office@infrastructure-ni.gov.uk.
The Executive Information Service operates an out of hours service for media enquiries between 1800hrs and 0800hrs Monday to Friday and at weekends and public holidays. The duty press officer can be contacted on 028 9037 8110.
Follow the Department on Twitter @deptinfra and on Facebook @DepartmentforInfrastructure
Changes to public transport fares 08 February 2023
Resurfacing scheme for A28 Armagh Road, Loughgilly 07 February 2023
Resurfacing scheme for B30, Newry Road, Lislea 03 February 2023
Road improvement scheme for A2 Clooney Road, Caw to Gransha Roundabouts 03 February 2023


Learn more about the O2 Digital Products and how these can help you to serve the businesses of the future.
Nowadays businesses increasingly work flexible time from remote locations. Millennials expect to access their work on the go from any mobile device. While this increases productivity, it can also create holes in an organisation's cyber security and endanger confidential data.
To future proof your customers' businesses and guarantee a steady revenue stream yourself, we recommend O2's digital products. These applications enable collaboration, help companies to comply to law and regulations, and improve employee's work satisfaction.
To find out more about O2's products you can watch the webinar recording, or call us on 0330 102 5610.


On arrival, you will receive a glass of Bimbadgen Sparkling wine as well as a small gift for Mum.
2-course Menu at $70 per person.
3-course Menu at $85 per person.
Price includes $5pp Sunday surcharge.
Cancellation Policy – Cancellation of your reservation within 24 hours will result in a $50pp charge to the credit card on file.


Our simple cosmetic bag in its handsome diamond stripe handwoven fabric is the ideal size to hold your essential cosmetics, toss into your handbag, and everything is accessible while you are on the go. The contrasting lining is waterproof providing the ultimate protection for your other things! The exterior is 100% cotton. Lined with ripstop nylon and closes with a long pull zipper. Machine wash and dry.
8" long x 4 ½" high and 3" wide at the base.


Mindful parenting is definitely popular right now. But sometimes it's difficult to get past the buzz when it comes to child-rearing approaches and figure out what you're actually supposed to do in order to become a calmer and less distracted mom.
Thankfully, we had the opportunity to talk with Dr. Kirsten Kuzirian, a licensed clinical psychologist about what it means to parent mindfully. Not only does she guide families with these techniques in her private practice, but she also hosts an amazing Podcast, Wide Awake Parenting.
Ready for some strategies to be more present, patient, and positive with your children in any situation? Let's take a look.


The Girls' Brigade is a world wide Christian organisation and each week the company endeavours to fulfil our aim "to help girls to become followers of the Lord Jesus Christ, and through self-control, reverence and a sense of responsibility, to find true enrichment of life" and we try and show the girls in a practical way how to live out our moto which is "seek, serve and follow Christ".
Explorer section meets on Monday nights from 5:45pm to 7pm in the Sam Boyd Hall.
Junior, Senior and Brigader sections meet on Monday nights from 7pm to 9pm in the Samuel Donaldson Hall.


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AMLA Announces Madison County Recipients of 2022 PEAK Awards
By Alabama Mountain Lakes Tourist Association on Sep. 22, 2022
CONTACT: Alabama Mountain Lakes Tourist Association
PHONE: 256.350.3500 / 800.648.5381
EMAIL: info@northalabama.org
Decatur, Ala. (September 22, 2022) – The Alabama Mountain Lakes Tourist Association (AMLA) announced the recipients of the 2022 PEAK Awards today at the AMLA Annual Meeting held at Mars Music Hall in downtown Huntsville, Ala. A Madison County attraction, two tourism professionals and a brand ambassador were recognized for their hard work and contribution to North Alabama's $3.7 billion tourism industry.
Representing the highest achievement in North Alabama tourism, the PEAK Awards honor individuals, events and organizations that have had a major impact on tourism in North Alabama.
AMLA has honored the Orion Amphitheater as the 2022 Attraction of the Year. The Attraction of the Year award is presented to the AMLA member attraction that has made a great contribution to tourism in the past year. The Greek-style Orion Amphitheater is the beginning of a new era for entertainment in North Alabama. The 8,000 capacity venue offers a state-of-the-art sound system, upscale bars on every level and a bowl-like structure that maximizes the view and sound quality for every seat in the house. More than 120,000 tickets have been sold in the past five months. The venue's mission goes beyond music as it will provide cultural experiences throughout the year including farmer's markets, art displays and educational events. Ben Lovett, a member of the Grammy-winning Mumford and Sons band, spearheaded the project and his team worked closely with Huntsville city officials to develop the world-class facility.
AMLA has honored Pam Williams, tourism sales manager of the Huntsville/Madison County Convention and Visitors Bureau, with the 2022 Tourism Professional of the Year award. The Tourism Professional of the Year award is presented to an individual or employee of an AMLA member organization who has made a great contribution to tourism in the past year. Williams has been employed with the Huntsville/Madison County Convention and Visitors Bureau for over 15 years. As tourism sales manager, she is responsible for promoting Huntsville to the group tour market and coordinating special projects. She oversees the Bureau's Educational Escapes, a program that offers teachers and students curriculum-based experiences that are interactive, fun and educational for student groups. She has directed the Huntsville Restaurant Week campaign since its inception 10 years ago and was instrumental in the installation of the Lucky Duck Scavenger Hunt many years ago in downtown Huntsville and the Trains on Main Scavenger Hunt in downtown Madison. This required a great deal of organization and interaction for Williams with the attractions and city departments to ensure the installation went smoothly and the trails were successful. Both programs continue to bring visitors to the Visitor Center.
AMLA has honored Jabria Finch, assistant general manager of DoubleTree Suites Huntsville South, with the 2022 Northern Star Award. History tells us that tourism really began by the navigation of the stars. The Northern Star Award is presented to an individual, such as a ranger, guide, and a front line employee, who has gone "above and beyond" expectations. Starting as a front desk agent while a college student at Alabama A&M University, Finch quickly climbed the ranks to front office supervisor and now serves as the assistant general manager, all in three years. During the Covid-19 pandemic, she became a jack of all trades and helped in all areas of the hotel. Despite ever changing brand protocols due to the pandemic, she focused on happy guests and happy co-workers. She volunteered for two training programs. During Flawless Delivery, a two-day customer service training program, she further refined her guest service skills and sought to improve her conflict management and resolution skills to negotiate only positive outcomes. She participated in the Tourism All-A-Bama training program aimed at assisting hoteliers in becoming more autism/sensory aware. Using the skills learned from the class, she spearheaded her hotel's commitment to a sensory friendly hotel experience. The effort was recognized by a travel blogger and a mom of an autistic ten-year-old.
The Top North Alabama Ambassador of the Year award recognizes the work and achievements of the top performing North Alabama Ambassador in the past year. Launched in 2019, the North Alabama Ambassadors program is an online influencer program where bloggers and content creators produce compelling material highlighting north Alabama. These enthusiasts work with AMLA to help inspire and motivate other outdoor fans to visit North Alabama through social media, the AMLA blog and on the ground in their communities. AMLA has selected David Parham for the 2022 Top North Alabama Ambassador of the Year award. Parham lives in Huntsville and works as an engineer. When he's not at home, at work or at Jiu Jitsu, he is exploring the outdoors with his two daughters and wife. They love spending their time exploring North Alabama, seeing all the different views, caves and waterfalls that the region has to offer and sharing the adventures with his 17K followers on Instagram and nearly 13K followers on Facebook. Parham is known as Huntsville Adventurer on social media and has a website that is loaded with valuable advice and tips on exploring North Alabama as well as other places all over the country and world.
"With the PEAK Awards, we celebrate and recognize the many talented individuals and organizations who offer our visitors a unique and memorable experience. These individuals and organizations continue to show their commitment, resilience and ingenuity, even during the uncertain times a pandemic brings with it. Because of these tourism leaders, attractions, festivals and front-line workers, the tourism and travel industry continues to be a driving force in North Alabama's economy," said AMLA President/CEO Tami Reist.
Eleven awards were handed out at the awards presentation, held during the annual meeting. This year's award winners are:
Organization of the Year
The Organization of the Year award is presented to the AMLA member organization that has contributed significantly to the betterment and development of tourism in North Alabama in the past year.
Winner: Marshall County Tourism and Sports
Event of the Year
The Event of the Year award is presented to the AMLA member special event or festival that has made a great contribution to tourism in the past year.
Winner: Lexington Truck and Tractor Pull
Attraction of the Year
The Attraction of the Year award is presented to the AMLA member attraction that has made a great contribution to tourism in the past year.
Winner: Orion Amphitheater
Good Neighbor Award
The "Good Neighbor" Friend of Tourism Award is presented to a person or organization whose body of work has had a significant impact on tourism in the past year.
Winner: Mayor Kerry Underwood, City of Tuscumbia
Northern Star Award
The Northern Star Award is presented to an individual, such as a ranger, guide and a front line employee, who has gone "above and beyond" expectations.
Winner: Jabria Finch, Double Tree Suites by Hilton South Parkway
Young Professional of the Year Award
The Young Professional of the Year "Green Apple Award" is awarded to a person who has been in the tourism business less than seven years and has exhibited extraordinary leadership skills and initiative associated with their position.
Winner: Blake Williams, Blake Williams Communications
Tourism Professional of the Year
The Tourism Professional of the Year award is presented to an individual or employee of an AMLA member organization who has made a great contribution to tourism in the past year.
Winner: Pam Williams, Huntsville/Madison County Convention and Visitors Bureau
Lasting Impression Award
The Lasting Impression Award is presented to an attraction, event, person or organization that has provided a "lasting impression" on tourists thereby improving the overall image of tourism in North Alabama.
Winner: Scottsboro Three Arts Club
Chairman's Cup
The Chairman's Cup Award is given annually to a travel writer or media person who has over time demonstrated support of AMLA, the region and its tourism activities.
Winner: Carrie Banks, Communications Director, Alabama League of Municipalities and editor, Alabama Municipal Journal
The President's Award is presented to any person or organization whose tourism-related project created awareness or presented a positive image for North Alabama in the past year.
Winner: The late William "Bill" Jarnigan, University of North Alabama
Top North Alabama Ambassador of the Year
The award recognizes the work and achievements of the top performing North Alabama Ambassador in the past year.
Winner: David Parham, Huntsville, Ala.
Images of award recipients are available upon request by calling AMLA at 800.648.5381 or by sending an email to angie@northalabama.org.
About Alabama Mountain Lakes Tourist Association (AMLA)
AMLA is a nonprofit organization dedicated to the promotion and development of the travel industry within the 16 northernmost counties of the state. It is supported by 500-plus members consisting of chambers of commerce, Convention & Visitors Bureaus, attractions, campgrounds, festivals, communities, counties, golf courses, restaurants, tour operators, accommodations, vendors, financial institutions and individuals. Counties included within the AMLA region are Blount, Cherokee, Colbert, Cullman, DeKalb, Etowah, Franklin, Jackson, Lauderdale, Lawrence, Limestone, Madison, Marion, Marshall, Morgan and Winston. Additional information on North Alabama destinations, accommodations and special events is available by calling 800.648.5381 or by visiting www.NorthAlabama.org. Follow us on Twitter and Instagram @VisitNorthAL and like us at www.facebook.com/VisitNorthAL.
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© 2023 Alabama Mountain Lakes Tourist Association


War with China won't be won by deep strikes. Distant, defensive deterrence and blockades suit us better.
In contrast, Offshore Control is strategically and operationally offensive but tactically defensive. Tactical actions are guided by an operational campaign to attack Chinese strategic assets.
Colby also has some misconceptions about escalation. He seems to think U.S. concerns are limited to nuclear escalation. However, as outlined above, much of the concern is about the initial escalation that moves from crisis to open warfare. Colby states that AirSea Battle, if done properly, can effectively manage escalation. Colby assumes the U.S. can conduct a precision campaign without stimulating escalation. This assumes a level of precise intelligence that has been missing in our last few conflicts. Given the fact we mistakenly bombed the Chinese embassy in Belgrade in 1999, the Chinese may not be as confident we will limit our attacks to tactical targets.
I am also concerned about potential nuclear escalation. Four thousand years of recorded military history show people are poor at controlling escalation once a war starts. In fact, Clausewitz cautioned that once war starts, passion tends to dominate the primary trinity of passion, chance, and reason. Given that the United States does not understand China's decision-making process for the employment of nuclear weapons, this seems an egregious risk. The risk is magnified by the low return even if the strikes succeed. If the strikes succeed, we gain the ability to continue to strike into China. Unless we believe Douhet will finally be right and bombing will defeat China, what is the theory of victory for an AirSea Battle-based campaign? Do we risk nuclear conflict in order to gain the "advantage" of conducting a conventional-bombing campaign against a continent-sized state?
As a clinching argument, Colby suggests that the larger U.S. nuclear arsenal means China cannot "win" an exchange. I am not sure the American people are willing to risk the destruction of a dozen or so major U.S. cities to find out.
It is an old adage that the best way to prevent war is to prepare for it. Like many adages it assumes a certain level of common sense. You must be able to pay for the preparation. If you cannot afford the plan you have for war, you are not preparing by pretending you can afford it.
We are facing massive defense-budget cuts. Polls indicate the American people want even deeper cuts. Strategists have to balance ends, ways and means. If the means are restricted, then the ways must be altered. Thus it is vital the United States develop an affordable, executable strategy for the unlikely event of a conflict with China. It must deter China, assure allies, guide investment and, if needed, achieve a favorable conflict resolution. I proposed Offshore Control as a way to fill those requirements. I hope others will propose different strategies. Only by comparing strategies to each other can we evaluate them. As the Germans learned, an operational concept simply won't do.
T.X. Hammes is a Distinguished Research Fellow at National Defense University. These views are his own and do not reflect the views of the National Defense University, the Department of Defense, or the U.S. Government.


Even though there have been some revaluations of the Länder in the last two decades German debates on federalism hardly take subnational constitutional politics into account. For example, textbooks on federalism deal with amendments of the German constitution, i.e. the Basic Law, but they mostly fail to address constitutional adjustments at the subnational level or causal interrelations between the two constitutional levels.


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Total Subsidies
Pick a county Minnesota State Total Aitkin County, Minnesota Anoka County, Minnesota Becker County, Minnesota Beltrami County, Minnesota Benton County, Minnesota Big Stone County, Minnesota Blue Earth County, Minnesota Brown County, Minnesota Carlton County, Minnesota Carver County, Minnesota Cass County, Minnesota Chippewa County, Minnesota Chisago County, Minnesota Clay County, Minnesota Clearwater County, Minnesota Cook County, Minnesota Cottonwood County, Minnesota Crow Wing County, Minnesota Dakota County, Minnesota Dodge County, Minnesota Douglas County, Minnesota Faribault County, Minnesota Fillmore County, Minnesota Freeborn County, Minnesota Goodhue County, Minnesota Grant County, Minnesota Hennepin County, Minnesota Houston County, Minnesota Hubbard County, Minnesota Isanti County, Minnesota Itasca County, Minnesota Jackson County, Minnesota Kanabec County, Minnesota Kandiyohi County, Minnesota Kittson County, Minnesota Koochiching County, Minnesota Lac qui Parle County, Minnesota Lake County, Minnesota Lake of the Woods County, Minnesota Le Sueur County, Minnesota Lincoln County, Minnesota Lyon County, Minnesota McLeod County, Minnesota Mahnomen County, Minnesota Marshall County, Minnesota Martin County, Minnesota Meeker County, Minnesota Mille Lacs County, Minnesota Morrison County, Minnesota Mower County, Minnesota Murray County, Minnesota Nicollet County, Minnesota Nobles County, Minnesota Norman County, Minnesota Olmsted County, Minnesota Otter Tail County, Minnesota Pennington County, Minnesota Pine County, Minnesota Pipestone County, Minnesota Polk County, Minnesota Pope County, Minnesota Ramsey County, Minnesota Red Lake County, Minnesota Redwood County, Minnesota Renville County, Minnesota Rice County, Minnesota Rock County, Minnesota Roseau County, Minnesota Saint Louis County, Minnesota Scott County, Minnesota Sherburne County, Minnesota Sibley County, Minnesota Stearns County, Minnesota Steele County, Minnesota Stevens County, Minnesota Swift County, Minnesota Todd County, Minnesota Traverse County, Minnesota Wabasha County, Minnesota Wadena County, Minnesota Waseca County, Minnesota Washington County, Minnesota Watonwan County, Minnesota Wilkin County, Minnesota Winona County, Minnesota Wright County, Minnesota Yellow Medicine County, Minnesota Minnesota NRCS
Pick a district 1st District of Minnesota (Rep. Jim Hagedorn) 2nd District of Minnesota (Rep. Angie Craig) 3rd District of Minnesota (Rep. Dean Phillips) 4th District of Minnesota (Rep. Betty McCollum) 5th District of Minnesota (Rep. Ilhan Omar) 6th District of Minnesota (Rep. Tom Emmer) 7th District of Minnesota (Rep. Collin Peterson) 8th District of Minnesota (Rep. Pete Stauber)
Total Subsidies payments in Minnesota totaled $28.1 billion from 1995-2021.
Total Subsidies in Minnesota by county, 1995-2021
1 Marshall County, Minnesota $960,901,446 3.4% 3.4%
2 Renville County, Minnesota $790,015,330 2.8% 6.2%
3 Stearns County, Minnesota $686,179,551 2.4% 8.7%
4 Redwood County, Minnesota $667,816,591 2.4% 11.0%
5 Martin County, Minnesota $636,268,612 2.3% 13.3%
6 Nobles County, Minnesota $606,195,338 2.2% 15.5%
7 Clay County, Minnesota $580,089,438 2.1% 17.5%
8 Mower County, Minnesota $579,710,690 2.1% 19.6%
9 Faribault County, Minnesota $577,513,764 2.1% 21.6%
10 Norman County, Minnesota $575,399,092 2.0% 23.7%
11 Polk County, Minnesota $568,654,240 2.0% 25.7%
12 Kittson County, Minnesota $564,560,112 2.0% 27.7%
13 Blue Earth County, Minnesota $562,958,570 2.0% 29.7%
14 Yellow Medicine County, Minnesota $561,229,259 2.0% 31.7%
15 Freeborn County, Minnesota $551,049,259 2.0% 33.7%
16 27120 $538,711,583 1.9% 35.6%
17 Murray County, Minnesota $535,490,674 1.9% 37.5%
18 Jackson County, Minnesota $530,556,355 1.9% 39.4%
19 Swift County, Minnesota $530,526,357 1.9% 41.3%
20 Lac qui Parle County, Minnesota $519,487,440 1.8% 43.1%
21 Roseau County, Minnesota $513,240,584 1.8% 45.0%
22 Cottonwood County, Minnesota $511,259,169 1.8% 46.8%
23 Kandiyohi County, Minnesota $510,450,188 1.8% 48.6%
24 Fillmore County, Minnesota $509,866,498 1.8% 50.4%
25 Lyon County, Minnesota $508,237,265 1.8% 52.2%
26 Brown County, Minnesota $471,138,564 1.7% 53.9%
27 Chippewa County, Minnesota $438,000,821 1.6% 55.4%
28 Traverse County, Minnesota $435,343,458 1.5% 57.0%
29 Stevens County, Minnesota $431,506,103 1.5% 58.5%
30 Wilkin County, Minnesota $431,040,993 1.5% 60.1%
31 Goodhue County, Minnesota $427,558,118 1.5% 61.6%
32 Sibley County, Minnesota $407,306,459 1.4% 63.0%
33 Grant County, Minnesota $401,290,136 1.4% 64.5%
34 Rock County, Minnesota $389,459,072 1.4% 65.8%
35 Lincoln County, Minnesota $387,132,509 1.4% 67.2%
36 Otter Tail County, Minnesota $367,381,521 1.3% 68.5%
37 Watonwan County, Minnesota $367,267,468 1.3% 69.8%
38 Olmsted County, Minnesota $358,692,026 1.3% 71.1%
39 Meeker County, Minnesota $356,767,134 1.3% 72.4%
40 Waseca County, Minnesota $347,906,412 1.2% 73.6%
41 Pope County, Minnesota $346,178,714 1.2% 74.9%
42 Steele County, Minnesota $337,240,234 1.2% 76.1%
43 Dodge County, Minnesota $335,542,116 1.2% 77.2%
44 Le Sueur County, Minnesota $327,600,208 1.2% 78.4%
45 Pennington County, Minnesota $320,285,541 1.1% 79.6%
46 Nicollet County, Minnesota $308,590,924 1.1% 80.6%
47 McLeod County, Minnesota $306,501,237 1.1% 81.7%
48 Pipestone County, Minnesota $305,737,255 1.1% 82.8%
49 Becker County, Minnesota $304,602,756 1.1% 83.9%
50 Big Stone County, Minnesota $282,078,155 1.0% 84.9%
51 Rice County, Minnesota $281,181,023 1.0% 85.9%
52 Morrison County, Minnesota $279,502,121 1.0% 86.9%
53 Wabasha County, Minnesota $258,220,094 0.9% 87.8%
54 Winona County, Minnesota $257,354,792 0.9% 88.7%
55 Dakota County, Minnesota $246,832,140 0.9% 89.6%
56 Wright County, Minnesota $243,998,973 0.9% 90.5%
57 Todd County, Minnesota $232,448,808 0.8% 91.3%
58 Douglas County, Minnesota $222,518,011 0.8% 92.1%
60 Red Lake County, Minnesota $208,538,651 0.7% 93.6%
61 Houston County, Minnesota $196,602,655 0.7% 94.3%
62 Mahnomen County, Minnesota $193,450,071 0.7% 95.0%
63 Carver County, Minnesota $176,606,365 0.6% 95.6%
64 Benton County, Minnesota $173,925,037 0.6% 96.3%
65 Scott County, Minnesota $105,062,812 0.4% 96.6%
66 Isanti County, Minnesota $83,762,261 0.3% 96.9%
67 Sherburne County, Minnesota $74,044,229 0.3% 97.2%
68 Chisago County, Minnesota $73,786,482 0.3% 97.5%
69 Wadena County, Minnesota $66,357,673 0.2% 97.7%
70 Mille Lacs County, Minnesota $64,252,639 0.2% 97.9%
71 Beltrami County, Minnesota $63,428,196 0.2% 98.1%
72 Washington County, Minnesota $59,221,202 0.2% 98.4%
73 Pine County, Minnesota $58,155,648 0.2% 98.6%
74 Clearwater County, Minnesota $53,596,227 0.2% 98.7%
75 Lake of the Woods County, Minnesota $52,894,269 0.2% 98.9%
76 Hennepin County, Minnesota $48,712,964 0.2% 99.1%
77 Kanabec County, Minnesota $44,148,570 0.2% 99.3%
78 Hubbard County, Minnesota $41,720,192 0.1% 99.4%
79 Cass County, Minnesota $24,221,060 0.1% 99.5%
80 Crow Wing County, Minnesota $22,675,206 0.1% 99.6%
81 Anoka County, Minnesota $19,726,248 0.1% 99.7%
82 Aitkin County, Minnesota $14,996,849 0.1% 99.7%
83 Koochiching County, Minnesota $10,618,250 0.0% 99.7%
84 Carlton County, Minnesota $6,298,072 0.0% 99.8%
85 Itasca County, Minnesota $6,066,600 0.0% 99.8%
86 27138 $3,611,240 0.0% 99.8%
87 Saint Louis County, Minnesota $3,063,873 0.0% 99.8%
88 Lake County, Minnesota $289,042 0.0% 99.8%
89 Ramsey County, Minnesota $278,655 0.0% 99.8%
90 Cook County, Minnesota $187,901 0.0% 99.8%


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« In 1945, Cardinals hit into fewest double plays
Danny Litwhiler: Perfect fit for champion Cardinals »
Joe Medwick set a doubles pace that is hard to beat
September 18, 2011 by retrosimba
(Updated Aug. 26, 2018)
Cardinals left fielder Joe Medwick was a prolific doubles hitter.
Medwick hit 305 doubles in his first seven seasons with the Cardinals.
No other big-league player has hit as many doubles in his first seven seasons as Medwick did, according to MLB.com. (The Cardinals' Albert Pujols came close to matching Medwick, hitting 298 in his first seven seasons).
Here are the doubles totals in Medwick's first seven seasons:
YEAR……………….DOUBLES
1932…………………12 in 26 games
1933…………………40 in 148 games
Medwick rapped so many doubles because he was strong (he preferred the nickname "Muscles" instead of the better known "Ducky"), aggressive and could rip line drives to all fields.
In describing Medwick's swing, St. Louis sports editor Bob Broeg wrote that the right-handed slugger "stepped in the bucket but kept his hips and shoulders into the pitch so that he hit savagely to all fields."
"I just wonder what he would have hit if he didn't go for those bad pitches," Cardinals manager Frankie Frisch said.
Yankees catcher Yogi Berra, who also had a reputation as a bad-pitch hitter, grew up in St. Louis and was influenced by Medwick, who was his favorite Cardinals player.
In the April 1968 Baseball Digest magazine, Broeg wrote, "Joe was such a notorious bad-ball hitter that the only safe place to pitch him was under the plate."
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