text:
\section{Introduction}

The first excited state of the nucleon --- the $\Delta(1232)$ resonance ---
dominates pion-production phenomena and plays an important role in
our understanding of the low-energy nucleon structure.
High-precision measurements of the nucleon-to-$\De$
transition by means of electromagnetic probes became possible with the
advent of the new generation of electron scattering facilities, such as
BATES, MAMI, and JLab,  many measurements being completed
in recent years~\cite{Mainz97,LEGS97,Bates01,Jlab}.

The {\it electromagnetic}  nucleon-to-$\De$ (or, in short $\gamma} \def\Ga{{\it\Gamma} N \De$)
transition is predominantly
of the magnetic dipole ($M1$) type. In a simple quark-model picture,
this $M1$ transition
is described by a spin flip of a quark in the $s$-wave state.
Any $d$-wave admixture
in the nucleon {\it or} the $\Delta$ wave-functions allows for the electric ($E2$) and Coulomb ($C2$)
quadrupole transitions. Therefore by measuring these one is able to assess the presence
of the $d$-wave components and hence quantify to which extent the nucleon or the $\De$ wave-function
deviates from the spherical shape, {\it i.e.}, to which extent they are
``deformed''~\cite{deformation}.
The  $d$-wave component of $\De$'s wave-function can be separately
assessed by measuring its electric quadrupole moment. However,
this would be extremely difficult because of the tiny lifetime of the $\De$.
The $\gamma} \def\Ga{{\it\Gamma} N\Delta$ transition, on the other hand,  was
accurately measured in the pion photo- and electro-production reactions
in the $\De$-resonance energy region. The $E2$ and $C2$ transitions
 were found to be relatively small at moderate  momentum-transfers ($Q^2$),
the ratios $R_{EM}=E2/M1$ and
$R_{SM}=C2/M1 $ are at the level of a few percent.

Traditionally, the resonance parameters are extracted
by using {\it unitary isobar models}~\cite{ELAs,SpainModel,Drechsel:2000um,KVItheory,GiessenModel,Drechsel:1998hk,Inna}, which
in essence are unitarized tree-level calculations based on
phenomenological Lagrangians. However,
at low  $Q^2$ the $\gamma N \Delta$-transition
shows great sensitivity to the ``pion
cloud'', which until recently could only be
comprehensively studied within
{\it dynamical models}~\cite{Sato2, GrS96, DMT,Fuda03,PaT00,Caia04}.
(Unlike the isobar models, dynamical models include quantum effects
due to pion loops.)

With the advent of the chiral effective field theory ($\chi$EFT)
of QCD~\cite{Weinberg:1978kz,Gasser:1983yg}
and its extensions to the $\De$-resonance region~\cite{isospurion,PP03},
it has become possible to study the nucleon and $\Delta$-resonance properties
in a profoundly different way. Recently,
we have been able to perform  first $\chi$EFT\  studies of the
$\gamma N \Delta$-transition
in pion electroproduction~\cite{Pascalutsa:2005ts} and
of the $\De$-resonance magnetic moment in the radiative
pion photoproduction~\cite{PV05}. The advantages over the
previous dynamical approaches are apparent: $\chi$EFT\ is
a low-energy effective field theory of QCD and as such
it provides a firm theoretical foundation, with all
the relevant symmetries and scales of QCD built in consistently.
Moreover, we find that already at next-to-leading order
(NLO) in the ``$\delta} \def\De{\Delta$-expansion'' of Ref.~\cite{PP03},  the observables
for pion electroproduction in the $\De$-resonance region
are described remarkably well. The $\chi$EFT, therefore,
provides a theoretically
consistent and phenomenologically viable  framework, which, in particular,
will allow for a model-independent extraction of the resonance parameters.

Tremendous progress has recently been achieved as well in the
lattice QCD simulations of the $\gamma} \def\Ga{{\it\Gamma} N\De$ transition. The present
state-of-the-art results~\cite{Ale05} are ``quenched'' and are obtained
for pion masses above 300 MeV.
These results can only  be confronted with experiment after
an extrapolation  down to the physical pion mass of $140$ MeV.
A {\it linear} in the
quark mass ($m_q \sim m_\pi^2$) extrapolation was used in Ref.~\cite{Ale05}.
The thus obtained $R_{SM}$ ratio, at low $Q^2$, was found to be
in major disagreement with experiment.
The apparent caveat of this result is that the extrapolation
in the quark mass needs not to be linear. The
non-analytic dependencies, such as $\sqrt{m_q}$
and $\ln m_q$, are known to be important as one approaches the
small physical value of $m_q$.
These  non-analytic terms can also be obtained from $\chi$EFT,
and, as we have demonstrated~\cite{Pascalutsa:2005ts},
the  $R_{EM}$ and $R_{SM}$ ratios
do exhibit a pronounced non-analytic quark-mass dependence, such that
the lattice results~\cite{Ale05} can be reconciled with experiment.
Here we shall refine our calculation by consistently including
the quark-mass dependence of the nucleon and $\De$-isobar masses
in the same $\chi$EFT\ framework~\cite{PV05self}.

 A brief account of this work has recently been
published~\cite{Pascalutsa:2005ts}. In
 this paper we present an extensive description of the work,
improve on our theoretical error analysis, and
present some new results that could
not be included in the brief publication.
This paper is organized as follows.

In Sec.~\ref{sec2} we recall the relevant chiral Lagrangians,
while Sec.~\ref{sec3} explains the power counting in the $\delta} \def\De{\Delta$-expansion
scheme based on which
the leading- and next-to-leading-order contributions are selected.
The chiral-loop contributions
to the $\gamma N \Delta$-transition
 form factors are evaluated in Sec.~\ref{sec5} by using two
different techniques.
In Sec.~\ref{sec4} we discuss the theoretical uncertainty of
our calculation due to the neglect of higher-order effects.
In Sec.~\ref{sec6} we present the results for
pion photo- and electroproduction observables, multipoles,
and the extracted $\gamma} \def\Ga{{\it\Gamma} N\De$ form factors. In that section we
also discuss the $\chi$EFT\ predictions for
the $m_\pi$-dependence of the $\gamma} \def\Ga{{\it\Gamma} N\De$ transition and compare them
with available lattice results.
Sec.~\ref{sec7} lists the main points and conclusions of the paper.
The two Appendices contain technical details concerning the
Feynman-parameter and dispersion integrals, respectively.

\section{The effective chiral Lagrangian}
\label{sec2}

In this section we define the relevant effective Lagrangian
of low-energy QCD, where we restrict ourselves to the two flavor,
isospin symmetric case ($m_u = m_d\equiv m_q$). The guiding principle
is the symmetry under the SU(2)$_L\times$SU(2)$_R$ chiral rotations,
see \cite{Weinberg:1978kz,Gasser:1983yg,GSS89}.
The lowest-order Lagrangian of the Goldstone-boson (pion) isovector field
$\pi^a$ ($a=\ol{1,3}$)
\begin{equation}
{\mathcal L}_\pi^{(2)} = \mbox{\small{$\frac{1}{4}$}} f_\pi^2\, \mbox{Tr}[ \,\partial_\mu  U
\partial^\mu U^\dagger +  ( U + U^\dagger )\,\chi\, ]
= \mbox{\small{$\frac{1}{2}$}} \partial_\mu \pi^a \partial^\mu \pi^a - \mbox{\small{$\frac{1}{2}$}} m_\pi^2 [\, \pi^2 + O(\pi^4)\,] ,
\end{equation}
where
\begin{equation}
U(x) = e^{i\pi^a(x)\,\tau^a/f_\pi} \equiv u^2(x),
\,\,\,\, \mbox{with} \,\,\,
\pi^a \tau^a =
\left(\begin{array}{cc} \pi^0  &\sqrt{2}\,\pi^+ \\
\sqrt{2}\,\pi^- &  - \pi^0  \end{array}\right),
\end{equation}
and the linear in $U$, explicit
symmetry-breaking term is proportional to the quark (or, pion)
masses:  $\chi = 2B m_q = m_\pi^2 + O(m_\pi^4)$.
Furthermore, at this lowest order, $f_\pi \simeq 92.4$ MeV is the pion decay
constant and $B$ is related to the scalar quark condensate as $B=
-\left< \bar q q\right>/f_\pi^2 $.
In the notation ${\mathcal L}^{(i)} $, the superscript stands for the number of
derivatives of Goldstone-boson fields and insertions of their mass.

The chirally symmetric Lagrangian for the nucleon isodoublet
field, $N=(p,n)^T$, can conveniently
be written by using the SU(2) vector and axial-vector currents,
\begin{subequations}
\eqlab{currents}
\begin{eqnarray}
\mbox{\small{$\frac{1}{2}$}}\, \tau^a v_\mu^a(x) & \equiv &  \frac{1}{2i} \left(u \,
\partial_\mu u^\dagger+u^\dagger \partial_\mu  u \right)
= \frac{1}{4 f_\pi^2}  \, \varepsilon}  \def\eps{\epsilon^{abc} \, \tau^a \pi^b \, \partial_\mu \pi^c
+ O(\pi^4), \\
\mbox{\small{$\frac{1}{2}$}} \,\tau^a a^{\,a}_\mu(x) & \equiv &  \frac{1}{2i} \left(u^\dagger \,
\partial_\mu  u- u \,\partial_\mu  u^\dagger \right)
= \frac{1}{2 f_\pi} \tau^a \,\partial_\mu  \pi^a  + O( \pi^3),
\end{eqnarray}
\end{subequations}
and the chiral covariant derivative acting on the nucleon field,
\begin{equation}
D_\mu N = \partial_\mu N  + \mbox{\small{$\frac{1}{2}$}} i \tau^a v_\mu^a\, N .
\end{equation}
In terms of these definitions the nucleon Lagrangian begins at
\begin{eqnarray}
\eqlab{Nlagran}
{\mathcal L}^{(1)}_N &=& \ol N( i \sla{D} -{M}_{N} +  \mbox{\small{$\frac{1}{2}$}} g_A \,
\tau^c a \hspace{-2.25mm} \slash^{\,c} \gamma} \def\Ga{{\it\Gamma}_5 ) N\,,
\end{eqnarray}
where $g_A\simeq 1.267$ is the nucleon axial-coupling constant.

In the presence of the electromagnetic field ($A_\mu$), the charge
of the pions is accounted for by making the ``minimal substitution'':
$\partial_\mu \pi^a  \to \partial_\mu \pi^a + e  \,\varepsilon}  \def\eps{\epsilon^{ab3}A_\mu\pi^b $,
in the above expressions.  Similarly, the proton charge is included
by the minimal substitution in the chiral derivative as:
$D_\mu N \to D_\mu N + ie \mbox{\small{$\frac{1}{2}$}} (1+\tau^3) A_\mu N$.

The Lagrangian for the spin-3/2 isospin-3/2 $\De$-isobar can be written
in terms of the Rarita-Schwinger (vector-spinor) {\it isoquartet} field\footnote{Frequently chiral Lagrangians for the $\De$-isobar are
written in terms of the isovector-isodoublet field~\cite{isospurion}:
$\De_\mu^{a}  =  T^a \De_\mu$, with $T$ isospin-1/2-to-3/2
transition matrices. It is an  alternative to the isoquartet
representation adopted in this work. The two representations, however,
are equivalent at the level of observables.},
$\De_\mu = (\De_\mu^{++},\De_\mu^{+}, \De_\mu^{0}, \De_\mu^{-})^T $,
by using the vector and axial-vector currents \Eqref{currents} in the
isospin-3/2 representation of SU(2). The corresponding generators ${\mathcal T}^a$
have the following matrix representation:
\begin{subequations}
\begin{eqnarray}
{\mathcal T}^1 & = & \frac{2}{3} \left(\begin{array}{cccc}
0 & \sqrt{3}/2  & 0 & 0 \\
\sqrt{3}/2 & 0 & 1 & 0 \\
0 & 1 & 0 & \sqrt{3}/2\\
0 & 0 & \sqrt{3}/2 & 0
\end{array}\right), \\
{\mathcal T}^2 & = & \frac{2i}{3} \left(\begin{array}{cccc}
0 & -\sqrt{3}/2  & 0 & 0 \\
\sqrt{3}/2 & 0 &-1 & 0 \\
0 & 1 & 0 & -\sqrt{3}/2\\
0 & 0 & \sqrt{3}/2 & 0
\end{array}\right),\\
{\mathcal T}^3 & = & \mbox{diag} (1, \mbox{\small{$\frac{1}{3}$}}, -\mbox{\small{$\frac{1}{3}$}}, -1),
\end{eqnarray}
\end{subequations}
and satisfy ${\mathcal T}^a {\mathcal T}^a = 5/3$.
The chiral derivative is then given by:
\begin{equation}
D_\mu \De_\nu = \partial_\mu \De_\nu  +
 i \,v_\mu^a\,{\mathcal T}^a  \De_\nu ,
\end{equation}
and the  first-order chiral Lagrangian can be written as:
\begin{equation}
\eqlab{deltaLag}
{\mathcal L}^{(1)}_\De = \ol\De_\mu \left(i\gamma} \def\Ga{{\it\Gamma}^{\mu\nu\rho}\,D_\rho -
M_\De\,\gamma} \def\Ga{{\it\Gamma}^{\mu\nu} \right) \De_\nu -
 \mbox{\small{$\frac{1}{2}$}} H_A  \,\ol\De_\mu
\, a \hspace{-2.25mm} \slash^{\,c}\,{\mathcal T}^c \gamma} \def\Ga{{\it\Gamma}_5\, \De^\mu,
\end{equation}
where  $M_\De\simeq 1.232 $ GeV is the mass of the $\De$-isobar,
$H_A$ is the axial coupling constant given,
in the large-$N_C$ limit~\cite{Dashen:1993jt},
by $H_A=(9/5) g_A$ and this value is known to be consistent
with the empirical information, see, e.g.,~\cite{Butler:1992pn}.
The totally-antisymmetric products
of $\gamma} \def\Ga{{\it\Gamma}$-matrices are defined as: $\gamma} \def\Ga{{\it\Gamma}^{\mu\nu}=\mbox{\small{$\frac{1}{2}$}}[\gamma} \def\Ga{{\it\Gamma}^\mu,\gamma} \def\Ga{{\it\Gamma}^\nu]$,
$\gamma} \def\Ga{{\it\Gamma}^{\mu\nu\alpha}= i\varepsilon}  \def\eps{\epsilon^{\mu\nu\alpha\beta}\gamma} \def\Ga{{\it\Gamma}_\beta\gamma} \def\Ga{{\it\Gamma}_5$. The electric
charge of the $\De$ is accounted for by the following
minimal substitution: $D_\mu \to D_\mu + ie \mbox{\small{$\frac{1}{2}$}}(1+3{\mathcal T}^3) A_\mu$.

The free Rarita-Schwinger field obeys the following field equation:
\begin{equation}
\eqlab{fieldeq}
i\gamma} \def\Ga{{\it\Gamma}^{\mu\nu\rho}\,\partial_\rho \De_\nu = M_\De  \gamma} \def\Ga{{\it\Gamma}^{\mu\nu} \De_\nu,
\end{equation}
which, in particular, yields  the constraints:
$ \gamma} \def\Ga{{\it\Gamma}^\mu \De_\mu = 0 =   \partial^\mu \De_\mu $. The constraints reduce the
number of  spin degrees of freedom of the vector-spinor
field to the physical number appropriate for a massive particle with
spin 3/2. These constraints arise essentially due to the invariance
of the free-field kinetic term [l.h.s.\ of \Eqref{fieldeq}],
upon the gauge transformation
of the spin-3/2 field:
\begin{equation}
\eqlab{gsym}
\De_\mu(x) \to \De_\mu(x) +\partial_\mu \eps(x),
\end{equation}
where $\eps$ is a spinor. In order for interactions to
support the number of the free-field constraints, they must be
symmetric with respect to the same (or, at least,  similar)
transformation~\cite{Pas98}. Now, the chiral interactions
in \Eqref{deltaLag} obviously do not have this symmetry. Therefore,
we redefine them by using the free field equation, or, equivalently,
redefine the spin 3/2 field, such that they become symmetric~\cite{Pas01}.
Then, e.g., the lowest-order axial coupling becomes
\begin{equation}
{\mathcal L}^{(1)}_{\De\De\pi} = -\frac{H_A}{2M_\De f_\pi}  \,
\varepsilon}  \def\eps{\epsilon^{\mu\nu\rho\sigma} \def\Si{{\it\Sigma}} \,\ol\De_\mu
\, {\mathcal T}^a \,(\partial_\rho \De_\nu )\,\partial_\sigma} \def\Si{{\it\Sigma} \pi^{a} + O(\pi^3).
\end{equation}
The difference with the original interaction term from \Eqref{deltaLag}
is of higher order (in the pion derivatives and mass).

We write the $N\De$-transition Lagrangian right away in such an expanded
form which exhibits the spin-3/2 gauge symmetry:
\begin{subequations}
\eqlab{lagran}
\begin{eqnarray}
{\mathcal L}^{(1)}_{N\De} &=&  \frac{i h_A}{2 f_\pi M_\De}
\ol N\, T^a \,\gamma} \def\Ga{{\it\Gamma}^{\mu\nu\lambda} \def\La{{\Lambda}}\, (\partial_\mu \De_\nu)\, \partial_\lambda} \def\La{{\Lambda} \pi^a
+ \mbox{H.c.}, \\
{\mathcal L}^{(2)}_{N\De} &=&   \frac{3 i e g_M}{2M_N (M_N + M_\Delta)}\,\ol N\, T^3
\,\partial_{\mu}\De_\nu \, \tilde F^{\mu\nu}  + \mbox{H.c.},\\
{\mathcal L}^{(3)}_{N\De} &=&  \frac{-3 e}{2M_N (M_N + M_\Delta)} \ol N \, T^3
\gamma} \def\Ga{{\it\Gamma}_5 \left[ g_E (\partial_{\mu}\De_\nu)
+  \frac{i g_C}{M_\De} \gamma} \def\Ga{{\it\Gamma}^\alpha
(\partial_{\alpha}\De_\nu-\partial_\nu\De_\alpha) \,\partial_\mu\right] F^{\mu\nu}+ \mbox{H.c.},
\;\;\;\;\;
\end{eqnarray}
\end{subequations}
where $F^{\mu\nu}$ and $\tilde F^{\mu\nu}$
are the electromagnetic field strength and its dual,
$T^a$ are the isospin-1/2-to-3/2 transition $(2\times 4$) matrices:
\begin{subequations}
\begin{eqnarray}
T^1 & = & \frac{1}{\sqrt{6}} \left(\begin{array}{cccc}
-\sqrt{3}   & 0 & 1 & 0 \\
0 & -1 & 0 & \sqrt{3}
\end{array}\right), \\
T^2 & = & \frac{-i}{\sqrt{6}} \left(\begin{array}{cccc}
\sqrt{3}  & 0 & 1 & 0 \\
 0 & 1 & 0 & \sqrt{3}
\end{array}\right),\\
T^3 & = & \sqrt{\frac{2}{3} } \left(\begin{array}{cccc}
 0 & 1 & 0 & 0\\
 0 & 0 & 1 & 0
\end{array}\right),
\end{eqnarray}
\end{subequations}
satisfying $T^a T^{b\dagger} = \delta} \def\De{\Delta^{ab} -\mbox{\small{$\frac{1}{3}$}} \tau^a \tau^b$.

Note that the electric and the Coulomb $\gamma} \def\Ga{{\it\Gamma} N\De$ couplings
are of one order higher than
the magnetic one, because of the $\gamma} \def\Ga{{\it\Gamma}_5$ which involves
the ``small components'' of the fermion fields and thus
introduces an extra power of the 3-momentum.

For the momentum-space $\De$ propagator we  use
\begin{equation}
S^{\alpha\beta} (p) = \frac{p \hspace{-2mm} \slash+M_\De}{M_\De^2 - p^2}\left[ g^{\alpha\beta} -\mbox{\small{$\frac{1}{3}$}}
\gamma} \def\Ga{{\it\Gamma}^\alpha \gamma} \def\Ga{{\it\Gamma}^\beta + \frac{(1-\zeta)(\zeta p \hspace{-2mm} \slash +M_\De)}{3(\zeta^2 p^2-M_\De^2)}
(\gamma} \def\Ga{{\it\Gamma}^\alpha p^\beta - \gamma} \def\Ga{{\it\Gamma}^\beta p^\alpha)
+ \frac{2 (1-\zeta^2)\,p^\alpha p^\beta}{3(\zeta^2 p^2-M_\De^2)} \right],
\end{equation}
where $\zeta$ is a gauge-fixing parameter,
see Refs.~\cite{PV05self,Pa98thesis}
for details. The analog of the Landau-gauge for spin-1 case here is,
$\zeta = \infty$:
\begin{equation}
 S^{\alpha\beta} (p)=
\frac{p \hspace{-2mm} \slash+M_\De}{M_\De^2 - p^2} \, {\mathcal P}^{(3/2)\,\alpha\beta}(p),
\end{equation}
with
\begin{equation}
{\mathcal P}^{(3/2)\,\alpha\beta}(p) =\frac{2}{3}
\left(g^{\alpha\beta} - \frac{p^\alpha p^\beta}{p^2} \right) +\frac{p \hspace{-2mm} \slash}{3 p^2}
\gamma} \def\Ga{{\it\Gamma}^{\alpha\beta\mu}\, p_\mu ,
\end{equation}
the covariant
spin-3/2 projection operator.
As long as the coupling of the spin-3/2 field are gauge-symmetric
with respect to the transformation \eref{gsym}, the results are independent
of the choice of the gauge-fixing parameter $\zeta$.

\section{Power counting and renormalization}
\label{sec3}

The inclusion of the
$\De$-resonance introduces another light scale in the theory,
the resonance excitation energy: $\De\equiv M_\De-M_N\sim 0.3$ GeV.
This energy scale is still relatively light in comparison to
the chiral symmetry breaking scale
$\La_{\chi SB} \sim 1$ GeV. Therefore,
$\delta} \def\De{\Delta = \De/\La_{\chi SB}$ can be treated as a small parameter.
The question is, how to compare this parameter
with the small parameter of chiral perturbation theory ($\chi$PT),
$\eps = m_\pi /\La_{\chi SB}$. After all, our aim is to organize
an expansion in {\it a} small parameter and to estimate the size of various
contributions based on power-counting rules.  So some
specific relation between $\delta} \def\De{\Delta$ and $\eps$ would be helpful.

In most of the literature (see, e.g.,~\cite{isospurion})
they are assumed to be of
comparable size, $\delta} \def\De{\Delta\approx \eps$.
This, however, leads to a somewhat unsatisfactory
result: the $\De$-resonance
contributions are always estimated to be of the
same size as the nucleon contributions. Thus, obviously,
the $\De$-contributions
are {\it overestimated} at lower energies and {\it underestimated}
at the resonance energies. To estimate the $\De$-resonance
contributions correctly, and depending on the energy region,
one needs to count $\delta} \def\De{\Delta$ and $\eps$ {\it differently}.

A relation $\eps =\delta} \def\De{\Delta^2$ was suggested and explored in Ref.~\cite{PP03},
and is adopted in this work.
The second power is indeed the closest integer power for
the relation of these parameters in the real world.
We should stress that this relation is used for power-counting purposes
only. It is not imposed in the actual evaluations of diagrams.
Each diagram is simply  characterized
by an overall $\delta$-counting index $n$,
which tells us that its contribution begins at O($\delta} \def\De{\Delta^n$).

Because of the distinction of $m_\pi$ and $\De$ the counting of a given diagram depends
on whether the characteristic momentum $p$ is
in the low-energy region ($p\sim m_\pi$) or in the resonance
region ($p\sim \De$).
In the low-energy region the index of a graph with $L$ loops, $N_\pi$ pion propagators, $N_N$
nucleon propagators, $N_{\De}$ $\Delta$-propagators, and $V_i$ vertices of
dimension $i$ is
\begin{equation}
n=
2 \,(\,\sum_i i V_i + 4 L  - N_N - 2 N_\pi )
-N_\De \equiv 2 n_{\chi{\mathrm PT}} - N_\De,
\end{equation}
where $ n_{\chi{\mathrm PT}} $
is the index in $\chi$PT with no $\De$'s \cite{GSS89}.
In the resonance region, one distinguishes the one-$\De$-reducible (O$\De$R) graphs~\cite{PP03}, see e.g.,
graph (a) in \Figref{diagrams}. Such graphs contain $\De$ propagators
which go as $1/(p-\De)$, and hence for $p\sim \De$ they are large and
all need to be included. This gives an incentive, {\it within
the power-counting scheme}, to
resum $\De$ contributions.
Their resummation amounts to
dressing the $\De$ propagators so that they behave as $1/(p-\De-\Si)$. The self-energy
$\Si$ begins at order $p^3$ and thus
a dressed O$\De$R propagator counts as $1/\delta} \def\De{\Delta^3$.
If the number of
such propagators in a graph is $N_{O\De R}$, the power-counting index of
this graph in the resonance region is given by
\begin{equation}
n=n_{\chi{\mathrm PT}} - N_\De - 2N_{O\De R},
\end{equation}
where $N_\De$ is the total number
of $\De$-propagators.

A word on the renormalization program, as it is an indivisible part
of power counting in a relativistic theory. Indeed, without some
kind of renormalization the loop graphs diverge as $\La^{\cal N} $,
where $\La$ is an ultraviolet cutoff, and ${\cal N}$ is a positive
power proportional to the power-counting index of the graph.
Also, contributions of heavy scales,
such as baryon masses, may appear as $M^{\cal N}$.
The renormalization
of the loop graphs can and should be performed so as to absorb
these large contributions into the available low-energy constants,
thus bringing the result in accordance
with power counting \cite{Gegelia:1999gf}.

To give an example, consider the one-$\pi N$-loop contribution to
the nucleon mass, \Figref{nucself}. For the $\pi NN$ vertex from
${\mathcal L}^{(1)}_N$ the power counting tells us that this contribution
begins at $O(m_\pi^3)$. An explicit calculation, however, will
show (e.g., \cite{GSS89}) that the loop produces
$O(m_\pi^0)$ and $O(m_\pi^2)$ terms, both of which are (infinitely) large.
{\it This is not a violation of power counting}, because there are two
low-energy constants: the nucleon mass in the chiral limit, $M^{(0)}$,
and $c_{1N}$, which enter at order  $O(m_\pi^0)$ and $O(m_\pi^2)$,
respectively, and {\it renormalize away}
the large contributions coming from the loop.
The renormalized relativistic result, up to and including
$O(m_\pi^3)$,  can be written as~\cite{PV05self}:
\begin{eqnarray}
M_N &=& M_N^{(0)} - 4 \, c_{1 N} \, m_\pi^2 \nonumber \\
&-& \frac{3 \, g_A^2}{(8 \pi f_\pi)^2} \, m_\pi^3 \,
\left\{ 4  \left( 1 - \frac{m_\pi^2}{4 M_N^2} \right)^{5/2}
\arccos\frac{m_\pi}{2 M_N}
+ \frac{17 m_\pi}{16 M_N}
- \left(\frac{m_\pi}{2 M_N}\right)^3 \right. \nonumber \\
&&\hspace{2.5cm} \left. + \, \frac{m_\pi}{8 M_N}
\left[ 30 - 10 \left(\frac{m_\pi}{M_N}\right)^2
+  \left(\frac{m_\pi}{M_N}\right)^4 \right]
\, \ln \frac{m_\pi}{M_N}
\right\} ,
\label{eq:nucpin}
\end{eqnarray}
and one can easily verify that the loop contribution begins
at $O(m_\pi^3)$ in agreement with power counting.

We now turn to the analysis of the
pion electroproduction process.
The pion electroproduction amplitude to NLO in the $\delta} \def\De{\Delta$ expansion, in the
resonance region, is given by graphs in  \Figref{diagrams}(a) and (b), where the shaded
blobs in graph (a) include  corrections depicted in \Figref{diagrams}(c--f). The hadronic part of
graph (a) begins at ${\cal O}(\delta} \def\De{\Delta^0)$ which here is the leading order.
The Born graphs \Figref{diagrams}(b) contribute at ${\cal O}(\delta} \def\De{\Delta)$.
We note that at NLO there are also vertex corrections of the type (e) and (f) with nucleon propagators in the loop
replaced by the $\De$-propagators. However, after the appropriate
renormalizations and $Q^2\ll\La\De$,
these graphs start to contribute at next-next-to-leading order.

The $\Delta$ self-energy \Figref{diagrams}(c) can, in the $\zeta=\infty$
gauge, be presented as
\begin{equation}
\Si_{\alpha\beta} (p) = \Si_\De (p \hspace{-2mm} \slash) \,{\mathcal P}^{(3/2)}_{\alpha\beta}(p),
\end{equation}
where $\Si_\De (p \hspace{-2mm} \slash)$ has the spin-1/2 Lorentz form,
computed by us earlier~\cite{PV05,PV05self}. To recall
these results, it is convenient to
 introduce the following dimensionless quantities:
\begin{eqnarray}
\eqlab{defs}
&& \mu = \frac{m_\pi}{M_\De}, \,\,\,
r = \frac{M_N}{M_\Delta}, \,\,\, \delta =  \frac{\De}{M_\Delta} = 1-r, \nonumber\\
&& \beta =\mbox{\small{$\frac{1}{2}$}}  (1-r^2+\mu^2 ), \,\,\,
\alpha = 1 - \beta, \\
&& \lambda} \def\La{{\Lambda}^2 = \mbox{\small{$\frac{1}{4}$}}(\delta} \def\De{\Delta^2-\mu^2)[(1+r)^2-\mu^2]
=\beta^2-\mu^2 = \alpha^2 - r^2. \nonumber
\end{eqnarray}
The imaginary part of the $\Delta$ self-energy is related to the
resonance decay
width, which at the leading order (LO) gives:
\begin{equation}
\Gamma_\Delta = -
2\, \mathrm{Im}\, \Si_\Delta(M_\Delta) =
(4 \pi/3) \,M_\Delta \,  C_\Delta^2 \,
\lambda^3\, (\alpha + r).
\end{equation}
with $C_\Delta = h_A M_\Delta / (8 \pi f_\pi)$.
The experimental value for the
$\De$-resonance width, $\Gamma_\Delta  = 0.115$~GeV,
fixes $h_A\simeq 2.85$.
At NLO, the residue of the $\De$ propagator receives a correction:
\begin{equation}
{\rm Im}\,\Si_\De'(M_\De) = - 2\pi C_\De^2\,
\lambda} \def\La{{\Lambda} \left[ \alpha\,\beta\,  (\alpha+r) -\mbox{\small{$\frac{1}{3}$}} \lambda} \def\La{{\Lambda}^2 (r+r^2-\mu^2)\right].
\end{equation}
The renormalized $\De$-propagator, in the $\zeta=\infty$ gauge, at
NLO reads:
\begin{equation}
S_{\alpha\beta}(p) =
\frac{-{\mathcal P}^{(3/2)}_{\alpha\beta}(p)}{(p \hspace{-2mm} \slash -M_\De)[1-i\,\mbox{Im}\,\Si_\De'(M_\De)]
+ \mbox{\small{$\frac{1}{2}$}} i\Ga_\De} \,.
\end{equation}

The real part of the renormalized $\Delta$ self-energy,
$\tilde \Si_\Delta$,  contributes to the mass as:
\begin{eqnarray}
M_\Delta &=& M_\Delta^{(0)} - 4 \, c_{1 \Delta} \, m_\pi^2
+ \mathrm{Re} \tilde \Si_\Delta^{(\pi N)}(M_\Delta) ,
\label{eq:delpin1}
\end{eqnarray}
where
\begin{eqnarray}
\tilde \Si_\Delta (M_\Delta)
= - \mbox{\small{$\frac{1}{2}$}} C_\Delta^2 \, M_\Delta \,
\left[ V(\mu, \delta) - V(0, \delta) - \mu^2 \, V^\prime(0, \delta) \right] .
\label{eq:delpin2}
\end{eqnarray}
The $\pi N$ loop integral $V(\mu, \delta)$ is given in analytical form
by~\cite{PV05self}:
\begin{eqnarray}
V(\mu, \delta} \def\De{\Delta )
&=& \mbox{\small{$\frac{1}{3}$}} (r+\alpha) \left[ \beta \, (\mu^2-2\lambda} \def\La{{\Lambda}^2) \,\ln\mu^2
+ \alpha \, (r^2-2\lambda} \def\La{{\Lambda}^2) \,\ln r^2  -\mbox{$\frac{2}{3}$} (\alpha^3 + \beta^3)
\right. \nonumber \\
&+ & \left.  4\lambda} \def\La{{\Lambda}^2
+ 4\lambda} \def\La{{\Lambda}^4 \,\Omega(\lambda} \def\La{{\Lambda}) \,  \right]
+\mbox{\small{$\frac{1}{4}$}} \mu^4 (\ln \mu^2 -\mbox{\small{$\frac{1}{2}$}} ) - \mbox{\small{$\frac{1}{4}$}} r^4 (\ln r^2 -\mbox{\small{$\frac{1}{2}$}} )
\,,
\label{eq:delpin3}
\end{eqnarray}
with the elementary function $\Omega$ defined as:
\begin{equation}
\Omega(\lambda} \def\La{{\Lambda}) = \left\{ \begin{array}{lc}
\frac{1}{2\lambda} \def\La{{\Lambda}}
\ln \mbox{$\frac{\beta-\mu^2-\lambda} \def\La{{\Lambda}}{\beta-\mu^2+\lambda} \def\La{{\Lambda}}$}\,, & \lambda} \def\La{{\Lambda}^2 \ge 0\\
-\frac{1}{\sqrt{-\lambda} \def\La{{\Lambda}^2}}
 \arctan\frac{\sqrt{-\lambda} \def\La{{\Lambda}^2}}{\alpha\beta+\lambda} \def\La{{\Lambda}^2}
\, . & \lambda} \def\La{{\Lambda}^2 < 0
\end{array} \right.
\end{equation}
The region $\lambda^2 > 0$,
corresponds with $m_\pi < M_\Delta - M_N$, where the $\Delta$ is unstable.
The region $\lambda^2 < 0$,
corresponds with $m_\pi > M_\Delta - M_N$, where the $\Delta$ is stable.

Furthermore, from Eq.~(\ref{eq:delpin3}) we read off
the $m_\pi^0$ and $m_\pi^2$ terms, which enter
Eq.~(\ref{eq:delpin2}) and need to be absorbed by the renormalization of the
low-energy constants,
\begin{eqnarray}
V(0, \delta) &=& \frac{1}{4}
\left[\frac{r}{3} (2 - r^2) + 2 (1 - \frac{r^2}{3}) \right]
r^5 \ln r^2
- \frac{1}{12} (1 + r)^2 (1 - r^2)^3 \ln (1 - r^2) \nonumber \\
&+& \left[ \frac{5}{36} - \frac{7 r^2}{36} - \frac{r^4}{8}
+ \frac{r^6}{12} + \frac{r}{18} (5 - 12 r^2 + 3 r^4) \right], \\
V^\prime(0, \delta) &=& \frac{1}{2}  (1 + \frac{2 r}{3}) \, r^5 \ln r^2
+ \left[ \frac{1}{3} (1 + r^2 + r^4) + \frac{r}{2} (1 + r^2) \right]
(1 - r^2) \ln (1 - r^2) \nonumber \\
&-& \left[\frac{7}{18} + \frac{r}{2} + \frac{r^2}{6} + \frac{r^3}{2}
+ \frac{r^4}{3} \right].
\label{eq:delpin5}
\end{eqnarray}

The vector-meson diagram,  \Figref{diagrams}(d), contributes to NLO for $Q^2\sim \La\De$. We include
it effectively by giving the $g_M$-term a dipole $Q^2$-dependence (in analogy to how it is usually done
for the nucleon isovector form factor):
\begin{equation}
g_M\to \frac{g_M}{(1+Q^2/0.71\,\mbox{GeV}^2)^{2}}.
\end{equation}
Analogous
effect for the $g_E$ and $g_C$ couplings begins at N$^2$LO. The loop
corrections to the $\gamma} \def\Ga{{\it\Gamma} N\De$-vertex are discussed separately,
in the following section.

\section{Evaluation of the $\gamma} \def\Ga{{\it\Gamma} N\De$ form factors}
\label{sec5}

In this section we present a detailed analysis of the $\gamma} \def\Ga{{\it\Gamma} N\De$-vertex
corrections that appear at NLO, see \Figref{diagrams}(e) and (f).
To the order we work, the {\it pseudovector} $\pi NN$ interaction from
the Lagrangian \eref{Nlagran} is equivalent to the {\it pseudoscalar}
one, and we use the latter in the actual calculations of the graphs
\Figref{diagrams}(e, f).

The rest of the section is organized as follows.
First we will comment on the electromagnetic gauge invariance
of these contributions. Then,
after a brief summary of different decompositions
of the $\gamma N \Delta$ vertex into Lorentz-invariant form factors,
we evaluate the chiral corrections to those form factors at NLO.
As a cross-check, we do the calculations using
two different techniques: the Feynman-parameter method
and the sideways dispersion relations.

\subsection{Electromagnetic current conservation}

The addition of the one-loop graphs \Figref{diagrams}(e, f)
to the ``tree-level'' $\gamma} \def\Ga{{\it\Gamma} N\De$ vertex from \Eqref{lagran}, should not spoil the  e.m.\ current conservation:
\begin{equation}
q_\mu \, \bar u_\alpha(p' ) \, \Gamma^{\alpha \mu}_{\gamma} \def\Ga{{\it\Gamma} N\De} \, u(p) =0\,,
\end{equation}
where $u_\alpha$ is the free Rarita-Schwinger vector-spinor of the $\De$,
$u$ is the free Dirac spinor of the nucleon,
and momenta are defined as in \Figref{treevertex}.
This sort of conditions are most conveniently checked
by using the Ward-Takahashi (WT) identities
for the e.m.\ couplings of the nucleon, pion and  $\De$ fields:
\begin{subequations}
\begin{eqnarray}
&& q_\mu \,  \Ga_{\gamma} \def\Ga{{\it\Gamma} NN}^\mu = e_N\, q\cdot \gamma} \def\Ga{{\it\Gamma} = e_N\,[ S^{-1}_N(p') - S^{-1}_N(p)]\,,\\
&& q_\mu \,  \Ga_{\gamma} \def\Ga{{\it\Gamma}\pi\pi}^\mu = e_\pi\, q\cdot (p+p')= e_\pi [S^{-1}_\pi (p') - S^{-1}_\pi(p)]\,,\\
&& q_\mu \,  \Ga_{\gamma} \def\Ga{{\it\Gamma}\De\De}^{\mu\alpha\beta} = e_\De\, \gamma} \def\Ga{{\it\Gamma}^{\mu\alpha\beta} q_\mu
 = e_\De [S^{-1\,\alpha\beta}_\De (p') - S^{-1\,\alpha\beta}_\De (p)]\,,
\end{eqnarray}
\end{subequations}
where $q=p'-p$, $e_N=\mbox{\small{$\frac{1}{2}$}} e (1+\tau_3) $, $e_\pi=ie \eps^{abc} \tau_c$, $e_\De=\mbox{\small{$\frac{1}{2}$}} e (1+3{\mathcal T}^3) $ and $S^{-1}$ denote the corresponding inverse
propagators. Applying these identities to the diagrams  \Figref{diagrams}(e, f)
we obtain:
\begin{equation}
\eqlab{cccond}
q_\mu \, \bar u_\alpha(p' ) \, \Gamma^{\alpha \mu}_{\gamma} \def\Ga{{\it\Gamma} N\De} \, u(p) =
e\, \bar u_\alpha(p' )
\,\left[\Si_{N\De}^\alpha(p)- \Si_{N\De}^\alpha (p')\right] \,u(p),
\end{equation}
where $\Si_{N\De}^\alpha$ is the one-loop $N\to \De$ self-energy, which, due
to Lorentz covariance has the following general form:
\begin{equation}
\eqlab{Lstructure}
\Si_{N\De}^\alpha(p) = A(p^2) \,p^\alpha + B(p^2) \,\gamma} \def\Ga{{\it\Gamma}^\alpha \,.
\end{equation}
The scalar functions $A$ and $B$ do not vanish, however we may use
the Rarita-Schwinger conditions, $p\cdot u(p) = 0 = \gamma} \def\Ga{{\it\Gamma} \cdot  u(p)$,
to show that the second term on the r.h.s.\ of ~\Eqref{cccond} vanishes.
The first term can only be canceled by the contribution
of the diagram in \Figref{extra}.

Thus, the condition of
the e.m.\ current conservation requires
the diagram of \Figref{extra} to be included in addition to
the NLO contributions~\Figref{diagrams}(e, f). When computing this diagram, we
discover that it is actually of N$^2$LO, because the Lorentz form
\Eqref{Lstructure}
 cuts out the spin-3/2 part of the $\De$-propagator and hence also the
pole of the propagator. The $\De$-propagator in \Figref{extra},
therefore, counts as $\delta} \def\De{\Delta^0$.

To NLO, the sum of \Figref{diagrams}(e) and \Figref{diagrams}(f)
conserves the e.m.\ current.

\subsection{Form-factor decompositions}

The $\gamma N \Delta$ vertex can in general be decomposed into three
Lorentz covariants. For instance,
\begin{eqnarray}
\eqlab{newform}
\bar u_{\alpha}(p') \, \Gamma^{\alpha \mu}_{\gamma} \def\Ga{{\it\Gamma} N\De} \, u(p)   &=&
\sqrt{\frac{3}{2}} \frac{ M_\Delta + M_N}{M_N \, [(M_\Delta + M_N)^2+Q^2]}
\nonumber \\
&\times&  \bar u_{\alpha}(p^\prime) \, \left\{\,
g_M(Q^2) \, \varepsilon^{\alpha \mu \kappa \lambda} \,
p^{\prime}_\kappa \,  q_\lambda  \right. \nonumber\\
&& \hspace{1.75cm} + \; g_E(Q^2)
\left( q^\alpha \, p^{\prime \, \mu} -q \cdot p^\prime  \, g^{\alpha \mu} \right) i \gamma_5   \\
&& \hspace{1.75cm} + \; \left.  g_C(Q^2)
\left( q^\alpha \, q^\mu - q^2 \, g^{\alpha \mu}  \right) i \gamma_5
\right\} \; u (p) , \nonumber
\label{eq:diagndel1}
\end{eqnarray}
where $q=p'-p$ is the
photon 4-momentum, $Q^2=-q^2$.
Furthermore, $g_M(Q^2)$, $g_E(Q^2)$, and $g_C(Q^2)$
are the magnetic dipole, electric quadrupole, and
Coulomb quadrupole form factors, as defined in Ref.~\cite{Caia04}.
In the limit $Q^2=0$ they are equal to the
physical values of the corresponding parameters
in the Lagrangian \eref{lagran}. These
form factors relate to the conventional magnetic ($G_M^\ast$),
electric ($G_E^\ast$) and Coulomb  ($G_C^\ast$) form factors
of Jones and Scadron~\cite{Jones:1972ky} as follows:
\begin{eqnarray}
\eqlab{JS}
G_M^\ast &=& g_M \,+\frac{M_\De^2}{Q_+^2} \left(-\beta_\gamma} \def\Ga{{\it\Gamma} \,g_E+  \bar Q^2 g_C\right), \nonumber\\
G_E^\ast &=& \frac{M_\De^2}{Q_+^2} \left(-\beta_\gamma} \def\Ga{{\it\Gamma} \,g_E+ \bar Q^2  g_C\right), \\
G_C^\ast &=&-\frac{2 M_\De^2}{Q_+^2} \left(g_E +\beta_\gamma} \def\Ga{{\it\Gamma}\,  g_C\right),\nonumber
\end{eqnarray}
where $Q_\pm=\sqrt{(M_\De\pm M_N)^2 +Q^2}$, $\bar Q^2 = Q^2/M_\De^2$,
$\beta_\gamma} \def\Ga{{\it\Gamma} =\mbox{\small{$\frac{1}{2}$}}  (1-r^2-\bar Q^2)$.
The multipole ratios $E2/M1$ and $C2/M1$ at the $\Delta$-resonance
position can be expressed in terms of these form factors as:
\begin{equation}
\eqlab{ratios}
R_{EM}=- \frac{G_E^\ast}{G_M^\ast} \,,\;\;\;\;
R_{SM}=- \frac{Q_+Q_-}{4M_\De^2} \, \frac{G_C^\ast}{G_M^\ast} .
\end{equation}

Alternatively, the on-shell $\gamma N \Delta$ vertex
is also often expressed in the following form~:
\begin{eqnarray}
\bar u_{\alpha}(p') \, \Gamma^{\alpha \mu}_{\gamma N \Delta} \, u(p)
&=&\, \mbox{$\sqrt{\frac{2}{3}}$} \, \bar u_{\alpha}(p' ) \,
\left\{
\, \left( \gamma^\mu q^\alpha \, -\, \gamma \cdot q \, g^{\alpha \mu} \right)
\, g_1(Q^2) \right. \nonumber \\
&&\hspace{2.5cm} + \left(
q \cdot p^\prime \, g^{\alpha \mu} - q^\alpha \, p^{\prime \, \mu} \right)
\, g_2(Q^2)
\\
&&\left. \hspace{2.5cm}
+ \left(q^\alpha \, q^\mu - q^2 \, g^{\alpha \mu} \right) \,
g_3(Q^2) \; \right\}  i \, \gamma_5 \, u(p) ,\nonumber
\label{eq:diagndel0}
\end{eqnarray}
which is defined for the $p \to \Delta^+$ transition,
yielding the isospin factor $\sqrt{2/3}$.
The form factors $g_M, g_E$ and $g_C$ can then be
expressed in terms of $g_i(Q^2) (i = 1, 2, 3)$
through the relations~:
\begin{eqnarray}
g_M(Q^2) \,&=& \, - \frac{2}{3} \,
\frac{M_N \, Q_+^2}{M_\Delta (M_\Delta + M_N)} \, g_1(Q^2) ,
\\
g_E(Q^2) \,&=& \, \frac{2}{3} \,
\frac{M_N \, Q_+^2}{M_\Delta (M_\Delta + M_N)} \,
\left\{ g_1(Q^2) - M_\Delta \, g_2(Q^2) \right\} ,
\nonumber \\
g_C(Q^2) \,&=& \, \frac{2}{3} \,
\frac{M_N \, Q_+^2}{(M_\Delta + M_N)} \, g_3(Q^2) .\nonumber
\label{eq:gmecgi}
\end{eqnarray}

\subsection{Feynman-parameter method}
The one-loop corrections to the $\gamma N \Delta$ form factors are given by
the graphs in \Figref{diagrams}(e) and (f).
As they contain three propagators, we apply the Feynman-parameter trick~:
\begin{equation}
\frac{1}{ABC} = 2 \int_0^1 \! dx \int_0^x \!dy \frac{1}{[Ay+B(x-y)+C(1-x)]^3}.
\end{equation}
For the diagram  \Figref{diagrams}(e) we have
 $A=k^2-m_\pi^2+i\eps$, $B=(k+q)^2-m_\pi^2+i\eps$,
$C=(p-k)^2-M_N^2+i\eps$, where $k$ is the integration 4-momentum.
After the shift of the integration momentum, $k\rightarrow k-(x-y)q+(1-x)p$,
we obtain~:
$Ay+B(x-y)+C(1-x)\rightarrow k^2-{\mathcal M}^2 $,
where
\begin{equation}
{\mathcal M}^2(x,y) = m_\pi^2 x+ (M_N^2 -p^2 x)(1-x)- 2p\cdot q (1-x)(x-y)
-q^2 (x-y)(1-x-y) -i\eps.
\end{equation}

For the case of the $N \to \De$ transition we may use:
$p^2=M_N^2$, ${p'}^2=(p+q)^2=M_\De^2$, and hence
$2p\cdot q= M_\De^2 - M_N^2 -q^2$, and
\begin{equation}
{\mathcal M}^2(x,y) = m_\pi^2 x+ [M_N^2- xM^2_\De +(M_\De^2-M_N^2) y] (1-x)
+q^2 y (x-y) -i\eps.
\end{equation}

The next step is to perform the integration over 4-momentum which can be done
by using the following rules:
\begin{subequations}
\begin{eqnarray}
\eqlab{jnmunu}
&&\int \!\frac{d^4 k}{(2\pi)^4} \frac{1}{(k^2 - m^2)^{n}} \equiv J_n(m^2)
 =  i \frac{(-1)^n}{ (4 \pi)^{2} }
\frac{\Gamma (n-2)}{\Gamma (n)} \, m^{-2(n - 2)}\,,\\
&& \int \!\frac{d^4 k}{(2\pi)^4} \frac{k_\mu k_\nu}{(k^2 - m^2)^{n}}
 = \frac{1}{2(n-1)} J_{n-1}(m^2) \, g_{\mu\nu}\,,
\end{eqnarray}
\end{subequations}
Integrals with an odd number of 4-vectors $k$ in the numerator vanish.
The integral $J_n$ diverges for $n=1$ and 2, but can be defined via
dimensional regularization as:
\begin{subequations}
\begin{eqnarray}
J_1(m^2) & = & \frac{-i m^2}{(4\pi)^2} \left(
- \frac{2}{4-d} + \gamma} \def\Ga{{\it\Gamma}_E -1  + \ln \frac{m^2}{4\pi}  \right), \\
J_2(m^2) & = & \frac{-i }{(4\pi)^2} \left(
- \frac{2}{4-d} + \gamma} \def\Ga{{\it\Gamma}_E   + \ln \frac{m^2}{4\pi}  \right),
\end{eqnarray}
\end{subequations}
where $d\to 4^-$ is the number of dimensions and $\gamma} \def\Ga{{\it\Gamma}_E = - \Gamma'(1) \simeq 0.5772 $
is the Euler constant.

Then, the  ($\ol {MS}$-subtracted) result for the graph of
\Figref{diagrams}(e), where the photon couples to the charged pion in the
loop, can be decomposed into the three Lorentz covariants \Eqref{newform}
and cast in the form~:
\begin{subequations}
\begin{eqnarray}
g_M^{(e)} &=&  - C_{N\De} \! \int\limits_0^1 dy \, y \!\int\limits_0^{1-y} \!dx\,\ln {\cal M}_e^2 ,
\label{eq:gmpionloop} \\
g_E^{(e)} &=&  + C_{N\De}  \! \int\limits_0^1 \!dy \, y \! \int\limits_0^{1-y} \!\!dx\,
\left\{ \ln {\cal M}_e^2
\, - 2 x \, [(1-y)\,(1+r)-x ]\,  {\cal M}_e^{-2} \right\} ,
\label{eq:gepionloop} \\
g_C^{(e)} &=&  -C_{N\De}  \! \int\limits_0^1 dy \, y \,(2 y-1)\!\!\int\limits_0^{1-y} \!\! dx
\, [(1-y)\,(1+r)-x ]\, {\cal M}_e^{-2} ,
\label{eq:gcpionloop} \\
\mbox{with} && {\cal M}_e^2 \equiv (x-\beta )^2 -\lambda} \def\La{{\Lambda}^2 +2\beta_\gamma} \def\Ga{{\it\Gamma}  x y + \bar Q^2 y(1-y)- i\varepsilon}  \def\eps{\epsilon ,
\end{eqnarray}
\end{subequations}
while the analogous contribution of the graph \Figref{diagrams}(f), where the photon
couples to the electric charge of the nucleon in the loop, is given by~:
\begin{subequations}
\begin{eqnarray}
g_M^{(f)} &=&  - C_{N\De} \!\int\limits_0^1 dy \, y \!\int\limits_0^{1-y} \!dx\,\left\{ 2\ln {\cal M}_f^2  \right. \nonumber\\
&& \left. + \,
{\cal M}_f^{-2}
\left[ 2\alpha_\gamma} \def\Ga{{\it\Gamma}  x y + \bar Q^2 y(1-y)-(x+r)(1-x)\right]\right\},
\label{eq:gmnucloop} \\
g_E^{(f)} &=& -g_M^{(f)} + 2 C_{N\De} \! \int\limits_0^1 \!dy \, y \!
\int\limits_0^{1-y} \!\!dx\,{\cal M}_f^{-2}
\, \left[ (x+r)(1-x)-xy(1+r)\right] ,
\label{eq:genucloop} \\
g_C^{(f)} &=&  -2 C_{N\De}  \int\limits_0^1 dy \, y^2
\int\limits_0^{1-y} dx\,[y(1+r)-1+x]\,  {\cal M}_f^{-2},
\label{eq:gcnucloop} \\
\mbox{with} && {\cal M}_f^2
\equiv  (x-\alpha)^2 -\lambda} \def\La{{\Lambda}^2 +2\alpha_\gamma} \def\Ga{{\it\Gamma}  x y + \bar Q^2 y(1-y)- i\varepsilon}  \def\eps{\epsilon,
\end{eqnarray}
\end{subequations}
where we have used the definitions of \Eqref{defs} and introduced,
\begin{equation}
C_{N\De} = \frac{4 g_A h_A}{(8 \pi f_\pi)^2}
\frac{Q_+^2 r^2}{3 \, (1+r)}.
\end{equation}

Finally, the integration over $x$ and some of the $y$-integration can be
done analytically, see Appendix A.

\subsection{Dispersion method}

Alternatively, we can  compute
the correction to the $\gamma N \Delta$ vertex
by exploiting the {\it analyticity} of the loop
contributions. We use the so-called
{\it sideways dispersion relations}~\cite{bincer60,HPV05},
and hence start with calculating
the absorptive (imaginary) part of the $\gamma N \Delta$ vertex
from the cut in the $\pi N$-loop diagrams of Fig.~\ref{fig:gandelabs}.
Subsequently, the real part of the on-shell
$\gamma N \Delta$ vertex is computed through the dispersion relation
in the $\Delta$-resonance 4-momentum squared ${p'}^2$.

We first express the half off-shell $\gamma N \Delta$
vertex (where we allow the $\Delta$ to be off-shell) in terms of
the invariants $g_i(s = {p'}^2, Q^2)$, for $i = 1, 2, 3$, as~:
\begin{eqnarray}
\Gamma^{\alpha \mu}  \, u(p)
&=&\, \left({\cal P}^{3/2}_+ (p^\prime)\right)^\alpha_{\; \; \beta} \,
\left\{\,
\, \left( \gamma^\mu q^\beta \, -\, \gamma \cdot q \, g^{\beta \mu} \right)
\, g_1^+(s,\, Q^2) \right. \nonumber \\
&&\hspace{2.5cm} + \left(
q \cdot p^\prime \, g^{\beta \mu} - q^\beta \, p^{\prime \, \mu} \right)
\, g_2^+(s,\, Q^2)
\nonumber \\
&&\left. \hspace{2.5cm}
+ \left(q^\beta \, q^\mu - q^2 \, g^{\beta \mu} \right) \,
g_3^+(s,\, Q^2) \; \right\}  i \, \gamma_5 \, u (p) \nonumber \\
&+& \left({\cal P}^{3/2}_- (p^\prime)\right)^\alpha_{\; \; \beta}  \,
\left\{\,
\, \left( \gamma^\mu q^\beta \, -\, \gamma \cdot q \, g^{\beta \mu} \right)
\, g_1^-(s,\, Q^2) \right. \nonumber \\
&&\hspace{2.5cm} + \left(
q \cdot p^\prime \, g^{\beta \mu} - q^\beta \, p^{\prime \, \mu} \right)
\, g_2^-(s,\, Q^2)
\nonumber \\
&&\left. \hspace{2.5cm}
+ \left(q^\beta \, q^\mu - q^2 \, g^{\beta \mu} \right) \,
g_3^-(s, \,Q^2) \; \right\}  i \, \gamma_5 \, u(p) ,
\label{eq:regndel}
\end{eqnarray}
where the spin-3/2 positive (negative) energy projectors are given by
\begin{eqnarray}
\left( {\cal P}^{3/2}_\pm (p^\prime) \right)_{\alpha \beta} =
\frac{(\gamma \cdot p^\prime \pm M_\Delta)}{2 \, M_\Delta} \,
\left\{ g_{\alpha \beta} - \frac{1}{3} \gamma_\alpha \gamma_\beta
- \frac{1}{3 \, {p'}^2 }
\left( \gamma \cdot p^\prime \gamma_\alpha p^{\prime}_{\beta}
+ p^{\prime}_\alpha \gamma_\beta \gamma \cdot p^\prime  \right)
\right\}.
\end{eqnarray}

We next determine the real parts of the invariants $g_i^+$
for $s = M_\Delta^2$ (i.e., the $\Delta$ is on-shell)
through a dispersion relation in $s$ as~:
\begin{eqnarray}
\mathrm{Re} \, g_i^+ (M_\Delta^2, Q^2) = \frac{1}{\pi} \,
{\mathrm P} \!\int\limits_{s_{th}}^\infty \! d s \,
\frac{\mathrm{Im} \, g_i^+ (s, Q^2)}{s - M_\Delta^2},
\end{eqnarray}
where the integration starts from the $\pi N$ threshold
$s_{th} =(M_N + m_\pi)^2$, and
where ${\mathrm P}$ denotes the principal value integration.
These unsubstracted dispersion integrals do not converge. However,
the dispersion relations for the following differences
\begin{eqnarray}
\eqlab{subdr}
\mathrm{Re} \, g_i^+ (M_\Delta^2, Q^2) - \mathrm{Re} \, g_i^+ (M_\Delta^2, 0)
= \frac{1}{\pi} \,
{\mathrm P}\!\int\limits_{s_{th}}^\infty \!
 d s \,
\frac{\mathrm{Im} \, g_i^+ (s, Q^2)
- \mathrm{Im} \, g_i^+ (s, 0)}{s - M_\Delta^2},
\label{eq:subtdr}
\end{eqnarray}
do converge, for all the considered contributions. This subtraction can be put in correspondence with the
renormalization of appropriate low-energy constants from the Lagrangian~\eref{lagran}.

The expressions for the
absorptive (imaginary) parts of these form factors are calculated
in Appendix \ref{sec:phase} with the results given
by Eqs.~\eref{gipi}, \eref{giN}.
The dispersion integrals in \Eqref{subdr} are evaluated numerically.
The physical form factors are then simply given by
$g_i(Q^2) = g_i^+(M_\Delta^2, Q^2)$.

We compared the thus obtained results with
the  Feynman-parameter method results.
We found perfect agreement between both methods
for all three $\gamma N \Delta$ transition form factors, which
provides a cross-check on our calculations.

\section{Errors due to neglect of N$^2$LO effects}
\label{sec4}

Prior to presenting the results of our NLO calculation,
we would like to make an estimate of the theoretical uncertainty
due to the neglect of higher-order effects. Some of the
next-next-to-leading order (N$^2$LO)
contributions are shown in \Figref{NNLOgraphs}. Of course,
there is no substitute for an actual calculation of those
effects, but at present we only know that they must be suppressed
by at least one power of $\delta} \def\De{\Delta$ ($=\De/\La_{\chi SB}$)
as compared to the NLO and two powers of $\delta} \def\De{\Delta$
as compared to the LO contributions. Therefore, we can {\it estimate}
the size of the N$^2$LO contribution to an amplitude $A$ as:
$A_{\mathrm{NLO}}\, \delta} \def\De{\Delta $, or $A_{\mathrm{LO}}\, \delta} \def\De{\Delta^2 $. The theoretical
uncertainty of a calculation up to and including NLO can thus be
estimated as:
\begin{subequations}
\eqlab{naive}
\begin{equation}
A_{err} = | A_{\mathrm{LO}} + A_{\mathrm{NLO}}| \, \delta} \def\De{\Delta^2.
\end{equation}
In cases where the amplitude does not receive any LO contributions,
we have no other option than,
\begin{equation}
A_{err} = | A_{\mathrm{NLO}}| \, \delta} \def\De{\Delta.
\end{equation}
\end{subequations}

This looks nice and simple, however, there are a few caveats in the
implementation of such an estimate. First of all, although we have
introduced $\delta} \def\De{\Delta$ as $\De/\La_{\chi SB}$, it counts as well
the other small scales of the theory, $m_\pi$ and $Q^2$.
Therefore, we shall estimate the error using the following
expansion parameter (assuming $\La_{\chi SB}\sim M_N$):
\begin{equation}
\tilde \delta} \def\De{\Delta = \frac{1}{3}
 \left[\frac{\De}{M_N} +
\left(\frac{m_\pi}{M_N}\right)^{1/2}
+ \left(\frac{Q^2}{M_N^2}\right)^{1/2} \right],
\end{equation}
where all the light scales are treated on equal footing and
hence are averaged over.

Secondly, what if the amplitude happens to vanish at some
kinematical point. According to \Eqref{naive} the theoretical calculation
at that point would be perfect, which is of course  unlikely to be true
in reality. So, when considering dependencies on kinematical variable(s),
we shall take an {\it average of the error} over some appropriate
region of that variable.

Given these two points, we are led to the following
formula for the theoretical uncertainty of the NLO
calculation for an amplitude $A$,
\begin{equation}
\eqlab{Aerror}
A_{err} = \left\{ \begin{array}{cc}
|A|_{av}  \, \tilde\delta} \def\De{\Delta^2, & \mbox{LO}\neq 0 \\
|A|_{av}  \, \tilde\delta} \def\De{\Delta, & \mbox{LO} = 0,\,
\end{array}\right.
\end{equation}
and the subscript ``{\it av}'' indicates
that the appropriate averaging is performed.

The theoretical uncertainty of the NLO
calculation
of an observable $O$ is:
\begin{equation}
\eqlab{Oerror}
O_{err} = \left\{ \begin{array}{cc}
2 |O|_{av}  \, \tilde\delta} \def\De{\Delta^2, & \mbox{LO}\neq 0 \\
2 |O|_{av}  \, \tilde\delta} \def\De{\Delta, & \mbox{LO} = 0,\,
\end{array}\right.
\end{equation}
where the factor of 2 takes into account that an observable
is a product of two amplitudes.

Note that this error estimate differs from the one presented
in our earlier paper~\cite{Pascalutsa:2005ts}, where we overlooked the case
of vanishing LO contributions. This underestimated the
error in some observables, as will be discussed in more detail below.

\section{Results and discussion}
\label{sec6}

We are now in position to discuss the NLO results for
pion photo- and electroproduction amplitudes and observables.
We begin with the multipole-analysis for pion photoproduction.
The resonant photoproduction multipoles are well-established,
and we use them to determine the two low-energy constants:
$g_M$ and $g_E$, the strength of the $M1$ and $E2$
$\gamma N \Delta$ transitions.
Subsequently, we discuss the results for pion
{\it electro}production: cross sections,
multipoles and the $Q^2$ dependence of the $R_{EM}$ and $R_{SM}$ ratios.
The comparison with pion electroproduction observables allows to determine
the third low energy constant in the $\chi$EFT\ framework, related to the
strength of the $C2$ $\gamma N \Delta$ transition.
Once the three low energy constants are determined,
the $Q^2$- and $m_\pi$-dependencies follow as a prediction of the NLO result.
We discuss the  predictions for the $m_\pi$-dependence of the magnetic
dipole $\gamma N \Delta$ form factor, the $R_{EM}$ and $R_{SM}$ ratios,
and compare with the recent lattice-QCD results.

\subsection{Pion photoproduction}

In Fig.~\ref{fig:gap_pin_mult}, we show the result of the $\chi$EFT\  calculations
for the pion photoproduction resonant multipoles
$M_{1+}^{(3/2)}$ and $E_{1+}^{(3/2)}$, around the
resonance position, as function of the total {\it c.m.} energy $W$
of the $\pi N$ system.
These two multipoles are well established by the MAID~\cite{Drechsel:1998hk}
and SAID~\cite{Arndt:2002xv} partial-wave solutions which allow us to fit the two low-energy constants
of the chiral Lagrangian
\Eqref{lagran} as~:~$g_M = 2.9$, $g_E = -1.0$.   As is seen from the figure,
with these values the NLO results (solid lines) give a good description of the energy
dependence of the resonant multipoles in
a window of 100 MeV around the $\Delta$-resonance position.
Also, these values yield $R_{EM}= -2.3$ \%,
in a nice agreement with experiment~\cite{Mainz97}.

The dashed curves in Fig.~\ref{fig:gap_pin_mult}
show the contribution of the $\Delta$-resonant diagram of \Figref{diagrams}(a)
{\it without} the NLO vertex corrections \Figref{diagrams}(e, f).
For the $M_{1+}$ multipole this is the LO contribution.
For the $E_{1+}$ multipole
the LO contribution is absent [recall that the $g_E$ coupling
is of one order higher than $g_M$, see \Eqref{lagran}].
Hence,  the dashed curve represents
a partial NLO contribution to $E_{1+}$ therein.

Upon adding the non-resonant  Born graphs \Figref{diagrams}(b) to the
dashed curves we obtain the dotted curves in
Fig.~\ref{fig:gap_pin_mult}. These non-resonant contributions are purely
real at this order and do not affect the imaginary part of the multipoles.
One sees that the resulting calculation is flawed because the real
parts of the resonant multipoles now fail to cross zero at the resonance
position and hence unitarity, in the sense of Watson's theorem~\cite{Wat54},
is violated.\footnote{Recall that Watson's
theorem is a simple statement of unitarity based on a coupled-channel
scattering equation to leading order in the electromagnetic
interaction. The theorem relates the phase of a
photoproduction multipole $M_l^{(\gamma} \def\Ga{{\it\Gamma} N)}$
to a corresponding (in spin and parity)
$\pi N$-scattering phase-shift $\delta} \def\De{\Delta^{(\pi N)}_l$:
$$
M_l^{(\gamma} \def\Ga{{\it\Gamma} N)} = | M_l^{(\gamma} \def\Ga{{\it\Gamma} N)} | \, e^{i\delta} \def\De{\Delta^{(\pi N)}_l}\,.
$$
For the resonant channel the corresponding phase-shift
crosses 90 degrees
at the resonance position, and thus the real
part of the resonant multipoles must vanish.}
The complete NLO calculation, shown by the solid curves in the figure
includes in addition the vertex
corrections \Figref{diagrams}(e, f), which restore unitarity
{\it exactly}. This may come as a surprise, since we are doing a perturbative
calculation, without a resummation of rescattering contributions.
Nevertheless, it not difficult to see that our
numerical result is not a fluke. {\it Watson's theorem is satisfied exactly by
the NLO, up to-one-loop  amplitude}
given the graphs in \Figref{diagrams}.

It is also interesting to examine our calculation for the non-resonant
multipoles, which all receive contributions of NLO only.
In Fig.~\ref{fig:gap_pin_nonresmult}, we show the NLO calculations for
the real parts of the non-resonant $s$-, $p$- and $d$-wave
pion photoproduction multipoles in the $\Delta(1232)$ region in
comparison with the
two state-of-the-art phenomenological multipole
solutions, MAID and SAID.
Note that at NLO in the $\delta$-expansion, the non-resonant
multipoles are purely real. The multipole solutions show
indeed that the imaginary
parts of non-resonant multipoles, around the $\Delta$ resonance,
are negligibly smaller than their real parts.
From the figure one thus sees that for most of the non-resonant
multipoles, the parameter-free NLO results,
agree fairly well with the phenomenological
multipole solutions.
The largest differences are observed for $M_{1-}^{(1/2)p}$ multipole.
This multipole corresponds with nucleon quantum numbers. The cause
of the appreciable difference in this channel is largely
due to the nucleon anomalous-magnetic-moment
contributions, which are not included in our calculation (since
they appear at N$^2$LO in the
$\delta$-expansion), but which are included in the
phenomenological solutions.

\subsection{Pion electroproduction}

We now  will present the electroproduction observables obtained from
the NLO amplitude of \Figref{diagrams}.
The five-fold pion electroproduction cross section can be
expressed as~:
\begin{eqnarray}
\frac{d \sigma}{(d E_e^\prime \, d \Omega_e^\prime)^{lab} \,
d \Omega_\pi^{c.m.}} \,=\, \Gamma_v \, \frac{d \sigma}{d \Omega_\pi^{c.m.}},
\end{eqnarray}
where the virtual photon flux factor $\Gamma_v$ is defined as~:
\begin{eqnarray}
\Gamma_v = \frac{e^2}{(2 \pi)^3}\, \frac{E_e^\prime}{E_e}\,
\frac{(W^2 - M_N^2)}{2 M_N} \,
\frac{1}{ (1 - \varepsilon)\,Q^2},
\end{eqnarray}
where  $W$ is  the invariant mass of the final $\pi N$ system,
$E_e$ (${E_e}'$) are the initial (final) electron {\it lab}
energies, and $\varepsilon$ denotes the photon polarization parameter.

The $\gamma^* N \to \pi N$ cross section
for unpolarized nucleons is expressed in terms of 5 response functions
as~:
\begin{eqnarray}
\eqlab{xsecn}
\frac{d \sigma}{d \Omega_\pi} &=&
\frac{d \sigma_T}{d \Omega_\pi}
+ \varepsilon \, \frac{d \sigma_L}{d \Omega_\pi} + \varepsilon \, \cos 2 \Phi \,
\frac{d \sigma_{TT}}{d \Omega_\pi} \nonumber \\
&+&
\sqrt{2 \varepsilon  (1 + \varepsilon) }\, \cos \Phi \,
\frac{d \sigma_{LT}}{d \Omega_\pi}
+ h \sqrt{2 \varepsilon  (1 - \varepsilon) }\, \sin \Phi \,
\frac{d \sigma_{LT}^ \prime}{d \Omega_\pi} ,
\end{eqnarray}
where $\Th_\pi$ and $\Phi$ are the pion polar and azimuthal
{\it c.m.} angles, respectively,
and $h$ denotes the electron helicity.

In \Figref{crossections} we show the NLO results for the
different virtual photon absorption cross sections
entering \Eqref{xsecn} at the resonance position, and
for $Q^2 \simeq 0.127$~GeV$^2$, where recent precision data are available.
Besides the low-energy constants $g_M$ and $g_E$, which were fixed
from the resonant multipoles in Fig.~\ref{fig:gap_pin_mult},
the only other low-energy constant from \Eqref{lagran}
entering the NLO electroproduction calculation is $g_C$.
In \Eqref{xsecn}, the main sensitivity on $g_C$ enters in
$\sigma_{LT}$. A best description of the $\sigma} \def\Si{{\it\Sigma}_{LT}$ data in
\Figref{crossections} is obtained by choosing
$g_C = -2.36$.

The theoretical uncertainty
of the NLO result is estimated by using \Eqref{Oerror}, where
the average is taken over the range of  $\Theta_\pi$.
Note that $\sigma} \def\Si{{\it\Sigma}_{LT}$ and $\sigma} \def\Si{{\it\Sigma}_{LT'}$ do not receive any LO
contributions and therefore the LO$=0$ case in \Eqref{Oerror}
must be applied in the estimate. This point was overlooked
in our first error estimates of the NLO calculation~\cite{Pascalutsa:2005ts},
which clearly led to an underestimate of the theory error
for  $\sigma} \def\Si{{\it\Sigma}_{LT}$ and $\sigma} \def\Si{{\it\Sigma}_{LT'}$.

From \Figref{crossections}, one
sees that the NLO $\chi$EFT\ calculation, within its accuracy,
is consistent with the experimental data for these observables.

The reliability of the present $\chi$EFT\ calculation
can also be tested by comparing its predictions for the non-resonant
multipoles with the phenomenological multipole solutions MAID and SAID. In
\Figref{ep_epin_nonresmult}, we show this comparison for the
non-resonant $s$- and $p$-wave pion electroproduction
multipoles at the resonance position as function of $Q^2$.
Note that there is
a considerable uncertainty in the phenomenologically extracted
$s$-wave scalar multipoles $S_{0+}^{(3/2)}$ and $S_{0+}^{(1/2)p}$ at low $Q^2$.
As one can see from the figure, our calculation is in a reasonable
agreement with the phenomenological solutions for
most of the $s$- and $p$-wave non-resonant multipoles.
 The largest discrepancy is observed in the
$Q^2$-dependence of both $E_{0+}^{(3/2)}$ and $E_{0+}^{(1/2)p}$.
This discrepancy will need to be resolved by
$s$-wave N$^2$LO corrections which grow with $Q^2$.

In \Figref{ratiosQ2} we show the $Q^2$ dependence of the ratios $R_{EM}$ and $R_{SM}$. Having
fixed the low energy constants $g_M$, $g_E$ and $g_C$, this $Q^2$ dependence follows as a prediction.
The theoretical uncertainty here (shown by
the error bands) is estimated according to \Eqref{Aerror}, for LO=$0$
case, and the average
taken over the range  of $Q^2$ from 0 to 0.2~GeV$^2$.
From the figure one sees that the
NLO calculations are consistent with the experimental data for both
of the ratios.

To see how higher-order effects may affect the $Q^2$ dependence,
we include the vector-meson type of dependence for the electric
$\gamma} \def\Ga{{\it\Gamma} N\De$ transition by the following replacement of the
low-energy constant:
\begin{equation}
\eqlab{geqsqr}
g_E \to \frac{g_E}{\left(1 + Q^2 / \Lambda_E^2 \right)^2}
\end{equation}
In contrast to the analogous effect for $g_M$, which is of NLO,
the inclusion of the $Q^2$ dependence in $g_E$ is a N$^2$LO effect.
We choose $\Lambda_E$ so as to satisfy the asymptotic condition~\cite{pQCD}: $R_{EM} \to 1$,
for $Q^2 \to \infty$. This is achieved by
the choice~: $\Lambda_E^2 = \sqrt{ - g_M(0) / g_E(0) } \, 0.71$GeV$^2$.
The dashed curve in \Figref{ratiosQ2} shows the resulting
effect of the replacement \eref{geqsqr}, as imposed on the
complete NLO result shown by the solid curves with error bands.
The fact that the dashed curves go out of the error bands at some point
indicates that our error estimate is not designed for such high $Q^2$ values.

\subsection{Chiral behavior and chiral extrapolations }

Since the low-energy constants $g_M$, $g_E$, and $g_C$ have been fixed, our
calculation can provide a prediction for the $m_\pi$ dependence of the
$\gamma N \Delta$ transition form factors. The study of the $m_\pi$-dependence
is crucial to connect to lattice QCD results, which at present
can only be obtained for larger pion masses (typically $m_\pi \gtrsim 300$ MeV).

The $m_\pi$ dependence of the nucleon and $\De$-resonance
masses, given above by  Eq.~(\ref{eq:nucpin}) and (\ref{eq:delpin2}),
are compared with lattice results in
\Figref{nucdelmass}. We constrain one of the two parameters
in Eq.~(\ref{eq:nucpin})
by the physical nucleon mass value at $m_\pi = 0.139$ GeV,
while the other parameter
is fit to the lattice data shown in the figure.
This yields~: $M_N^{(0)} = 0.883$~GeV and $c_{1N} = -0.87$~GeV$^{-1}$.
As is seen from the figure, with this two-parameter form for $M_N$,
a good description of lattice results is obtained
up to $m_\pi^2 \simeq 0.5$~GeV$^2$.

Analogously to the nucleon case,
we fix one parameter in Eq.~(\ref{eq:delpin2})
from the physical value of the $\Delta$ mass,
while the second parameter is fit to the lattice data
shown in \Figref{nucdelmass},
yielding~: $M_\Delta^{(0)} = 1.20$~GeV and
$c_{1 \Delta} = -0.40$~GeV$^{-1}$.
As well as for the nucleon,
this two-parameter form for $M_\Delta$ yields a fairly good description
of the lattice results up to $m_\pi^2 \simeq 0.5$~GeV$^2$.

In Fig.~\ref{fig:regmmpi} we examine the $m_\pi$-dependence of the
{\it magnetic} $\gamma N \Delta$-transition form factor $G_M^\ast$, in
the convention of Jones and Scadron, see \Eqref{JS}.
At the physical pion mass, this form
factor can be obtained from the imaginary part of the $M_{1+}^{3/2}$ multipole
at $W = M_\Delta$ (where the real part is zero by Watson's theorem) as~:
\begin{eqnarray}
{\rm Re} \, G_M^\ast (Q^2) =
\left( \frac{8 \, M_N^2 \, |{\bf p}_\pi^\ast| \, \Gamma_\Delta }{3 \, \alpha_{em} \, |{\bf q}^\ast|^2}  \right)^{1/2} \,
{\rm Im} \, M_{1+}^{3/2} (W = M_\Delta, Q^2),
\end{eqnarray}
where $|{\bf p}_\pi^\ast|$ ($|{\bf q}_\pi^\ast|$) denote the
pion (virtual photon) {\it c.m.} three-momenta respectively
at the resonance position, i.e. for $W = M_\Delta$.
Recall that the value of $G_M^\ast$ at $Q^2 = 0$ is
determined by the low-energy constant $g_M$.
The $Q^2$-dependence then follows as a prediction of the NLO
result, and Fig.~\ref{fig:regmmpi} shows that
this prediction is consistent with the experimental
value at $Q^2 = 0.127$~GeV$^2$ and physical pion mass.

The $m_\pi$-dependence
of $G_M^\ast$ is also completely fixed at NLO, no new parameters appear.
In Fig.~\ref{fig:regmmpi}, the result for $G_M^\ast$ at
$Q^2 = 0.127$~GeV$^2$ is shown both when the $m_\pi$-dependence of
the nucleon and $\Delta$ masses is included and when it is not.
Accounting for the $m_\pi$-dependence in $M_N$ and $M_\Delta$,
shown in Fig.~\ref{fig:nucdelmass}, apparently
changes the result for $G_M^\ast$ quite significantly.
The $\chi$EFT\  calculation,
with the $m_\pi$ dependence of $M_N$ and $M_\Delta$ included, is in
a qualitatively good agreement with the lattice data shown in the figure.
The $\chi$EFT\ result also follows
an approximately linear behavior in $m_\pi^2$,
although it falls about 10 - 15 \% below the lattice data.
This is just within the uncertainty of the NLO results.
One should also keep in mind that the present lattice simulations
are not done in full QCD, but are ``quenched'', so discrepancies
are not unexpected.

In \Figref{ratios}, we show the $m_\pi$-dependence of the ratios
$R_{EM}$ and $R_{SM}$ and compare them to lattice QCD calculations.
The recent state-of-the-art lattice calculations of
$R_{EM}$ and $R_{SM}$~\cite{Ale05} use a {\it linear},
in the quark mass ($m_q\propto m_\pi^2$), {\it extrapolation}
to the physical point,
thus assuming that the non-analytic $m_q$-dependencies are  negligible.
The thus obtained value for $R_{SM}$ at the physical
$m_\pi$ value displays a large
discrepancy with the  experimental result, as seen in \Figref{ratios}.
Our calculation, on the other hand, shows  that the non-analytic dependencies
are {\it not} negligible. While
at larger values of $m_\pi$,
where the $\Delta$ is stable, the ratios display a smooth
$m_\pi$ dependence, at $m_\pi =\De $ there is an inflection point, and
for  $m_\pi \leq \Delta$ the non-analytic effects are crucial,
as was also observed for the $\De$-resonance
magnetic moment~\cite{Cloet03,PV05}.

One also sees from \Figref{ratios} that, unlike the result for
$G_M^\ast$, there is only little difference between the $\chi$EFT\
calculations with the $m_\pi$-dependence of
$M_N$ and $M_\Delta$ accounted for, and our earlier calculation
\cite{Pascalutsa:2005ts},
where the ratios were evaluated neglecting the $m_\pi$-dependence of the masses.
This is easily understood, as the main effect due to the $m_\pi$-dependence
of $M_N$ and $M_\Delta$ arises due to a common factor in the evaluation
of the $\gamma N \Delta$ form factors, which drops out of the ratios.
One can speculate that the ``quenching'' effects
drop out, at least partially, from the ratios as well.

In \Figref{ratios} we also show the $m_\pi$-dependence of the $\gamma N \De$ transition ratios,
with the theoretical uncertainty  estimated according to \Eqref{Aerror},
for the case LO$=0$, and with
the average
taken over the range  of $m_\pi^2 $ from 0 to 0.15~GeV$^2$.
The $m_\pi$ dependence obtained here from $\chi$EFT\  clearly shows that
the lattice results for $R_{SM}$ may in fact be consistent with experiment.

\section{Conclusion}
\label{sec7}

Let us briefly go over the main points and results presented in this paper,
which is the first one in a series devoted to $\chi$EFT\ in the
$\De$(1232)-resonance region.
\begin{itemize}
\item[(i)]
We  develop an extension of chiral perturbation theory to the
$\Delta$(1232)-resonance energy region, based
on the $\delta} \def\De{\Delta$-expansion of Ref.~\cite{PP03}.
In this $\chi$EFT\ framework the expansion is done in
the small parameter $\delta} \def\De{\Delta$ equal to the excitation
energy of the resonance over the chiral symmetry-breaking scale.
The other low-energy scale of the theory, the pion mass,
counts as $\delta} \def\De{\Delta^2$, which is a crucial point for an adequate counting
of the $\De$-resonance contributions in both the low-energy
and the resonance energy regions.
\item[(ii)]
This framework has been applied here to the process of
pion electroproduction.
This is a first $\chi$EFT\ study of this reaction
in the $\Delta(1232)$-resonance region.
We have performed a complete calculation of this process in the
resonance region up to,
and including, next-to-leading order in the  $\delta} \def\De{\Delta$-expansion.
The power counting in $\delta} \def\De{\Delta$ has only been used to establish which graphs
contribute at the leading and next-to-leading order,
no actual expansion of the diagrams themselves is being done.
Therefore, some higher-order in $\delta} \def\De{\Delta$ effects, required
by relativity and analyticity, are automatically included.
Such effects are known to
improve the convergence and
extend the region of applicability of $\chi$EFT\ calculations.
\item[(iii)] Our NLO calculation of pion electroproduction
satisfies {\it gauge and chiral symmetries} perturbatively, and
{\it Lorentz-covariance, analyticity, unitarity (Watson's theorem)} exactly.
\item[(iv)] The chiral-loop contributions to the
$\gamma N \Delta$ transition have been evaluated
using two independent techniques: the
Feynman parameter method and  the
sideways dispersion relations.
Both methods yield the same result.
\item[(v)] The only free parameters entering at this order
are the $\gamma} \def\Ga{{\it\Gamma} N\De$ couplings $g_M$, $g_E$, $g_C$ characterizing
the $M1$, $E2$,  $C2$ transitions, respectively.
By comparing our NLO results with the standard
multipole solutions (MAID and SAID) for the photoproduction multipoles
we have extracted $g_M$ = 2.9 and
$g_E = -1.0$, corresponding to $R_{EM} = -2.3$~\%.
The NLO $\chi$EFT\
result was also found to give a good description of the
energy-dependence of most non-resonant $s$, $p$ and $d$-wave
photoproduction multipoles in a 100 MeV window
around the $\Delta$-resonance position.
From the pion electroproduction cross-section $\sigma} \def\Si{{\it\Sigma}_{LT}$
we have extracted $g_C = -2.36$, which yields $R_{SM} \simeq -7$~\%
near $Q^2 = 0.1$~GeV$^2$. In overall, the NLO
results are consistent with the experimental data
of the recent high-precision measurements at MAMI and BATES.
\item[(vi)]
The $\chi$EFT\ framework plays a {\it dual role}
in that it allows for
an extraction of resonance parameters from observables {\em and}
predicts their pion-mass dependence. In this way it may provide
 a crucial connection of present lattice QCD results (obtained
at unphysical values of $m_\pi$) to the experiment.
We have shown here that the opening of the
$\De\to \pi N$ decay channel at $m_\pi = M_\De-M_N$
induces a pronounced  non-analytic
behavior of the $R_{EM}$ and $R_{SM}$ ratios.
While the linearly-extrapolated lattice QCD results
for $R_{SM}$ are in disagreement with experimental data, the
$\chi$EFT\ prediction of the non-analytic dependencies suggests that
these results are in fact consistent with experiment.
\item[(vii)]
The present calculation is systematically improvable.
We have indicated what are the next-next-to-leading order effects,
however,  at present we could
only estimate the theoretical uncertainty of our calculations
due to such effects. We have defined and provided a corresponding
error band on our NLO results. An actual calculation of
N$^2$LO effects is a worthwhile topic for a future work.
\end{itemize}

As high-precision data for low-$Q^2$ pion electroproduction
in the $\Delta$-resonance region become available
from BATES, MAMI and JLab,
and the  next-generation lattice calculations of the
$\gamma N \Delta$ transition are on the way~\cite{Alexandrou:2005em},
the $\chi$EFT\ presented here makes a promise
to be the theoretical framework to
examine and connect these results.

redpajama_set_name: RedPajamaArXiv


text:
Anne Frank Unbound
Media, Imagination, Memory
Edited by Barbara Kirshenblatt-Gimblett and Jeffrey Shandler
Series: The Modern Jewish Experience
Format: Paperback 15 color illus., 42 b&w illus.
As millions of people around the world who have read her diary attest, Anne Frank, the most familiar victim of the Holocaust, has a remarkable place in contemporary memory. Anne Frank Unbound looks beyond this young girl's words at the numerous ways people have engaged her life and writing. Apart from officially sanctioned works and organizations, there exists a prodigious amount of cultural production, which encompasses literature, art, music, film, television, blogs, pedagogy, scholarship, religious ritual, and comedy. Created by both artists and amateurs, these responses to Anne Frank range from veneration to irreverence. Although at times they challenge conventional perceptions of her significance, these works testify to the power of Anne Frank, the writer, and Anne Frank, the cultural phenomenon, as people worldwide forge their own connections with the diary and its author.
Barbara Kirshenblatt-Gimblett is University Professor of Performance Studies and Affiliated Professor of Hebrew and Judaic Studies at New York University. Her books include (with Mayer Kirshenblatt) They Called Me Mayer July: Painted Memories of a Jewish Childhood in Poland before the Holocaust and The Art of Being Jewish in Modern Times. She currently leads the exhibition development team for the Museum of the History of Polish Jews in Warsaw.
Jeffrey Shandler is Professor of Jewish Studies at Rutgers University. He is author of Adventures in Yiddishland: Postvernacular Language and Culture and While America Watches: Televising the Holocaust, editor of Awakening Lives: Autobiographies of Jewish Youth in Poland before the Holocaust, and editor (with Hasia R. Diner and Beth S. Wenger) of Remembering the Lower East Side (IUP, 2000).
"Looking beyond the most familiar victim of the Holocaust at the numerous ways people have engaged her life and writing."
"This astute collection of essays unpacks the complexity of Anne Frank as the source of numerous editions, artworks, musical compositions, plays, films, novels, souvenirs, memorials, museums, and so on. It is a superb pedagogical tool for examining the particular impact of Anne Frank's story as well as the larger issue of how cultural icons are constructed, circulate, and inspire engagement."
— Marita Sturken, New York University
"In their highly readable, illuminating essays, a cutting-edge cast of scholars explores the Anne Frank Phenomenon--the myriad mediations that one little plaid notebook has inspired over the last seven decades. . . . Rich and penetrating, Anne Frank Unbound brings into focus the world's ongoing engagement with a girl who has been constantly reinvented as paradigm, paragon, and even parody."
— Alisa Solomon, Columbia University
"A brilliantly conceived and long overdue opening up [or deconstruction] of the Anne Frank story."
— James Clifford, Professor Emeritus, History of Consciousness Department University of California, Santa Cruz
"These essays provide fresh takes on Anne Frank, her diary, and historical magnitude. The volume leaves the reader in a new place of discovery and awareness."
— Shelley Hornstein, coeditor of Image and Remembrance: Representation and the Holocaust (IUP, 2002).
"This collection of brilliant essays offers fascinating and unexpected insights into the significance of Anne Frank's iconic Holocaust-era diary from many disciplinary perspectives in the arts and humanities."
— Jan T. Gross, Princeton University
"This engrossing collection of 12 interdisciplinary essays covers multiple aspects of 'the Anne Frank phenomenon' . . . The overall aim is to provide a greater understanding of the general and particular engagement with Anne Frank as a person, a symbol, an icon, an inspiration, and perhaps most polarizing, as one victim, not the victim of the Nazi holocaust."
— Broadside
"Principally the work of senior international scholars in history, literature, Hebraic and Judaic studies, and performance and film studies, and of museum curators, this volume is a major contribution to scholarship regarding Anne Frank's diary and its cultural influence. . . . Highly recommended. "
— Choice
"Anne Frank Unbound . . . tell[s] us a great deal about the myriad uses to which one individual story has been and can be put. . . . In addition to these ethical and political questions, the essays engage productively with the aesthetic choices made by writers, visual artists, filmmakers, performance artists, and comedians, who recast Anne Frank in a variety of media and situations. . . . If Anne Frank Unbound is any indication, the diary will certainly continue . . . to raise a set of persistent ethical, political, and aesthetic questions that have been with us since its first publication. "
— Women's Review of Books
Introduction: Anne Frank, The Phenomenon
Part I: Mediating
1. From Diary to Book: Text, Object, Structure Jeffrey Shandler
2. From Page to Stage Edna Nahshon
3. In Moving Images Leshu Torchin
Part II: Remembering
4. Hauntings and Sitings in Germany Henri Lustiger Thaler and Wilfried Wiedemann
5. Teaching Anne Frank in America Ilana Abramovitch
6. Anne Frank as Icon, from Human Rights to Holocaust Denial Brigitte Sion
7. Anne Frank, a Guest at the Seder Liora Gubkin
Part III: Imagining
8. Literary Afterlives of Anne Frank Sara R. Horowitz
9. Suturing In: Anne Frank as Conceptual Model for Visual Art Daniel Belasco
10. Sounds from the Secret Annex: Composing a Young Girl's Thoughts Judah M. Cohen
Part IV: Contesting
11. Critical Thinking: Scholars Reread the Diary Sally Charnow
12. Anne Frank on Crank: Comic Anxieties Edward Portnoy
Epilogue: The Anne Frank Tree—A Life of Its Own Barbara Kirshenblatt-Gimblett
Musicography Judah M. Cohen
Videography Aviva Weintraub
Connect with the Anne Frank Unbound Facebook page
Read an excerpt from the book
Listen to an IU Press podcast with Jeffrey Shandler:
Today I Am a Woman
The End of the Holocaust

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text:
Home North Carolina Wendell
Wendell, NC Florists
Find Florists in Wendell
Designs By Donna
36 N MAIN ST, Wendell, NC 27591
Flowers By Mike
18 E 3rd St, Wendell, NC 27591
Flynt's Florist
53 Hillsboro Street
Pittsboro , NC 27312
Dandelions All Things Wedding and Events
226 N. Main St
Walnut Cove , NC 27052
All Occasions Florist
1205 MECKLENBURG HWY
Mooresville , NC 28115
The Floral Boutique
385 Maple Springs Lane
Bear Creek , NC 27207
Wendell Flower Shop News
The 5 best florists in Kansas City - Hoodline
Baltimore Ave., is another top choice, with Yelpers giving the florist and event planning spot 4.5 stars out of 21 reviews.3. Toblers FlowersPhoto: randy m./YelpToblers Flowers, a florist in Wendell Phillips, is another go-to, with four stars out of 30 Yelp reviews. Head over to 2010 E. 19th St. to see for yourself.4. Trapp And CompanyPhoto: trapp and company/YelpOver in Old Westport, check out Trapp And Company, which has earned four stars out of 19 reviews on Yelp. You can find the florist, which offers women's clothing, home decor and more, at 4110 Main St. 5. Studio Dan MeinersPhoto: studio dan meiners/YelpLast but not least, there's Studio Dan Meiners, a Westside South favorite with four stars out of 15 reviews. Stop by 2500 W. Pennway St. to hit up the florist and event space next time you're looking to score some flowers.This story was created automatically using local business data, then reviewed and augmented by an editor. Click here for more about what we're doing. Got thoughts? Go here to share your feedback. https://hoodline.com/2019/04/the-5-best-florists-in-kansas-city
Vandals flipping flower pots in Augusta - Ledger Independent
One council member, during a recent meeting, asked if there were any ordinances on the books that might help to curb the behavior.Augusta Mayor Wendell High said there is an ordinance requiring anyone under the age of 16 to be off the city streets by a certain hour."I know it's later on weekends than weekdays," he said. "But, I know we have something on the books about people under 16 having to be off the streets by a certain time."Anyone with information on the vandalism is asked to contact the Augusta Police Department at 606-756-3296. http://www.maysville-online.com/news/local/vandals-flipping-flower-pots-in-augusta/article_3b749db4-1cab-5e44-8797-881b3099a0b3.html
Lancaster doctor quilts a vintage flower garden - LancasterOnline
It's not cozy, but it's colorful. The stiffness comes the fact that each tiny fabric hexagon was wrapped around paper or cardstock. Wendell Zercher, a Lancasterhistory.org curator, highlighted the quilt in a talk on the history of quilting in Lancaster County.The pattern is called grandmother's flower garden, but it's also known as mosaic or grandmother's garden. Hexagon patterns became one of the most popular quilt styles in England by 1830 and in the U.S. a century later, according to the Illinois State Museum.The quilts used English paper piecing, where each strip of fabric is pinned to a paper pattern. The fabric's wrapped around the template, basted together and then pieced together to make a flower, Zercher said.Unlike many of the quilts in the museum's collection, this one was made by a man. Jacob Trexler was a doctor and surgeon in Lancaster who also worked with X-rays.He had an office on North Prince Street, near the Fulton Theater."The office was on the ground floor," Zercher said. "The second floor was where he lived. The third floor was where he quilted."The quilt has a large T at the center for Trexler. Several patches are decorated with embroidered letters, probably of family members because they end with "T."The detail to make the small patches took a lot of... http://lancasteronline.com/features/together/lancaster-doctor-quilts-a-vintage-flower-garden/article_685a26a0-535e-11e6-b956-8b19a36c7847.html
Flowers in bloom at Jersey City park make for an odd scene on Christmas Eve - NJ.com
But many of the park's visitors agreed that they'll be enjoying the holidays regardless of how unusual the weather may be. "It's not the weather that makes the holidays, it's the spirit," said Wendell Holbrook, who lives near the park on Jersey Avenue. "The festivity, family, dinner, and exchanging gifts." ... http://www.nj.com/hudson/index.ssf/2015/12/jersey_city_residents_speak_on_unusual_christmas_w.html
Sounding Off: Readers discuss Flower Mound's sex offender ordinance - Dallas Morning News
Flower Mound can't let them live life again. It's kind of stupid if you ask me.Daver Lapka, Southlake: Oliver Wendell Holmes Jr. stated that a man's right to swing his arm ends when it infringes on another person's nose — and that seems very fair. For a person to be a registered sex offender, they have to have committed a crime that violated the above rule. Government's role is to create environments that are safe and secure for all of the residents that pay taxes to live there. Since psychologists tell us they can never cure the affliction that causes individuals to behave in illegal sexual behavior, the government has a responsibility to keep those people away from those of us who haven't and might not feel safe for our families to be around them. If they don't, they are in effect sanctioning individuals to enact their own protections and retributions for criminal trespass. http://www.dallasnews.com/news/community-news/northwest-dallas-county/headlines/20151120-sounding-off-readers-discuss-flower-mounds-sex-offender-ordinance.ece


On 15 August I handed over a 105,328-word document to someone behind a welcome desk in Durham University's Palatine Centre. It was a rather unceremonious act in form.
Not in reality. And the good folks at Flat White Coffee supplied a memorable scene that was ceremonious enough (note the "receipt" of my submission next to the espresso drink).
As my wife pointed out—with a bound copy of the PhD thesis sitting between us on the dining table—that PDF is the most expensive thing we own… at least in a sense. Our lives have been hinged to the conviction that my vocational service to the church includes an academic slant. Multiple moves and costly degree programs have defined our past decade… along with the adventure of rearing 4 precious children amidst the pressures.
I will post more on the topic and arguments of the thesis later. For fun, though I would just like to point out that 17 years ago I was sitting at a desk in the O'Callaghan house reading John's Gospel and noticing a few threads that seemed worthy of further exploration.
14 years ago I started checking out PhD programs from the computer lab at Beeson Divinity School.
8 years ago I made my first (of two) exploratory trips to Durham.
And then a few weeks ago I turned in my third "book." (Thanks for indulging the historical sketch).
Pyrrhus was a Greek king who soldiered valiantly into the might and muscle of Rome in the 2nd century BC. After a brutalizing series of particular engagements, the battle dust began to settle and someone gave him the news that he was the victor.
Pyrrhus did not feel very victorious.
In fact, he felt messed up, broken down, and demoralized. To gain this "victory" he had sustained massive losses. Though most of the 15,000 corpses lying across the outskirts of Asculum belonged to the Romans, the Greek body count was grievously high (and the Romans had been much better resourced).
"Pyrrhus replied to one that gave him joy of his victory that one more such victory would utterly undo him. For he had lost a great part of the forces he brought with him, and almost all his particular friends and principal commanders…." (see here for Plutarch's biographical sketch).
I have been a bit of a drama queen over the difficulties of pursuing the PhD. (Please forgive me). But to be absolutely clear: it is miserably hard (though the academic stuff was often less difficult than the circumstantial).
Though a keen sense of the sacrifice in writing that these certainly accompanies its submission, I am delighted to report that over the following days it gradually occurred to me that breathing was a bit easier (figuratively), as if I had been sucking in air for years with a boulder on my chest only recently lifted. The increased oxygen supply has been wonderful. Unburdened by the thesis, my lungs have been able to expand a bit.
Amidst sleeping a bit extra and reading fiction mostly guilt-free, I am now in quest for a job. And the viva looms nearer each day, for which I feel I must have John's Gospel memorized in the Greek, along with all the writings of Alexandrian theologians in the first few centuries of the church. These scholarly endeavors are impossible feats, of course (at least for me).
But overall I am pleased to report a general sense of relief, and possibly an increased degree of sanity.
I feel almost 105,328 words lighter.
Still reeling from todays news! Very proud of you, Son-in-Law!


text:
info@linksinc.org
National Presidents
The Links, Incorporated
The Links Foundation
Program Reports
Program Collaborators
Brand Vendor Application
Certified Vendors List
Gwendolyn B. Lee
The Links, Incorporated / Gwendolyn B. Lee
14th National President
Gwendolyn B. Lee has over 38 years experience in secondary education. She retired in December 2007 from Thornton Township High School, District 205, where she served as the Associate Superintendent for Curriculum and Instruction and Professional Development and as the Director for Special Education and the District Coordinator for Small Learning Communities. She was named Principal of the Year for the State of Illinois in 1998 and Administrator of the Year in 2003. She received the Dare to Be Great Award in 2006 from the Illinois Women Administrators Organization. Dr. Lee currently serves as a Leadership Coach for New Leaders for New Schools.
Dr. Lee has a 4o plus year record of active participation in The Links, Incorporated. She is a charter member of the South Suburban Chicago (IL) Chapter located in the Central Area. In 1994, Dr. Lee was appointed as Director of National Trends and Services and served in that capacity until 1998. She has held four National offices; National Member-At-Large (1986-1990), National Nominating Committee Chairperson (1990-1994), National Recording Secretary (1998-2002), and National Vice President (2002-2006). Additional career highlights of Dr. Lee are listed below;
Adjunct Professor Governors State University 2002-Present
Adjunct Professor National-Louis University 2001-Present
Associate Superintendent Thornton Township High School, District 205 1999-Present
Principal Thornridge High School, Dolton, IL 1994-1999
Principal Hillcrest High School, Country Club Hills, IL 1992-1994
Assistant Principal Thornridge High School 1985-1992
Dean of Students Thornridge High School 1981-1985
Teacher Thornridge High School 1971-1981
EDUCATIONAL BACKGROUND:
Ph.D. – Fall 1998 Loyola University (Chicago) – Education Administration
Administrative Certificate, 1985 – Purdue University, West Lafayette, IN
Master of Science Degree in Education, 1975 – Purdue University, West Lafayette, IN
Bachelor of Science Degree in History and Political Science, 1971 – Ball State University, Muncie, IN
IASCD Board of Directors, 2001- Present
South Suburban IASCD, President, Treasurer – Present
South Suburban Curriculum Council, 2002 – Present
American Association of School Administrators, 1999 – Present
State Superintendent's Task Force School Designation, 2001
Executive Leadership Institute (Motorola) I & II Design Team Member, 1996-2001
Illinois Principal's Association State Director, 1997-2000
National Staff Development Council, 1994 – Present
National Association of Secondary School Principals, 1992 – 1999
Association for Supervision and Curriculum Development, 1992 – Present
Illinois Principal's Association – Region VII, Publicity Director, 1992; Director, 1996
State Supt. Advisory Board, 2001-2002
LEADERSHIP ACTIVITIES:
Leadership Coach – New Leaders for New Schools, 2008-Present
Resident Coach – New Leaders for New Schools, 2007-2008
National Staff Development – Consultant for School Professional Development Audits, JoEllen Killion, Chair, 2001-present
IL Gates Coalition for Systems Change and Technology Integration in Schools, State Trainer with Dr. Diane Asbey and Lynda Irvin, 2001- present
Illinois Principal's Association Aspiring Principals Workshop – Presenter since 1994
H.O.P.E. Foundation Site Coordinator for Alton High School, 2002-2004 (Harnessing Optimism and Potential Through Education)
Thornton Township High School's Leadership Academy, Developer and Facilitator, 1999-present
Institute Day Presenter, Hillcrest High School, Teaming and How to Write Smart Goals
Consultant for QLD Learning, 2004 – present
High Schools That Work – Presenter, National Conference, 2004
Connections Conference, Presenter, The Administrative Side of Implementing High Schools That Work and Small Learning Communities, 2004
U.S. Department of Education and Office of Vocational Education, Presenter and Coach, Smaller Learning Communities Leadership Institute, San Francisco, California and Atlanta, Georgia, 2005
Illinois Women Administrators 2005 Dare To Be Great Honoree
ISC4 Administrator of the Year – Excellence in Education Award 2003
Herman Graves Service Award (IPA, South Cook) -1999
State of Illinois Principal of the Year – 1998-99
Outstanding Staff Developer – Thornridge High School Faculty -1998
Exceptional Performance & Contributions, presented by Educational Talent Search – 1995
I.P.A. Region 7 Nominee for Met Life Principal of the Year Award – 1994 & 1996
Outstanding Young Woman of America – 1984
Honored Faculty Member – 1983
COMMUNITY SERVICE:
The South Suburban Family Shelter Board of Directors
The Links, Incorporated Board of Directors and National President
South Suburban Chicago Chapter of The Links, Incorporated
Alpha Kappa Alpha Sorority
University of Illinois Cooperative Extension, State Advisory Committee
The Society for The Arts
Thornton Township High Schools' Leadership Academy Training Manual
The Study of Protégé's Perceptions of Mentoring Relationships in Secondary Schools – dissertation
Constructing a Mentoring Program – soon to be released.
The Links, Incorporated is an international, not-for-profit corporation, established in 1946. It is one of the nation's oldest and largest volunteer service organizations.
Visit Us: 1200 Massachusetts Avenue, NW | Washington, DC 20005
Civil Rights Icon John Lewis Lauds The Links, Incorporated and Issues Voting Rights Call to Action


text:
Providence's Parkside restaurant must pay $79,000-plus in back wages, overtime
Katie Mulvaney
kmulvane@providencejournal.com
The Parkside Rotisserie & Bar and its owner must pay more than $79,000 in back wages and damages for failing to pay its workers the federal minimum wage and overtime pay.
Steven Davenport and S&D Restaurant, doing business as the Parkside on South Main Street in Providence, this week entered a consent judgment and order with the federal government to pay $79,246, including $39,623 in gross back wages and $39,623 in damages to more than 30 employees.
The order specifies that the owner agreed not to report or threaten to report workers to immigration or law-enforcement authorities, or threaten to have them deported or investigated by immigration or law-enforcement as a means to deter them from raising fair-wage complaints.
U.S. Secretary of Labor Eugene Scalia filed a complaint and consent judgment against the Parkside in U.S. District Court on Thursday following a year-long investigation.
The restaurant, which is popular with lawyers working nearby at the Licht Judicial Complex, failed to pay workers federal minimum wage of $7.25 an hour and overtime wages from Oct. 15, 2017, through Jan. 12, according to the U.S. Department of Labor.
Federal authorities also accuse Davenport of discriminating against his employees by threatening to retaliate against them if they spoke to investigators about their pay and the hours they worked. They say he unlawfully fired a sous chef as well.
The restaurant failed to maintain accurate financial records documenting the hours worked and pay received by the employees.
Joseph J. Pezza, Davenport's lawyer, declined comment.
On Twitter:@kmulvane


text:
Oil slides as Trump impeachment prospect, Saudi supply weighs
Business Reuters Sep 27, 2019 00:06:05 IST
By Laila Kearney
NEW YORK (Reuters) - Oil fell on Thursday as new developments in an inquiry into the impeachment of U.S. President Donald Trump weighed on demand sentiment while moves to quickly restore Saudi output after attacks on its oil installations promised more oil supply.
The U.S. House Intelligence Committee released a declassified version of a whistleblower report alleging Trump used his office to solicit interference in the 2020 presidential election from a foreign country.
"When the odds of impeachment go down, the market goes up. When the odds of impeachment goes up, it goes down," said Phil Flynn, an analyst with Price Futures Group in Chicago. "The market doesn't like the prospect of impeachment - that's going to be a negative for the U.S. economy, it's going to be a negative on U.S.-China trade."
Both benchmarks fell for a third straight day, with Brent crude futures down 39 cents, or 0.6%, at $62.00 a barrel and U.S. West Texas Intermediate (WTI) crude 65 cents, or 1.2%, lower at $55.84 a barrel by 11:47 a.m. EDT (1547 GMT).
Prices have been weighed down by the faster-than-expected recovery of Saudi output after the drone and missile strikes on two of its oil-processing plants, as well as a surprise 2.4-million-barrel build in U.S. crude inventories last week.
Saudi Arabia, the world's top oil exporter, has restored its production capacity to 11.3 million barrels per day, sources briefed on state oil company Saudi Aramco's operations told Reuters.
Comments by Trump on Wednesday, which signalled that a resolution to the U.S. trade dispute with China might be near, helped limit losses.
A day after delivering a stinging rebuke to China over its trade policies, Trump said Beijing wanted to make a deal and it "could happen sooner than you think."
Trump and Japanese Prime Minister Shinzo Abe also signed a limited trade deal that would open Japanese markets to $7 billion of U.S. products annually.
"The oil market has seemingly returned to business as usual," said Norbert Ruecker, head of economics and next-generation research at Julius Baer.
"Instead of the attack-related fallout including disruption and geopolitical risks, the soft economy and stagnant oil demand are back in focus."
Crude futures were pressured by sluggish economic data in leading European economies and Japan.
"There's not too much to be cheery about on oil markets today," said Jeffrey Halley, senior market analyst for Asia Pacific at OANDA.
(Additional reporting by Noah Browning in London and Roslan Khasawneh in Singapore; Editing by Marguerita Choy and Mark POtter)
1Oil slides as Trump impeachment prospect, Saudi supply weighs