text:
\section{INTRODUCTION}

Aspects of the $k_T$ factorization theorem
\cite{CCH,CE,LRS,BS,LS,HS} in perturbative QCD have been
investigated intensively. One of the important issues is about the
derivation of a $k_T$-dependent hard kernel at subleading level,
which is defined as the difference between QCD diagrams and
effective diagrams for transverse-momentum-dependent (TMD) hadron
wave functions. We have explained that partons in both sets of
diagrams should remain off mass shell by $k_T^2$ in the $k_T$
factorization theorem \cite{NL2}. The same statement has been made
in the application of the $k_T$ factorization theorem to inclusive
processes such as prompt photon production \cite{LMZ11}. The
off-shellness of partons may cause concern of the gauge invariance
\cite{FMW08,FMW11}\footnote{Criticisms raised in \cite{FMW08,FMW11}
have been responded in \cite{LM09,NL09}.}. However, we have shown
that the gauge dependence cancels between the above two sets of
diagrams, and a $k_T$-dependent hard kernel is gauge invariant
\cite{LM09,NL07}. Following this prescription, the
next-to-leading-order (NLO) correction to the pion transition
(electromagnetic) form factor associated with the process
$\pi\gamma^*\to \gamma(\pi)$ has been calculated at leading twist,
i.e., twist 2 \cite{NL07,LWS10}. Here we shall study the
power-suppressed three-parton contribution to the pion
electromagnetic form factor in the $k_T$ factorization theorem. The
three-parton contribution in the collinear factorization theorem
\cite{LB80} to a simpler process, the $\rho$ meson transition form
factor, has been evaluated recently \cite{AIP08}.

We shall first demonstrate the gauge invariance of the three-parton
contribution to the pion electromagnetic form factor in the $k_T$
factorization theorem. There are two sources of gauge dependence for
this power correction \cite{Q90}: the first source is proportional
to parton transverse momenta from the two-parton Fock state. The
corresponding hadronic matrix element is written as
\begin{eqnarray} \langle
0|{\bar q}(z)\Gamma i\partial_\alpha q(0) |\pi\rangle, \label{d3}
\end{eqnarray}
where $z$ is the coordinate of the anti-quark field $\bar q$,
and $\Gamma$ represents a combination of Gamma matrices.
The second source is associated with the three-parton Fock state with an
additional valence gluon. The corresponding matrix element is given
by
\begin{eqnarray}
\langle 0|{\bar q}(z)\Gamma
gT^aA_\alpha^a(z') q(0) |\pi\rangle, \label{a3}
\end{eqnarray}
with a color matrix $T^a$, and the gluon field $A_\alpha^a$ at the
coordinate $z'$. The gauge dependences from the above two sources
cancel each other, when Eqs.~(\ref{d3}) and (\ref{a3}) are combined
to form the gauge-invariant matrix element
\begin{eqnarray}
\langle 0|{\bar q}(z)\Gamma iD_\alpha(z') q(0)
|\pi\rangle,\label{t3}
\end{eqnarray}
with the covariant derivative $iD_\alpha \equiv
i\partial_\alpha+gT^aA_\alpha^a$. The cancellation of the gauge
dependences is similar to that occurring in the collinear
factorization theorem \cite{Q90}. In the $k_T$ factorization theorem
we just keep the transverse momentum dependence in denominators of
particle propagators \cite{NL2}. Hence, it is natural that the gauge
dependence disappears at higher twists in the same way as in the
collinear factorization theorem.

Our formalism implies that contributions proportional to transverse
momenta in numerators of hard kernels must be combined with
contributions from three-parton Fock states in order to guarantee
gauge invariance. Therefore, there is concern on the study of the
pion transition form factor in \cite{CHM96}, where only the former
was included. It has been pointed out explicitly that the
leading-order (LO) hard kernel for the pion electromagnetic form
factor becomes gauge-dependent, if one simply considers parton
transverse momenta in numerators \cite{FMW11}. However, the
contribution from the three-parton Fock state was still missing in
\cite{FMW11}, such that the false postulation on the gauge
dependence of the $k_T$ factorization theorem was made.

Since both the initial- and final-state pions involve higher-twist matrix
elements like that in Eq.~(\ref{t3}), the three-parton contribution
to the pion electromagnetic form factor is suppressed at least by
$1/Q^2$, $Q^2$ being the momentum transfer squared. After examining
the gauge invariance, we calculate the $k_T$-dependent hard kernel
for the three-parton-to-three-parton scattering in the Feynman
gauge, and convolute it with the three-parton pion wave functions.
We observe that the diagrams with a four-gluon vertex dominate this
power correction. It will be shown that the three-parton
contribution is only about 5\% of the sum of those which have been
investigated before, including the LO twist-2, NLO twist-2, and LO
two-parton twist-3 ones \cite{LWS10}. That is, the three-parton
contribution is not crucial for accommodating experimental data of
the pion electromagnetic form factor. At the same power of $1/Q^2$,
one should also take into account the scattering of two (four)
partons into four (two) partons in principle. This piece has been
analyzed in light-cone sum rules \cite{BK02}, and found to be less
important than other contributions. With this work, we conclude that
the chirally enhanced two-parton twist-3 correction is the most
important $1/Q^2$ correction to the pion electromagnetic form
factor.

In Sec.~II we verify the gauge invariance of the LO three-parton
contribution to the pion electromagnetic form factor by combining
the gauge-dependent hard kernels corresponding to Eqs.~(\ref{d3})
and (\ref{a3}). The $k_T$-dependent hard kernel is then derived from
the three-parton-to-three-parton scattering diagrams, and convoluted
with the three-parton pion wave functions numerically in Sec.~III.
Section IV is the conclusion. Detailed calculations of the
gauge-dependent hard kernels corresponding to Eq.~(\ref{a3}) are
presented in Appendix A, and the expressions of the
three-parton-to-three-parton hard kernels are collected in Appendix
B.

\section{GAUGE INVARIANCE}

Consider the pion electromagnetic form factor involved in the
process $\pi(P_1)\gamma^*\to\pi(P_2)$, whose LO diagrams are
displayed in Fig.~\ref{fig1}. The momentum $P_1$ of the
initial-state pion and $P_2$ of the final-state pion are
parameterized as
\begin{eqnarray}
P_1=(P_1^+,0,{\bf 0}_T)=\frac{Q}{\sqrt{2}}(1,0,{\bf 0}_T),\;\;\;\;
P_2=(0,P_2^-,{\bf 0}_T)=\frac{Q}{\sqrt{2}}(0,1,{\bf 0}_T),
\label{mpp}
\end{eqnarray}
with $Q^2=-q^2$, $q=P_2-P_1$ being the virtual photon momentum. The
gluon propagator of momentum $l$ is written as
\begin{eqnarray}
\frac{-i}{l^2}\left(g^{\sigma\nu} - \lambda \frac{l^\sigma
l^\nu}{l^2}\right),\label{tensor}
\end{eqnarray}
in the covariant gauge, where the parameter $\lambda$ will be used
to identify sources of gauge dependence. We assume that the
anti-quarks in the initial- and final-state pions, represented by
lower fermion line, carry the parton momenta
\begin{eqnarray}
k_1=(x_1P_1^+,0,{\bf k}_{1T}),\;\;\;\; k_2=(0,x_2P_2^-,{\bf
k}_{2T}),
\end{eqnarray}
respectively, $x_1$ and $x_2$ being the momentum fractions. It is
understood that the components $k_1^-$ and $k_2^+$ have been dropped
in hard kernels, and integrated out of the TMD pion wave functions.

\begin{figure}[t]
\begin{center}
\includegraphics[height=3.3cm]{kt3f1.eps}
\caption{LO diagrams for the pion electromagnetic form factor, where
the symbol $\times$ represents the virtual photon vertex.}
\label{fig1} \vspace{-0.5cm}
\end{center}
\end{figure}

We employ the Fierz identity
\begin{eqnarray}
I_{ij}I_{lk}=\frac{1}{4}I_{ik}I_{lj}
+\frac{1}{4}(\gamma_5)_{ik}(\gamma_5)_{lj}
+\frac{1}{4}(\gamma^\alpha)_{ik}(\gamma_\alpha)_{lj}
+\frac{1}{4}(\gamma_5\gamma^\alpha)_{ik}(\gamma_\alpha\gamma_5)_{lj}
+\frac{1}{8}(\sigma^{\alpha\beta}\gamma_5)_{ik}
(\sigma_{\alpha\beta}\gamma_5)_{lj},\label{fie}
\end{eqnarray}
to factorize the fermion flow, where $I$ denotes the $4\times 4$
identity matrix. The structure $\gamma_\alpha\gamma_5$ in the above
identity contributes at twist 2 and higher twists, and $\gamma_5$
and $\sigma_{\alpha\beta}\gamma_5$ contribute at twist 3 and higher
twists at two-parton level. The matrix $\Gamma$ in Eqs.~(\ref{d3})
and (\ref{a3}) can pick up one of the above structures, among which
we focus on the first one $\gamma_{\alpha}\gamma_5$ as an example
below. We also insert the identity
\begin{eqnarray}
I_{ij}I_{lk}=\frac{1}{N_c}I_{lj}I_{ik}+2
(T^c)_{lj}(T^c)_{ik},\label{col}
\end{eqnarray}
to factorize the color flow, where $N_c=3$ is the number of colors,
$I$ denotes the $3\times 3$ identity matrix, and $T^c$ is a color
matrix. The first (second) term in Eq.~(\ref{col}) will be
associated with a color-singlet (color-octet) state of the valence
quark and anti-quark.

The first source of gauge dependence is extracted from the diagrams
in Fig.~\ref{fig1}, where the quark and anti-quark pair forms a
color-singlet state. Combining the decompositions in
Eqs.~(\ref{fie}) and (\ref{col}), we sandwich Fig.~\ref{fig1} with
the structures
\begin{eqnarray}
\frac{1}{4N_c}\gamma_{\alpha}\gamma_5,\;\;\;\;
\frac{1}{4N_c}\gamma_5\gamma_{\beta},\label{str}
\end{eqnarray}
from the initial and final states, respectively, where the
subscripts $\alpha$ and $\beta$ can take arbitrary components. The
LO hard kernel from Fig.~\ref{fig1}(a) contains the gauge-dependent
piece
\begin{eqnarray}
H^{a\lambda}=-ieg^2\lambda \frac{C_F}{16N_c}\frac{tr[ \gamma^\sigma
\gamma_5\gamma_{\beta}\gamma_\mu (\not P_1-\not
k_2)\gamma^\nu\gamma_{\alpha}\gamma_5]}
{(P_1-k_2)^2(k_1-k_2)^2}\frac{(k_1-k_2)_\sigma
(k_1-k_2)_\nu}{(k_1-k_2)^2},\label{hlam}
\end{eqnarray}
with $C_F$ being a color factor. The above expression diminishes
with the substitution $k_1= x_1P_1$, $k_2= x_2P_2$,
$\gamma_\alpha=\gamma^-$ (proportional to $\not\! P_1$), and
$\gamma_\beta=\gamma^+$ (proportional to $\not\! P_2$) into the
numerator, implying that Eq.~(\ref{hlam}) does not contribute at
leading power in the $k_T$ factorization theorem.

To obtain the gauge-dependent hard kernel from Eq.~(\ref{hlam}) at
$1/Q^2$, we insert the identity
$(k_1-k_2)_\nu=(P_1-k_2)_\nu-(P_1-k_1)_\nu$. It can be shown that
the contribution from the $(P_1-k_2)_\nu$ term is cancelled by the
corresponding one in Fig.~\ref{fig1}(b). The second term, with
$P_1-k_1$ being the momentum of the incoming valence quark,
corresponds to the matrix element with the derivative of the quark
field in the initial-state pion. This term can be picked up by
differentiating Eq.~(\ref{hlam}) with respect to $k_1$. Once
Eq.~(\ref{d3}) for the initial-state pion is identified, we
pick up the $k_{2\sigma}$ term in $(k_1-k_2)_\sigma$ via
differentiation, which
corresponds to the derivative of the valence anti-quark field in the
final-state pion. Note that denominators of particle propagators,
depending on $k_1$ and $k_2$, will be differentiated too. However,
their differentiation gives rise to even higher-twist matrix elements,
and can be neglected. We then extract
\begin{eqnarray}
H^{a\lambda}_{TT}(k_1,k_2)\equiv\frac{\partial^2 H^{a\lambda}}{\partial
k_{1\alpha}\partial k_{2\beta}} &=&ieg^2\lambda
\frac{C_F}{16N_c}\frac{tr[ \gamma^\beta \gamma_5\gamma_{\beta}\gamma_\mu
(\not P_1-x_2\not P_2)\gamma^\alpha\gamma_{\alpha}\gamma_5]}
{(P_1-k_2)^2(k_1-k_2)^4},\label{tt}
\end{eqnarray}
associated with $\langle 0|{\bar q}(z)\gamma_5\gamma^{\alpha}
i\partial_\alpha q(0) |\pi(P_1)\rangle$ and the similar matrix
element for the final-state pion.

The LO hard kernel from Fig.~\ref{fig1}(b) contains
\begin{eqnarray}
H^{b\lambda}=-ieg^2 \lambda\frac{C_F}{16N_c}\frac{tr[ \gamma^\sigma
\gamma_5\gamma_{\beta}\gamma^\nu(\not P_2-\not
k_1)\gamma_\mu \gamma_{\alpha}\gamma_5]}{(P_2-k_1)^2(k_1-k_2)^2}\frac{(k_1-k_2)_\sigma
(k_1-k_2)_\nu}{(k_1-k_2)^2},\label{hlbm}
\end{eqnarray}
whose differentiation with respect to $k_{1\alpha}$ and $k_{2\beta}$
leads to
\begin{eqnarray}
H^{b\lambda}_{TT}(k_1,k_2)&=&ieg^2\lambda \frac{C_F}{16N_c}\frac{tr[
\gamma^\alpha \gamma_5\gamma_{\beta}\gamma^\beta(\not
P_2-x_1\not P_1)\gamma_\mu\gamma_{\alpha}\gamma_5]}
{(P_2-k_1)^2(k_1-k_2)^4}.\label{ttb}
\end{eqnarray}
This hard kernel corresponds to the matrix element of the
initial-state pion with the derivative of the anti-quark field $\bar
q(z)$. Equations~(\ref{tt}) and (\ref{ttb}) represent the gauge
dependence in LO two-parton-to-two-parton scattering at power of
$1/Q^2$, which was also observed in \cite{FMW11}.

\begin{figure}[t]
\begin{center}
\includegraphics[height=2.8cm]{kt3f2a.eps}\hspace{1.0cm}
\includegraphics[height=2.8cm]{kt3f2b.eps}
\caption{Diagrams with three partons from the initial state, where
letters $A$, $B$, $\cdots$ denote the attachments of the additional
valence gluon.} \label{fig2}
\end{center}
\end{figure}

The second source of gauge dependence arises from Fig.~\ref{fig2},
where three partons appear in the initial state as indicated by
Eq.~(\ref{a3}). All possible attachments of the additional valence
gluon to the lines other than the valence quark and anti-quark in
the initial state are labeled by letters $A$, $B$, $\cdots$. In
principle, the diagrams with the attachments to the valence quark
and anti-quark in the initial state should be included in order to
respect $U(1)$ gauge symmetry for the electromagnetic
interaction\footnote{We thank V. Braun for pointing out this $U(1)$
gauge symmetry.}. These diagrams contribute to higher Gegenbauer
terms in the two-parton twist-3 pion distribution amplitudes.
Equations of motion can then be constructed to relate the
coefficients of the higher Gegenbauer terms in the two-parton
twist-3 and three-parton twist-3 pion distribution amplitudes
\cite{BF90}. Hence, one should pay attention to the consistency
between the models for these two sets of distribution amplitudes in
a numerical analysis. We shall adopt the non-asymptotic models for
both sets of distribution amplitudes in Sec.~III, when estimating
the importance of the three-parton contribution relative to other
two-parton contributions in the pion electromagnetic form factor.

According to Eq.~(\ref{col}), we absorb a color matrix $T^c$ and the
coupling constant $g$ associated with an attachment into the matrix
element for the initial-state pion, and another $T^c$ goes into the
evaluation of gauge-dependent hard kernels. For example, the color
factor corresponding to the attachment $A$ of the valence gluon to
the virtual quark line is given by
\begin{eqnarray}
tr[T^bT^aT^bT^c] =-\frac{1}{4N_c} \delta^{ac},\label{ca}
\end{eqnarray}
where the color matrix $T^a$ ($T^b$) comes from the valence (hard)
gluon vertex. After summing over $c$, the tensor $\delta^{ac}$ sets
$c=a$ in the matrix element for the initial-state pion, leading to
Eq.~(\ref{a3}). Including the coefficient 2 in Eq.~(\ref{col}), we
adopt the structure $\gamma_{\alpha}\gamma_5/2$ for the initial
state in the calculation of Fig.~\ref{fig2}, whose details can be
found in Appendix A. The results are collected as follows:
\begin{eqnarray}
H_{AT}^{\lambda}&=&-ieg^2\lambda\frac{1}{32N_c^2}
\frac{tr[\gamma^\beta \gamma_5\gamma_{\beta}\gamma_\mu (\not
P_1-x_2\not P_2)\gamma^\alpha\gamma_{\alpha}\gamma_5]}
{(P_1-k_2)^2(k_1-k_2)^4},\label{aA}\\
H_{BT}^{\lambda}
&=&ieg^2\lambda \frac{1}{32}\frac{tr[\gamma^\beta\gamma_5
\gamma_{\beta}\gamma_\mu (\not P_1-x_2\not P_2)\gamma^\alpha
\gamma_{\alpha}\gamma_5]}{(P_1-k_2)^2(k_1-k_2)^4},\label{B}\\
H^{\lambda}_{CT}&=&0,\label{C}\\
H^{\lambda}_{DT}&=&-ieg^2\lambda\frac{1}{32N_c^2}
\frac{tr[\gamma^\beta \gamma_5
\gamma_{\beta}\gamma^\alpha(\not P_2-x_2\not P_2-y_1\not
P_1)\gamma_\mu
\gamma_{\alpha}\gamma_5]}{(P_2-k_2-l_1)^2(k_1-k_2)^4},\label{D}\\
H^{\lambda}_{ET}&=&-ieg^2\lambda\frac{C_F}{16N_c}
\frac{tr[\gamma^\beta \gamma_5\gamma_{\beta} \gamma^\alpha (\not
P_2- x_1\not P_1-y_1\not P_1) \gamma_\mu
\gamma_{\alpha}\gamma_5]}{(P_2-k_1-l_1)^2(k_1-k_2)^4},\label{E}\\
H^{\lambda}_{FT}&=&ieg^2\lambda\frac{1}{32N_c^2}
\frac{tr[\gamma^\beta \gamma_5\gamma_{\beta}\gamma^\alpha(
\not P_2- x_2\not P_2-y_1\not P_1)\gamma_\mu
\gamma_{\alpha}\gamma_5]}
{(P_2-k_2-l_1)^2(k_1-k_2)^4}\nonumber\\
& &-ieg^2\lambda\frac{1}{32N_c^2} \frac{tr[\gamma^\beta
\gamma_5\gamma_{\beta} \gamma^\alpha(\not P_2-x_1\not P_1-y_1\not
P_1)\gamma_\mu
\gamma_{\alpha}\gamma_5]}{(P_2-k_1-l_1)^2(k_1-k_2)^4},\label{F}\\
H^{\lambda}_{GT} &=& ieg^2\lambda \frac{1}{32}\frac{tr[\gamma^\beta
\gamma_5\gamma_{\beta}\gamma^\alpha (\not P_2-x_1\not P_1-y_1\not
P_1)\gamma_\mu
\gamma_{\alpha}\gamma_5]}{(P_2-k_1-l_1)^2(k_1-k_2)^4}\nonumber\\
& &+ieg^2 \lambda\frac{1}{32}\frac{tr[\gamma^\alpha\gamma_5
\gamma_{\beta} \gamma^\beta(\not P_2-x_1\not P_1-y_1\not P_1)\gamma_\mu
\gamma_{\alpha}\gamma_5]}{(P_2-k_1-l_1)^2(k_1+l_1-k_2)^4},\label{G}\\
H^{\lambda}_{HT} &=&-ieg^2\lambda\frac{1}{32N_c^2} \frac{tr[
\gamma^\alpha\gamma_5\gamma_{\beta}\gamma^\beta(\not
P_2-x_1\not P_1-y_1\not P_1)\gamma_\mu
\gamma_{\alpha}\gamma_5]}{(P_2-k_1-l_1)^2 (k_1+l_1-k_2)^4},\label{H}
\end{eqnarray}
with the gluon momentum fraction $y_1=l_1^+/P_1^+$.

Summing the above expressions, we arrive at
\begin{eqnarray}
\sum_{i=A}^H
H_{iT}^\lambda=H^{a\lambda}_{gT}+H^{b\lambda}_{gT},
\end{eqnarray}
with
\begin{eqnarray}
H^{a\lambda}_{gT}=H^{a\lambda}_{TT}(k_1,k_2),\;\;\;\;
H^{b\lambda}_{gT}=H^{b\lambda}_{TT}(k_1+l_1,k_2).\label{aB}
\end{eqnarray}
The contributions from the attachments $A$ and $B$ are added into
$H^{a\lambda}_{gT}$ with the desired color factor $C_F$. The
contribution from the attachment $D$ and the first term in the
attachment $F$ cancels each other. The second term from the
attachment $F$ and the first term from the attachment $G$ are
combined into the expression with the color factor $C_F$, which then
cancels the contribution from the attachment $E$. The second term of
$G$ and the contribution from the attachment $H$ are added into
$H^{b\lambda}_{gT}$. Note that $H^{a\lambda}_{gT}$ does not depend
on the valence gluon momentum $l_1$, which can then be integrated
out of the matrix element, giving $\langle 0|{\bar
q}(z)\gamma_5\gamma^{\alpha} gT^aA_\alpha^{a}(0) q(0)
|\pi(P_1)\rangle$. The hard kernel $H^{b\lambda}_{gT}$, depending on
the combination $k_1+l_1$, corresponds to the matrix element
$\langle 0|{\bar q}(z)\gamma_5\gamma^{\alpha} gT^aA_\alpha^a(z)
q(0)|\pi(P_1)\rangle$. Because of the symmetry under the exchange of
the initial- and final-state kinematic variables, the
gauge-dependent hard kernels with three partons from the final state
are written as
\begin{eqnarray}
H^{a\lambda}_{Tg}=H^{a\lambda}_{TT}(k_1,k_2+l_2),\;\;\;\;
H^{b\lambda}_{Tg}=H^{b\lambda}_{TT}(k_1,k_2),\label{bB}
\end{eqnarray}
where $l_2$ is the momentum carried by the outgoing valence gluon.

\begin{figure}[t]
\begin{center}
\includegraphics[height=3.5cm]{kt3ggC.eps}
\caption{One of the three-parton-to-three-parton diagrams, where the
two valence gluons scatter via a three-gluon vertex.} \label{figC}
\end{center}
\end{figure}

At last, we extract the gauge dependence from the
three-parton-to-three-parton diagrams.
In this case the indices $\alpha$ and $\beta$, associated with the
initial and final valence gluons, respectively, must be carried by
gluon vertices, instead of by parton momenta. Focusing on the
gauge-dependent piece, we can apply the Ward identity to hard
gluons. It is easy to find that Fig.~\ref{figC}, where the two
valence gluons scatter via a three-gluon vertex, does not
contribute: if the gauge dependence arises from the lower hard
gluon, the results, being proportional to the parton momenta after
applying the Ward identity, should be dropped. If the gauge
dependence arises from the upper hard gluon, the Ward identity
diminishes the amplitude for a similar reason. It is also easy to
see that Fig.~\ref{fig9}(a) with a four-gluon vertex does not
contribute to a gauge-dependent hard kernel. If both the valence
gluons attach to the quark line, the gauge-dependent contribution
vanishes because of the Ward identity applied to the anti-quark
line. If both the valence gluons attach to the anti-quark line, the
gauge-dependent contribution vanishes too.

\begin{figure}[t]
\begin{center}

\includegraphics[height=2.5cm]{kt3f3c.eps}\hspace{0.5cm}
\includegraphics[height=2.5cm]{kt3f3d.eps}\hspace{0.5cm}
\includegraphics[height=2.5cm]{kt3f3e.eps}\hspace{0.5cm}
\includegraphics[height=2.5cm]{kt3f3f.eps}\vskip 0.5cm

\includegraphics[height=3.2cm]{kt3f3g.eps}\hspace{0.5cm}
\includegraphics[height=3.2cm]{kt3f3h.eps}\hspace{0.5cm}
\includegraphics[height=3.2cm]{kt3f3i.eps}\hspace{0.5cm}
\includegraphics[height=2.5cm]{kt3f3j.eps}\vskip 0.5cm

\includegraphics[height=3.2cm]{kt3f3a.eps}\hspace{0.5cm}
\includegraphics[height=2.5cm]{kt3f3b.eps}\hspace{0.5cm}
\includegraphics[height=2.5cm]{kt3f3k.eps}\hspace{0.5cm}
\includegraphics[height=2.5cm]{kt3f3l.eps}
\caption{Three-parton-to-three-parton diagrams, where the valence
gluon from the initial state is shown, and the possible attachments
of the valence gluon from the final state are represented by dots.}
\label{fig3}
\end{center}
\end{figure}

The other diagrams are classified into several sets, which are
formed by all possible attachments of a hard gluon as displayed in
Figs.~\ref{fig3}(a)-\ref{fig3}(l). Cancellation occurs in each set
of diagrams after applying the Ward identity. Neglecting those
pieces proportional to the parton momenta, we have
\begin{eqnarray}
& &(a)+(b)=(c)+(d)=0,\nonumber\\
& &(e)+(f)+(g)+(h)=0.
\end{eqnarray}
A finite gauge-dependent contribution comes only from
Figs.~\ref{fig3}(i)-\ref{fig3}(l), given by
\begin{eqnarray}
& &(i)+(j)=H^{a\lambda}_{gg}=H^{a\lambda}_{TT}(k_1,k_2+l_2),\nonumber\\
& &(k)+(l)=H^{b\lambda}_{gg}=H^{b\lambda}_{TT}(k_1+l_1,k_2).
\end{eqnarray}

Due to the same expressions of $H^{a\lambda}_{TT}$ and
$H^{a\lambda}_{gT}$, their corresponding matrix elements are
combined into
\begin{eqnarray}
\langle 0|{\bar q}(z)\gamma_5i\gamma^\alpha D_\alpha(0) q(0)
|\pi(P_1)\rangle=0, \label{eq}
\end{eqnarray}
with the equation of motion for the quark field, $\not\!D(0)q(0)=0$.
That is, the gauge invariance holds, when the contributions from
Figs.~\ref{fig1} and \ref{fig2} are combined. The combination of the
matrix elements for $H^{a\lambda}_{Tg}$ and $H^{a\lambda}_{gg}$ also
vanishes according to Eq.~(\ref{eq}). A similar reasoning applies to
the gauge-dependent hard kernels $H^{b\lambda}_{TT}$ and
$H^{b\lambda}_{Tg}$ and to $H^{b\lambda}_{gT}$ and
$H^{b\lambda}_{gg}$: the combination of their matrix elements
vanishes due to the equation of the motion for the quark field in
the final-state pion. This observation completes the proof of the
gauge invariance of the LO three-parton contribution to the pion
electromagnetic form factor at power of $1/Q^2$ in the $k_T$
factorization theorem. The extension of the proof to all orders can
follow the steps outlined in \cite{NL07}. Note that our proof
applies to the collinear factorization theorem too: simply
neglecting transverse momenta in denominators, one can show the
gauge invariance of the three-parton contribution to the pion
electromagnetic form factor in the collinear factorization theorem.

\section{THREE-PARTON CONTRIBUTION}

In this section we calculate the three-parton contribution to the
pion electromagnetic form factor. Start with the gauge-invariant
twist-3 matrix element
\begin{eqnarray}
\langle 0|{\bar q}(z)\sigma^{+}_{\hspace{0.2cm}\alpha'}\gamma_5
iD_\alpha(z') q(0) |\pi(P_1)\rangle,
\label{gim}
\end{eqnarray}
where the subscript $\alpha$ is associated with the vertex the
valence gluon attaches to. The power behavior will not be changed,
and the gauge invariance will not be broken by inserting another
covariant derivative $D^+$. We then exchange $D^+$ and $D_\alpha$,
take the difference of $D^+ D_\alpha$ and $D_\alpha D^+$, and apply
the identity $[D^+,D_\alpha]=-igG^+_{\hspace{0.2cm}\alpha}$. It is
then equivalent to employ the following alternative matrix element
\cite{EKT06}, which defines the three-parton twist-3 pion wave
function $T(z,z')$,
\begin{eqnarray}
\langle 0|{\bar q}(z)\sigma^+_{\hspace{0.2cm}\alpha'}\gamma_5
gG^+_{\hspace{0.2cm}\alpha}(z') q(0) |\pi(P_1)\rangle=if_\pi
m_0(P_1^+)^2g^T_{\alpha\alpha'}T(z,z'),\label{gi3}
\end{eqnarray}
with the chiral scale $m_0=m_\pi^2/(m_u+m_d)$, $m_\pi$, $m_u$, and
$m_d$ being the pion, $u$ quark and $d$ quark masses, respectively.
The operators with other spin structures contribute at higher
twists: for example, the operator $\gamma_\mu\gamma_5
G_{\alpha\beta}$ gives a three-parton twist-4 contribution, and
$\gamma_5 G_{\alpha\beta}$ does not contribute
\cite{BF90,PB3,BBL06}. With Eq.~(\ref{gi3}), one may verify the
gauge invariance of a hard kernel in the $k_T$ factorization theorem
by demonstrating the cancellation between the gauge-dependent
contributions from the operators $\partial^+ A_\alpha$ and
$\partial_\alpha A^+$ \cite{EKT06}.

Below we derive the hard kernels from the
three-parton-to-three-parton diagrams corresponding to
Eq.~(\ref{gi3}) in the Feynman gauge ($\lambda=0$). Choosing this
gauge, the operator $\partial_\alpha A^+$ does not contribute, so
only $\partial^+ A_\alpha$ is relevant. The three momenta
$P_1-k_1-l_1, k_1$, and $l_1$ are assigned to the initial-state
quark, antiquark, and gluon, respectively, and $P_2-k_2-l_2, k_2$,
and $l_2$ to the final-state quark, anti-quark, and gluon,
respectively. We have the structures for the initial- and
final-state pions
\begin{eqnarray}
& &\frac{1}{4}\sigma^{-\alpha'}\gamma_5
\frac{i}{l_1^+}if_\pi m_0 (P_1^+)^2 g^T_{\alpha\alpha'}=
-\frac{i}{4y_1}\not P_1\gamma^T_{\alpha}\gamma_5f_\pi m_0,\nonumber\\
& &-\gamma_5\frac{1}{4}\sigma^{+\beta'} \frac{-i}{l_2^-}(-i)f_\pi
m_0 (P_2^-)^2 g^T_{\beta\beta'}= \frac{i}{4y_2}\gamma_5\not
P_2\gamma^T_{\beta}f_\pi m_0,
\end{eqnarray}
where the gluon momentum fraction $y_2$ is defined by $y_2=l_2^-/P_2^-$,
and the gamma matrix $\gamma^T$ involves only transverse components.

There are totally 196 diagrams for the three-parton-to-three-parton
scattering, which can be divided into four categories\footnote{We
thank the Referee for suggesting this classification of diagrams.}.
Category A
contains 20 quark-gluon configurations, in which neither of the valence
gluons attaches to the hard gluon line. Each configuration allows 6
different attachments for the photon line. We further divide this
category into two groups as shown in Fig.~\ref{fig6}, where both
valence gluons attach to the same quark or anti-quark line, and in
Fig.~\ref{fig7}, where one valence gluon attaches to the quark line
and another to the anti-quark line. Only the diagrams giving nonvanishing
contributions are displayed. As observed in Appendix B, the
amplitudes with both the valence gluons attaching to the quark line
are power-suppressed. Category B contains 8 quark-gluon
configurations, in which one of the valence gluons attaches to the hard
gluon line. Each configuration allows 5 different attachments for the
photon line, among which those with nonvanishing
contributions are displayed in Fig.~\ref{fig8}. Category C contains
4 quark-gluon configurations, where both the valence gluons are connected
to the hard gluon. Each configuration allows 4 different attachments
for the photon line, among which those with nonvanishing
contributions are displayed in Fig.~\ref{fig9}. Category D contains
4 quark-gluon configurations, where the two valence gluons scatter
via a three-gluon vertex as shown in Fig.~\ref{figC}. Each
configuration allows 5 different attachments for the photon line.
Since this category of diagrams does not contribute, we shall not
discuss them further. Besides, when a valence gluon
attaches to a valence quark, the diagram should be regarded as
being from an effective two-parton Fock state, and will not
be calculated.

We extract the hard kernels proportional to the final-state momentum
$P_{2\mu}$. The hard kernels proportional to $P_{1\mu}$ can be
obtained by exchanging the kinetic variables of the initial- and
final-state pions. Adopting the electric charge $e$, instead of the
quark charge $e_u$ or $e_d$, we have taken into account the diagrams
with the virtual photon attaching to the anti-quark line.
Figure~\ref{fig9}(a) with a four-gluon vertex gives the dominant
three-parton contribution
\begin{eqnarray}
H_{3}& =& ieg^2\frac{f_\pi^2 m_0^2}{16y_1y_2}
\frac{N_c^2}{8(N_c^2-1)} \frac{tr[ \gamma^\sigma \gamma_5\not
P_2\gamma^{T\beta}\gamma_\mu (\not P_1-\not k_2-\not
l_2)\gamma^\nu\not
P_1\gamma^{T\alpha}\gamma_5]}{(P_1-k_2-l_2)^2(k_1-k_2)^2(k_1-k_2+l_1-l_2)^2}
(g_{\alpha \nu} g_{\beta \sigma} +g_{\alpha \sigma} g_{\beta \nu}-2
g_{\alpha\beta} g_{\sigma\nu})\nonumber\\
& = & -i eg^2\frac{f_\pi^2 m_0^2}{16y_1y_2} \frac{ N_c^2 }{N_c^2-1}
\frac{tr[ \not P_2\gamma_\mu \not P_2\not P_1]}{
(P_1-P_2)^2(k_1-k_2+l_1-l_2)^2(k_1-k_2)^2}.\label{aa1}
\end{eqnarray}
To arrive at the second line, we have made an approximation
according to the power counting $Q^2\gg xQ^2, yQ^2\gg k_T^2$
\cite{LWS10}, under which the TMD term in the following denominator
is neglected,
\begin{eqnarray}
\frac{(\not k_2+\not l_2)}{(P_1-k_2-l_2)^2}=
\frac{(k_2+l_2)^-\gamma^+}{-2P_1^+(k_2+l_2)^-}=
\frac{P_2^-\gamma^+}{-2P_1^+P_2^-}=\frac{\not P_2}{(P_1-P_2)^2}.
\end{eqnarray}
The expressions for other three-parton-to-three-parton diagrams are
collected in Appendix B.

Since the Sudakov factor for exclusive QCD processes was derived in
the space of impact parameters \cite{BS,LS}, we Fourier transform
Eq.~(\ref{aa1}). The $k_T$ factorization formula for the pion
electromagnetic pion form factor from Fig.~\ref{fig9}(a) is then
written as
\begin{eqnarray}
F_{3}(Q^2)&=& \pi\alpha_s f_\pi^2 m_0^2\frac{ N_c^2}{N_c^2-1
}\int_0^1 dx_1\int_0^{1-x_1}  \frac{d y_1}{y_1} \int_0^1
dx_2\int_0^{1-x_2}
\frac{d y_2}{y_2}\nonumber\\
& &\times \Phi(x_1,y_1)\Phi(x_2,y_2)
K\left(\sqrt{(x_1+y_1)(x_2+y_2)}Q\right)K(\sqrt{x_1
x_2}Q),\label{f3}
\end{eqnarray}
where the three-parton pion distribution amplitude $\Phi(x_1,y_1)$
corresponds to $T(z,z')$ in Eq.~(\ref{gi3}) in the space of momentum
fractions. The functions $K$, arising from the Fourier
transformation of the TMD denominators $(k_1-k_2+l_1-l_2)^2$ and
$(k_1-k_2)^2$ in Eq.~(\ref{aa1}), are defined by
\begin{eqnarray}
K(t) = \int_0^{1/\Lambda} b db K_0(tb)\exp[-s(P_1^+,b)],\label{kt}
\end{eqnarray}
in which $K_0$ is the modified Bessel function, and the explicit expression
of the Sudakov exponent $s(P_1^+,b)$ is referred to \cite{LS,KLS}.
We have kept only the most effective piece of the Sudakov evolution
in the small $x$ region, that results from the gluon exchanges
between the energetic valence quark and the Wilson line associated
with it. Because the Sudakov factor $\exp[-s(P_1^+,b)]$ diminishes
at $b=1/\Lambda$, with the QCD scale $\Lambda\approx 0.3$ GeV, the
upper bound of the integration variable has been set to $1/\Lambda$
in Eq.~(\ref{kt}). For order-of-magnitude estimate and for
demonstrating the smallness of the three-parton contribution, we do
not consider the renormalization-group evolution from the low scale,
at which $\Phi$ is defined, to the scale of the hard kernel. The
coupling constant is also assumed to be a constant $\alpha_s=0.5$.

\begin{figure}[t]
\begin{center}
\includegraphics[height=15.0cm]{KT.eps
\caption{$Q^2$ dependence of the three-parton contribution to the
pion electromagnetic form factor. The LO twist-2, NLO twist-2, and
LO two-parton twist-3 contributions are quoted from Fig.~8 in
\cite{LWS10}, which were derived using the non-asymptotic models for
the pion distribution amplitudes.}
\label{figd}
\end{center}
\end{figure}

Fourier transforming the hard kernels in Appendix B into the
impact-parameter space, we construct the corresponding $k_T$
factorization formulas similar to Eq.~(\ref{f3}). We then add all
the contributions to the pion electromagnetic form factor, and
employ the model of the three-parton twist-3 pion distribution
amplitude \cite{BF90,PB3}
\begin{eqnarray}
\Phi(x_1,y_1)=360\eta_3 x_1(1-x_1-y_1)y_1^2
\left[1+\frac{\omega_3}{2}(7y_1-3)\right],
\end{eqnarray}
for a numerical analysis, with the parameters $\eta_3=0.015$ and
$\omega_3=-3$, and $1-x_1-y_1$ being the momentum fraction of the
valence quark. The total three-parton contribution to $Q^2F(Q^2)$,
$F(Q^2)$ being the pion electromagnetic form fact, is displayed in
Fig.~\ref{figd}. The curve exhibits a decrease in $Q^2$ (though not
obvious in the figure) compared to the LO and NLO twist-2
contributions, indicating that this contribution is
power-suppressed. It is only about 5\% of the sum of those evaluated
in \cite{LWS10,CKO09}, including the LO twist-2, NLO twist-2, and LO
two-parton twist-3 pieces. That is, the three-parton contribution is
not crucial for accommodating the experimental data of $Q^2F(Q^2)$
\cite{Huber:2008id,Bebek:1977pe} in the whole accessible range of
$Q^2$ up to 10 GeV$^2$. The only important subleading contribution
to the pion electromagnetic form factor that have been investigated
so far comes from the chirally enhanced two-parton twist-3 one.

\section{CONCLUSION}

In this paper we have applied the $k_T$ factorization theorem to the
study of the power-suppressed three-parton contribution to the pion
electromagnetic form factor. It was demonstrated that the gauge
invariance of the $k_T$-dependent hard kernel holds for this power
correction: the gauge dependence proportional to parton transverse
momenta from the two-parton Fock state and the gauge dependence
associated with the three-parton Fock state cancel each other. We
have calculated the three-parton-to-three-parton hard kernel at LO,
and found that the three-parton contribution is about 5\% of the sum
of the LO twist-2, NLO twist-2, and LO two-parton twist-3 ones in
the whole range of experimentally accessible $Q^2$. Our analysis
shows that the power expansion for this exclusive process might be
reliable in the $k_T$ factorization theorem. At the same power of
$1/Q^2$, the two-parton twist-4 contribution should be taken into
account, which has been studied in the framework of light-cone sum
rules \cite{BK02}. We shall calculate this correction in the $k_T$
factorization theorem in the future, which involves a twist-2
distribution amplitude from one side and a two-parton twist-4
distribution amplitude from the other side. We shall also extend our
framework to exclusive $B$ meson decays, for which three-parton
contributions have been analyzed in light-cone sum rules,
\cite{K01}, in the QCD (collinear) factorization \cite{Yeh08} and in
the soft-collinear effective theory \cite{ARS06}.

We thank Z.T. Wei for his suggestion, which initiates this project.
We also thank V. Braun, S. Brodsky, J.P. Ma and Y.M. Wang for useful
discussions. This work was supported in part by the National Science
Council of R.O.C. under Grant No. NSC-98-2112-M-001-015-MY3, and by
the National Center for Theoretical Sciences.

redpajama_set_name: RedPajamaArXiv


lem area for traffic traveling west on MLK and wanting to turn south. With no turn lane in the single southbound lane, vehicles wanting to continue west are often stuck behind vehicles trying to turn south onto 15th as sometimes heavy traffic travels east. Drivers can sit through two or more light changes trying to travel west through the intersection.
Ward 5 Councilman Harold Byrd Jr. attended the June meeting with FDOT and heard a lot of concern about the proposal to install a roundabout, which tends to be unpopular. Byrd asked FDOT to conduct a second public meeting, which Reck said also was well attended.
"This is something new to this area and I wanted to make sure the public had the opportunity to have their questions answered," said Byrd. "The thing I've pushed for the most, which everybody wants, is to see improvements in the aesthetics of that intersection."
Reck said the roundabout will feature red bricks and a center island landscaped with yellow flowering peanut grass and palm trees.
Byrd said he wasn't convinced southbound traffic on MLK would not still have issues, but that FDOT's explanation and presentations have given him more confidence. Reck said nationwide, more than 5,000 intersections have made the transition from traditional signal lighting to roundabouts.
According to Robin Stublen, FDOT communications specialist, the MLK and 15th Street East project is a separate project from the ongoing planning for a complete overhaul of 15th Street East. Stublen said that project is part of a Manatee County project FDOT is helping design.
"This one is all about traffic flow and correcting the backups that take place at that intersection," said Stublen. "It is scheduled for 2018 as part of our five-year work program, which means it's not set in stone. It is in design now so it is likely."
FDOT plans to shift the intersection to make room for the roundabout. That means some land acquisition work is now starting, Stublen said.
"That's what will take the most time, which is why it's scheduled for 2018," he said.
Reck said land acquisitions could cost as much as $690,000 bringing the project's total costs closer to $1.7 million.


These HDPE traveler endstops from JCD measure 2 in x 2-5/8 in x 3/4 in. With a deep recessed hole, your knot hides flush while keeping it snug in the cavity. Edges are smoothed and rounded for your crews comfort!


text:
#ifndef WGRS_FP_Sweeps_h
#define WGRS_FP_Sweeps_h

#include "V_WGRS.h"

/** @file   WGRS_FP_Sweeps.h
  *   @author pmaginot
  *   @brief Solve for within group scattering (and scattering like if this is grey) using fixed point iteration and sweeps only
 */

class WGRS_FP_Sweeps : public V_WGRS
{
public:
  WGRS_FP_Sweeps(const Input_Reader& input_reader,
    const Fem_Quadrature& fem_quadrature,
    const Cell_Data& cell_data,
    Materials& materials,
    const Angular_Quadrature& angular_quadrature,
    const int n_stages,
    const Temperature_Data& t_old,
    const Intensity_Data& i_old,
    const K_Temperature& kt,
    K_Intensity& ki,
    const Temperature_Data& t_star,
    std::vector<double>& phi_ref_norm);

  virtual ~WGRS_FP_Sweeps(){}

  void kill_petsc_objects() override;

  int solve(Intensity_Moment_Data& phi_new, bool& update_success) override;
protected:

  const int m_n_groups;
  const double m_wg_tolerance;
  const int m_max_sweeps;

  Intensity_Moment_Data m_phi_old;
  Err_Phi m_err;

};

#endif

redpajama_set_name: RedPajamaGithub


text:
Home/March 2017, Opinions/Supremes Overturn Gorsuch Ruling, And Dems Vow To Filibuster
Supremes Overturn Gorsuch Ruling, And Dems Vow To Filibuster
By Kelleigh Nelson|2017-03-28T05:14:31-05:00March 28th, 2017|
True freedom requires the rule of law and justice, and a judicial system in which the rights of some are not secured by the denial of rights to others. Jonathan Sacks
The people in general ought to have regard to the moral character of those whom they invest with authority either in the legislative, executive, or judicial branches. John Witherspoon
Judge Gorsuch's 2008 Decision
Judge Gorsuch's 2008 decision in the U.S. Court of Appeals for the 10th Circuit was the touchstone for Endrew F. v. Douglas County School District, the case the justices heard in January, and have just now unanimously decided for the plaintiff. Believe me folks, this is not good for our school districts or for taxpayers.
At issue was an autistic youngster who had been in public school from kindergarten to fourth grade. The parents decided the child was not progressing as they expected in fifth grade, so they put him in a private school costing $70,000 a year.
They then sued the school to pay for the private school cost because they claimed he wasn't reaching his potential in the public school despite 37 hours of special-ed services per week.
This case first went to an Administrative Law Judge (ALJ) for review, and the ALJ rejected Endrew's parent's request for reimbursement concluding that Endrew's public school had provided him with "free appropriate public education" (FAPE) as required by the Individuals with Disabilities Education Act (IDEA), an act signed into law in 1990 by George H.W. Bush, a law which needs repealing.
Then the district court affirmed their ruling, and the U.S. Court of Appeals for the Tenth Circuit also affirmed, as did Judge Gorsuch who ruled the school had reached their "de minimis" standard. The Individuals with Disabilities Education Act (IDEA) requires States to provide "merely more than de minimis" in providing "some educational benefit" to students under the IDEA.
Supremes Overturn Gorsuch Ruling
On March 22, 2017, the Supreme Court unanimously overturned the 10th circuit court decision, ruling in favor of higher standards for disabled children via IDEA in the case brought by plaintiff child, Endrew.
Sadly, Judge Gorsuch apologized in the confirmation hearings for his ruling against this autistic child, saying that he was bound by precedent, which Gorsuch does not like to go against. In this case, he should have held his ground, because he was right! And, apologies win nothing!
So, from now on when parents of a disabled child decide (all of which is subjective) that their child is not making the proper advances in a public school, they can arbitrarily send their child to a costly private school and have the school district pay for it. The school district is funded by TAXPAYERS!!!
Do you understand what this will cost the American taxpayer in real estate taxes to support the local school districts who will have to pay high costs when a parent decides the school isn't doing the job they expect? Just who will decide who gets taxpayer dollars for their child's special education, because the school district doesn't meet their objectives? More unelected councils?
Think about the Education Czar, Betsy DeVos, who wants to privatize our public schools and have corporations running them for profit! The public schools will either go broke or tax us to death for the costs which will also affect private, and religious schools. Betsy Devos is anathema to academic education, but few people understand how very dangerous she truly is to our children. She will love this decision, because it plays right into her hands.
Dems Confirm Gorsuch Filibuster
The Democrats just confirmed they will filibuster Gorsuch, requiring that he be withdrawn and replaced.
Democratic Sen. Bob Casey, son of the famous Pennsylvania pro-life Gov. Bob Casey, who is up for reelection, and who pretends to be pro-life, announced he will join the filibuster. Gorsuch and the pro-choice backed and funded Federalist Society made that easy for Sen. Casey to do. It might even help his reelection in light of Gorsuch's pro-choice performance.
Another Democratic Senator up for reelection, who also pretends to be pro-life, is Sen. Joe Donnelly of Indiana, and he may join the filibuster. The Democrats may have enough votes to block Gorsuch.
This truly creates a second chance for 30 million unborn babies, for Trump's advisers, and for the Republican Party to nominate someone who is actually pro-life, like Charles Canady of Florida. Second chances do not occur often in politics, but here it is, so time to pray and work, because big money is for Gorsuch.
Koch Brothers Fund Gorsuch Nomination
The Koch brothers, David and Charles, and other billionaires who are pro-abortion are clamoring for the Senate to change their rules to try to overcome the filibuster.
David and Charles Koch
The Koch brothers (pronounced "Coke") are the billionaire co-owners of Koch Industries. As two of the richest people in the world, they are key funders of the neo-con infrastructure, including the pro-Constitutional Convention, American Legislative Exchange Council (ALEC), and the State Policy Network (mini-Heritage Foundations in each state). Key articles on the Kochs include: Koch Brothers, Koch Industries, Americans for Prosperity, and Freedom Partners.
David Koch is on the board of Trustees of Rockefeller University. He is also on the board of Earthwatch Institute, an environmental, sustainability, Agenda 21 group, and the board of the globalist new age Aspen Institute. See this map of Koch's many connections. David Koch also supports abortion, gay marriage, and stem-cell research.
The secret donors who are financing ads for Gorsuch don't care about the unborn. They're just fine with the prospect of relegating pro-life senators to irrelevancy in future confirmation battles. Many senators, both Republican and conservative Democrats, have spoken out against changing their rules to confirm Gorsuch. Changing the rules to confirm pro-choice Gorsuch would be a colossal mistake for the pro-life movement, for the GOP, and for our country.
All President Trump has to do is pick a pro-life nominee who has a base of support, which Gorsuch never had. The sooner Trump does this, the sooner his own polling will improve.
It is essential that we pray for our President, for the unborn, and for our country. We need to help him keep all his promises to the American people. We also need to pray that he has discernment as to those who surround and guide him.
Never in my lifetime have I seen so many advertisements for a Supreme Court nominee, but then the Koch billionaires are allegedly paying for them. And, Gorsuch, a term limit CFR member will join two other CFR members on the court, Breyer and Ginsberg. Here is a short tutorial on the CFR.
The first week after his nomination, Neil Gorsuch told a US senator that President Donald Trump's tweets about the judiciary, i.e., Judge Robard's restraining order against Trump's immigration ban, are "demoralizing" and "disheartening." He also said in his confirmation hearing that "Judges have to "start with a heavy, heavy presumption in favor of precedent in our system," which is why he loves stare decisis, and will probably never vote to overturn Roe v. Wade. Stare decisis is Latin for "to stand by things decided," and is the doctrine of precedent.
During the confirmation hearings, while being questioned by Sen. Dick Durbin, D-Ill., on the destruction of unborn children, Gorsuch stated "the Supreme Court of the United States has held in Roe v. Wade that a fetus is not a person for purposes of the Fourteenth Amendment."
When Durbin asked if he accepted that, the judge replied, "That's the law of the land. I accept the law of the land, senator, yes."
And his statement on "same-sex marriage," Gorsuch acknowledged the Supreme Court had ruled such marriages are protected by the Constitution. He declared Obergefell v. Hodges, which struck down "same-sex marriage" bans nationwide in 2015, to be "absolutely settled law."
I believe Gorsuch will be confirmed, but I am praying he is not. The theatre being played out by the democrats and some of the left leaning neo-conservative Republicans is just that, theatre for the dumbed down believing masses, who don't understand the game.
One need only remember that in 2006, when Neil Gorsuch was nominated by George W. Bush, "not a single Democrat opposed his confirmation in 2006," tweeted Mitch McConnell on February 1st. Both Bush families stated they were pro-choice after they were out of office. It is likely the Democrat Senators knew the families were really pro-choice and thus confirmed their nominee. Here's the list of the leftists who confirmed him.
The Senators most likely know that Neil Gorsuch is another John Roberts or Kennedy, and is no Antonin Scalia.
P.S. NewsWithViews has launched a new website to help get the truth out to even more Americans. Be patient while we smooth out the bugs. Our country needs citizens who will stand up for her and for her Constitution. This is the call of the editor of NewsWithViews. He has put his reputation and finances on the line for our country. Please help him keep the truth arriving daily in your mailbox. You can donate here, even if it's just a small amount every month, it will definitely help. Thank you for your patience and love of the truth.
About the Author: Kelleigh Nelson
Kelleigh Nelson has been researching the Christian right and their connections to the left, the new age, and cults since 1975. Formerly an executive producer for three different national radio talk show hosts, she was adept at finding and scheduling a variety of wonderful guests for her radio hosts. She and her husband live in Knoxville, TN, and she has owned her own wholesale commercial bakery since 1990. Prior to moving to Tennessee, Kelleigh was marketing communications and advertising manager for a fortune 100 company in Ohio. Born and raised in Chicago, Illinois, she was a Goldwater girl with high school classmate, Hillary Rodham, in Park Ridge, Illinois. Kelleigh is well acquainted with Chicago politics and was working in downtown Chicago during the 1968 Democratic convention riots. Kelleigh is presently the secretary for Rocky Top Freedom Campaign, a strong freedom advocate group. Email: Proverbs133@bellsouth.net

redpajama_set_name: RedPajamaCommonCrawl


text:
General Cigar Co.
Habanos S.A.
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ELs & ERs
2019 Release List
IPCPR
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Cigar Locations
The Consensus
•November 23, 2019
•Patrick Lagreid
•Accessories, Lighters, Reviews
Bugatti Vulcan
It's pretty rare to use the term game changing when it comes to cigar accessories, but those are exactly the words that Derrick Hinnendael used to describe the two new lighters that Bugatti released at the 2019 IPCPR Convention & Trade Show, the Vulcan and Mirage.
Both are dual flame torch lighters, with the Mirage being a more pocket-friendly design using a flip-top lid, while the Vulcan is much longer, doesn't have a lid, and offers a flame that can be angled a bit. But what makes them unique are two features, one with the fuel tank and the other with the burner.
Instead of using a large canister of butane to refill an internal tank, a small tank of fuel gets inserted into the base of the lighter, and then swapped out for a new one when empty. As for the replaceable burner component, while the two models require different designs, but both allow the user to replace the part that is commonly the source of problems: clogged jets and faulty ignitions.
The Bugatti Vulcan ($99.99) is a dual flame torch lighter that features an adjustable neck design, and is available in six color choices: gun metal with black, green or red accents, or chrome with, blue, purple or yellow accents. It measures just under 5 1/4 inches long, 1 1/4 inches wide and just over an inch thick, and weighs a hefty 5.8 ounces (166g) due in large part to its solid metal case. It also offers a flame adjustment lever on the side opposite of the push-button ignition, and comes with a two-year warranty.
Both the Bugatti Vulcan and Mirage were released with the note that the swappable burner piece would allow customers to replace one of the more commonly serviced parts of a lighter without having to send it in for repair and thus be without the lighter for a few weeks. These pieces are not seemingly not on sale, so I would have to assume that they are covered by the warranty, though have not received final confirmation on that, nor have I been through the warranty process. As for the fuel cylinder, it's a different approach to an issue that all butane-fueled lighters face: having to be refilled. Hinnendael said that a fresh cylinder would last 30 days, though that is highly dependent on how frequently the lighter is used. In my case, that a pretty big overstatement, as I went through a cylinder in about two weeks, and more than two in the month or so that I've been using the Vulcan on a daily basis.
The Flame — It's both powerful and accurate, packing enough punch to quickly torch any cigar but rarely resulting in any scorched sides. The two torches produce flames that meet at a single point, making this a handy tool for precise touch-ups.
It's Sturdy — If you like a heftier lighter, the Bugatti Vulcan should satisfy. It's a beefy, substantial lighter in the hand but that remains easy to use.
It Won't Slip — It also offers plenty of tactile sensation with its many ridges, making it feel very secure in the hand.
An Easy to Adjust Lighter — The adjustable head of the Vulcan is a small but appreciated feature, making it possible to point the flame in the direction you find most helpful. Similarly, the flame adjustment level requires no tools to use.
Large Fuel Capacity — When you have a fresh butane tank, you've likely got the most butane of anyone in the room, save for the guy using the Alec Bradley Mega Burner, except he can't share that butane like you can.
An Interesting Possibility — This isn't an available option now, but the possibility could seemingly exist that you could swap the dual torch insert for a single or triple, which would really add to the versatility of this lighter.
The Butane is Expensive — The cost of butane for the Bugatti Vulcan—and Mirage, for that matter—is appreciably more than the typical lighter. A 12-pack of 18 mL cans has an MSRP of $36, though is often found a few bucks cheaper, about $32.50. Regardless, for 216 mL—a 12 pack of 18 mL cans—it works out to 16.6 cents per mL at full MSRP. Meanwhile, a 400 mL can of XIKAR's High Performance butane has an MSRP of $9.99, about 2.5 cents per mL. That's a pretty significant premium to pay to have fuel for this lighter.
The Butane is Also Annoying — You have to use Bugatti's butane cylinders for this lighter, so you will need to make sure you have a fresh can in, or have a backup available. Once that cylinder gets empty, there is no other way to fill up the Vulcan, rendering it useless. Bugatti's distribution isn't that of XIKAR, so there's also a chance that your local shop might not carry it.
No Fuel Window — There is also no way to tell how much fuel remains in the cylinder, setting up a constant guessing game.
Waste — Similar to K-cups for coffee makers, all those butane cans could add up to a lot of waste.
The Flame is Powerful, Maybe Too Powerful — With a reasonably full fuel can in the cylinder, even at its lowest flame setting, the Vulcan puts out enough flame that you need to be six inches away from the foot of the cigar to avoid scorching it. The lowest setting on this lighter feels like fully open on most pocket lighters, which leads me to wonder how fast it is going through fuel. The fuel flow improves as the fuel tank depletes a bit, but those first lights are definitely overkill.
Not Good Around Kids — There is no safety mechanism whatsoever, something that will be of varying concerns based on your situation. If you have kids around, I could see this being a concern. For me, I noticed it when I needed to put it in my backpack and wondered if something might be able to depress the ignition and fire it up unintentionally.
It's Big — While it's fine for table top use, it is a bit big for portability. It will chew up a spot in your travel humidor and would definitely be noticeable in a pocket.
The Bugatti Vulcan occupies a unique spot that makes it a bit more challenging to find direct competition, as it depends on which aspect matters most, the double flame, the physical shape, or the hardest to compare, the swappable fuel tank system and user-replaceable burner.
JetLine R-200 ($49.99) — Released in January 2019, this long, slender lighter offers dual jet torch flames if that is the priority. It also features a push-up ignition and a fuel window, though lacks the adjustable head. A single flame version, the R-100, was added over the summer.
Vector EPEAK ($35) — One of the first lighters that came to mind is this one, mainly due to its shape, though there are a number of differences. It's a single flame and doesn't offer the adjustable head, let alone the fuel cylinder and burner aspects.
XIKAR 5×64 Turrim ($84.99) — A 2015 release, this dual-flame lighter offers a cylindrical shape and all the standard features of a premium lighter, but not the replaceable burner or cylindrical fuel tank option. It does come with a lifetime warranty, however.
JetLine Pocket Torch Dual Flame ($12.99) — This particular model and numerous ones just like it are the standby of many a cigar smoker, and they offer much of what the Vulcan does—a big fuel tank and dual flame torch—for a fraction of the price, plus it resolves my issue with refilling the lighter and a fuel window. It's also available in single and triple flame models, and there are some manufacturers that even offer them with angled heads.
No. As much as I want to like the Bugatti Vulcan, I just couldn't come to a place where it really made sense, at least not given how everything stands at the moment. The premium on butane, the lack of a fuel window or other way to see how much fuel remains, and some sporadic issues with the ignition all led me to a point where I'd keep my credit card in my pocket. The size is a matter of personal preference, but I don't see it delivering any additional benefit; rather it is simply needed to accommodate the fuel canister, an aspect I'm not crazy about. The Bugatti Vulcan is certainly innovative, and I commend those involved with its development as possibly a subsequent version will bring all the best aspects together. But for now, it's a pass for me.
The lighter used for this review was purchased by halfwheel.
accessoriesBugattiBugatti VulcanDerrick HinnendaelLighters
Patrick Lagreid
I strive to capture the essence of a cigar and the people behind them in my work – every cigar you light up is the culmination of the work of countless people and often represents generations of struggle and stories. For me, it's about so much more than the cigar – it's about the story behind it, the experience of enjoying the work of artisans and the way that a good cigar can bring people together. In addition to my work with halfwheel, I'm the public address announcer for the Colorado Rockies and Arizona Diamondbacks during spring training, as well as for the Salt River Rafters of the Arizona Fall League, the WNBA's Phoenix Mercury and the Arizona Rattlers of the Indoor Football League. I also work in a number of roles for MLB.com, plus I'm a voice over artist. I previously covered the Phoenix and national cigar scene for Examiner.com, and was an editor for Cigar Snob magazine.
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