text:
\section{Introduction}

\label{sec:introduction}

Tackling the infamous UV/IR mixing problem~\cite{Minwalla:1999, Susskind:2000} plaguing Moyal-deformed QFTs has been one of the main research interests in the f\/ield for almost a decade (see \cite{Tanasa:2008d,Rivasseau:2007a,Douglas:2001} for reviews of the topic). It is accepted on a broad basis that non-commutativity necessitates additional terms in the action to reobtain renormalizability. Several interesting approaches have been worked out~\cite{Grosse:2003,Grosse:2008a}, and proofs of renormalizability have been achieved mainly by utilizing Multiscale Analysis (MSA)~\cite{Rivasseau:2005a,Rivasseau:2005b}, or formally in the matrix base~\cite{Grosse:2004b}.

In the line of these developments Gurau et al.~\cite{Gurau:2009} introduced a term of type $\phi\star\frac{a}{\square}\phi$ into the Lagrangian which, in a natural way, provides a counter term for the inevitable $\inv{p^2}$ divergence inherently tied to the deformation of the product. In this way the theory is altered in the infrared region which breaks the {UV/IR mixing} and renders the theory renormalizable. This latter fact has been proven up to all orders by the authors using MSA. Motivated by the inherent translation invariance and simplicity of this model (referred to as $\inv{p^2}$ model), a thorough study of the divergence structure and explicit renormalization at one-loop level~\cite{Blaschke:2008b}, as well as a~computation of the beta functions~\cite{Tanasa:2008a} have been carried out.

In the present article we work on Euclidean $\mathbb{R}_\th^4$ with the Moyal-deformed product (also referred to as `star product') $\starco{x_\mu}{x_\nu} \equiv  x_{\mu} \star x_{\nu} -x_{\nu} \star x_{\mu} = \ri \th_{\mu \nu}$ of regular commuting coordinates $x_\mu$. In the simplest case, the real parameters $\th_{\mu\nu}=-\th_{\nu\mu}$ form the block-diagonal tensor
\begin{gather*
( \th_{\mu\nu} )
=\th\left(\begin{array}{cccc}
0&1&0&0\\
-1&0&0&0\\
0&0&0&1\\
0&0&-1&0
\end{array}\right) ,  \qquad \text{with} \ \ \th \in \mathbb{R}  ,
\end{gather*}
obeying the practical relation $\theta_{\mu\r}\theta_{\r\nu}=-\th^2\d_{\mu\nu}$, where $\dim \th=-2$.
With these def\/initions we use the abbreviations $\tilde{v}_\mu \equiv \th_{\mu\nu}v_\nu$ for vectors $v$ and $\tilde{M} \equiv \th_{\mu\nu}M_{\mu\nu}$ for matrices $M$.

Further research focused on the generalization of the scalar $\inv{p^2}$ model to $U_{\star}(1)$ gauge theory\footnote{Notice, that the star product modif\/ies the initial $U(1)$ algebra in a way that
it becomes non-Abelian. Hence, we call the resulting algebra $U_\star(1)$.} which was f\/irst proposed in~\cite{Blaschke:2008a} yielding the action
\begin{gather}
 S =S_{\text{inv}}[A]+S_{\text{gf}}[A,b,c,\bc]\nonumber\\
\phantom{S}{} = \intx\Big[\inv{4}F_{\mu\nu}\star F_{\mu\nu}\!+F_{\mu\nu}\star\inv{D^2\Dt^2}\star F_{\mu\nu}\Big]\!+\!\intx \Big[b\star\partial\cdot A-\frac{\a}{2}b\star b - \bc\star\partial_\mu D_\mu c\Big],\!\!\!
 \label{eq:act_old_complete}
\end{gather}
with the usual gauge boson $A_\mu$, ghost and antighost f\/ields $c$ and $\bc$ respectively, the Lagrange multiplier f\/ield $b$ implementing the gauge f\/ixing, and a real $U_{\star}(1)$ gauge parameter $\a$. The antisymmetric f\/ield strength tensor $F_{\mu\nu}$ and the covariant derivative $D_\mu$ are def\/ined by
\begin{gather*}
 F_{\mu\nu}=\partial_\mu A_\nu -\partial_\nu A_\mu -\ri g \starco{A_{\mu}}{A_{\nu}}, \qquad \text{and} \qquad D_{\mu}\varphi= \partial_\mu\varphi-\ri g \starco{A_\mu}{\varphi},
\end{gather*}
for arbitrary $\varphi$. The non-local term
\begin{align}
S_{\text{nloc}}=\intx\, F_{\mu\nu}\star\inv{D^2\Dt^2}\star F_{\mu\nu},
\label{eq:act_nl_old}
\end{align}
implements the damping mechanism of the $\inv{p^2}$ model by Gurau~et al.~\cite{Gurau:2009} in a gauge covariant way. It has been described in~\cite{Blaschke:2008b} that the new operator can only be interpreted in a physically sensible way if it is cast into an inf\/inite series which, however, corresponds to an inf\/inite number of gauge vertices. A f\/irst attempt to localize the new operator by introducing a real valued auxiliary tensor f\/ield~\cite{Blaschke:2009a} led to additional degrees of freedom. However, this was considered to be dissatisfactory. Following the ideas of Vilar et al.~\cite{Vilar:2009} we enhanced our approach by coupling gauge and auxiliary sectors via complex conjugated pairs of f\/ields together with associated pairs of ghosts in such a way, that BRST doublet structures were formed \cite{Blaschke:2009b}. Such a mechanism has already been applied successfully for the Gribov--Zwanziger action of QCD~\cite{Zwanziger:1989,Zwanziger:1993,Dudal:2008} where a~similar damping mechanism is applied.

Starting from a recapitulation of our recently presented localized model in \secref{sec:loc_p2inv} we give explicit one-loop calculations in \secref{sec:one_loop}, and undertake the attempt of one-loop renor\-ma\-lization. Subsequently, the results and their implications for higher loop orders are analyzed in \secref{sec:higher_loop_calc}, and f\/inally we give a concluding discussion of the lessons learned in \secref{sec:discussion}.

\section[The localized $p^{-2}$ $U(1)$ gauge model]{The localized $\boldsymbol{\inv{p^2}}$ $\boldsymbol{U(1)}$ gauge model}

\label{sec:loc_p2inv}

\subsection{Review: the construction of the model}\label{sec:review}

As mentioned in \secref{sec:introduction}, the non-local term of the action \eqref{eq:act_old_complete} leads to an inf\/inite number of vertices: it formally consists of the inverse of covariant derivatives acting on f\/ield strength tensors, and therefore stands for an inf\/inite power series (cf.~\cite{Blaschke:2008a}) making explicit calculations impossible. Considering only the f\/irst few orders of this power series is not an option as this would destroy gauge invariance. Yet, the present problem can be circumvented by the localization of the term under consideration.
In this sense, in a f\/irst approach described in~\cite{Blaschke:2009a}, the introduction of an additional real antisymmetric f\/ield $\mathcal{B}_{\mu\nu}$ of mass dimension two led to the following localized version of the non-local term \eqref{eq:act_nl_old}:
\begin{gather}
S_{\text{nloc}}\toS_{\text{loc}}=\intx\big[a'\mathcal{B}_{\mu\nu}\star F_{\mu\nu}-\mathcal{B}_{\mu\nu}\star \Dt^2D^2\star \mathcal{B}_{\mu\nu}\big].
\label{eq:gauge_act_damping_replacement}
\end{gather}
However, the $\mathcal{B}_{\mu\nu}$-f\/ield appears to have its own dynamical properties leading to new physical degrees of freedom which can only be avoided if the new terms in the action are written as an exact BRST variation. In order for such a mechanism to work, further unphysical f\/ields are required.

Following the ideas of Vilar et al.~\cite{Vilar:2009}, the localized action \eqref{eq:gauge_act_damping_replacement} was further developed in~\cite{Blaschke:2009b} by replacing $\mathcal{B}_{\mu\nu}$ with a complex conjugated pair of f\/ields ($B_{\mu\nu}$, $\bar{B}_{\mu\nu}$) and by the introduction of an additional pair of ghost and antighost f\/ields $\psi_{\mu\nu}$ and $\bar{\psi}_{\mu\nu}$ (all of mass dimension~1), thus leading to
\begin{gather}\label{act-loc}
S_{\text{loc}} =\intx\left[\frac{\l}{2}\left(B_{\mu\nu}+
\bar{B}_{\mu\nu}\right)F_{\mu\nu}-\mu^2\bar{B}_{\mu\nu}D^2\Dt^2B_{\mu\nu}+\mu^2\bar{\psi}_{\mu\nu}D^2\Dt^2\psi_{\mu\nu}\right].
\end{gather}
In this expression, as well as throughout the remainder of this section, all f\/ield products are~con\-sidered to be star products. The new parameters $\l$ and $\mu$ both have mass dimension 1 and replace the former dimensionless parameter $a'$ by $a'=\l/\mu$. The proof of the equivalence between the non-local action \eqref{eq:act_nl_old} and equation~\eqref{act-loc} can be found in~\cite{Blaschke:2009b}. With the addition of a f\/ixing term to the action one has BRST invariance, and for simplicity, we choose the Landau gauge
\begin{gather*}
S_{\phi\pi}=\intx\left(b\partial^\mu A_\mu-\bc\partial^\mu D_\mu c\right).
\end{gather*}
The BRST transformation laws for the f\/ields read:
\begin{alignat}{3}
 &sA_\mu=D_\mu c,\qquad && sc=\ri g{c}{c} ,& \nonumber\\
 &s\bc=b,     \qquad                                                  && sb=0 ,&  \nonumber\\
& s\bar{\psi}_{\mu\nu}=\bar{B}_{\mu\nu}+\ri g\aco{c}{\bar{\psi}_{\mu\nu}}, \qquad                && s\bar{B}_{\mu\nu}=\ri g\co{c}{\bar{B}_{\mu\nu}}, & \nonumber\\
& sB_{\mu\nu}=\psi_{\mu\nu}+\ri g\co{c}{B_{\mu\nu}}, \qquad                   && s\psi_{\mu\nu}=\ri g\aco{c}{\psi_{\mu\nu}},& \nonumber\\
&s^2\varphi=0\quad \forall\; \varphi\in\left\{A_\mu,b,c,\bc,B_{\mu\nu},\bar{B}_{\mu\nu},\psi_{\mu\nu},\bar{\psi}_{\mu\nu}\right\}.\quad&&&\label{eq:BRST_all_fields}
\end{alignat}
With \eqref{eq:BRST_all_fields} one can see that the localized part of the action can be written as the sum of a BRST exact and a so-called soft breaking term:
\begin{gather*
S_{\text{loc}}=\intx\left[s\left(\frac{\l}{2}\bar{\psi}_{\mu\nu}F^{\mu\nu}
-\mu^2\bar{\psi}_{\mu\nu}D^2\Dt^2B^{\mu\nu}\right)+\frac{\l}{2}B_{\mu\nu}F^{\mu\nu}\right],
\end{gather*}
where
\begin{gather}\label{eq:act_break_wo_source}
S_{\text{break}}=\intx\, \frac{\l}{2}B_{\mu\nu}F^{\mu\nu}, \qquad \text{with }\quad sS_{\text{break}} =\intx\, \frac{\l}{2}\psi_{\mu\nu}F^{\mu\nu}.
\end{gather}
As discussed in detail in \cite{Blaschke:2009b}, the breaking is considered to be soft, since the mass dimension of the f\/ield dependent part is $<D=4$ and the term only modif\/ies the infrared regime of the model. As has been shown by Zwanziger~\cite{Zwanziger:1993} terms of this type therefore do not spoil renormalizability. In order to restore BRST invariance in the UV region (as is a prerequisite for the application of algebraic renormalization) an additional set of sources{\samepage
\begin{alignat*}{3}
&s \bar{Q}_{\mu\nu\a\b}=\bar{J}_{\mu\nu\a\b}+\ri g \aco{c}{\bar{Q}_{\mu\nu\a\b}},\qquad && s \bar{J}_{\mu\nu\a\b}=\ri g \co{c}{\bar{J}_{\mu\nu\a\b}}, &
\nonumber\\
&s Q_{\mu\nu\a\b}=J_{\mu\nu\a\b}+\ri g \aco{c}{Q_{\mu\nu\a\b}},\qquad  && s J_{\mu\nu\a\b}=\ri g \co{c}{J_{\mu\nu\a\b}}, &
\end{alignat*}}

\noindent
is introduced, and coupled to the breaking term which then takes the (BRST exact) form
\begin{gather*}
S_{\text{break}} =\intx s\left(\bar{Q}_{\mu\nu\a\b}B^{\mu\nu}F^{\a\b}\right)
 =\intx \left(\bar{J}_{\mu\nu\a\b}B^{\mu\nu}F^{\a\b}-\bar{Q}_{\mu\nu\a\b}\psi^{\mu\nu}F^{\a\b}\right) .
\end{gather*}
Equation~\eqref{eq:act_break_wo_source} is reobtained if the sources  $\bar{Q}$ and $\bar{J}$ take their `physical values'
\begin{alignat}{3}
& \bar{Q}_{\mu\nu\a\b}\big|_{\text{phys}}=0,\qquad  && \bar{J}_{\mu\nu\a\b}\big|_{\text{phys}}=\frac{\l}{4}\left(\d_{\mu\a}\d_{\nu\b}-\d_{\mu\b}\d_{\nu\a}\right),& \nonumber\\
& Q_{\mu\nu\a\b}\big|_{\text{phys}}=0,\qquad && J_{\mu\nu\a\b}\big|_{\text{phys}}=\frac{\l}{4}\left(\d_{\mu\a}\d_{\nu\b}-\d_{\mu\b}\d_{\nu\a}\right).& \label{JQ-phys}
\end{alignat}
Note that the Hermitian conjugate of the counter term $S_{\text{break}}$ in equation~\eqref{act-loc} (i.e.\ the term $\intx\bar{B}_{\mu\nu}F^{\mu\nu}$) may also be coupled to external sources which, however, is not required for BRST invariance but restores Hermiticity of the action:
\begin{gather*}
\frac{\l}{2}\intx \, \bar{B}_{\mu\nu}F^{\mu\nu}\ \longrightarrow \ \intx\,s\big(J_{\mu\nu\a\b}\bar{\psi}^{\mu\nu}F^{\a\b}\big)=\intx\,J_{\mu\nu\a\b}\bar{B}^{\mu\nu}F^{\a\b}.
\end{gather*}

Including external sources $\Omega^\phi$, $\phi\in\{A,c,B,\bar{B},\psi,\bar{\psi},J,\bar{J},Q,\bar{Q}\}$ for the non-linear BRST transformations the complete action with Landau gauge $\pa^\mu A_\mu=0$ and general $Q/\bar{Q}$ and $J/\bar{J}$ reads:
\begin{gather}
S =S_{\text{inv}}+S_{\phi\pi}+S_{\text{new}}+S_{\text{break}}+S_{\text{ext}} ,\qquad \text{with}\nonumber\\
S_{\text{inv}} =\intx\inv{4}F_{\mu\nu}F_{\mu\nu} ,\nonumber\\
S_{\phi\pi} =\intx\,s\left(\bc\,\pa_\mu A_\mu\right)=\intx\left(b\,\pa_\mu A_\mu-\bc\,\pa_\mu D_\mu c\right) ,
\nonumber\\
S_{\text{new}} =\intx\,s\big(J_{\mu\nu\a\b}\bar{\psi}_{\mu\nu}F_{\a\b}-\mu^2\bar{\psi}_{\mu\nu}D^2\Dt^2B_{\mu\nu}\big)\nonumber\\
\phantom{S_{\text{new}}}{}
=\intx\big(J_{\mu\nu\a\b}\bar{B}_{\mu\nu}F_{\a\b}-\mu^2\bar{B}_{\mu\nu}D^2\Dt^2B_{\mu\nu}+\mu^2\bar{\psi}_{\mu\nu}D^2\Dt^2\psi_{\mu\nu}\big),
\nonumber\\
S_{\text{break}} =\intx\,s\big(\bar{Q}_{\mu\nu\a\b}B_{\mu\nu}F_{\a\b}\big)
=\intx\big(\bar{J}_{\mu\nu\a\b}B_{\mu\nu}F_{\a\b}-\bar{Q}_{\mu\nu\a\b}\psi_{\mu\nu}F_{\a\b}\big),\nonumber\\
S_{\text{ext}} =\intx\left(\Omega^A_\mu D_\mu c+{\rm i}g\, \Omega^c c c+\Omega^{B}_{\mu\nu}\big(\psi_{\mu\nu}+{\rm i}g\co{c}{B_{\mu\nu}}\big) +{\rm i}g\,\Omega^{\bar{B}}_{\mu\nu}\co{c}{\bar{B}_{\mu\nu}}\right.\nonumber\\
 \phantom{S_{\text{ext}} =}{}
 +{\rm i}g\, \Omega^\psi_{\mu\nu}\aco{c}{\psi_{\mu\nu}}+ \Omega^{\bar{\psi}}_{\mu\nu}\big(\bar{B}_{\mu\nu}+{\rm i}g \aco{c}{\bar{\psi}_{\mu\nu}}\big)+\Omega^{Q}_{\mu\nu\a\b}\big( J_{\mu\nu\a\b}+{\rm i}g \aco{c}{Q_{\mu\nu\a\b}}\big)  \nonumber\\
\left.
\phantom{S_{\text{ext}} =}{}
+{\rm i}g\, \Omega^J_{\mu\nu\a\b}\co{c}{J_{\mu\nu\a\b}}+\Omega^{\bar{Q}}_{\mu\nu\a\b}\big( \bar{J}_{\mu\nu\a\b}+{\rm i}g \aco{c}{\bar{Q}_{\mu\nu\a\b}}\big) +{\rm i}g\, \Omega^{\bar{J}}_{\mu\nu\a\b}\co{c}{\bar{J}_{\mu\nu\a\b}}\right).\!\!
\label{eq:act_complete}
\end{gather}
Table~\ref{tab:field_prop} summarizes properties of the f\/ields and sources contained in the model~\eqref{eq:act_complete}.

\begin{table}[!th]\centering
\caption{Properties of f\/ields and sources.}
\label{tab:field_prop}
\vspace{1mm}

\begin{tabular}{l c c c c c c c c c c c}
\hline
\hline
\rule[12pt]{0pt}{0.1pt}
Field       & $A_\mu$ & $c$ & $\bc$ & $B_{\mu\nu}$ & $\bar{B}_{\mu\nu}$ & $\psi_{\mu\nu}$ & $\bar{\psi}_{\mu\nu}$ & $J_{\a\b\mu\nu}$ & $\bar{J}_{\a\b\mu\nu}$ & $Q_{\a\b\mu\nu}$ & $\bar{Q}_{\a\b\mu\nu}$\\[2pt]
\hline
$g_\sharp$  &    0   &  1  &   $-1$  &     0      &    0         &        1      &      $-1$        &   0            &       0              &  $-1$            &  $-1$\tsep{1pt}\\
Mass dim.   &    1   &  0  &   2   &     1      &    1         &        1      &       1        &   1            &       1              &  1             &  1\\
Statistics  &    b   &  f  &   f   &     b      &    b         &        f      &       f        &   b            &       b              &  f             &  f\\
\hline
\rule[14pt]{0pt}{0.1pt}
Source      & $\Omega^A_\mu$ & $\Omega^c$ & $b$ & $\Omega^B_{\mu\nu}$ & $\Omega^{\bar{B}}_{\mu\nu}$ & $\Omega^{\psi}_{\mu\nu}$ & $\Omega^{\bar{\psi}}_{\mu\nu}$ & $\Omega^J_{\a\b\mu\nu}$ & $\Omega^{\bar{J}}_{\a\b\mu\nu}$ &$\Omega^Q_{\a\b\mu\nu}$ & $\Omega^{\bar{Q}}_{\a\b\mu\nu}$\\[2pt]
\hline
$g_\sharp$  &   $-1$   &  $-2$ &   0   &    $-1$      &   $-1$         &       $-2$      &       0        &  $-1$            &      $-1$              & 0             &  0\tsep{1pt}\\
Mass dim.   &    3   &  4  &   2   &     3      &    3         &        3      &       3        &   3            &       3              &  3            &  3\\
Statistics  &    f   &  b  &   b   &    f     &    f         &        b      &      b        &   f            &       f              &  b            &  b\\
\hline
\hline
\end{tabular}
\end{table}

Notice that the mass $\mu$ is a physical parameter despite the fact that the variation of the action $\diff{S}{\mu^2}=s\big(\bar{\psi}_{\mu\nu}D^2\Dt^2B^{\mu\nu}\big)$ yields an exact BRST form. Following the argumentation in~\cite{Baulieu:2009} this is a consequence of the introduction of a soft breaking term. For vanishing Gribov-like parameter $\l$ the contributions to the path integral of the $\mu$ dependent sectors of $S_{\text{new}}$ in \eqref{eq:act_complete} cancel each other. If $\l\neq0$ one has to consider the additional breaking term which couples the gauge f\/ield $A_\mu$ to the auxiliary f\/ield $B_{\mu\nu}$ and the associated ghost $\psi_{\mu\nu}$. This mixing is ref\/lected by the appearance of $a'=\l/\mu$ in the damping factor $\nok$ featured by all f\/ield propagators \eqref{eq:prop_aa}--\eqref{eq:prop_bb} below.

\subsection{Feynman rules}

\subsubsection{Propagators}
From the action \eqref{eq:act_complete} with $J/\bar J$ and $Q/\bar Q$ set to their physical values given by \eqref{JQ-phys} one f\/inds the propagators
\begin{subequations}
\begin{gather}
G^{\bc c}(k) =-\inv{k^2} ,\label{eq:prop_cc}\\
G^{\bar{\psi}\psi}_{\mu\nu,\r\sigma}(k) =\frac{\left(\d_{\mu\r}\d_{\nu\sigma}-\d_{\mu\sigma}\d_{\nu\r}\right)}{2\mu^2k^2\k^2}\,,\label{eq:prop_psipsi}\\
G^{AA}_{\mu\nu}(k) =\inv{\nok}\left(\d_{\mu\nu}-\frac{k_\mu k_\nu}{k^2}\right) ,\label{eq:prop_aa}\\
G^{AB}_{\mu,\r\sigma}(k) =\frac{\ri a'}{2\mu}\frac{\left(k_\r\d_{\mu\sigma}-k_\sigma\d_{\mu\r}\right)}{k^2\k^2\nok}=G^{A\bar{B}}_{\mu,\r\sigma}(k)=-G^{\bar{B} A}_{\r\sigma,\mu}(k) ,\label{eq:prop_ab}\\
G^{\bar{B} B}_{\mu\nu,\r\sigma}(k) =\frac{-1}{2\mu^2k^2\k^2}\left[\d_{\mu\r}\d_{\nu\sigma}-\d_{\mu\sigma}\d_{\nu\r}-a'^2\frac{k_\mu k_\r\d_{\nu\sigma}{+}k_\nu k_\sigma\d_{\mu\r}{-}k_\mu k_\sigma\d_{\nu\r}{-}k_\nu k_\r\d_{\mu\sigma}}{2k^2\k^2\nok}\right]\!,\!\!\!\!\label{eq:prop_bbarb}\\
G^{BB}_{\mu\nu,\r\sigma}(k) =\frac{a'^2}{2\mu^2k^2\k^2}\left[\frac{k_\mu k_\r\d_{\nu\sigma}+k_\nu k_\sigma\d_{\mu\r}-k_\mu k_\sigma\d_{\nu\r}-k_\nu k_\r\d_{\mu\sigma}}{2k^2\k^2\nok}\right]=G^{\bar{B}\bB}_{\mu\nu,\r\sigma}(k) ,\label{eq:prop_bb}
\end{gather}
\end{subequations}
where the abbreviation $a'\equiv\l/\mu$ is used. Notice, that they obey the following symmetries and relations:
\begin{subequations}
\label{proprel}
\begin{gather}
 G^{AB}_{\mu,\r\sigma}(k) =G^{A\bar{B}}_{\mu,\r\sigma}(k)=-G^{BA}_{\r\sigma,\mu}(k)=-G^{\bar{B} A}_{\r\sigma,\mu}(k),\label{prop-rel_a}\\
 G^{\phi}_{\mu\nu,\r\sigma}(k) =-G^{\phi}_{\nu\mu,\r\sigma}=-G^{\phi}_{\mu\nu,\sigma\r}(k)=G^{\phi}_{\nu\mu,\sigma\r}(k),\qquad
 \text{for} \ \ \phi \mathrel{\in}\{\bar{\psi}\psi,\bar{B} B, BB, \bar{B} \bar{B}\},\!\!\! \\
 2k^2\k^2 G^{AB}_{\r,\mu\nu}(k) =\ri\frac{a'}{\mu}\left(k_\mu G^{AA}_{\r\nu}(k)-k_\nu G^{AA}_{\r\mu}(k)\right),\label{prop-rel_c}\\
  \inv{\mu^2}\left(\d_{\mu\rho}\d_{\nu\sigma}-\d_{\mu\sigma}\d_{\nu\rho}\right) =\ri\frac{a'}{\mu}\left(k_\mu G^{BA}_{\r\sigma,\nu}(k)-k_\nu G^{BA}_{\r\sigma,\mu}(k)\right)-2k^2\k^2 G^{B\bar{B}}_{\mu\nu,\rho\sigma}(k),\\
  0 =\ri\frac{a'}{\mu}\left(k_\mu G^{BA}_{\r\sigma,\nu}(k)-k_\nu G^{BA}_{\r\sigma,\mu}(k)\right)-2k^2\k^2G^{BB}_{\mu\nu,\rho\sigma}(k),\label{prop-rel_f}\\
  G^{B\bar{B}}_{\mu\nu,\r\sigma}(k) =G^{\bar{\psi}\psi}_{\mu\nu,\r\sigma}(k)+G^{BB}_{\mu\nu,\r\sigma}(k).
\end{gather}
\end{subequations}

\subsubsection{Vertices}
The action \eqref{eq:act_complete} leads to 13 tree level vertices whose rather lengthy expressions are listed in Appendix~\ref{app:vertices}. One immediately f\/inds the following vertex relation:
\begin{gather*
\widetilde{V}^{\bar{\psi} \psi (n\times A)}_{\mu\nu,\rho\sigma,\xi_1\ldots\xi_n}(q_1, q_2, k_{\xi_1},\ldots, k_{\xi_n}) =
-\widetilde{V}^{\bar{B} B (n\times A)}_{\mu\nu,\rho\sigma,\xi_1\ldots\xi_n}(q_1, q_2, k_{\xi_1},\ldots, k_{\xi_n}),
\end{gather*}
i.e.\ all vertices with one $B$, one $\bar{B}$ and an arbitrary number of $A$ legs have exactly the same form as the ones with one $\psi$, one $\bar{\psi}$ and an arbitrary number of $A$ legs. This is due to the fact that the $\bar{\psi} \psi nA$ and $\bar{B} B nA$ vertices stem from terms in the action which are of the same structure, and are thus equal in their form.

Finally, the vertices obey the following additional relations:
\begin{gather*}
 \widetilde{V}^{\bar{\psi} \psi (n\times A)}_{\mu\nu,\rho\sigma,\xi_1\ldots\xi_n}(q_1, q_2, k_{\xi_1},\ldots, k_{\xi_n}) =-\widetilde{V}^{\psi \bar{\psi} (n\times A)}_{\rho\sigma,\mu\nu,\xi_1\ldots\xi_n}(q_2,q_1, k_{\xi_1}, \ldots, k_{\xi_n})\nonumber\\
\qquad {} =-\widetilde{V}^{\bar{\psi} \psi (n\times A)}_{\nu\mu,\r\sigma,\xi_1\ldots\xi_n}(q_1,q_2, k_{\xi_1},\ldots, k_{\xi_n})
 =-\widetilde{V}^{\bar{\psi} \psi (n\times A)}_{\mu\nu,\sigma\r,\xi_1\ldots\xi_n}(q_1,q_2, k_{\xi_1},\ldots, k_{\xi_n}),
\end{gather*}
and
\begin{gather*}
 \widetilde{V}^{\bar{B} B (n\times A)}_{\mu\nu,\rho\sigma,\xi_1\ldots\xi_n}(q_1, q_2, k_{\xi_1},\ldots, k_{\xi_n}) =+\widetilde{V}^{B \bar{B} (n\times A)}_{\rho\sigma,\mu\nu,\xi_1\ldots \xi_n}(q_2,q_1, k_{\xi_1},\ldots, k_{\xi_n})\\
 =-\widetilde{V}^{\bar{B} B (n\times A)}_{\nu\mu,\r\sigma,\xi_1\ldots\xi_n}(q_1,q_2, k_{\xi_1},\ldots, k_{\xi_n})
 =-\widetilde{V}^{\bar{B} B (n\times A)}_{\mu\nu,\sigma\r,\xi_1\ldots\xi_n}(q_1,q_2, k_{\xi_1},\ldots, k_{\xi_n}),\quad
\text{for}  \  n \in {1,2,3,4} .\nonumber
\end{gather*}

\subsection{Symmetries}

Before moving on to explicit one-loop calculations, let us brief\/ly discuss the symmetries of our action equation~\eqref{eq:act_complete}. The Slavnov--Taylor identity is given by
\begin{gather}
 \mathcal{B}(S)=\intx\Bigg[\var{S}{\Omega^A_\mu}\var{S}{A_\mu}+\var{S}{\Omega^c}\var{S}{c}+ b \var{S}{\bc}+\var{S}{\Omega^B_{\mu\nu}}\var{S}{B_{\mu\nu}}+\var{S}{\Omega^{\bar{B}}_{\mu\nu}}\var{S}{\bar{B}_{\mu\nu}} \nonumber\\
\phantom{\mathcal{B}(S)=}{}+\var{S}{\Omega^\psi_{\mu\nu}}\var{S}{\psi_{\mu\nu}}
+\var{S}{\Omega^{\bar{\psi}}_{\mu\nu}}\var{S}{\bar{\psi}_{\mu\nu}}+\var{S}{\Omega^{Q}_{\mu\nu\a\b}}\var{S}{Q_{\mu\nu\a\b}}
+\var{S}{\Omega^J_{\mu\nu\a\b}}\var{S}{J_{\mu\nu\a\b}}\nonumber\\
 \phantom{\mathcal{B}(S)=}{}+\var{S}{\Omega^{\bar{Q}}_{\mu\nu\a\b}}\var{S}{\bar{Q}_{\mu\nu\a\b}}
+\var{S}{\Omega^{\bar{J}}_{\mu\nu\a\b}}\var{S}{\bar{J}_{\mu\nu\a\b}}\Bigg]=0.\label{mod2-SL-id}
\end{gather}
Furthermore we have the gauge f\/ixing condition
\begin{gather*}
\var{S}{b}=\pa_\mu A_\mu=0,
\end{gather*}
the ghost equation
\begin{gather*}
\mathcal{G}(S)=\pa_\mu\var{S}{\Omega^A_\mu}+\var{S}{\bc}=0,
\end{gather*}
and the antighost equation
\begin{gather*}
\bar{\mathcal{G}}(S)=\intx \var{S}{c}=0.
\end{gather*}

Following the notation of~\cite{Vilar:2009} the identity associated to the BRST doublet structure is given~by
\begin{gather*}
\mathcal{U}^{(1)}_{\a\b\mu\nu}(S)=\intx\Bigg(\bar{B}_{\a\b}\var{S}{\bar{\psi}_{\mu\nu}}+\Omega^{\bar{\psi}}_{\mu\nu}\var{S}{\Omega^{\bar{B}}_{\a\b}}
+\psi_{\mu\nu}\var{S}{B_{\a\b}}-\Omega^B_{\a\b}\var{S}{\Omega^{\psi}_{\mu\nu}} \nonumber\\
\phantom{\mathcal{U}^{(1)}_{\a\b\mu\nu}(S)=}{}
+J_{\mu\nu\r\sigma}\var{S}{Q_{\a\b\r\sigma}}+\Omega^{Q}_{\a\b\r\sigma}\var{S}{\Omega^J_{\mu\nu\r\sigma}}
+\bar{J}_{\a\b\r\sigma}\var{S}{\bar{Q}_{\mu\nu\r\sigma}}+\Omega^{\bar{Q}}_{\mu\nu\r\sigma}\var{S}{\Omega^{\bar{J}}_{\a\b\r\sigma}}\Bigg)=0.
\end{gather*}
Note that the f\/irst two terms of the second line,
\[
\intx\left( J_{\mu\nu\r\sigma}\var{S}{Q_{\a\b\r\sigma}}+\Omega^{Q}_{\a\b\r\sigma}\var{S}{\Omega^J_{\mu\nu\r\sigma}}\right)=0,
\]
constitute a symmetry by themselves. These terms stem from the insertion of conjugated f\/ield partners $J$ and $Q$ for $\bar{J}$ and $\bar{Q}$, respectively, which are not necessarily required as discussed above in~\secref{sec:review}.

Furthermore, we have the linearly broken symmetries $\mathcal{U}^{(0)}$ and $\tilde{\mathcal{U}}^{(0)}$:
\begin{gather*}
\mathcal{U}^{(0)}_{\a\b\mu\nu}(S)=-\Theta^{(0)}_{\a\b\mu\nu}=-\tilde{\mathcal{U}}^{(0)}_{\a\b\mu\nu}(S),
\end{gather*}
with
\begin{gather*}
\mathcal{U}^{(0)}_{\a\b\mu\nu}(S)=\intx\Bigg[B_{\a\b}\var{S}{B_{\mu\nu}}-\bar{B}_{\mu\nu}\var{S}{\bar{B}_{\a\b}}
-\Omega^B_{\mu\nu}\var{S}{\Omega^B_{\a\b}}+\Omega^{\bar{B}}_{\a\b}\var{S}{\Omega^{\bar{B}}_{\mu\nu}} \nonumber\\
\phantom{\mathcal{U}^{(0)}_{\a\b\mu\nu}(S)=}{} +J_{\a\b\r\sigma}\var{S}{J_{\mu\nu\r\sigma}}-\bar{J}_{\mu\nu\r\sigma}\var{S}{\bar{J}_{\a\b\r\sigma}}
-\Omega^J_{\mu\nu\r\sigma}\var{S}{\Omega^J_{\a\b\r\sigma}}+\Omega^{\bar{J}}_{\a\b\r\sigma}\var{S}{\Omega^{\bar{J}}_{\mu\nu\r\sigma}}\Bigg],\nonumber\\
\tilde{\mathcal{U}}^{(0)}_{\a\b\mu\nu}(S)= \intx\Bigg[\psi_{\a\b}\var{S}{\psi_{\mu\nu}}-\bar{\psi}_{\mu\nu}\var{S}{\bar{\psi}_{\a\b}}
-\Omega^{\psi}_{\mu\nu}\var{S}{\Omega^{\psi}_{\a\b}}+\Omega^{\bar{\psi}}_{\a\b}\var{S}{\Omega^{\bar{\psi}}_{\mu\nu}} \nonumber\\
\phantom{\tilde{\mathcal{U}}^{(0)}_{\a\b\mu\nu}(S)=}{} +Q_{\a\b\r\sigma}\var{S}{Q_{\mu\nu\r\sigma}}-\bar{Q}_{\mu\nu\r\sigma}\var{S}{\bar{Q}_{\a\b\r\sigma}}
-\Omega^Q_{\mu\nu\r\sigma}\var{S}{\Omega^Q_{\a\b\r\sigma}}+\Omega^{\bar{Q}}_{\a\b\r\sigma}\var{S}{\Omega^{\bar{Q}}_{\mu\nu\r\sigma}}\Bigg],\nonumber\\
\Theta^{(0)}_{\a\b\mu\nu}=\intx\left[\bar{B}_{\mu\nu}\Omega^{\bar{\psi}}_{\a\b}-\psi_{\a\b}\Omega^B_{\mu\nu}
+\bar{J}_{\mu\nu\r\sigma}\Omega^{\bar{Q}}_{\a\b\r\sigma}-J_{\a\b\r\sigma}\Omega^{Q}_{\mu\nu\r\sigma}\right].
\end{gather*}
The above relations would, if applicable, form the starting point for the algebraic renormalization procedure. In order to assure the completeness of the set of symmetries it has to be assured that the algebra generated by them closes. From the Slavnov--Taylor identity \eqref{mod2-SL-id} one derives the linearized Slavnov operator
\begin{gather*}
 \mathcal{B}_{S}= \intx\Bigg[\var{S}{\Omega^A_\mu}\var{\ }{A_\mu}+\var{S}{A_\mu}\var{\ }{\Omega^A_\mu}+\var{S}{c}\var{\ }{\Omega^c}+\var{S}{\Omega^c}\var{\ }{c}+ b \var{S}{\bc}+\var{S}{\Omega^B_{\mu\nu}}\var{\ }{B_{\mu\nu}}+\var{S}{B_{\mu\nu}}\var{\ }{\Omega^B_{\mu\nu}}\nonumber\\
\phantom{\mathcal{B}_{S}=}{} +\var{S}{\Omega^{\bar{B}}_{\mu\nu}}\var{\ }{\bar{B}_{\mu\nu}}+\var{S}{\bar{B}_{\mu\nu}}\var{\ }{\Omega^{\bar{B}}_{\mu\nu}}+\var{S}{\Omega^\psi_{\mu\nu}}\var{\ }{\psi_{\mu\nu}}+\var{S}{\psi_{\mu\nu}}\var{\ }{\Omega^\psi_{\mu\nu}}+\var{S}{\Omega^{\bar{\psi}}_{\mu\nu}}\var{\ }{\bar{\psi}_{\mu\nu}}+\var{S}{\bar{\psi}_{\mu\nu}}\var{\ }{\Omega^{\bar{\psi}}_{\mu\nu}}\nonumber\\
\phantom{\mathcal{B}_{S}=}{}+\var{S}{\Omega^{Q}_{\mu\nu\a\b}}\var{\ }{Q_{\mu\nu\a\b}}+\var{S}{Q_{\mu\nu\a\b}}\var{\ }{\Omega^{Q}_{\mu\nu\a\b}}+\var{S}{\Omega^J_{\mu\nu\a\b}}\var{\ }{J_{\mu\nu\a\b}}+\var{S}{J_{\mu\nu\a\b}}\var{\ }{\Omega^J_{\mu\nu\a\b}}\nonumber\\
\phantom{\mathcal{B}_{S}=}{} +\var{S}{\Omega^{\bar{Q}}_{\mu\nu\a\b}}\var{\ }{\bar{Q}_{\mu\nu\a\b}}+\var{S}{\bar{Q}_{\mu\nu\a\b}}\var{\ }{\Omega^{\bar{Q}}_{\mu\nu\a\b}}+\var{S}{\Omega^{\bar{J}}_{\mu\nu\a\b}}\var{\ }{\bar{J}_{\mu\nu\a\b}}+\var{S}{\bar{J}_{\mu\nu\a\b}}\var{\ }{\Omega^{\bar{J}}_{\mu\nu\a\b}}\Bigg]
\end{gather*}
Furthermore, the $\mathcal{U}^{(0)}$ and $\tilde{\mathcal{U}}^{(0)}$ symmetries are combined to def\/ine the operator $\mathcal{Q}$ as
\begin{gather*}
\mathcal{Q}\equiv\d_{\a\mu}\d_{\b\nu}\big(\mathcal{U}^{(0)}_{\a\b\mu\nu}+\tilde{\mathcal{U}}^{(0)}_{\a\b\mu\nu}\big).
\end{gather*}
Notice that the action is invariant under $\mathcal{Q}$, i.e.\ $\mathcal{Q}(S)=0$ because of $\mathcal{U}^{(0)}_{\a\b\mu\nu}(S)=-\tilde{\mathcal{U}}^{(0)}_{\a\b\mu\nu}(S)$.

Having def\/ined the operators $\mathcal{B}_{S}$, $\bar{\mathcal{G}}$, $\mathcal{Q}$ and $\mathcal{U}^{(1)}$ we may derive the following set of graded commutators:
\begin{alignat*}{4}
& \aco{\bar{\mathcal{G}}}{\bar{\mathcal{G}}}=0,\qquad & & \aco{\mathcal{B}_{S}}{\mathcal{B}_{S}}=0,\qquad  & &
\{\bar{\mathcal{G}},\mathcal{B}_{S}\}=0, & \nonumber\\
& \co{\bar{\mathcal{G}}}{\mathcal{Q}}=0, \qquad & & \co{\mathcal{Q}}{\mathcal{Q}}=0, \qquad & &
\big\{\bar{\mathcal{G}},\mathcal{U}^{(1)}_{\mu\nu\a\b}\big\}=0,  & \nonumber\\
&
\big\{\mathcal{B}_{S},\mathcal{U}^{(1)}_{\mu\nu\a\b}\big\}=0,\qquad & &
\big\{\mathcal{U}^{(1)}_{\mu\nu\a\b},\mathcal{U}^{(1)}_{\mu'\nu'\a'\b'}\big\}=0,
\qquad  & &
\big[\mathcal{U}^{(1)}_{\mu\nu\a\b},\mathcal{Q}\big]=0,
& \nonumber\\
& \co{\mathcal{B}_{S}}{\mathcal{Q}}=0,&& &&
\end{alignat*}
which shows that the algebra of symmetries closes.

Having derived the symmetry content of the model, we would now be ready to apply the method of Algebraic Renormalization (AR). The latter requires locality which, however, is not given in the present case and generally for all {non-com\-mu\-ta\-tive} QFTs, due to the inherent non-locality of the star product. Hence, before the application of AR it would be required to establish the foundations of this method also for {non-com\-mu\-ta\-tive} theories. For a detailed discussion we would like to refer to our recent article~\cite{Blaschke:2009c}.

\section{One-loop calculations}
\label{sec:one_loop}

In this section we shall present the calculations relevant for the one-loop correction to the gauge boson propagator. Due to the existence of the mixed propagators $G^{\text{AB}}$, $G^{\text{A}\bar{B}}$, and their mirrored counterparts, the two point function $\langle A_\mu A_\nu\rangle$ receives contributions not only from graphs with external gauge boson legs, but also from those featuring external $B$ and/or $\bar{B}$ f\/ields.

In the following (i.e.~in Sections~\ref{sec:1loop_vacpol}--\ref{sec:1loop_BbB}), we will present a detailed analysis of all truncated two-point functions relevant for the calculation of the full one-loop $AA$-propagator. Every type of correction, being characterized by its amputated external legs (i.e.~$A$, $B$ or $\bar{B}$), is discussed in a separate subsection. Finally, in \secref{sec:1loop_analysis} the dressed $AA$-propagator and the attempt for its one-loop renormalization will be given explicitly.

\subsection{Vacuum polarization}
\label{sec:1loop_vacpol}

The model \eqref{eq:act_complete} gives rise to 23 graphs contributing to the two-point function $G^{AA}_{\mu\nu}(p)$. Omitting convergent expressions, there are 11 graphs left depicted in \figref{fig:1loop_vacpol_all}.
Being interested in the divergent contributions one can apply the expansion~\cite{Blaschke:2009a}
\begin{gather}
\Pi_{\mu\nu}= \int\limits_{-\infty}^{+\infty}\!\! \frac{{\rm d}^4k}{\left(2\pi\right)^4}\, \mathcal{I}_{\mu\nu}(p,k) \sin^2\left(\tfrac{k\p}{2}\right)\approx
 \int\limits_{-\infty}^{+\infty}\!\! \frac{{\rm d}^4k}{\left(2\pi\right)^4}\sin^2\left(\tfrac{k\p}{2}\right)\bigg\{\mathcal{I}_{\mu\nu}(0,k)+p_\rho\left[\partial_{p_\rho}\mathcal{I}_{\mu\nu}(p,k)\right]_{p=0}\nonumber\\
\phantom{\Pi_{\mu\nu}=}{} +\frac{p_\rho p_\sigma}{2}\left[\partial_{p_\rho}\partial_{p_\sigma}\mathcal{I}_{\mu\nu}(p,k)\right]_{p=0}
+\mathcal{O}\left(p^3\right)\bigg\},\label{eq:1-l_expansion}
\end{gather}
where the integrand $\mathcal{I}_{\mu\nu}(p,k)$ has been separated from the phase factor in order to keep the regularizing ef\/fects in the non-planar parts due to rapid oscillations for large $k$. Summing up the contributions of the graphs in \figref{fig:1loop_vacpol_all} and denoting the result at order~$i$ for the planar (p) part by $\Pi_{\mu\nu}^{(i),\text{p}}$, one is left with
\begin{gather*}
 \Pi_{\mu\nu}^{(0),\text{p}}(p)= \frac{g^2}{16 \pi ^2} \L^2 \d _{\mu\nu}\left(-10 s_{\rm c}-96 s_{\rm h}-96 s_{\rm j}+12 s_{\rm a}+s_{\rm b}+96 s_{\rm d}+96 s_{\rm f}\right) =0, \\
\Pi_{\mu\nu}^{(2),\text{p}}(p)= -\frac{1}{3} \frac{g^2}{16 \pi ^2} \Big[\d _{\mu\nu} p^2\left(22 s_{\rm a}+s_{\rm b}+48(s_{\rm d}+s_{\rm f})\right)\nonumber\\
 \phantom{\Pi_{\mu\nu}^{(2),\text{p}}(p)=}{}
 +2 p_{\mu} p_{\nu} \left(72 (s_{\rm h}+s_{\rm j})-8 s_{\rm a}+s_{\rm b}-96(s_{\rm d}+s_{\rm f})\right)\Big] {\rm K}_0\left(2 \sqrt{\frac{M^2}{\L ^2}}\right)\nonumber\\
 \phantom{\Pi_{\mu\nu}^{(2),\text{p}}(p)}{}= -\frac{5 g^2}{12 \pi ^2} \left(p^2 \d _{\mu\nu}-p_{\mu} p_{\nu}\right) K_0\left(2 \sqrt{\frac{M^2}{\L ^2}}\right) \nonumber\\
\phantom{\Pi_{\mu\nu}^{(2),\text{p}}(p)}{} \approx -\frac{5 g^2}{24 \pi ^2} \left(p^2 \d _{\mu\nu}-p_{\mu} p_{\nu}\right) \ln\left(\frac{\L^2}{M^2}\right)+\text{f\/inite}
\end{gather*}
where the symmetry factors in Table~\ref{tab:1loop_vacpol_sf} have been inserted and the approximation
\[
\mathrm{K}_0(x)\underset{x\ll1}{\approx}\ln\tfrac{2}{x}-\gamma_E+\mathcal{O}\left(x^2\right),
\]
for the modif\/ied Bessel function ${\rm K}_0$ can be utilized for small arguments, i.e.\ vanishing regulator cutof\/fs\footnote{The cutof\/fs are introduced via a factor $\exp\left[-M^2\alpha-\inv{\Lambda^2\alpha}\right]$ to regularize parameter integrals $\int_0^\infty{\rm d}\alpha$. See~\cite{Blaschke:2009a} for a more extensive description of the mathematical details underlying these computations.} $\Lambda\to \infty$ and $M\to0$. Finally, $\gamma_E$ denotes the Euler--Mascheroni constant. Note that the f\/irst order vanishes identically due to an odd power of $k$ in the integrand which leads to a~cancellation under the symmetric integration over the momenta.
\begin{figure}[t]
 \centering
 \includegraphics[scale=0.8]{Blaschke-Fig1}
 \caption{One loop corrections for the gauge boson propagator.}
 \label{fig:1loop_vacpol_all}
\end{figure}

\begin{table}[t]
 \centering
 \caption{Symmetry factors for the one loop vacuum polarization (where the factor $(-1)$ for fermionic loops has been included).}
 \begin{tabular}{l r | l r | l r}
\hline
 \hline
 $s_{\rm a}$ &$\inv{2}$ & $s_{\rm e}$ &  1& $s_{\rm i}$ & 1\tsep{2pt}\\
 $s_{\rm b}$ & $-1$ 	& $s_{\rm f}$ & $-1$& $s_{\rm j}$ & $-1$\\
 $s_{\rm c}$ &$\inv{2}$ & $s_{\rm g}$ & $-1$& $s_{\rm k}$ & $-1$\\
 $s_{\rm d}$ &1 	& $s_{\rm h}$ &  1&       &      \\
\hline
\hline
 \end{tabular}
 \label{tab:1loop_vacpol_sf}
\end{table}

Of particular interest is the non-planar part (np) which for small $p$ results to:
\begin{subequations}
\begin{gather}
 \Pi_{\mu\nu}^{(0),\text{np}}(p)= \frac{g^2}{4\pi ^2\p^2} \Big[\d _{\mu\nu}\left(96(s_{\rm h}+s_{\rm j}- s_{\rm d}- s_{\rm f})-12 s_{\rm a}-s_{\rm b}+10 s_{\rm c}\right)\nonumber\\*
 \phantom{\Pi_{\mu\nu}^{(0),\text{np}}(p)=}{}
 -2 \frac{\p_{\mu} \p_{\nu}}{\p^2} \left(48(s_{\rm h}+ s_{\rm j})-96 (s_{\rm d}+s_{\rm f})-12 s_{\rm a}-s_{\rm b}+2 s_{\rm c}\right)\Big]
= \frac{2g^2}{\pi^2}\frac{\p_\mu \p_\nu}{(\p^2)^2} , \label{eq:1loop_AA_o0_np}\\
   \Pi_{\mu\nu}^{(2),\text{np}}(p)= \frac{g^2}{48\pi^2\p^2} \bigg\{2 \th ^2 p_\mu p_\nu p^2\left(72 (s_{\rm h}+s_{\rm j})-8 s_{\rm a}+s_{\rm b}- 96(s_{\rm d}+ s_{\rm f})\right) \mathop{{\rm K}_0}\left(\sqrt{M^2 \p^2}\right)\nonumber\\ \phantom{\Pi_{\mu\nu}^{(2),\text{np}}(p)=}{}
   +\sqrt{\frac{\p^2}{M^2}}p^2\bigg[\sqrt{\frac{\p^2}{M^2}}\left(22 s_{\rm a}+s_{\rm b}+48(s_{\rm d}+s_{\rm f})\right)M^2 \d_{\mu\nu} \mathop{{\rm K}_0}\left(\sqrt{M^2 \p^2}\right)\nonumber\\
\phantom{\Pi_{\mu\nu}^{(2),\text{np}}(p)=}{}
+2 M^2 \left(13 s_{\rm a}+s_{\rm b}+120(s_{\rm d}+s_{\rm f})\right) \p_{\mu} \p_\nu \mathop{{\rm K}_1}\left(\sqrt{M^2 \p^2}\right)\nonumber\\
\phantom{\Pi_{\mu\nu}^{(2),\text{np}}(p)=}{}
-3 \sqrt{\frac{M^2}{\p^2}} \left(16 s_{\rm a}+s_{\rm b}+96(s_{\rm d}+ s_{\rm f})\right) \p_\mu \p_\nu\bigg]\bigg\}\nonumber\\
\phantom{\Pi_{\mu\nu}^{(2),\text{np}}(p)}{}= -\frac{g^2}{48\pi^2} \bigg[\p_\mu\p_\nu\bigg(\frac{21}{\th^2}-11 p^2 \sqrt{\frac{M^2}{\p^2}}\mathop{{\rm K}_1}\left(\sqrt{M^2\p^2}\right)\bigg)\nonumber\\
\phantom{\Pi_{\mu\nu}^{(2),\text{np}}(p)=}{}
-10 \mathop{{\rm K}_0}\left(\sqrt{M^2 \p^2}\right) \left(p^2\d_{\mu\nu} -p_\mu p_\nu \right)\bigg].
 \label{eq:1loop_AA_o2_np}
\end{gather}
\end{subequations}
Considering the limit $\p^2\to0$ rectif\/ies application of the approximation
\[
\mathrm{K}_1(x)\underset{x\ll1}{\approx}\tfrac{1}{x}
+\tfrac{x}{2}\left(\gamma_E-\tfrac{1}{2}+\ln\tfrac{x}{2}\right)+\mathcal{O}\left(x^2\right),
\]
which reveals that the second order is IR f\/inite (which is immediately clear from the fact that the terms of lowest order in $p$ are $\mathcal{O}\left(p^2\right)$), apart from a $\ln (M^2)$-term which cancels in the sum of planar and non-planar contributions. Hence, collecting all divergent terms one is left with (in the limit $M\to0$ and $\L\to\infty$),
\begin{gather}\label{eq:complete-1loop-sum}
\Pi_{\mu\nu}(p) =\frac{2g^2}{\pi^2}\frac{\p_\mu \p_\nu}{(\p^2)^2}-\lim\limits_{\L\to\infty}\frac{5 g^2}{24 \pi ^2} \left(p^2 \delta _{\mu\nu}-p_{\mu} p_{\nu}\right) \ln\left(\Lambda^2\right)+\text{f\/inite terms},
\end{gather}
which is independent of the IR-cutof\/f $M$. As expected\footnote{In fact, equation~\eqref{eq:complete-1loop-sum} qualitatively resembles the result of the ``na\"ive'' {non-com\-mu\-ta\-tive} gauge model discussed e.g.\ in~\cite{Hayakawa:1999, Armoni:2000xr,Ruiz:2000,Blaschke:2005b}. The dif\/ferent numerical factor in front of the logarithmic UV divergence is a~consequence of the contribution of additional f\/ields in the current model.}, equation~\eqref{eq:complete-1loop-sum} exhibits a quadratic IR divergence in $\p^2$ and a logarithmic divergence in the cutof\/f $\L$. Furthermore, the transversality condition $p_\mu\Pi_{\mu\nu}(p)=0$ is fulf\/illed, which serves as a consistency check for the symmetry factors.

\subsection[Corrections to the $AB$ propagator]{Corrections to the $\boldsymbol{AB}$ propagator}
\label{sec:1loop_AB}

\begin{figure}[t]
 \centering
 \includegraphics[scale=0.8]{Blaschke-Fig2}
 \caption{One loop corrections for $\langle A_\mu B_{\nu_1\nu_2} \rangle$ (with amputated external legs).}
 \label{fig:1loop_AB}
\end{figure}
\begin{table}[t]
 \centering
 \caption{Symmetry factors for the graphs depicted in \figref{fig:1loop_AB}.}
 \vspace{1mm}

 \begin{tabular}{l r | l r }
\hline
 \hline
 $(a)$ & 1/2 &  $(e)$ & 1\tsep{2pt}\\
 $(b)$ & 1 & $(f)$ & 1 \\
 $(c)$ & 1 &  $(g)$ & 1 \\
 $(d)$ & 1 &  $(h)$ & 1\\
\hline
\hline
 \end{tabular}
 \label{tab:1loop_AB_div}
\end{table}

The action \eqref{eq:act_complete} gives rise to eight divergent graphs with one external $A_\mu$ and one $B_{\mu\nu}$ which are depicted in~\figref{fig:1loop_AB}.
Applying an expansion of type~\eqref{eq:1-l_expansion} for small external momenta $p$ and summing up the divergent contributions of all graphs (all orders of an expansion similar to equation~\eqref{eq:1-l_expansion}) one ends up with,
\begin{gather*}
\Sigma_{\m1,\n1 \n2}^{\text{p,AB}}(p)=-\frac{3\ri g^2}{32 \pi^2} \lambda  \left(p_{\nu 1} \d_{\m1 \n2}-p_{\n2} \d_{\m1 \n1}\right) \mathop{{\rm K}_0}\left(2 \sqrt{\frac{M^2}{\Lambda^2}}\right)+{\rm f\/inite}, \nonumber\\
\Sigma_{\m1,\n1 \n2}^{\text{np,AB}}(p)= \frac{3\ri g^2}{32 \pi^2} \lambda \mathop{{\rm K}_0}\left(\sqrt{M^2 \p^2}\right) \left(p_{\nu1} \delta_{\mu1 \nu2}-p_{\nu2} \delta_{\mu1 \nu1}\right)+{\rm f\/inite}.
\end{gather*}
Approximating the Bessel functions as in \secref{sec:1loop_vacpol} and summing up planar and non-planar parts one f\/inds the expression
\begin{gather*}
\Sigma_{\m1,\n1\n2}^{\text{AB}}(p)= \frac{3\ri g^2}{32 \pi^2} \lambda \left(p_{\nu 1} \d_{\m1 \n2}-p_{\n2} \d_{\m1 \n1}\right)\left(\ln\Lambda+\ln|\p|\right)+{\rm f\/inite},
\end{gather*}
where the IR cutof\/f $M$ has cancelled, and which shows a logarithmic divergence for $\Lambda\to \infty$.

Due to the symmetry between $B$ and $\bar{B}$ in the sense that both have identical interactions with the gauge f\/ield, it is obvious that $\Sigma_{\m1,\n1\n2}^{\rm AB}\equiv\Sigma_{\m1,\n1\n2}^{{\rm A}\bar{B}}$ and as implied by equation~\eqref{prop-rel_a} it also holds that $\Sigma_{\m1\m2,\n1}^{\rm BA}\equiv-\Sigma_{\n1,\m1\m2}^{\rm AB}$.

\subsection[Corrections to the $BB$ propagator]{Corrections to the $\boldsymbol{BB}$ propagator}
\label{sec:1loop_BB}

\begin{figure}[t]
 \centering
 \includegraphics[scale=0.8]{Blaschke-Fig3}
 \caption{One loop corrections for $\langle  B_{\mu_1\mu_2} B_{\nu_1\nu_2} \rangle$ (with amputated external legs).}
 \label{fig:1loop_BB}
\end{figure}
\begin{table}[t]
 \centering
 \caption{Symmetry factors for the graphs depicted in \figref{fig:1loop_BB}.}
 \vspace{1mm}

 \begin{tabular}{l c | l c | l c}
\hline
 \hline
$(a)$ & 1/2 &  $(d)$ & 1 &  $(g)$ & 1\tsep{2pt}\\
$(b)$ & 1 &  $(e)$ & 1 &  $(h)$ & 1\\
$(c)$ & 1 &  $(f)$ & 1 &  $(i)$ & 1\\
\hline
\hline
 \end{tabular}
 \label{tab:1loop_BB_div}
\end{table}

The set of divergent graphs contributing to $\langle B_{\m1\m2}B_{\n1\n2} \rangle$ consists of those depicted in \figref{fig:1loop_BB}.
Making an expansion of type \eqref{eq:1-l_expansion} for small external momenta $p$ and summing up the contributions of all nine graphs yields
\begin{gather*}
\Sigma_{\m1\m2,\n1 \n2}^{\text{p,BB}}(p)= \frac{g^2 \lambda ^2}{32 \pi^2} \left(\d _{\m1 \n1} \d_{\m2 \n2}-\d _{\m2 \n1} \d _{\m1 \n2}\right)  \mathop{{\rm K}_0}\left(2 \sqrt{\frac{M^2}{\L ^2}}\right)+{\rm f\/inite}, \nonumber\\
\Sigma_{\m1\m2,\n1 \n2}^{\text{np,BB}}(p)= \frac{g^2 \lambda ^2}{64 \pi^2} \bigg(\frac{\d_{\m1\n2} \p_{\m2}\p_{\n1}-\d_{\m1 \n1}\p_{\m2}\p_{\n2}-\d_{\mu 2 \nu 2} \p_{\m1} \p_{\n1}+\d_{\m2\n1}\p_{\m1}\p_{\n2}}{\p^2}\nonumber\\
 \phantom{\Sigma_{\m1\m2,\n1 \n2}^{\text{np,BB}}(p)= }{}
+2  \mathop{{\rm K}_0}\left(\sqrt{M^2 \p^2}\right) \left(\d_{\m1 \n2} \d_{\m2 \n1}-\d_{\m1 \n1} \d _{\m2 \n2}\right)\bigg)+{\rm f\/inite},
\end{gather*}
for the planar/non-planar part, respectively.
Approximating the Bessel functions as in \secref{sec:1loop_vacpol} reveals cancellations of contributions depending on $M$ in the f\/inal sum. Hence, the divergent part boils down to
\begin{gather}
\Sigma_{\m1\m2,\n1\n2}^{\text{BB}}(p)= \frac{g^2 \l^2}{64 \pi^2}\left(\d_{\m1 \n1} \d_{\m2 \n2}-\d_{\m2 \n1} \d_{\m1 \n2}\right)\left(\ln\L^2+\ln\p^2\right)+{\rm f\/inite},
\label{eq:1loop_BB_final}
\end{gather}
leaving a logarithmic divergence for both the planar and the non-planar part.
Due to symmetry reasons this result is also equal to the according correction to the $\bar{B}\bB$ propagator, i.e.
\begin{gather*}
 \Sigma_{\m1\m2,\n1\n2}^{\rm \bar{B}\bB}(p)=\Sigma_{\m1\m2,\n1\n2}^{\text{BB}}(p).
\end{gather*}

\subsection[Corrections to the $B\bar{B}$ propagator]{Corrections to the $\boldsymbol{B\bar{B}}$ propagator}
\label{sec:1loop_BbB}

\begin{figure}[t]
 \centering
 \includegraphics[scale=0.8]{Blaschke-Fig4}
 \caption{One loop corrections for $\langle   B_{\mu_1\mu_2} \bar{B}_{\nu_1\nu_2}  \rangle$ (with amputated external legs).}
 \label{fig:1loop_BbB}
\end{figure}

\begin{table}[t]
 \centering
 \caption{Symmetry factors for the graphs depicted in \figref{fig:1loop_BbB}.}
 \vspace{1mm}

 \begin{tabular}{l c | l c | l c}
 \hline
 \hline
$(a)$ & 1/2 &  $(e)$ & 1 &  $(i)$ & 1\tsep{2pt}\\
$(b)$ & 1 &  $(f)$ & 1 &  $(j)$ & 1/2\\
$(c)$ & 1 &  $(g)$ & 1 &  & \\
$(d)$ & 1 &  $(h)$ & 1 &  & \\
\hline
\hline
 \end{tabular}
 \label{tab:1loop_BbB_div}
\end{table}

For the correction to $\langle B_{\m1\m2}\bar{B}_{\n1\n2} \rangle$ one f\/inds the ten divergent graphs depicted in \figref{fig:1loop_BbB}.
Expansion for small external momenta $p$ and summation of the integrated results yields
\begin{gather*}
\Sigma_{\m1\m2,\n1 \n2}^{{\rm p,B\bar{B}}}(p)= \frac{g^2}{2 \pi^2} \L ^2 \mu ^2 \p^2\left(\d_{\mu2\nu1} \d _{\mu 1 \n2}-\d _{\mu 1 \n1} \d _{\mu 2 \n2}\right)\nonumber\\
\phantom{\Sigma_{\m1\m2,\n1 \n2}^{{\rm p,B\bar{B}}}(p)=}{}
+\frac{g^2 \l^2}{32 \pi^2} \left(\d _{\m1 \n1} \d_{\m2 \n2}-\d _{\m2 \n1} \d _{\m1 \n2}\right)  \mathop{{\rm K}_0}\left(2 \sqrt{\frac{M^2}{\L ^2}}\right)+{\rm f\/inite}, \nonumber\\
\Sigma_{\m1\m2,\n1 \n2}^{{\rm np,B\bar{B}}}(p)= \frac{g^2 \l^2}{64 \pi^2} \bigg(\frac{\d_{\m1\n2} \p_{\m2}\p_{\n1}-\d_{\m1 \n1}\p_{\m2}\p_{\n2}-\d_{\mu 2 \nu 2} \p_{\m1} \p_{\n1}+\d_{\m2\n1}\p_{\m1}\p_{\n2}}{\p^2}\nonumber\\
 \phantom{\Sigma_{\m1\m2,\n1 \n2}^{{\rm np,B\bar{B}}}(p)=}{}
 +2  \mathop{{\rm K}_0}\left(\sqrt{M^2 \p^2}\right) \left(\d_{\m1 \n2} \d_{\m2 \n1}-\d_{\m1 \n1} \d _{\m2 \n2}\right)\bigg)+{\rm f\/inite} ,
\end{gather*}
Hence, the divergent part is given by
\begin{gather*}
\Sigma_{\m1\m2,\n1\n2}^{\text{B}\bar{B}}(p)=\frac{g^2}{2 \pi^2} \L^2 \mu ^2 \p^2\left(\d_{\mu2\nu1} \d_{\mu 1 \n2}-\d_{\mu 1 \n1} \d_{\mu 2 \n2}\right)\nonumber\\
\phantom{\Sigma_{\m1\m2,\n1\n2}^{\text{B}\bar{B}}(p)=}{} +\frac{g^2 \l^2}{64 \pi^2}\left(\d_{\m1 \n1} \d_{\m2 \n2}-\d_{\m2 \n1} \d_{\m1 \n2}\right)\left(\ln\L^2+\ln\p^2\right)+{\rm f\/inite},
\end{gather*}
which is logarithmically divergent in $\p^2$ and quadratically in $\Lambda$. Once more, $M$ has dropped out in the sum of planar and non-planar contributions.
Furthermore, note that $\Sigma_{\m1\m2,\n1\n2}^{\rm B\bar{B}}\equiv\Sigma_{\n1\n2,\m1\m2}^{\rm \bar{B} B}$ as is obvious from the result \eqref{eq:1loop_BB_final}.

\subsection{Dressed gauge boson propagator and analysis}
\label{sec:1loop_analysis}

In the standard renormalization procedure, the dressed propagator at one-loop level is given by
\begin{gather}\label{eq:dressed1}
\raisebox{-11pt}[0pt][0pt]{\includegraphics[scale=0.23]{Blaschke-P1}}
\equiv \Delta'(p) = \inv{\mathcal A} + \inv{\mathcal A}
\Sigma(\Lambda,p)  \inv{\mathcal A}  ,
\end{gather}
where
\begin{gather*}
  \inv{\mathcal A} \equiv  G^{\rm AA}_{\mu\nu}(p) ,
\qquad
 \Sigma(\Lambda,p)\equiv \big( \Pi^{\text{plan}}
\big)_{{\rm regul.}} (\Lambda,p) +\Pi^{\text{n-pl}}(p) .
\end{gather*}
For $\mathcal A\neq 0$, one can apply the formula
\begin{gather}
\label{eq:Expansion}
\inv{\mathcal A+\mathcal B}=\inv{\mathcal A}-\inv{\mathcal A}\, \mathcal B \, \inv{\mathcal A+\mathcal B} = \inv{\mathcal A}-\inv{\mathcal A}\, \mathcal B
\, \inv{\mathcal A} + {\mathcal O} (\mathcal B^2)   ,
\end{gather}
which allows one to rewrite expression \eqref{eq:dressed1} to order
$\Sigma$ as
\begin{gather*}
\Delta'(p) =\inv{\mathcal A-\Sigma(\Lambda,p)}  ,
\end{gather*}
and thus (in the case of renormalizability) to absorb any divergences in the appropriate parameters of the theory present in $\mathcal A$ (see \cite{Blaschke:2008b} for an example).

However, in our case \eqref{eq:Expansion} cannot be applied directly, as the complete one loop correction to the gauge boson propagator is given by the sum of all the results of Sections~\ref{sec:1loop_vacpol}--\ref{sec:1loop_BbB} after multiplication with {\em appropriate, i.e.\ different} external legs:
\begin{gather}
G^{\rm AA,1l-ren}_{\mu\nu}(p)=G^{\rm AA}_{\mu\nu}(p)+G^{\rm AA}_{\mu\r}(p)\Pi_{\r\sigma}(p)G^{\rm AA}_{\sigma\nu}(p)
  + G^{\rm AA}_{\mu\r}(p)2\Sigma^{\rm AB}_{\r,\s1\s2}(p)G^{\rm BA}_{\s1\s2,\nu}(p)\nonumber\\
\phantom{G^{\rm AA,1l-ren}_{\mu\nu}(p)=}{}  + G^{\rm AA}_{\mu\r}(p)2\Sigma^{\rm A\bar{B}}_{\r,\s1\s2}(p)G^{\rm \bar{B} A}_{\s1\s2,\nu}(p)
  + G^{\rm AB}_{\mu,\r1\r2}(p)\Sigma^{\rm BB}_{\r1\r2,\s1\s2}(p)G^{\rm BA}_{\s1\s2,\nu}(p)\nonumber\\
\phantom{G^{\rm AA,1l-ren}_{\mu\nu}(p)=}{}  + G^{\rm AB}_{\mu,\r1\r2}(p)2\Sigma^{\rm B\bar{B}}_{\r1\r2,\s1\s2}(p)G^{\rm \bar{B} A}_{\s1\s2,\nu}(p)
  + G^{\rm A\bar{B}}_{\mu,\r1\r2}(p)\Sigma^{\rm \bar{B}\bB}_{\r1\r2,\s1\s2}(p)G^{\rm \bar{B} A}_{\s1\s2,\nu}(p)\nonumber\\
 \phantom{G^{\rm AA,1l-ren}_{\mu\nu}(p)=}{}
  +\mathcal{O}\left(g^4\right).\label{eq:DressedPropComplete1}
\end{gather}
Note, that the factors 2 stem from the (not explicitly written) mirrored contributions $AB\leftrightarrow BA$, $A\bar{B}\leftrightarrow \bar{B} A$, and $B\bar{B}\leftrightarrow \bar{B} B$.
Since the factor $\mathcal A$ must be the same for all summands we have to use the Ward Identities
\eqref{prop-rel_a} and \eqref{prop-rel_c}, i.e.
\begin{gather}
 G^{AB}_{\mu,\r\sigma}(k) =G^{A\bar{B}}_{\mu,\r\sigma}(k)=-G^{BA}_{\r\sigma,\mu}(k)=-G^{\bar{B} A}_{\r\sigma,\mu}(k),\nonumber\\
 2k^2\k^2 G^{AB}_{\r,\mu\nu}(k) =\ri\frac{a'}{\mu}\left(k_\mu G^{AA}_{\r\nu}(k)-k_\nu G^{AA}_{\r\mu}(k)\right),\label{eq:ABtoAA}
\end{gather}
which allow us to express the (tree level) $AB$ and $A\bar{B}$ propagators uniquely in terms of $AA$-propagators.
This leads (in analogy to \eqref{eq:Expansion}) to the following representation for the dressed one-loop gauge boson propagator:
\begin{gather*}
G^{\rm AA,1l-ren}_{\mu\nu}(p)=\inv{\mathcal A}-\inv{\mathcal A} \Big(\sum \mathcal B_i\Big) \inv{\mathcal A},
\end{gather*}
where $1/\mathcal A$ once more stands for the tree level gauge boson propagator. The $\mathcal B_i$'s are given by the one-loop corrections (with amputated external legs) of the two-point functions relevant for the dressed gauge boson propagator, multiplied by any prefactors coming from \eqref{eq:ABtoAA} and the factor $2$ where needed (c.f.~\eqref{eq:DressedPropComplete1}). Thus, the full propagator is given by
\begin{gather*}
 G^{\rm AA,1l-ren}_{\mu\nu}(p)=G^{\rm AA}_{\mu\nu}(p)+G^{\rm AA}_{\mu\r}(p)\Pi_{\r\sigma}(p)G^{\rm AA}_{\sigma\nu}(p)\nonumber\\
 \quad{} + \left(\frac{\ri a'}{\mu p^2\p^2} \right)\Bigg\{2G^{\rm AA}_{\mu\r}(p)\left(\Sigma^{\rm AB}_{\r,\s1\s2}(p)+\Sigma^{\rm A\bar{B}}_{\r,\s1\s2}(p)\right)p_{\s2}G^{\rm AA}_{\nu\s1}(p) \\
 \quad{}+ \left(\tfrac{\ri a'}{\mu p^2\p^2} \right) p_{\r1}G^{\rm AA}_{\mu\r2}(p)
\left(\Sigma^{\rm BB}_{\r1\r2,\s1\s2}(p)+2\Sigma^{\rm B\bar{B}}_{\r1\r2,\s1\s2}(p)+\Sigma^{\rm \bar{B}\bB}_{\r1\r2,\s1\s2}(p)\right)p_{\s2}G^{\rm AA}_{\nu\s1}(p)\Bigg\}.\nonumber
\end{gather*}
The expression $\mathcal{B}=\sum\limits_i\mathcal{B}_i$ for $M\to0$ is explicitly given by
\begin{gather*}
 \mathcal{B}= \frac{g^2}{8\pi^2\mu^4} \Bigg\{\p_\mu\p_\nu\left(\frac{16\mu^4}{(\p^2)^2}+\frac{\theta^4\lambda^4}{2(\p^2)^4}\right)-7\l^2\mu^2\frac{\theta^4}{(\p^2)^4}\left(p^2\d_{\mu\nu}-p_\mu p_\nu\right)\left(4-\p^2\Lambda^2\right)\nonumber\\
\phantom{\mathcal{B}=}{}
+ \left(p^2 \delta _{\mu \nu}-p_{\mu} p_{\nu}\right) \left[\ln2-\ln|\p|-\ln\L\right]\left(\frac{5}{3} \mu^4+\frac{3\lambda^2 \mu^2\theta^2}{(\p^2)^2}+\frac{\lambda^4\theta^4}{(\p^2)^4}\right)\Bigg\}
 +{\rm f\/inite},
\end{gather*}
and shows us two things: In contrast to commutative gauge models and even though the vacuum polarization tensor $\Pi_{\mu\nu}$ only had a logarithmic UV divergence, the full $\mathcal{B}$ diverges quadratically in the UV cutof\/f $\L$. Secondly, despite the fact that $\Pi_{\mu\nu}$ exhibited the usual quadratic IR divergence, $\mathcal B$ behaves like $\inv{(\p^2)^3}$ in the IR limit. Both properties arise due to the existence (and the form) of the mixed $AB$ and $A\bar{B}$ propagators, and seem problematic concerning renormalization for two reasons: On the one hand, the form of the propagator is modif\/ied implying new counter terms in the ef\/fective action. On the other hand, higher loop insertions of this expression can lead to IR divergent integrals, as will be discussed in the next section.

\section{Higher loop calculations}
\label{sec:higher_loop_calc}

In the light of higher loop calculations it is important to investigate the IR behaviour of expected integrands with insertions of the one-loop corrections being discussed in \secref{sec:one_loop}. The aim is to identify possible poles at $\p^2=0$. Hence, we consider a chain of $n$ non-planar insertions denoted by $\Xi^{\phi_1\phi_2}(p,n)$, which may be part of a higher loop graph. Every insertion $\Xi$ represents the sum of all divergent one-loop contributions with external f\/ields $\phi_1$ and $\phi_2$ (cf. Sections~\ref{sec:1loop_vacpol}--\ref{sec:1loop_BbB}). Due to the numerous possibilities of constructing such graphs, we will examine only a few exemplary conf\/igurations in this section~-- especially those for which one expects the worst IR behaviour.

To start with, let us state that amongst all types of two point functions, the vacuum polarization shows the highest, namely a quadratic divergence. Amongst the propagators those with two external double-indexed legs, e.g.~$B$ or~$\bar{B}$ feature the highest (quartic) divergence in the limit of vanishing external momenta. A chain of $n$ vacuum polarizations $\Pi_{\mu\nu}^{\text{np}}(p)$ (see equations~\eqref{eq:1loop_AA_o0_np} and \eqref{eq:1loop_AA_o2_np}) with $(n+1)$ $AA$-propagators ($(n-1)$ between the individual vacuum polarization graphs, and one at each end) leads to the following expression (for a graphical representation, see \figref{fig:nloop_AA}):
\begin{gather*}
\Xi^{A A}_{\mu\nu}(p,n) =\left(G^{AA}(p)\Pi^{\text{np}}(p)\right)^n_{\mu\r}G^{AA}_{\r\nu}(p)
 =\left(\frac{2g^2}{\pi^2}\right)^n\frac{1}{\left(p^2+\frac{a'^2}{\p^2}\right)^{n+1}}
 \frac{\p_{\mu}\p_{\nu}}{(\p^2)^{n+1}} .
\end{gather*}
Note that due to transversality, from the propagator \eqref{eq:prop_aa} only the term with the Kronecker delta enters the calculation.
For vanishing momenta, i.e.\ in the limit $\p^2\to0$ the expression reduces to
\begin{gather*}
\lim\limits_{\p^2\to0}\Xi^{A A}_{\mu\nu}(p,n)=\left(\frac{2g^2}{\pi^2}\right)^n\frac{\p_{\mu}\p_{\nu}}{a'^{2(n+1)}}  ,
\end{gather*}
exhibiting IR f\/initeness which is independent from the number of inserted loops.

\begin{figure}[t]
  \centering
 \includegraphics[scale=0.64]{Blaschke-Fig5}
 \caption{A chain of $n$ non-planar insertions, concatenated by gauge f\/ield propagators.}
 \label{fig:nloop_AA}
\end{figure}

Another representative is the chain
 \[\Xi^{A \phi}(p,n)\equiv G^{A\phi}(p)\left(\Sigma^{\rm np,\phi A}(p)G^{A\phi}(p)\right)^n, \qquad\text{where}\quad\phi\in\{B,\bar{B}\},
 \]
 which could replace any single $G^{A B}$ (or $G^{A \bar{B}}$) line. Obviously, one has
\begin{gather*}
 \Xi^{A\phi}_{\mu,\n1\n2}(p,n)= \frac{\ri a'}{2\mu}\left(-\frac{3 g^2}{32 \pi^2} a'^2\right)^n \frac{\left(p_{\n1}\d_{\mu\,\n2}-p_{\n2}\d_{\mu\,\n1}\right)}{p^2\left[\p^2\left(p^2+\frac{a'^2}{\p^2}\right)\right]^{n+1}} n \ln\p^2 ,
 \end{gather*}
 which for $\p^2\ll1$ (and neglecting dimensionless prefactors) behaves like
 \begin{gather*}
\Xi^{A\phi}_{\mu,\n1\n2}(p,n)\approx n\frac{\left(p_{\n1}\d_{\mu\,\n2}-p_{\n2}\d_{\mu\,\n1}\right)}{\mu p^2}\ln \p^2.
\end{gather*}
The latter insertion can be regularized since the pole at $p=0$ is independent of $n$. In contrast, higher divergences are expected for chain graphs being concatenated by propagators with four indices, i.e.\ $G^{\bar{B} B}_{\mu\nu,\r\sigma}$, $G^{B B}_{\mu\nu,\r\sigma}$, $G^{\bar{\psi} \psi}_{\mu\nu,\r\sigma}$, due to the inherent quartic IR singularities. Let us start with the combination $\Xi^{\bar{B} B}(p,n)\equiv\big(G^{\bar{B} B}(p)\Sigma^{{\rm p,B\bar{B}}}(p)\big)^nG^{\bar{B} B}(p)$. As before, we can approximate for $\p^2\ll1$ and, omitting dimensionless prefactors and indices, f\/ind
\begin{gather*}
 \Xi^{A \phi}(p,n)\underset{\p^2\ll 1}{\propto}\frac{n}{\mu^2}\frac{\ln\p^2}{\left(p^2\p^2\right)^n},
\end{gather*}
which represents a singularity $\forall\,n>1$ (since in any graph, at $n=0$, the divergence is regularized by the phase factor being a sine function which behaves like $p$ for small momenta). Regarding the index structures, no cancellations can be expected since the product of an arbitrary number of contracted, completely antisymmetric tensors is again an antisymmetric tensor with the outermost indices of the chain being free.

Exactly the same result is obtained for $\Xi^{B B}(p)\equiv\left(G^{B B}(p)\Sigma^{{\rm p,B B}}(p)\right)^nG^{B B}(p)$. From this it is clear that the damping mechanism seen in $\Xi^{AA}(p,n)$ fails for higher insertions of $B/\bar{B}$ (and also $\psi/\bar{\psi})$ f\/ields).

\section{Discussion}
\label{sec:discussion}

We have elaborated on our recently introduced {non-com\-mu\-ta\-tive} gauge model~\cite{Blaschke:2009b}. Initially, the intent was to apply Algebraic Renormalization (AR), as was suggested by Vilar {et al.}~\cite{Vilar:2009}. In the light of that renormalization scheme it is most important to maximise the symmetry content of the theory which is the basis for the generation of constraints to potential counter terms. Therefore, after recapitulating general properties of our model, we studied the resulting algebra of symmetries.
However, as we exposed recently~\cite{Blaschke:2009c}, the foundations of AR are only proved to be valid in \emph{local} QFTs so far, and hence may not be applicable in {non-com\-mu\-ta\-tive} f\/ield theories, as the deformation inherently implies non-locality. In order to f\/ind a way out of this dilemma, explicit loop-calculations were presented, and our hope was to show renormalizability~-- at least at the one-loop level. In this respect, unexpected dif\/f\/iculties appeared. The soft breaking term, being required to implement the IR damping behaviour of the $1/p^2$ model in a way being compatible with the Quantum Action Principle of AR, gives rise to mixed propagators~$G^{AB}$ and~$G^{A\bar{B}}$. These, in turn, allow the insertion of one-loop corrections with external $B$-f\/ields into the dressed $AA$ propagator (see \secref{sec:higher_loop_calc}) and, therefore, enter the renormalization. Despite all corrections featuring the expected~$\inv{\p^2}$ IR behaviour, the dressed propagators with external~$AB$ or~$A\bar{B}$ legs multiplicatively receive higher poles due to the inherent quadratic divergences in $G^{AB}(p)$ (and~$G^{A\bar{B}}(p)$) for $p\to 0$.
As a consequence, the resulting corrections cannot be absorbed in a straightforward manner.

However, renormalizability of the non-local model \eqref{eq:act_old_complete} cannot depend on how it is localized due to equivalence of the respective path integrals (see~\cite{Blaschke:2009b}). Therefore, we expect the same problems to appear in \emph{all} localized versions of \eqref{eq:act_old_complete}, including the one of Vilar {et al.}~\cite{Vilar:2009}. In fact, from the discussion in Appendix~\ref{app:vilar}, one notices that the propagators \eqref{eq:vilar_prop-bBbB}--\eqref{eq:vilar_prop-BbB} and \eqref{eq:vilar_prop-xibxi} of their action all exhibit the same quartic IR divergences as those of our present model~\eqref{eq:act_complete}, even though the operator $D_\mu$ appears at most quadratically as $D^2$ in the according action \eqref{eq:vilar-action}. Nonetheless, the authors claim to have shown renormalizability using Algebraic Renormalization, which as we have discussed in~\cite{Blaschke:2009c} may not be applicable in {non-com\-mu\-ta\-tive} theories.

In this respect it has to be noted that in commutative space the model of Vilar {et al.}~\cite{Vilar:2009} should indeed be renormalizable, since the action, apart from the star product, is completely local and provides the necessary symmetries for the Quantum Action Principle. Since the propagators are the same in both spaces, and hence show the same quartic IR divergences, one may expect related IR problems to cancel when considering the sum of bosonic and fermionic sectors (i.e.~$B/\chi$ and~$\psi/\xi$). These cancellations should also take place in {non-com\-mu\-ta\-tive} space (in both models), but the problem of proving renormalization remains (cf.\ \secref{sec:1loop_analysis}).

Coming back to the problem of IR divergent propagators we have also investigated the structure of singularities in higher-loop integrands by studying chain graphs consisting of interleaving tree-level propagators, and one-loop corrections of various types. It turned out that chains containing gauge f\/ields benef\/it from the damping of the propagator \eqref{eq:prop_aa} while those consisting (solely) of concatenated $B$ and $\bar{B}$ f\/ields and insertions do (expectedly) not. Hence, at f\/irst sight, there exist divergences which increase order by order, which would indicate non-renormalizability. However, we may point out that, due to the symmetry between the $B/\bar{B}$ and $\psi/\bar{\psi}$ sectors, cancellations can be expected. These already appear in our one-loop calculations, and there is strong evidence that they appear to all orders. An intuitive argument can be given when considering the action~\eqref{act-loc} for $\l\to0$, i.e.\ vanishing damping. In this case, the $B/\bar{B}$ and $\psi/\bar{\psi}$ f\/ields may simply be integrated out in the path integral formalism (see~\cite{Blaschke:2009b}), and the contributions cancel exactly.
An alternative approach which avoids these uncertainties is in preparation.

redpajama_set_name: RedPajamaArXiv


In a blender, puree the sorrel with 3/4 cup of the chicken stock until smooth. In a small bowl, sprinkle the gelatin over the remaining 1/4 cup of chicken stock and let stand until softened, about 2 minutes. Microwave the gelatin at medium power just until melted, 5 to 10 seconds. Add the gelatin mixture to the blender along with 2 tablespoons of the lemon juice and the red wine vinegar. Season with salt and white pepper and puree until very smooth. Add 1 cup of the heavy cream and pulse just until very lightly whipped, about 1 minute. Transfer the sorrel mousse to a measuring cup and pour into 6 small glasses. Refrigerate the mousse until set, about 4 hours.
Meanwhile, in a bowl, whisk the remaining 1/4 cup of cream and 1 tablespoon of lemon juice with the crème fraîche to soft peaks. Season with salt and white pepper. Dollop the lemon cream on the mousse and top with the caviar. Serve chilled.


If at any time you have a medical emergency, do not stay without knowing where to go. Hospitalsworldguide recommends you Radiation Oncology Victoria, because this hospital has good reputation and satisfied patients, especially with the care provided by health professionals.
Have you been in Radiation Oncology Victoria? Have you had a good experience with the staff working in this hospital? Tell us what you thought! Your comment will be very useful to other users who now need to decide which hospital to go to.
A great team of professionals conforms Radiation Oncology Victoria. If you are in doubt about which hospital to go in case of an emergency, we recommend this center. Please, if you find any errors in this information, let us know so that we can correct it as soon as possible. Thank you!.
35-47 green st, wangaratta vic 3677, australia 3677 Wangaratta ,Wangaratta ,Victoria ,Australia .
45-53 coster st, benalla vic 3672, australia 3672 Benalla ,Benalla ,Victoria ,Australia .
36 hardy ave, wagga wagga nsw 2650, australia 2650 Wagga ,Wagga Wagga ,New South Wales ,Australia .
88 fenaughty st, kyabram vic 3620, australia 3620 Kyabram ,Campaspe ,Victoria ,Australia .
main road, gembrook vic 3783, australia 3783 Gembrook ,Cardinia ,Victoria ,Australia .


text:
Q: Ant task for easier deployment We have quite a big tree of source code, parts of it are deployed as two separate jar files. We need an easy control of what goes to which jar.
So far we do this by <exclude name="" /> and <include name="" /> tags, but this is quite inconvenient. The best option would be a separate config file with all the packages listed which we could comment out when needed, say  with a '#' character.
Does something similar exist or do we have to write a new ant task that reads such a file and runs a <jar> task?

A: The best option would bee to seperate the code into different modules which can be build on their own (of course with dependencies to each other). Doing this also makes cyclic dependencies obvious and gives you the chance to optimize your code base.

A: ANT includes and excludes can be managed with external files and referenced in a fileset using includesfile and excludesfile attributes.

includesfile      the name of a file; each
  line of this file is taken to be an
  include pattern.
excludesfile  the
  name of a file; each line of this file
  is taken to be an exclude pattern.

For example:
 <jar destfile="${dist}/lib/app1.jar">
   <fileset dir=".">
    <includesfile name="app1.properties"/>
   </fileset>
  </jar>
 <jar destfile="${dist}/lib/app2.jar">
   <fileset dir=".">
    <includesfile name="app2.properties"/>
   </fileset>
  </jar>

redpajama_set_name: RedPajamaStackExchange


Image Title: Open Gable Patio Designs Covers Full Intended For Porch Roof Ideas Decor 7. Post Title: Porch Roof Ideas. Filename: open-gable-patio-designs-covers-full-intended-for-porch-roof-ideas-decor-7.jpg. Image Dimension: 800 x 600 pixels. Images Format: jpg/jpeg. Publisher/Author: Jairo Hilll. Uploaded Date: Wednesday - October 17th. 2018 17:01:15 PM. Category: Architecture. Image Source: lowes.com. Porch Roof Designs Front Flat With Ideas 4. Build A Porch In Minute By Front Ideas YouTube Pertaining To Roof Plans 10. Porch Roof Ideas 27 Best Back Images On Pinterest Inside Remodel 15. Porch Roof Ideas Designs With Aluminum Outdoor Pub And Throughout 11. New Front Porch Roof Designs Ideas Why Is Some With Remodel 18. Porch Roof Construction House Pinterest Concrete Patios Regarding Ideas Plans 3. Porch Roof Construction How To Build Designs Intended For Ideas Design 12. Back Porch Roofs Designs Porches Ideas For Roof Plan 6. Tin Roof Porch Plans Porches Ideas In Decor 1. Porch Roof Ideas Back Patio With Regard To 14.


Car finance deals are becoming increasingly more popular own the UK as opposed to buying and owning a car outright. Ownership of a car is becoming less attractive due to the rising cost of motoring and intense depreciation some cars experience almost immediately.
It makes some financial sense to not shell out thousands of pounds to buy a new vehicle outright and instead pay for a car in monthly installments.
Ultimately these deals are flexible and can be tailored to each individual motorist depending on what they need, as long as they are clear about what they are signing up for.
Products such as hire purchase and PCP also have the added benefits of consumer protection under the Consumer Credit Act 1974 – the same protection is not available when making cash payments.
James Fairclough, CEO of AA Cars, digs into some of the less obvious benefits of taking out motor finance.
When using your own money to buy a car it will ultimately massively limit exactly what vehicle you can purchase. The make, model, engine size etc may have to be compromised as a result.
Motor finance often removes this ceiling as instead of a one-off lump sum, the cost of the car is split into more manageable monthly repayments.
Breaking down the costs like this can help to bring a lot more cars within range of the buyer's budget, meaning you probably don't have to resign yourself to an old car with thousands and thousands of miles on the clock.
In the fullness of time, taking out motor finance can actually have a positive impact on your credit history. Of course, there's a big caveat to all of this – paying instalments on time can build up a great picture of your reliability to future lenders on your credit rating, but missing repayments will have the opposite effect.
It's worth noting that your credit score might go down, on initially being accepted for a loan but, after a few months of repayments, this picture will start to look rosier.
It's also worth considering that just enquiring about your eligibility for car finance deals can leave a footprint on your score. Certain websites, such as AA Cars, actually run a "soft credit check" so you can find out what products are available to you without leaving a mark on your credit history.
With a lot of car finance products, lenders will need an estimate the value of the car at the end of the deal – which is typically calculated using your expected mileage to estimate the value depreciation.
This, again, can be a double-edged sword. Overshoot your anticipated mileage and you can end up paying some hefty per-mile fees.
Undershoot, and the lender might decide that there is actually equity in the car, since it hasn't depreciated as much as they initially thought. In many scenarios, this can mean you end up with money off your next car if you choose to upgrade, rather than buy the car outright.
Depending on the finance product, there are a variety of options as to what you do with your car once the term ends. If you're set on owning your car at the end of the contract, you might want to consider PCP or HP.
PCP (or Personal Contract Plans) allows the leasee to own the car once they've paid off the lump sum at the end of the deal; with HP (or Hire Purchase) you will own the car once all you've paid all of the repayments.
In the case of PCP, you can still choose to return the car to the dealer and upgrade to a new model, sometimes with equity (see above).
There's also the option of leasing the car with a PCH (or Personal Contract Hire), which might work better for you if you're happy just to hand the car back once the term is over.
January movies: New films out in January 2019 – What to watch at the cinema this month?


text:
Punahou continues tennis tradition with 2 team titles
It started with an entire island pitching in so the Carlsmith Ball/HHSAA Tennis Championships could be played Thursday. It ended yesterday with someone old, someone new, something rare and something blue — again.
Kapaa senior Chas Okamoto grabbed a title on his last opportunity and Mililani freshman Alyssa Tobita got one on her first try, at Royal Lahaina Tennis Ranch on Maui.
The rarity came when Interscholastic League of Honolulu champions Roman Kop and Marcel Chan captured Kamehameha's first boys doubles title in 46 years. They defeated the seventh-seeded 'Iolani team of Colin Tseng and Lawrence Ho 6-3, 6-3 in the final.
The blue was Punahou, which continued its team title domination. The Buffanblu clinched the boys title after Friday's quarterfinals and the girls title after the semifinals. That was a day after rain threatened to wash out Thursday's opening matches at Royal Lahaina and Lahaina Civic.
The word went out on Maui and resort courts were made available at Wailea, Makena and Kapalua. Every match was finished by 8 p.m.
The Buffanblu girls, with a lift from Tobita, edged 'Iolani 18-16 to run their streak to nine in a row.
The boys set the high school tennis national record last year when they won their 20th in a row. This year they amassed 22 points — 'Iolani was second with 10 — to break the Hawaii record for most consecutive titles in any sport. Rusty Komori has coached the last 18.
"Every year four or five players graduate, but it's OK because we have such a great tradition that the guys are mentoring the underclassmen — seventh- and eighth-graders," Komori said. "They are following our team and they know the standard of excellence, so it becomes a lot easier to mold them into the players we need them to be."
The only individual title Punahou won came in girls doubles, where top-seeded Jennifer Laws and Sarah Steele stopped teammates Ashley Nakaoka and Dani Young, 7-6 (7-4), 6-2.
Laws and Nakaoka — the lone seniors on the Buffanblu roster — won the doubles title together the past two years, but opted to split up and nurture younger teammates their final season. When both teams reached the final Friday, and Tobita upset 'Iolani's two seeded singles players, the Buffanblu had their title.
Okamoto stopped St. Anthony senior Kento Tanaka-Tamaki, 6-4, 6-2, in a boys singles final featuring the top two seeds.
"I was down 2-4 and letting my nerves get to me," Okamoto said. "The wind started to pick up and I got more topspin on my shots. I came back to win 6-4 and the momentum was all on my side. In the second set I broke him right off the bat. That was huge."
Okamoto is also the Kauai Interscholastic Federation's 800-meter champion. He credits his long-distance training for helping with stamina and focus. He also got an assist from Brad Lum-Tucker — the only other state tennis champion from Kauai — who hit with him last week.
Okamoto is headed to the University of Hawaii on an academic scholarship. He hopes to be teammates with Jared Spiker, who beat him in last year's semis on the way to his second straight title. Tanaka has been invited to walk on for the Pepperdine team in the fall.
Tobita and Sarah Dvorak, the top-seeded freshman she beat in yesterday's final, have lots of time to decide on a college. They played for 2-plus hours before Tobita won, 7-6 (7-3), 7-5. Tobita, a month younger, was down 4-5 in the first set and 0-3 in the second, but tamed the hard-hitting Dvorak with her variety and consistency.
"I was kind of nervous there," Tobita said of her second-set hole. "I told myself to remember how much I wanted it. I was thinking to myself I couldn't give up right there. Even though there was another set I just wanted to win it."
She knew going in that Erin Hoe, who played for Mililani from 1997 to 2000, is the only player to win four Hawaii state tennis titles.
"I want to do that," Tobita said. "I know it will be tough, but I took the first step today. It was all or nothing."
Punahou's boys lose five off their team –seniors Robin Kiyabu, Travis Yoshimoto, Colby Ing and Shaun Chow, and junior William Chen, who received early acceptance to Stanford. He will join Ing there, while Yoshimoto is headed to USC, Chow to UH and Kiyabu to Loyola Marymount, where he was recruited for tennis.
Mid-Pacific wins it all
Top seed Punahou still has something to prove
Las Vegas Raiders oust GM Mike Mayock after 3 seasons
VIDEO: Former UH Warrior and CFL great Chad Owens hosts Star-Advertiser sports show 'The CO2 RUN DWN'
University of Hawaii men's volleyball team shows off its talent, depth in sweep

redpajama_set_name: RedPajamaCommonCrawl


Tonight is the first installment of The Green Room. The Green Room is a "worship experience" held downtown at the Skinny Improv in Springfield. It's designed to be a backstage experience different from the type of worship that happens on a Sunday morning. It is a place where music, art, imagery, and emotion combine to bring us closer to God. If you're looking for worship that is more intimate, join us at The Green Room.


Research has shown SR-9009, stenabolic, stimulates the Rev-Erb protein which has an effect on circadian rhythms, fat-storing cells as well as glucose and lipid metabolism. Research has shown that activating Rev-erb protein increases cell production through mitochondria giving the muscle increased energy as well as giving the metabolism enhanced speed. Thus increasing the mitochondria in the muscle will result in strength gain as well as drastic energy increase to go along with strides in endurance. Research has shown Stenabolic can can increase all around endurance due to increases in mitochondria count. The biggest component that SR-9009 research shows is that it boosts metabolic activity and is known to mimic aerobic exercise without having to actually exercise hence being called "exercise in a bottle". Research shows SR9009 uses body fat that is stored and uses it as energy for exercise.


An efficient way to hold your guests gear.
Standard 10 foot long ski/board racks are designed to hold a combination of up to 25 pairs of skis/snowboards plus children's gear.Safety lugs for spacing ensure no sharp edges or points.
Lightweight, sturdy and durable.Racks break down easily for shipping.
Powder coating and overhead signs as options. Custom sizes and designs are available.
Clean up your Staging Area.
Economy 8 foot long ski/board racks are a cost effective solution with no maintenance. Made of lightweight aluminum and hold a combination of up to 25 pairs of skis/snowboards. Optional bottom bar for stability and kids skis available. Safety lugs for spacing ensure no sharp edges or points.
Powder coating and overhead signs as options.
Our 5 foot square ski racks are designed to hold a combination of up to 28 skis or snowboards plus children's gear. Safety lugs for spacing ensure no sharp edges or points.
Specially designed with your little ones in mind!
Made to hold small skis and boards, safety lugs ensure that there are no sharp edges or points.
Mount cut out characters or flags and banners for a fun, unique look.
Custom made gondola racks accommodate the larger 'powder' and 'shape' skis as well as snowboards. Perforated bottoms allow for drainage.
Made with lightweight aluminum; rack walls are thinner than classic plastic or steel models, allowing for extra room within same design.
All Racks are customized to fit specific requirements and existing mounting structures.
A more comfortable ride with gear on the outside!
Custom made bus racks accommodate the larger 'powder' and 'shape' skis as well as snowboards. Perforated bottoms allow for drainage.


text:
The Quickest Of Quick Jabs: Marquez-Casamayor, Campbell's Honeymoon, Malignaggi's Hair Extensions And More
Posted by Tim Starks on May 23, 2008 22:32
Some quick thoughts below, mentioning the following fighters: Oscar De La Hoya, Floyd Mayweather, Manny Pacquiao, Ricky Hatton, Miguel Cotto, Juan Manuel Marquez, Israel Vazquez, Rafael Marquez, Winky Wright, Cristian Mijares, Joel Casamayor, Nate Campbell, Joan Guzman, Antonio Margarito, Paulie Malignaggi and others. It's like a salad bar, but with fists. I quote the most overquoted movie since Austin Powers, Anchorman, to describe the war of words that has broken out between lightweights (135 lbs.) Nate Campbell and Joan Guzman this week: "God that really got out of hand fast. I mean that really escalated quickly." Any time one fighter verbally goes after another fighter's wife during his honeymoon, it isn't pretty. Read the final legs of the sequence here, here and here, or if you are sensitive to vulgarity, skip it. I agree with ESPN's Dan Rafael, (who quotes from the volley that started it all:) Guzman's in the wrong on this one… Speaking of lightweights — the division is maybe the hottest in boxing — there is now talk of Joel Casamayor and Juan Manuel Marquez fighting. The combined boxing IQ would be astounding in the ring that night. I think Marquez, not long ago a featherweight (126 lbs.), is going to struggle at lightweight, but Casamayor isn't as big as some other lightweights, so this might be the best match for him. It's all part of a Marquez plot to get Manny Pacquiao back in the ring, although Pacquiao, alas, has reportedly returned to his lazy-training ways for his lightweight debut against David Diaz next month. Not smart, because Diaz is deservedly the heavy underdog but he's a tough hombre who can narrow the gap against Pacquiao if he's fighting a non-optimal Pacquiao. Tsk tsk… I may have spoke too soon when I said Paulie Malignaggi's hair choices have gotten less irritating. Check out ESPN.com's front boxing page: The junior welterweight (140 lbs.) titlist is pictured as of this writing wearing some ridiculous hair extensions. There could be 70,000 fans on hand in Manchester to see his fight Saturday, so it's all the more embarrassing for him. It's fantastic, by the way, that once again, a boxing match somewhere is going to draw that kind of live gate. Ricky Hatton — A-side in the double-header, hero to the common man and the consensus junior welter champ — is the reason, of course, and kudos to him for insisting on rewarding his hometown fans for their loyalty. It's less clear to me whether 1. the card is broadcasting in the U.S. on Versus because, as Hatton's team insists, they wanted it at that date and that time but HBO wanted it to air as the undercard to the Oscar De La Hoya fight earlier this month; or 2. because it's not that sexy a card and nobody else really wanted it. I'm going to favor 1., because I have to imagine there's money, even in America, no matter who Hatton fights… Respect, part I: A judge in a court case brought by Floyd Mayweather, Jr., boxing's reigning pound-for-pound king, recently called the welterweight (147 lbs.) champ "one of the greatest technical boxers of all time" in a filing. Great, now Mayweather's ego has backing in jurisprudence… Respect, part II: In light of his highly-praised performance in defeating Alexander Munoz, 115-pound titlist Cristian Mijares has moved into the pound-for-pound top-10 lists of both Ring magazine and ESPN. I now have him there, too, kicking out 160-pound Winky Wright, who is a great fighter but has been so inactive I think he's got one of those Rip Van Winkle beards… Respect, part III: Tim Bradley, who pulled off perhaps the Upset of the Year when he toppled junior welterweight titlist Junior Witter this month, just got his own "Timothy Bradley Day" from his hometown of Palm Springs, California. Good for him… Respect, part IV: I know I've repeatedly made fun of his creepy chest chair shaved into the shape of a cross, but in all seriousness, I must give props to Richar-not-Richard Gutierrez for the manliness he showed in brawling — and almost beating — a naturally bigger, harder-hitting opponent in junior middleweight (154 lbs.) Alfredo Angulo last weekend. I'll always make fun of the chest hair, but I now have nothing but respect for his fight game, and I admire the fact that he wanted to please the fans; mission accomplished here, even in loss. I hope he gets his wish to fight welterweight contender Andre Berto — that's one helluva brawl, too… The junior featherweight (122 lbs.) division is nice and top-heavy, with Israel Vazquez and Rafael Marquez at the head of the class. Now, two others in the top five of the weight class, Steve Molitor and Celestino Caballero, are talking about rumbling. I'd love to see that one… Promoter Bob Arum says of rival promoter Golden Boy, "Golden Boy's fighters will not beat ours." Fears of a renewed Cold War between Arum's Top Rank and De La Hoya's self-named promotional company are stoked by things like this, of course, and such a renewal would plainly kill boxing's momentum dead, what with those two companies being the dominant in the sport. Stepping in and reffing a little: From a casual scan of each company's roster of fighters, I'm split fairly evenly in my predictions about who would win if I pick each company's best per division for a box-off. Play nice, fellas — and if you start to rub each other the wrong way, keep thinking of all the money you've made in the last year or so by doing business… Finishing where I started: I agree with ESPN's Rafael, too, in his suspicion about HBO's motive in not broadcasting a "Countdown to Miguel Cotto-Antonio Margarito" show, produced at no expense to the network by the fight's promoters. This is only the best match-up on the horizon, between two welterweight tanks, one has the potential to flourish into a boxing event if it's marketed properly. What is HBO thinking? I don't even have a good conspiracy theory. Instead, I'll be spending some extra energy on hyping it myself, the way I did Israel Vazquez-Rafael Marquez III in March and, recently, "Yuriorkis Gamboa Week." July 26. Watch, or the coming Apocalypse is YOUR fault…


text:
MAKING RELIABLE HEALTHCARE AND SCIENTIFIC DATA ACCESSIBLE IN A COOPERATIVE AND SECURE WAY!
Founded in 2009 as Target with roots in the Petabytes data regime.
Years of track record in building innovative and differentiating Artificial Intelligence & Machine Learning solutions for international customers.
The Spin-out Thunderbyte.AI was founded in 2019 with a focus on full-stack AI / ML services in the field of lifestyle, vitality and healthcare.
Thunderbyters
SOME OF OUR TEAMMEMBERS:
MARK VAN DER LINDENApp-developer
Working at Thunderbyte.AI is like entrepeneurship with a group of friends.
Working for Thunderbyte.AI? I would rather describe it as entrepreneurship with a group of friends. We have a lot of fun together, but because we know each other very well we can work really quickly as a well-coordinated team with the right dosis of fun. We are a group of colleagues who are very curious, like to think along with you and are happy to surprise you with what we can do for you! I've been working at Thunderbyte.AI for almost 2 years now and was given the opportunity to develop myself from a just graduated app developer to a team leader. This shows we don't think in boxes and challenge each other in developing our skillset. We are always looking for new challenges so feel free to come and meet us!
NICK BOSSCHERHealth Business Developer
Working on beautiful theme's with many different disciplines.
My name is Nick, I'm 24 years old and responsible for the WeFitter project within Thunderbyte AI. I have lots of passion and affinity with healthcar and lifestyle as a theme and therefore also with the focus of Thunderbyte AI on these types of topics. What I like the most in working in Thunderbyte AI is the freedom you get, the close team and informal atmosphere in the company. Everybody is extremely motivated to make the best out of their project with a lot of freedom in how to do it, that's why everybody is learning so much every day. In Thunderbyte AI we work on beautiful theme's with many different disciplines, a big dose of fun and really make a change in the health of people with our projects. That is making it very nice to be part of the Thunderbyte family!
JEROEN WILKENSMachine Learning Engineer
The freedom to decide how I want to approach and solve the project's challenges.
Working as a machine learning engineer at Thunderbyte.AI gives me the opportunity to work together with a group of highly skilled colleagues on some really cool projects. Currently I am working on projects involving data extraction from wearables and information extraction from websites and documents. What I really like about my job is that I have the freedom to decide how I want to approach and solve the challenges in the projects I am working on. Besides that, It feels good that I can bring in my experience I gained during my master studies and apply it on real projects that benefits our customers, and that all within cycling distance from the city center of Groningen..
JURJEN MELINGASolutions Architect
Head in the clouds but our feet planted firmly on the ground.
At Thunderbyte.AI we keep our eyes on the future, our head in the clouds but our feet planted firmly on the ground. This balance between innovation, creativity and at the same time a strong sense of realism enables us to really create beautiful things. With our partners and customers we solve real problems in different domains (health, care, energy). By employing our knowledge and experience on ai, machine learning, data science and software engineering we make proper use of the technology available. We're a diverse (background, age, coffee vs tea, etc.😊), but a tight "family" in which we work together, look after each other and have fun while doing that. We also make sure that everyone has the room to do "their own thing" as an individual. This makes Thunderbyte.AI. a fun, challenging, safe and great place to work, where we learn from each other, our customers and our partners every day.
Thunderbyte.AI
Sylviuslaan 2 | 9728 NS | Groningen
T: +31 653329718
E: info@thunderbyte.ai
Computer engineers are the wizards of our age…
© Thunderbyte.AI | Privacy and security


text:
<?xml version="1.0" encoding="utf-8"?>
<RelativeLayout xmlns:android="http://schemas.android.com/apk/res/android"
              android:orientation="vertical"
              android:layout_width="match_parent"
              android:layout_height="wrap_content"
              android:padding="@dimen/dp_5"
              android:background="@color/color_cc999999">

    <connect.view.roundedimageview.RoundedImageView
        android:id="@id/roundimg_head"
        android:layout_width="@dimen/dp_50"
        android:layout_height="@dimen/dp_50"/>

    <TextView
        android:id="@id/text_view"
        android:layout_width="wrap_content"
        android:layout_height="wrap_content"
        android:textColor="@color/color_ffffff"
        android:textSize="@dimen/sp_18"
        android:layout_centerInParent="true"/>

    <ImageView
        android:contentDescription="@string/app_name"
        android:id="@id/image_view"
        android:layout_width="wrap_content"
        android:layout_height="wrap_content"
        android:padding="@dimen/dp_10"
        android:layout_alignParentRight="true"
        android:layout_centerVertical="true"
        android:src="@mipmap/input_delete"/>

</RelativeLayout>

redpajama_set_name: RedPajamaGithub


text:
How to Reach Sri Lanka
Have Queries? Fill in the form below & get the best deals.
I accept Privacy Policy and I authorise Thomas Cook to contact me with details
international tourism
sri lanka tourism
The easiest way to reach Sri Lanka is by taking a flight from India and reaching there probably on the same day of your departure. Sri Lanka has a busy international airport and it is called the Bandaranaike Internal Airport. Located in Colombo, Sri Lankan flights from the major countries of the world arrive here. There are flights from India, Europe, Asia, Middle-East and Australia to Sri Lanka. However, as congestion increased, there was a need to tackle the increased air traffic and provide better immigration to tourists. That's why the Mattala Rajapaksa International Airport was inaugurated in the 2013 to cater to the increasing demand of tourist influx. The airport is located in the Mattala town in the southern most region of the country. However, the airport soon witnessed very less patronage and only a few flights from the Middle-East fly to this airport. Due to this, the airport is known as the emptiest international airport, making the airport in Colombo prominent among tourists. Flights from Indian cities like New Delhi, Kolkata, Mumbai, Bengaluru, Chennai and Hyderabad are aplenty with major airlines plying day in and day out. For better understanding, check out the table for the average cost per person from different airports and their respective durations. Frequent Flight Names: Air India, Jet Airways, Sri Lankan Airlines, Spice Jet and Vistara Biggest Airport: Bandaranaike International Airport
By Road & Train
Being a separate island country away from India, Sri Lanka is completely inaccessible by train and road. If you are traveling on a short budget, you can come down to any of the Indian cities that have flights to Sri Lanka by road or by train and board Sri Lankan flights from there. This will help you save on your expenses and do more when you are in the new land.
By Sea
There used to be ferry services from Chennai to Colombo that carried tourists from one country to the other at economic rates. However, the services were suspended indefinitely in 2011 for various reasons. Now, there are just a few ships that operate from Mumbai port that take you to Colombo on a luxurious cruise-liner. The 7-day cruise departs at Mumbai and reaches Colombo, taking you to all the essential destinations like Kandy and Nuwara Eliya along the way. The prices depend on the luxury class you intend to travel in and also feature a few adventure and water sports during the course of the cruise. The packages also feature a number of live shows and entertainment options on board and feature a fixed itinerary with preplanned places to see and things to do. Though cruise ships are fun, the best way to reach Sri Lanka is by taking a direct flight from the Indian cities. Since the seas also get violent during storms and monsoons, it is best to go for flights when you are looking for a fun holiday.
Transportation in sri lanka
Commuting within Srilanka
There are many ways to commute in Srilanka. Now that the transportation options are sorted, we will help you with your Sri Lanka Visa Application as well. . They have local buses that move you from one location to another.
Places To Visit in sri lanka
Best Time To Visit sri lanka
Photos & Videos of sri lanka
sri lanka FAQs
Recommended sri lanka Holidays Packages
View more sri lanka Packages


The sharp pain or the throbbing ache in the tooth is usually called toothache. It can be caused by many things like cavities or tooth decay, infection in the gums, broken or fractured teeth, abscessed tooth or a damaged filling. Severe toothache can lead to swelling around the tooth, headaches, earaches, fevers, foul odor from the mouth and terrible tasting drainage from the infected tooth. It is advisable to see a dentist right away in case you experience these symptoms or the pain is excruciating and/or lasts for more than a day.
Toothaches are mostly caused because the nerve of the tooth is being irritated. However, there are many other reasons that can lead to horrible toothaches.
Dental abscess is a severe form of tooth infection. It causes the area around the tooth to swell and can be extremely deadly. This kind of an infection arises when the teeth and are gums are neglected for a long period of time and they start to decay which worsens over time. The bacteria present in the decaying teeth causes the infection in the tooth and the surrounding area. Dental abscess is known to cause severe toothaches and also affect the rest of the body. The bacteria are capable of traveling from the mouth into the blood and then the heart where they can damage the cardiac tissue through infections.
Poor oral hygiene can cause tooth decay due to the buildup of bacteria in the mouth. A large amount of tooth decay can weaken the tooth as the bacteria penetrate the enamel of the tooth. This can cause extreme pain in the tooth.
Getting your wisdom teeth can be a painful experience. In case there is a limited amount of space at the back of your mouth for your wisdom teeth, they will push other teeth closer together to create space. There are some cases where the wisdom tooth does not emerge. This is called an impacted tooth. This can cause unbearable toothache and usually surgery is required to correct the problem.
The enamel may be a hard surface, but over time it can wear down. This can make it vulnerable to tiny cracks that grow bigger with time. A cracked, fractured or chipped tooth can result in severe toothache, but the fault in your tooth may not be visible to you. It is important for you to get it checked out by a dental specialist to prevent the crack from breaking the tooth and making matters much worse.
Toothache is not essentially caused by the tooth itself. Sometimes the actual reason for the ache can be the gums that support and secure the tooth in place. Gum disease is the result of bacteria buildup in and around the gums. This results in pockets that force the gums away from the tooth. Gum disease is a serious oral problem that can lead to the loss of teeth. So, make sure you visit a trained and qualified dentist immediately to find the cause of the pain and get proper treatment.
Tooth sensitivity may be the result of your enamel weakening. And the moment hot, cold, sweet or sour substances touch your teeth you experience sharp shooting pain in your teeth. The pain may not last an entire day, but you get, you should get your teeth checked out by a dentist.
Even knocked out fillings can cause tooth aches, so make sure you are careful. It is important to get medical assistance in the event of toothaches. A visit to the dentist can help alleviate the symptoms of a severe toothache and also prevent dental infections from spreading to the face, skull quite possibly even the bloodstream. For expert treatment of extreme toothaches schedule an appointment with the best in class dental specialist at Mann Dental Care.


Palm Centro coming in Pink?
SprintUsers forum member posted a picture of a pink Palm Centro marked with the Sprint logo on the top right corner. If the information given is true, this phone is definitely going to be a hit with the ladies and the suggested release dates (January 6 or 13, February 14) make it an easy choice for a gift for Valentine's Day. Palm has already released a black and a red Centro and we think that if the lady Centro comes out it will have the same price and specifications.