text:
\section{Introduction}

Ultraviolet fixed points are key for the renormalisability and  predictive power of quantum field theories.
 The classic example is given by asymptotic freedom of the strong nuclear force, where the
fixed point is non-interacting   \cite{Gross:1973id, Politzer:1973fx}. The possibility of interacting ultraviolet (UV) fixed points, often denoted   as asymptotic safety \cite{Weinberg:1980gg},
has  been conjectured  early on \cite{Bailin:1974bq}.
Recently, this field  has taken up some speed due to the  discovery  of UV conformal fixed points   in models of particle physics
\cite{Litim:2014uca,Bond:2018oco,Bond:2016dvk,Bond:2017sem,Bond:2019npq,Bond:2017lnq}.
Conditions under which asymptotic safety arises in weakly coupled four-dimensional quantum field theories (without gravity)
are by now well understood: Non-abelian gauge fields are central  \cite{Bond:2018oco}, alongside Yukawa and scalar
interactions and subject to a stable vacuum \cite{Bond:2016dvk,Bond:2017sem}. Templates
with strict perturbative control have been found for unitary \cite{Litim:2014uca},
orthogonal and symplectic \cite{Bond:2019npq}, or  product gauge groups  \cite{Bond:2017lnq}, and supersymmetry \cite{Bond:2017suy}. This has also triggered  new ideas for  model building \cite{Bond:2017wut,Kowalska:2017fzw}, asymptotically safe extensions of the Standard Model explaining the electron and muon $g-2$ anomalies \cite{Hiller:2019tvg,Hiller:2019mou,Hiller:2020fbu} whose new type of flavor phenomenology can be tested at colliders  \cite{Bissmann:2020lge}, and explanations of flavour anomalies as evidenced in rare $B$-meson decays~\cite{Bause:2021prv}.
Further results cover  vacuum stability  \cite{Litim:2015iea} including the Higgs \cite{Hiller:2019tvg,Hiller:2019mou,Hiller:2020fbu,Bause:2021prv}, abelian factors \cite{Kowalska:2017fzw},  global fixed points \cite{Buyukbese:2017ehm},
aspects of radiative symmetry breaking \cite{Abel:2017ujy}, UV conformal  windows  \cite{Bond:2017tbw}, and fixed point mergers \cite{Bond:2021tgu}.

Despite the vast body of weakly coupled fixed points at hand, it has remained an open challenge to understand  asymptotic safety of strongly coupled 4d quantum field theories from first principles. It would be desirable to have rigorous and explicit  examples at hand, if only as a proof of principle, and to clarify  whether large matter field anomalous dimensions  or new phenomena may become an obstacle for  safety in the UV.
Moreover, it is well known that weakly coupled fixed points often require a large number of gauge and matter fields, such as in a Veneziano large-$N$ limit,  while decreasing the number of matter fields turns fixed point interactions  stronger.
In the context of asymptotically safe model building where only finitely many new matter fields are added to the Standard Model \cite{Bond:2017wut,Kowalska:2017fzw,Hiller:2019tvg,Hiller:2019mou,Hiller:2019mou,Hiller:2020fbu,Bissmann:2020lge,Bause:2021prv}, it  becomes paramount to control the size  of UV conformal windows non-perturbatively, and to understand how few matter fields can sustain an underlying fixed point  \cite{Bond:2017tbw,Bond:2021tgu}.
It would be equally important to  understand whether or not qualitatively new types of fixed points  arise at strong coupling, beyond those discovered at weak coupling.

In this spirit, we put forward a non-perturbative search for interacting UV fixed points in conventional 4d quantum field theories, without gravity.
With asymptotically safe fixed points  being presently out of reach for lattice simulations or the conformal bootstrap, we turn, instead,  to $N=1$ global supersymmetry as the non-perturbative tool of choice:
Non-renormalisation theorems ensure that superpotential (Yukawa) couplings are renormalised non-perturbatively via the chiral superfield anomalous dimensions, and exact infinite order perturbative RG equations are available for gauge couplings \cite{Novikov:1983uc,Novikov:1985rd}. Further, at superconformal fixed points, anomalous dimensions of all chiral superfields can be determined unambigiously using the  method of $a$-maximisation \cite{Intriligator:2003jj}.
Finally, independent quartics do not  arise and vacuum stability is  automatically guaranteed provided   gauge and Yukawas  take viable fixed points.
 Taken together, these ingredients  prove sufficient  to determine  interacting  fixed points reliably, including at strong coupling.

\begin{figure}
\begin{center}
\vskip-0.2cm
\includegraphics[width=0.45\textwidth]{pASphaseGY1.pdf}
\vskip-0.3cm
\caption{
The basic setup  in the plane of gauge couplings $(\alpha_1,\alpha_2)$, showing the free Gaussian fixed point (G) and an interacting ultraviolet fixed point (UV), with arrows on RG trajectories pointing from the UV to the IR. Note that the Gaussian is a ``saddle" and asymptotic freedom is absent. \label{pSchematic}
}
\end{center}
\vskip-.6cm
\end{figure}

  Here, we apply this methodology exemplarily to an asymptotically non-free $SU(N_1)\times SU(N_2)$ supersymmetric gauge theory with massless chiral matter and a superpotential, and determine the entire conformal window of interacting UV fixed points. Our basic setup is illustrated in Fig.~\ref{pSchematic} which shows the phase diagram of a semi-simple gauge theory with superpotential interactions.  Notice that asymptotic freedom is absent because one of the gauge sector $(\alpha_2)$ is infrared free while the other not $(\alpha_1)$. The theory may nevertheless develop an asymptotically safe fixed point (UV) which allows as well-defined high energy limit and outgoing RG trajectories towards the IR, and which is the central topic of this study. Consistency with Seiberg duality, and constraints from the $a$-theorem, global charges, and  unitarity, are also observed  \cite{Mack:1975je}. Findings are confirmed independently using perturbation theory up to three loop order. Implications of our findings for the asymptotic safety conjecture, model building, and conformal field theory are indicated.

\begin{table}[b]
 \aboverulesep = 0mm
\belowrulesep = 0mm
\begin{center}
\begin{tabular}{`cc cc cc cc c`}
\toprule
\rowcolor{Yellow}
\ \
\bf Matter
&$\bm{\psi_L}$
&$\bm{\psi_R}$
&$\bm{\Psi_L}$
&$\bm{\Psi_R}$
&$\bm{\chi_L}$
&$\bm{\chi_R}$
&$\bm{Q_L}$
&$\bm{Q_R}$ \ \
\\
\midrule
&& && && && \\[-3mm]
$\ \ \bm{SU(N_1)}\ \ $
&$\overline{\Box}$
&$
\Box$
&$\Box$
&$\overline{\Box}$
&1
&1
&1
&1
\\
&& && && && \\[-3mm]
\rowcolor{LightGray}
&& && && && \\[-3mm]
\rowcolor{LightGray}
$\bm{SU(N_2)}$
&1
&1
&$\Box$
&$\overline{\Box}$
&$\overline{\Box}$
&$\Box$
&$\overline{\Box}$
&$\Box$
\\
&& && && && \\[-3mm]
&& && && && \\[-3mm]
{\bf Flavour}
&$N_F$
&$N_F$
&1
&1
&$N_F$
&$N_F$
&$N_Q$
&$N_Q$
\\[.4mm]
\bottomrule
\end{tabular}
\end{center}
 \vskip-.4cm
 \caption{\label{matter}Chiral matter with gauge charges and multiplicities.}
\end{table}

\section{Semi-Simple Gauge Theories with Matter}

We consider    semi-simple Yang-Mills theories
with  product gauge group $SU(N_1)\times SU(N_2)$  coupled to
chiral superfields $(\psi,\chi,\Psi,Q)$ with flavour multiplicities $(N_F,N_F,1,N_Q)$ and gauge charges  as in Tab.~\ref{matter}.
We require that the model has a global $N=1$ supersymmetry. It is then characterised by two gauge  couplings $g_1$ and $g_2$ and a Yukawa coupling $y$ via the  superpotential
\begin{equation}\label{W}
W= y\,{\rm Tr}\big[\psi_L\,\Psi_L\,\chi_L+\psi_R\,\Psi_R\,\chi_R\big]\,,
\end{equation}
where the trace sums over flavour and gauge indices.
The superfields $Q$ are not furnished with Yukawa interactions.
The theory has a  global
$SU(N_F)_L\times SU(N_F)_R\times SU(N_Q)_L\times SU(N_Q)_R$   flavour and a $U(1)_R$ symmetry and is characterised by the field multiplicities
\begin{equation}\label{Ns}
(N_1\,, \,N_2\,,\,  N_F\,,\,  N_Q)\,.
\end{equation}
Asymptotic freedom and interacting infrared fixed points arise for suitable matter field multiplicities.
For this study, the regime of interest is where
the $SU(N_1)$ gauge sector is asymptotically free while the $SU(N_2)$ gauge sector is infrared free.
Accordingly,
the free theory
corresponds to a ``saddle'' and asymptotic freedom cannot be achieved (Fig.~\ref{pSchematic}), very much like in the non-abelian gauge sectors
of the Minimal Supersymmetric Standard Model (MSSM).
 In this light, the gauge coupling $\alpha_2$ can be viewed as ``dangerously irrelevant'', in that it may to become relevant due to the gauge-Yukawa fixed point in the other gauge sector.

In the large $N$ Veneziano limit, the theory has previously been studied in perturbation theory, where it was found that interacting UV fixed points can arise at weak coupling with perturbatively small anomalous dimensions  \cite{Bond:2017suy}.
The main point of this study is to investigate the conditions under which theories with \eq{Bs} may develop strongly interacting ultraviolet fixed points where anomalous dimensions become large, of order unity.
 Ordinarily, this would require strong coupling methods such as lattice simulations to establish the claim. In supersymmetry, however,  powerful continuum methods  beyond perturbation theory  are available to which we turn next.\vspace{.5em}\step

\section{Renormalisation Group}

To achieve our claim, we must find the  renormalisation group equations for all couplings and identify their fixed points, and non-perturbative expressions for the chiral superfield anomalous dimensions. Here, we exploit  key features   of $N=1$ supersymmetic gauge theories which make this task feasible.

For supersymmetric gauge theories, closed  all-order expressions for perturbative $\beta$-functions   have been achieved by Novikov, Shifman, Vainstein, and Zakharov  (NSVZ) \cite{Novikov:1983uc,Novikov:1985rd} (see also \cite{Stepanyantz:2020uke,Korneev:2021zdz}).
We introduce the gauge and Yukawa couplings as
\begin{equation}
\alpha_{1,2}=
\left(\0{g_{1,2}}{4\pi}\right)^2\,,\quad\quad
\alpha_y=
\left(\0{y}{4\pi}\right)^2\,,
\end{equation}
and denote the chiral superfield anomalous dimensions   as $\gamma_a$   $(a=\psi,\Psi,\chi,Q)$,
with $\beta$-functions
$\beta_i\equiv d\alpha_i/d\ln \mu$ $(i=1,2,y)$   defined as usual.
The NSVZ beta functions for the gauge couplings of our models are then given by
\arraycolsep .1em \begin{eqnarray}\label{beta1}
\beta_1&=&\frac{2\alpha_1^2}{F(\alpha_1)}[N_F\left(1-2\gamma_\psi\right)+N_2\left(1-2\gamma_\Psi\right)-3N_1]
\\
\label{beta2}
\beta_2&=&\frac{2\alpha_2^2}{F(\alpha_2)}[N_F\left(1-2\gamma_\chi\right)
+N_1\left(1-2\gamma_\Psi\right)
+N_Q\left(1-2\gamma_Q\right)-3N_2]\,.
\end{eqnarray}
The scheme-dependent function $F(\alpha)$ is normalised to unity for vanishing coupling  \cite{Kutasov:2004xu} and reads $F(\alpha)=1-2C^G_2\alpha$ in the NSVZ scheme  \cite{Novikov:1983uc,Novikov:1985rd} with $C^G_2$ the quadratic Casimir in the adjoint.
Close to the Gaussian fixed point, the anomalous dimensions vanish and we can then read off the condition for a saddle
in terms of the  field multiplicities \eq{Ns},
\begin{equation}\label{Bs}
\begin{array}{lcl}
 3{N_1}&>&{N_2+N_F}\\[1ex]
3N_2 &\le &{N_1+N_F+N_Q}\,.
\end{array}
\end{equation}
This also covers the case where the one-loop coefficient of $\beta_2$ vanishes identically. For the Yukawa coupling, we exploit that  supersymmetry dictates strict non-renormalisation theorems which ensure that  superpotential couplings are only renormalised through field anomalous dimensions. Hence, the RG running  is given  by
\arraycolsep .1em \begin{eqnarray}
\label{betay}
\beta_y
&= &2\alpha_y\big[\gamma_\psi+ \gamma_{\Psi}+ \gamma_{\chi}\big]\,,
\end{eqnarray}
valid to all orders in perturbation theory. Renormalisation group fixed points $(\beta_i=0)$ correspond to superconformal field theories.

All perturbative  fixed points    $(\alpha^*_i\ll 1)$ for this theory have  previously been found in
\cite{Bond:2017suy}. In general, these are either Banks-Zaks fixed points where some of the gauge couplings are non-zero but $\alpha^*_y=0$, or gauge-Yukawa (GY) fixed points with one $({\rm GY}_1)$, the other $({\rm GY}_2)$, or both gauge couplings non-zero $({\rm GY}_{12})$ and  $\alpha^*_y>0$ \cite{Bond:2016dvk,Bond:2018oco}. It has also been shown that if ${\rm GY}_1$ is UV, its outgoing trajectory is connected with the IR fixed point $({\rm GY}_{12})$, and similarly for  ${\rm GY}_2$  \cite{Bond:2017suy}.

In this work, we focus on the  regime \eq{Bs} and search for  non-perturbative fixed points  with the property
\begin{equation}\label{GY1}
{\rm GY}_1:\quad\quad  \alpha_{1}^*>0\,,\quad \alpha_{y}^*>0\,,\quad \alpha_2^*=0\,.
\end{equation}
Our {\it ansatz} states that the $SU(N_2)$ gauge sector is infrared free, meaning $\beta_2>0$ in the vicinity of the Gaussian. If $\beta_2<0$ in the vicinity of the
interacting fixed point,  the $SU(N_2)$ gauge sector  suddenly becomes asymptotically free, and the  fixed point is ultraviolet.
In its vicinity, $\alpha_2$ is the only relevant coupling in the UV, which runs out of the fixed point as
\begin{equation}\label{relevant}
\alpha_2(\mu)=\frac{\delta\alpha_2(\Lambda)}{1+B_{2,\rm eff}\,\delta\alpha_2\,\ln(\mu/\Lambda)}\,,
\end{equation}
with $\delta\alpha_2(\Lambda)$ a small deviation at the high scale $\Lambda$, $B_{2,\rm eff}=B_{2} +N_F\,\gamma_\chi +4 N_1\,\gamma_\Psi>0$ the interaction-induced one loop coefficient, and $B_{2}<0$ given by (minus) the one-loop coefficient of \eq{beta2} at the Gaussian. The couplings $\alpha_1$ and $\alpha_y$ are irrelevant interactions in the UV and their running is fully determined by the one of $\alpha_2$ \cite{Bond:2017suy}. This  scenario is schematically depicted in Fig.~\ref{pSchematic}.

The  necessary and sufficient conditions for the partially interacting fixed point to turn the dangerously irrelevant coupling $\alpha_2$ into a relevant one are given by
\arraycolsep .1em \begin{eqnarray}
0&=&3N_1-N_F\left(1-2\gamma_\psi\right)-N_2\left(1-2\gamma_\Psi\right)
 \nonumber
 \label{NSVZ1}
\\
0&<&3N_2-N_F\,\left(1-2\gamma_\chi\right)
-N_1\left(1-2\gamma_\Psi\right)
-N_Q\left(1-2\gamma_Q\right)\,,
\label{NSVZ2}\\
0&= &\gamma_\psi+ \gamma_{\Psi}+ \gamma_{\chi}\,,
\label{NSVZy}
\nonumber
\end{eqnarray}
where the   two equations determine the  fixed point while the inequality ensures the sign flip from $\beta_2\ge 0$ close to the Gaussian to $\beta_2<0$ close to the   fixed point \eq{GY1}.

\section{a-Maximisation}

 The beta functions \eq{beta1}, \eq{beta2} and \eq{betay}, and the conditions \eq{NSVZ2}, still depend on the  superfield anomalous dimensions. These can  be determined either perturbatively, as we will do in Sec.~\ref{PT} below, or exactly, as we do here. To that end, we exploit that fixed points of the renormalisation group correspond to superconformal field theories. Hence superfields must transform in representations of the superconformal algebra. Focussing  on the bosonic part, we observe an extra global $U(1)_R$ symmetry in addition to the ordinary conformal algebra. The global and anomaly-free  $U(1)_R$ symmetry
 prescribes global charges $R_i$ for all chiral superfields at any superconformal fixed point and  thereby determines anomalous dimensions $\gamma_i$  of chiral superfields via
 %
\begin{equation}\label{Ri}
\gamma_i=\0{3}{2}R_i-1\,.
\end{equation}
The task then reduces to the determination of $R$-charges at superconformal fixed points for which we use the method
of  $a$-maximisation \cite{Intriligator:2003jj}.
Specifically, at  the fixed point GY${}_1$ with \eq{GY1} and characterised by \eq{NSVZ2},  we find
\begin{equation}\label{Rcharges}
\begin{array}{rl}
	R_\psi &\displaystyle=\frac23\left(1+\gamma_\psi\right)=
	\frac{N_F[N_F - (N_1 + N_2)]^2 - N_1^2\, N_2\,\Delta}{[N_2 - N_F][N_1(N_2 + N_F)
	+ (N_2-N_F)^2
     ]}\,,

	\\[2.5ex]
	R_\Psi &  \displaystyle
	=\frac23\left(1+\gamma_\Psi\right)=
	 \frac{N_2[N_2 - (N_1 + N_F)]^2 - N^2_1\,N_F\,  \Delta}{[N_F - N_2][N_1(N_2 + N_F)
	 	+ (N_2-N_F)^2
      ]}\,,
\\ [2.5ex]
	R_\chi &=\displaystyle
\frac23\left(1+\gamma_\chi\right)=
	 	 \frac{8}{9}\frac{(N_2 - N_F)^2}{ (N_2 - N_F)^2-(1-\Delta) N_1^2}\,,
\\ [2.5ex]
	R_Q &\displaystyle
	=\frac23\left(1+\gamma_Q\right)=\frac23\,,
\end{array}
\end{equation}
and $\Delta > 0$  the positive root of
\arraycolsep .1em \begin{eqnarray}
\Delta^2 &=& 1+ \left(\frac{N_2-N_F}{3N_1^2}\right)^2\left[ (4N_1+N_2+N_F)^2 - 34N_1^2-4N_2N_F\right]\,.
\end{eqnarray}
The $\psi_Q$ fermions do not interact at the fixed point \eq{GY1}, hence $\gamma_Q=0$ and none of the $R$-charges depends on $N_Q$. The results for the $R$-charges uniquely determine the remaining anomalous dimensions non-perturbatively, and provide    closure for the beta functions  \eq{beta1}, \eq{beta2}  and \eq{betay}.  At weak coupling,  anomalous dimensions are small and $R$-charges are close to their classical values $(R_i\approx \s023)$.
Moreover, unitarity mandates that scaling dimensions $D$ of spinless operators must satisfy $D\ge 1$  \cite{Mack:1975je}, additionally implying  $\gamma_i\ge -\s012$ ($R_i\ge \s013$)  for  gauge-invariant scalar operators  such as $\bar \psi \psi$.

The conditions  for unitary quantum field theories  with interacting UV fixed points  \eq{NSVZ2} only depend on ratios of field multiplicities.
Therefore, we can reduce the four-dimensional parameter space of field multiplicities \eq{Ns} to a three-dimensional one. Following  \cite{Bond:2017suy}, we do so by scaling-out  one of the field multiplicities, say $N_1$, and by introducing three suitable ratios  of field multiplicities instead,
 %
\begin{equation}\label{Peps}
\begin{array}{rcl}
R&=&
\displaystyle
\0{N_2}{N_1}\,,\quad
P=\0{N_1}{N_2}\0{N_Q+N_1+N_F-3N_2}{N_F+N_2-3N_1}\,,\quad
\epsilon=
\displaystyle
\0{N_F+N_2}{N_1}-3\,.
\end{array}
\end{equation}
The colour ratio  $R$ (not to be confused with the $R$-charges $R_i$) is sensitive to the relative size of gauge groups.
The parameter  $P<0$ is proportional to the ratio of one-loop gauge coefficients.
The parameter $\epsilon$, which in the region of interest is negative $\epsilon<0$, can be made arbitrarily small in a Veneziano limit where it controls  perturbative fixed points provided $|\epsilon|\ll 1$.
Since field multiplicities are semi-positive numbers, we further observe
\begin{equation}\label{phys}
0<R<3\,,\quad P <\frac{4(1-R)+\epsilon}{R\,\epsilon}\,,\quad R-3<\epsilon<0\,.
\end{equation}
The parameters \eq{Peps} take discrete values for integer field multiplicities except  in
a Veneziano large-$N$ limit where they become continuous.

\section{Fixed Points and Conformal Windows}

We are now in a position to investigate the range of field multiplicities for which the theory displays an interacting ultraviolet fixed point.
Exploiting all constraints
we find that the parameter $P$ is bounded from above and from below
\begin{equation}\label{Prange}
P_{\rm min}(R,\epsilon)<P\le P_{\rm max}(R,\epsilon)\,.
\end{equation}
The upper boundary $P_{\rm max}$ relates to \eq{Bs} and $P\le 0$, and to the physicality of
field multiplicities \eq{phys}, whichever is stronger. The lower boundary  $P_{\rm min}$ relates to the sign-flip
for induced asymptotic freedom  of $SU(N_2)$.  Explicitly,
\arraycolsep .1em \begin{eqnarray} \label{Pmax}
P_{\rm max}(R,\epsilon)&=&
\left\{
\begin{array}{cl}
\displaystyle
\frac{4(1-R)+\epsilon}{R\,\epsilon}&\ \  {\rm for}\ \ R<1+\0{\epsilon}{4}\\[2ex]
0& \ \ {\rm otherwise}
\end{array}
\right.\\[1ex]
\label{Pmin}
P_{\rm min}(R,\epsilon)&=& \frac{3R_\Psi-2}{R\,\epsilon}
+\frac{(3\,R_\chi-2)\,(3-R+\epsilon)}{R\,\epsilon}\,.
\end{eqnarray}
In the above,  the $R$-charges $R_\Psi$ and $R_\chi$ are   understood as functions of $(R,\epsilon)$ via  \eq{Rcharges} and  \eq{Peps}.
The parameters $R$  and $\epsilon$ are  constrained  globally by  the physicality of couplings
 ($\alpha\ge 0$) and unitarity,
\begin{equation} \label{window-eps}
\begin{array}{rcl}
\s014<&R&<2\quad{\rm and}\quad
-\s032\le \epsilon<0\,.
\end{array}
\end{equation}
Fig.~\ref{pConformalWindow} shows a contour plot of the
superconformal window \eq{Prange} and \eq{window-eps} in the $(\epsilon, R)$ plane.
The left panel shows the lower boundary $P_{\rm min}$ while the right panel shows the accessible range of $P$ values between $P_{\rm min}$ and $P_{\rm max}$. The lower boundary in both graphs is given by $R_{\rm min}=1+\s0{\epsilon}{2}$, corresponding to $N_1+N_2=N_F$.
Above the full white  line $R=1+\s0{\epsilon}{4}$ we have $P_{\rm max}=0$. Roughly speaking, the width in $P$ is largest around the full white line. At the upper border we have $R_{\rm max}=2+\s023\epsilon$ which translates into $N_2=N_F$. We notice that the width $P_{\rm max}-P_{\rm min}$ vanishes at $R_{\rm min}$, while it remains small but non-zero at $R_{\rm max}$, except at the endpoints.
Weakly coupled  fixed points correspond to parameters $(P,R,\epsilon)$  close to the boundary  where $|\epsilon|\ll1$.

\begin{figure}
\begin{center}
\includegraphics[width=.9\textwidth]{pConformalWindow.pdf}
\vskip-.2cm
\caption{Contour plot of  the superconformal window for asymptotic safety \eq{Prange} in the $(R,\epsilon)$ plane   showing the boundary $P_{\rm min}$ (left) and  the height $P_{\rm max}-P_{\rm min}$ (right); dashed and full white lines are explained in the main text.
 }\label{pConformalWindow}
\end{center}
\vskip-.6cm
\end{figure}

The conformal window is further illustrated in Fig.~\ref{pAS_strong} showing its projection onto the $(P,R)$ plane. Within the yellow-shaded area, weakly coupled fixed points arise  towards the right of the dashed line, while strongly-coupled fixed points can arise anywhere. Outside the yellow-shaded area, asymptotic safety is not available  and the corresponding quantum field theories must viewed as effective rather than fundamental.

It is interesting to discuss the  $N_F$ dependence of fixed points within the conformal window
while keeping $(N_1, N_2,P)$ fixed.  In    Fig.~\ref{pConformalWindow}, this  effectively corresponds to varying $\epsilon$  along horizontal cuts.
We notice that the conformal window splits into two distinct types of models, separated by  dashed  lines in Figs.~\ref{pConformalWindow} and \ref{pAS_strong},  to which we refer as ``mostly weakly" and ``mostly strongly" coupled.  Specifically, the first subset of models are those above the dashed line in Fig.~\ref{pConformalWindow} and on the right of the  dashed  line in Fig.~\ref{pAS_strong}. Their conformal window
is characterised by
   \begin{equation}\label{CW1}
\begin{array}{rcl}
1\le &R&<2\,.
 \end{array}
  \end{equation}
    Then, for any admissible value of $R$ within \eq{CW1}, there is a range of viable parameters $P$ within $(-1,0]$ such that the UV conformal window in $\epsilon$ covers the range
    \begin{equation} \label{CW1e}
    \epsilon_{\rm max}\le \epsilon<0\,.
    \end{equation}
    The lower bound $\epsilon_{\rm max}=\s032(R-2)$ may become as low as $-\s032$. The significance of this result is as follows. Perturbative fixed points are controlled by the parameter $|\epsilon|\ll 1$ in a large-$N$ Veneziano limit, meaning that this part of the UV conformal window
        contains all perturbatively controlled superconformal UV fixed points  found previously in \cite{Bond:2017suy}.
    Hence,  we observe     that all perturbative fixed points extend into  a non-perturbative conformal window for $\epsilon$ given precisely by the range \eq{CW1e}.
   The same considerations  apply  for  finite $N$, away from a Veneziano limit, the only difference being that  $(R,P,\epsilon)$ take discrete rather than continuous values. Since all theses fixed points are linked to weakly coupled ones, and for want of terminology, we   refer to the models  within  \eq{CW1}    as  ``mostly weakly" coupled.

 Next, we turn to the part of the conformal window
 below the dashed line in Fig.~\ref{pConformalWindow} and to the left of the  dashed  line in Fig.~\ref{pAS_strong}, characterised by
  \begin{equation}\label{CW2}
 \begin{array}{rcl}
 \s014<&R&<1\,.
\end{array} \end{equation}
For any admissible value of $R$ within \eq{CW2}, there is a range of viable parameters $P$ within $(-4,0]$ such that the UV conformal window in $\epsilon$ covers the range
    \begin{equation} \label{CW2e}
   -\s032\le \epsilon<\epsilon_{\rm min}<0\,.
    \end{equation}
    Here, the   lower bound $\epsilon_{\rm min}=2(R-1)<0$ ensures induced asymptotic freedom for the coupling $\alpha_2$.
 Most importantly, we observe the existence of a finite gap $[\epsilon_{\rm min},0]$ in $\epsilon$  within which no  asymptotically safe fixed points can be found.  This implies that none of the fixed points within \eq{CW2}  can be achieved with strict perturbative control $(\epsilon\to 0^-)$, not even in a Veneziano limit.
 It is in this sense that the  fixed points within \eq{CW2}  are non-perturbative and parametrically disconnected  from the free theory, quite different from  those in \eq{CW1}.  For these reasons, we refer to these superconformal fixed points  as ``mostly strongly" coupled.

  \begin{figure}
\begin{center}
\vskip-0.2cm
\includegraphics[width=0.45\textwidth]{pAS_strong2.pdf}
\vskip-0.3cm
\caption{
Projection of the superconformal window with interacting ultraviolet fixed points    onto the $(R,P)$ parameter plane.  Weakly coupled fixed points arise within the yellow-shaded area on the right of the dashed line.  Strongly-coupled fixed points arise within the entire yellow-shaded area for some range of $\epsilon$. The full dot  is discussed in the main text.
\label{pAS_strong}
}
\end{center}
\vskip-.6cm
\end{figure}

Another important aspect of the conformal window relates to the  saddle close to the Gaussian which needs to be overcome by the interacting fixed point.
Recall that the asymptotically free (infrared free) direction is characterised by the one-loop gauge coefficient $B_1=2 \,N_1\,\epsilon<0$  ($B_2=2\, N_2\,P\, \epsilon\ge 0$), with (minus)  their ratio  given by the imbalance parameter
\begin{equation}\label{I}
I=-B_2/B_1\equiv |P|\cdot R
\end{equation}
for which  we have $I\ge 0$. Quantum effects at the interacting fixed point overcome the positive or vanishing one-loop coefficient $B_2$ and turn it, effectively, into a negative one (see Fig.~\ref{pSchematic}). It is then important to understand the largest  imbalance
that can be achieved  without spoiling asymptotic safety. We find that the imbalance is bounded from above,
\begin{equation}\label{PR}
 0\le I=\0{3N_2-N_1-N_F-N_Q}{N_F+N_2-3N_1}<1\,,
 \end{equation}
where we recall that field multiplicities obey \eq{Bs}. In other words, as soon as one gauge sector is as or more infrared free at the Gaussian fixed point than the other gauge sector is ultraviolet free, asymptotic safety at an interacting fixed point cannot arise.

 In Figs.~\ref{pConformalWindow} and~\ref{pAS_strong}, the small imbalance region $0\le I\ll 1$ is realised close to the upper boundaries where $P\approx 0$.
Here, many ultraviolet fixed points can be found  in large parts of the parameter space  including perturbative and non-perturbative ones.
    With growing imbalance $I\to 1$, the set of perturbatively controlled fixed points shrinks to the vicinity of a single point $(P,R)=(-1,1)$.\footnote{In a Veneziano limit, this are the models with $N_1,N_2,N_F\to\infty$ while $N_F/N_1=N_F/N_2=2$, and $N_Q/N_1\to 0$.}
    Non-perturbatively, however, many more fixed points  realise a near-maximal imbalance, corresponding to  the lower boundary  in Fig.~\ref{pConformalWindow} with parameters  given by the  line  $R=R_{\rm min}(\epsilon)$,  $P=-1/R_{\rm min}$,  and  $\epsilon \in [-\s032,0)$.   In Fig.~\ref{pAS_strong}, the near-maximal imbalance is realised along the lower boundary to the left of the dashed line.
 We conclude that all models which may afford the near-maximal imbalance are contained  in the ``mostly strongly coupled" part of the conformal window \eq{CW2}.

\begin{figure}[t]
\begin{center}
\vskip-0.2cm
\includegraphics[width=0.45\textwidth]{pASphaseLine.pdf}
\vskip-0.3cm
\caption{
Schematic phase diagram of Leigh-Strassler type models in the plane of gauge couplings in the limit where the UV fixed point degenerates into a line of IR fixed points (see Fig.~\ref{pSchematic}).\label{pSchematicLine}
}
\end{center}
\vskip-.6cm
\end{figure}

We emphasize  that the boundary at $P_{\rm min}$ in \eq{Pmin}
is not part of the conformal window. The reason for this is that whenever $P\to P_{\rm min}$
the inequality in \eq{NSVZ2} becomes an equality, meaning that $\beta_2\to 0$ non-perturbatively at the fixed point \eq{GY1}. At this point the UV fixed point  ${\rm GY}_1$ merges with yet another fixed point (a non-perturbative IR fixed point ${\rm GY}_{12}$  \cite{Bond:2017suy}) in the limit $P\to P_{\rm min}$. In consequence, the fixed point \eq{GY1}  degenerates into a line of   fixed points for any $\alpha_2$, illustrated in Fig.~\ref{pSchematicLine}.
As such, this limit offers a manifold of Leigh-Strassler type models  \cite{Leigh:1995ep}, each of which is characterised by a
line
of interacting superconformal field theories, disconnected from the free theory, and generated by an exactly marginal operator. Further, each of these lines of fixed points corresponds to an infrared sink because the previously relevant perturbation,  given by $\alpha_2$, has become strictly marginal.
This  includes all settings with maximal imbalance $I=1$. Concrete examples for strongly-coupled Leigh-Strassler models
are delegated to Sec.~\ref{benchmarks} below.\footnote{The fate of an IR sink can be evaded by adding mass term perturbations for the $N_F$ superfields $\psi$ which may lift the degeneracy and allow RG trajectories to emanate from the line of fixed points. However, this  mechanism   defies the original setup \eq{Bs} in that the removal of the $\psi$ degrees of freedom would make the theory asymptotically free from the outset.}

 Fig.~\ref{pGamma} shows the superfield anomalous dimensions within the entire conformal window. Since the spectator fermions $\psi_Q$ are free at  the fixed point \eq{GY1}, the chiral superfield  anomalous dimensions   are only functions of $(R,\epsilon)$ and  independent of $P$.
 Overall, we find that anomalous dimensions grow with growing $|\epsilon|$. For models within \eq{CW1} or \eq{CW2}, the chiral anomalous dimensions cover the  range
 \begin{equation}\label{grange}
 0>\gamma_\psi,\gamma_\Psi\ge -\s012\quad{\rm and}\quad 0<\gamma_\chi\le 1
 \end{equation}
 with extremals reached at the $\epsilon=-\s032$ and $\epsilon=0$ boundaries of the conformal window. For models within \eq{CW2} anomalous dimensions tend to take larger values than for those in  \eq{CW1}. A comparison with  perturbation theory is given in Sec.~\ref{PT}. Incidentally, the monotonicity of $\gamma_i$ with growing $\epsilon$ establishes that anomalous dimensions do not vanish unless the fixed point is free.

\section{Central Charges and the $a$-Theorem}
Superconformal fixed points can also be characterised by  central charges $a$, $b$ and $c$,  the anomaly coefficients  \cite{Anselmi:1997am,Anselmi:1997ys}. Their
values have to satisfy certain conditions, and can be used to constrain viable fixed points.
The global charges $a$ and $c$ can be expressed in terms of the $R$-charges of chiral superfields,
 \begin{align}\label{eq:afunc}
	a &= \frac{3}{32}\left(3{\rm Tr} R^3 - {\rm Tr} R\right)\,\\
	c &= \frac{1}{32}\left(9{\rm Tr} R^3 - 5{\rm Tr} R\right)\,.
\end{align}
Given that the  positivity conditions $a,b,c>0$ are satisfied non-marginally for free theories,  they are guaranteed to be  satisfied for perturbative theories.
Similarly, the conformal collider bound \cite{Hofman:2008ar}
\begin{equation}
\012<\frac{a}{c}<\032
\end{equation}
cannot be violated for weakly interacting theories.
Here, we find that the  positivity conditions and the conformal collider bound are  satisfied  non-perturbatively,
in the entire conformal window, as they must.
 \begin{figure*}
\begin{center}
\includegraphics[width=.9\textwidth]{pAnomalousDimensions.pdf}
\vskip-.5cm
\caption{
Contour plot showing the non-perturbative anomalous dimensions of chiral superfields
at  the superconformal fixed points
of Fig.~\ref{pConformalWindow}.  The full dot and dashed line are explained in the main text.\label{pGamma}
}
\end{center}
\vskip-.6cm
\end{figure*}

We now turn to the $a$-theorem. It states that the central charge $a$ must be a decreasing function along RG trajectories in any 4d quantum field theory  \cite{Komargodski:2011vj,Komargodski:2011xv}. We find that
$a_{\rm UV}-a_{\rm Gauss} <0\,,$
which  confirms that none of the UV fixed points is connected by an RG trajectory with the free Gaussian fixed point, in accord with Fig.~\ref{pSchematic}. Further, we noted earlier that any interacting UV fixed point in the conformal window comes bundled with a
fully interacting non-perturbative IR fixed point $({\rm GY}_{12})$ \cite{Bond:2017suy}. We find
$a_{\rm UV}-a_{\rm IR} >0\,,$ meaning that  both conformal fixed points are connected by RG trajectories  flowing from  the former to the latter as in Fig.~\ref{pSchematic}.  We conclude that our results are consistent with the  $a$-theorem, as they must.

\section{Seiberg Duality}

 Next, we   comment on how our results relate to Seiberg's electric-magnetic duality \cite{Seiberg:1994pq}.
At the partially interacting fixed point \eq{GY1} where the $SU(N_2)$ gauge sector is free, the flavour symmetry  of the theory is enhanced. This manifests itself as an exchange symmetry under $N_2 \leftrightarrow N_F$, whereby the quarks $\psi$ and $\Psi$ interchange their roles while the fields $\chi$ remain unchanged,
\begin{equation}
	N_2 \leftrightarrow N_F\,, \quad\quad R_\psi \leftrightarrow R_\Psi\,,	\quad\quad	R_\chi \leftrightarrow R_\chi\,,
\end{equation}
see \eq{Rcharges}.  Hence, we observe an interacting ``magnetic'' $SU(N_1)$ gauge theory which has $N_F+N_2$ flavours of ``magnetic quarks'' $\psi$ and $\Psi$, alongside $(N_F+N_2)^2$ singlet mesons $(\psi_L\psi_R)$, $\chi_{L}$, $\chi_{R}$, and $(\Psi_L\Psi_R)$ \cite{Barnes:2005zn}. In this light, the gauge-Yukawa fixed point \eq{GY1} corresponds to a free non-Abelian magnetic phase whose superconformal window is known to cover the range
$\s032 N_1<N_F+N_2<3N_1$ \cite{Seiberg:1994pq}
which corresponds to
 the parameter range
$-\s032<\epsilon<0$ observed in \eq{window-eps}. Notice however that the conformal window in Fig.~\ref{pConformalWindow} has additional constraints  on $(R,P)$ which ensure that the $SU(N_2)$ gauge sector becomes a relevant perturbation at the fixed point.

Seiberg duality predicts the existence of a dual ``electric'' theory which must have gauge group $SU(N_F+N_2-N_1)$ coupled to $N_F+N_2$ flavours of electric quarks. The conformal window of the non-Abelian Coulomb phase is given by
 $\s032 (N_F+N_2-N_1)<N_F+N_2<3(N_F+N_2-N_1)\,.$  This corresponds to the exact same parameter band in $\epsilon$ as the one in \eq{window-eps}.
In the special case where $N_F=N_2$, given by  the upper boundary in Fig.~\ref{pConformalWindow}, the $R$-charges simplify and become
\begin{equation}
R_\psi=R_\Psi=1-\frac{N_1}{2N_F}\,,\quad R_\chi=\frac{N_1}{N_F}\,.
\end{equation}
One recognises the $R$-charges for the quarks and singlet mesons of supersymmetric magnetic QCD  \cite{Seiberg:1994pq}, whose Seiberg dual is characterised by electric quarks ${\widetilde\psi}$ and $\widetilde\Psi$ with
\begin{equation}
R_{\widetilde\psi}=R_{\widetilde\Psi}=\frac{N_1}{2N_F}\,.
\end{equation}
We refer to  \cite{Barnes:2005zn} for further aspects of Seiberg duality in theories with product gauge groups.

\section{Perturbation Theory} \label{PT}

In this section, we contrast the non-perturbative determination of anomalous dimensions with perturbation theory, using general expressions for perturbative beta functions  in the DRED regularisation scheme  up to three loop  \cite{Einhorn:1982pp,Machacek:1983tz}. The main addition are perturbative expressions for anomalous dimensions which  at superconformal fixed points can be used to cross-check the non-perturbative results from $a$-maximisation.

For the sake of this comparison, it is convenient to perform a Veneziano limit and rescale gauge couplings as $\alpha_i\to N_i\,\alpha_i$ and Yukawas as $\alpha_y\to N_1\,\alpha_y$. We also use the parametrisation \eq{Peps}.
The parameter $0<|\epsilon|\ll 1$ then serves as a small expansion parameter to ensure rigorous control of fixed points
in perturbation theory.
To find   fixed points, scaling exponents, and anomalous dimensions to first (second) order in $\epsilon$,
we must  retain  terms up to two (three) loop in the gauge coupling, and up to one (two) loop in the Yukawa beta functions \cite{Bond:2016dvk,Bond:2017tbw}.
We refer to these approximations as NLO and NNLO, respectively. Using the general results  of \cite{Einhorn:1982pp,Machacek:1983tz}
we find the gauge beta functions up to three loop for our models as
\arraycolsep .1em \begin{eqnarray}
	\beta_1^{(1)} &=&
	\displaystyle
	2\alpha_1^2\, \epsilon
\nonumber
\,,\\
	\beta_1 ^{(2)} &=&
	\displaystyle
	2\alpha_1^2\big[6\alpha_1 + 2R \alpha_2
	- 4R(3 +\epsilon- R)\alpha_y\big]
\label{beta13}
\,,\\ \nonumber
	\beta_1^{(3)} &=& 4\alpha_1^2\left[2\epsilon\alpha_1^2
			 - R\left(2\alpha_1\, \gam{\Psi}{1}+\gam{\Psi}{2}\right)
			 -(3 +\epsilon- R)\left(2\alpha_1\,\gam{\psi}{1} +\gam{\psi}{2}\right)\right]\,,
			 \end{eqnarray}
and
\arraycolsep .1em \begin{eqnarray}\nonumber
	\beta_2^{(1)}  &=&
		\displaystyle
2\alpha_2^2\,P\epsilon
\,,\\
\label{beta23}
	\beta_2^{(2)}  &=&
		\displaystyle
2\alpha_2^2\left[6\alpha_2 + \02R\alpha_1
		- \04R(3 - R+\epsilon)\alpha_y\right]
\,,\\ \nonumber
	\beta_2^{(3)} &=&
	4\alpha_2^2\left[2 P \epsilon\alpha_2^2 -\frac1R  \left(2\alpha_2\gam{\Psi}{1}+  \gam{\Psi}{2}\right)
\right.\\&&\nonumber 	\quad\quad \left.
	- \frac{3-R+\epsilon}{R}\left( 2\alpha_2\,\gam{\chi}{1} +\gam{\chi}{2}\right)
- \left(4 + P \epsilon - \frac{4+\epsilon}{R}\right)\left( 2\alpha_2\,\gam{Q}{1}+\gam{Q}{2}\right)\right]\,.
			\end{eqnarray}
The perturbative Yukawa beta function is given by \eq{betay} for any loop order.
The anomalous dimensions of the superfields are required up to two loop accuracy.
They read
\begin{equation}\label{gamma1}
\begin{array}{rcl}
	\gam{\psi}{1} &= &\displaystyle
	R\,\alpha_y - \alpha_1\,,\\[1ex]
	\gam{\Psi}{1} &=&\displaystyle (3 - R+\epsilon)\alpha_y - \alpha_1 -\alpha_2\,,\\[1ex]
	\gam{\chi}{1} &=&\displaystyle \alpha_y - \alpha_2\,,\\[1ex]
	\gam{Q}{1} &= &\displaystyle-\alpha_2\,,
\end{array}
\end{equation}
and
\begin{equation}\label{gamma2}
\begin{array}{rcl}
	\gam{\psi}{2} &= &\displaystyle-R\,\alpha_y\left(\gam{\Psi}{1} + \gam{\chi}{1}\right) - \alpha_1 \gam{\psi}{1} + 4\,\epsilon\,\alpha_1^2\,,\\[1ex]
	\gam{\Psi}{2} &=&\displaystyle -(3 - R+\epsilon)\,\alpha_y\left(\gam{\psi}{1} + \gam{\chi}{1}\right) - (\alpha_1 + \alpha_2)\gam{\Psi}{1}
				+ 4\,\epsilon\,\alpha_1^2 + 4 \,P\,\epsilon\,\alpha_2^2\,,\\[1ex]
	\gam{\chi}{2} &= &\displaystyle-\alpha_y\left(\gam{\psi}{1} + \gam{\Psi}{1}\right) - \alpha_2\gam{\chi}{1}
				+ 4\, P\,\epsilon\,\alpha_2^2\,,\\[1ex]
	\gam{Q}{2} &= &\displaystyle-\alpha_2\, \gam{Q}{1} + 4 \,P\,\epsilon \,\alpha_2^2\,,
\end{array}
\end{equation}
respectively. Then,  interacting fixed points with \eq{GY1} are determined via a Taylor expansion of couplings in $\epsilon$ around the zeros  of \eq{beta13}
and \eq{betay},  also using \eq{gamma1} and \eq{gamma2}. Notice that \eq{beta23} is only used to determine whether
the sign change from $\beta_2>0$ at the Gaussian to $\beta_2<0$ at \eq{GY1} has taken place.
Inserting the perturbative results for the fixed point into  \eq{gamma1} and \eq{gamma2} provides explicit expressions
for the anomalous dimensions of the form
    \begin{equation}\label{gammaAs}
     \gamma_i(\epsilon)=A^{(1)}_{i}\,\epsilon+A^{(2)}_{i}\,\epsilon^2+{\cal O}(\epsilon^3)\,.
     \end{equation}
The six coefficients $A^{(n)}_{i}$  for $i=\psi, \Psi, \chi$ which for $n=1,2$  arise from the perturbative fixed point solutions at NLO and NNLO accuracy  \cite{Bond:2017lnq,Bond:2017tbw} still depend on the parameter $R$ (but not on $P$). Their explicit expressions are not given because they do not offer further insights for the present purposes.

  The expressions \eq{gammaAs}  must be compared with the exact $R$-charges from $a$-maximisation \eq{Rcharges}.  Writing the exact anomalous dimensions in terms of the  $R$-charges, and expanding  the expressions to second order in $\epsilon$ with the help of  \eq{Peps}, we find the identical result for the coefficients $A^{(n)}_{i}$ in \eq{gammaAs}. This establishes consistency of findings between $a$-maximisation and perturbation theory, as it must.

       \begin{figure}
\begin{center}
\includegraphics[scale=.4]{pComparisonWeak.pdf}
\vskip-.3cm
\caption{
Exact anomalous dimensions and $U(1)_R$ charges  in comparison with 2-loop (NLO) and 3-loop (NNLO) results from perturbation theory. Shown are projections along $R=\s032$ (left panel) and $R=1$ (right panel).  Fixed points in the gray-shaded area do not show the required sign-flip in \eq{NSVZ2} for any $P$, and are therefore outside of the UV conformal window.}
\label{pComparison1}
\end{center}
\vskip-.7cm
\end{figure}

\section{Anomalous Dimensions}

      In  Fig.~\ref{pGamma} we have shown the exact anomalous dimensions across  the entire UV conformal window. Here, we have a closer look into anomalous dimensions and compare with the  NLO and NNLO predictions from perturbation theory. More specifically, in Figs.~\ref{pComparison1}  and~\ref{pComparison2}, we  compare results
      at fixed colour ratio $R=\s032, 1, \s023$ and  $\s013$, and  for any $\epsilon$ within $[-\s032,0]$, which corresponds to  horizontal cuts across Figs.~\ref{pConformalWindow} and~\ref{pGamma}.

       We begin with Fig.~\ref{pComparison1} where our results for $R=\s032$ (left panel) and $R=1$ (right panel) are shown. For small $|\epsilon|\ll 1$,  perturbation theory matches the exact results as it must. With growing $|\epsilon|$, all anomalous dimensions grow in magnitude. For $R=\s032$, the gray-shaded area indicates that fixed points are no longer in the UV conformal window, leading to a lower bound for $\epsilon$, see \eq{CW1e}. Also, the NLO results for $\gamma_\psi$ and $\gamma_\Psi$ coincide accidentally. At NNLO, perturbative results for all three anomalous dimensions are quite close to the exact results in the admissible range for $\epsilon$. For $R=1$,
      the extension of the conformal window is maximal.
      Here, anomalous dimensions
      correspond to
        the fixed points along the dashed lines in Figs.~\ref{pConformalWindow},~\ref{pAS_strong} and \ref{pGamma}, which marks the boundary between the ``mostly weakly" and ``mostly strongly" coupled quantum field theories, see  \eq{CW1} vs \eq{CW2}. Maximal values $\gamma_{\psi,\Psi}\to -\s012$ and $\gamma_\chi\to 1$  are reached for $\epsilon\to -\s032$.
              For $\gamma_\psi$, we observe that the NLO and NNLO results are close to the exact one over the entire range for $\epsilon$.
        For $\gamma_\Psi$, the NLO correction vanishes accidentally. Similarily, for $\gamma_\chi$, the NLO result strongly underestimates the exact value with growing $|\epsilon|$. However, at NNLO, perturbative results for all three anomalous dimensions are close to the exact findings in the entire range of $\epsilon$.  We conclude that differences between  NNLO      and exact results are  moderate over the entire interval. Note  that this does  not hold true in general, but when it does, one may use this near-coincidence to extract  estimates for  fixed points at strong coupling which elsewise are  not easily accessible.

        The same analysis is repeated in Fig.~\ref{pComparison2} for the parameter choices $R=\s023$ (left panel) and $R=\s013$ (right panel).  These horizontal cuts through Fig.~\ref{pConformalWindow} project onto the more strongly coupled fixed points in \eq{CW2}. For either of these  the small $\epsilon$ regions are excluded (grey-shaded areas) because the corresponding fixed points are not ultraviolet. In the increasingly narrow regions of interest \eq{CW2e} which are $\epsilon\in[-\s032,-\s023]$ and $\epsilon\in[-\s032,-\s043]$, respectively,  we  observe that differences between NLO, NNLO, and exact results become rather large for
        $\gamma_\chi$, $\gamma_\psi$ and $\gamma_\Psi$. Outside the UV conformal window, we also notice that $\gamma_\Psi$ is no longer monotonous with $\epsilon$,  but instead changes sign for non-vanishing $\epsilon$, and taking values outside the range \eq{grange}. Perhaps unsurprisingly, this confirms that the leading orders of perturbation theory cease to offer a good approximation for anomalous dimensions at strong coupling, once  $|\epsilon|$  is large.

         \begin{figure}
\begin{center}
\includegraphics[scale=.4]{pComparisonStrong.pdf}
\vskip-.3cm
\caption{
Same as   Fig.~\ref{pComparison1}, except that the  projections are along $R=\s023$ (left panel) and $R=\s013$ (right panel). Decreasing $R$ considerably narrows the UV conformal window. The excluded (gray-shaded) areas refer to regions where some matter field field multiplicities would be unphysical (negative).}\label{pComparison2}
\end{center}
\vskip-.7cm
\end{figure}

 \section{Benchmarks for Models of Particle Physics}\label{benchmarks}

At weak coupling, ultraviolet fixed points are often characterised by a large number of field multiplicities  \cite{Bond:2017suy}. At strong coupling, fixed points can arise with a much lower number matter fields. Therefore, we benchmark models with minimal numbers of gauge and matter fields  and  explore whether variants could serve as  templates  for asymptotically safe Standard Model extensions.

Our benchmarks of superconformal fixed points
with low numbers of fields are summarised in Tab.~\ref{tsolutions}.
They represent strongly coupled models in that they have large  $|\epsilon|$ and
anomalous dimensions. The benchmarks  also cover the full range of imbalance parameters $I$.
Note that because the $\psi_Q$ fermions are free at the superconformal fixed point, anomalous dimensions of chiral
superfields agree between different models as long as they only differ in the $N_Q$ multiplicity, see models 2--4, model 5 and 6,
or model 7 and 8.

We begin with model 1 which is the sole example with $SU(2)\times SU(2)$ gauge
symmetry, the simple reason being that the only other integer solution $(N_F,N_Q)=(3,2)$
to the constraint  \eq{Bs} does not lead to asymptotic safety. Here, anomalous dimensions are still small and well-approximated by the perturbative three loop result  (see Fig.~\ref{pComparison1}, right panel). Moreover, in models 1 and 2, the $SU(2)$ gauge sector has a vanishing one-loop coefficient.
In both cases a viable interacting UV fixed point is found, located at a boundary of the conformal window (Fig.~\ref{pConformalWindow})
where $I=0$.

For the gauge group $SU(3)\times SU(2)$, we find a total of five solutions (models 2--6).
For all of these  $SU(2)$ is infrared free at the Gaussian. Settings where $SU(3)$ is infrared free do not lead to asymptotic safety.
Also, all models are within the strongly-coupled domain where 3-loop perturbation theory does not offer an accurate approximation  (Fig.~\ref{pComparison2}, left panel). Model 3 has $N_F=3$ and $N_Q=1$ and its location in the conformal window is  indicated by a full dot in Figs.~\ref{pConformalWindow} and~\ref{pAS_strong}. The anomalous dimensions are of order unity
\begin{equation}
(\gamma_\psi,\gamma_\Psi,\gamma_\chi)\approx(-0.41, -0.38, 0.79)
\end{equation}
and shown by a full dot in Fig.~\ref{pGamma}. Enhancing $N_Q$ by one unit  gives another viable fixed point  (model 4) without changing  the anomalous dimensions. For $(N_F,N_Q)=(4,0)$ (model 5) and $(N_F,N_Q)=(4,1)$ (model 6), the main change is that $|\epsilon|$, and hence the anomalous dimensions come out slightly smaller than in model 2-4. Overall, increasing the number of matter fields charged under $SU(2)$ also increases the imbalance parameter.

Models $3-6$ have some similarities with the Minimal Supersymmetric Standard Model (MSSM). Unlike in the Standard Model, the  $SU(2)$ sector of the MSSM   becomes infrared free due to extra gauge charges from supersymmetry partners, while the $SU(3)$ sector  remains asymptotically free.
Hence, the Gaussian corresponds to a saddle with imbalance  $I_{\rm MSSM}=\s013$, just  as in model 5. The imbalance could even be made larger such as in model 4 or model 6, and still yield an asymptotically  safe fixed point. Hence, we see that supersymmetric gauge theories with the SM gauge groups (neglecting hypercharge) and  imbalances similar to the MSSM may very well become asymptotically safe
\cite{Hiller:2022hgt}.

\begin{table}[t]
 \aboverulesep = 0mm
\belowrulesep = 0mm
	\addtolength{\tabcolsep}{2pt}
	\setlength{\extrarowheight}{2pt}
\begin{center}
\begin{tabular}{`cc cc cc cc cc c`}
\toprule
\rowcolor{Yellow}
\bf \ Model\ \
&\bf \ Gauge Group \
&\ \ $\bm{N_F}$\ \
&\ \ $\bm{N_Q}$\ \
&\ \ $\bm{I}$ \ \
&\ \ $\bm{R}$ \ \
&\ \ $\bm{P}$ \ \
&\ \ $\bm{\epsilon}$ \ \
&\ \ $\bm{\gamma_\psi}$ \ \
&\ \ $\bm{\gamma_\Psi}$ \ \
&\ \ $\bm{\gamma_\chi}$ \ \
\\
\rowcolor{Yellow}
&& && && && &&\\[-4mm]
\rowcolor{white}
\midrule
1
&$SU(2)\times SU(2)$
&3
&1
&$0$
&$1$
&$0$
&$-\s012$
&$-0.126$
&$-0.061$
&$0.187$\\
2
&$SU(3)\times SU(2)$
&3
&0
&$0$
&$\s023$
&$0$
&$-\s043$
&$-0.412$
&$-0.382$
&$0.794$
\\
3
&$SU(3)\times SU(2)$
&3
&1
&$\s014$
&$\s023$
&$-\s038$
&$-\s043$
&$-0.412$
&$-0.382$
&$0.794$\\
4
&$SU(3)\times SU(2)$
&3
&2
&$\s012$
&$\s023$
&$-\s034$
&$-\s043$
&$-0.412$
&$-0.382$
&$0.794$
\\
5
&$SU(3)\times SU(2)$
&4
&0
&$\s013$
&$\s023$
&$-\s012$
&$-1$
&$-0.288$
&$-0.174$
&$0.462$
\\
6
&$SU(3)\times SU(2)$
&4
&1
&$\s023$
&$\s023$
&$-1$
&$-1$
&$-0.288$
&$-0.174$
&$0.462$
\\
7
&$SU(6)\times SU(2)$
&7
&0
&$\s079$
&$\s013$
&$-\s073$
&$-\s032$
&$-\s012$
&$-\s012$
&$1$
\\&& && && && &&\\[-4.5mm]
\midrule
\rowcolor{LightGray}
$8$
&$SU(6)\times SU(2)$
&7
&1
&$\s089$
&$\s013$
&$-\s083$
&$-\s032$
&$-\s012$
&$-\s012$
&$1$
\\
\rowcolor{LightGray}
$9$
&$SU(8)\times SU(2)$
&10
&0
&$1$
&$\s014$
&$-4$
&$-\s032$
&$-\s012$
&$-\s012$
&$1$
\\[.4mm]
\bottomrule
\end{tabular}
\end{center}
 \vskip-.4cm
 \caption{\label{tsolutions} Parameter and anomalous dimensions for a selection of benchmark models with superconformal fixed points and  a low number of field multiplicities, including Leigh-Strassler type models (8 and 9).}
\end{table}

The constraints  \eq{Bs}  imply that maximal values for  anomalous dimensions \eq{grange} cannot arise if gauge group factors are as small as $SU(2)$ or $SU(3)$. For larger gauge groups, however, we can find models realising maximal anomalous dimensions.
An example  is given by an $SU(6)\times SU(2)$ gauge theory with $N_F=7$ (model 7) which has a  near-maximal  imbalance parameter $(I=\s079$). The model is located at the $\epsilon=-\s032$ boundary of the conformal window and leads to an asymptotically safe fixed point with  non-perturbatively large
chiral field anomalous dimensions,
\begin{equation}
\gamma_\psi =-\012\,,\quad\gamma_\Psi=-\012\,,\quad\gamma_\chi=1\,.\end{equation}
We conclude that the benchmark models are low-field-multiplicity realisations of asymptotic safety with phase diagrams as in Fig.~\ref{pSchematic}. Most of them are non-perturbative with large $|\epsilon|$ and large anomalous dimensions, whose features are not captured reliably by a few leading orders of perturbation theory (Fig.~\ref{pComparison2}). With increasing sizes of gauge group factors  $(N_1,N_2)$, many more solutions $(N_F,N_Q)$ for the constraint \eq{Bs} arise, and a fair fraction of these lead to an asymptotically safe fixed point within the UV conformal window (Fig.~\ref{pConformalWindow}).

Finally, we discuss two   benchmark  models with $\epsilon=-\s032$  and maximal anomalous dimensions. These strongly-coupled models are of the Leigh-Strassler type and narrowly outside the UV conformal window. The first one is model 8 with $SU(6)\times SU(2)$ gauge symmetry and  $N_F=7$, which differs from model 7 only in that $N_Q=1$ instead of  $N_Q=0$,
thus leading to a larger imbalance ($I=\s089$).
In consequence, the  ultraviolet fixed point of model 7 degenerates into  a line of  fixed points due to $\beta_2|_*$ vanishing identically at the gauge-Yukawa fixed point. The  exactly marginal  coupling $\alpha_2$ becomes a free parameter and extends the  fixed point into an  infrared attractive line. The same phenomenon occurs for model 9, which has an $SU(8)\times SU(2)$ gauge symmetry with $N_F=10$ and imbalance $I=1$, situated at the lower-left corner of Fig.~\ref{pConformalWindow}.
Although  interesting in their own right, the models 8 and 9 no longer serve the purpose of ultraviolet fixed points. However, we stress that
the method of $a$-maximisation has been key in identifying  fixed points with exactly marginal directions. \vspace{.5em}\step

\section{Conclusions}

We have shown, as a proof of principle,  that interacting UV fixed points exist in strongly coupled quantum field theories
including away from    Veneziano (large-$N$) limits. Fixed point anomalous dimensions of matter fields  can grow large, taking values in the entire range  dictated by unitarity \eq{grange}.
Thereby, previously-found perturbative fixed points   \cite{Bond:2017suy} extend naturally into a non-perturbative conformal window
(Fig.~\ref{pComparison1}).
  Interestingly, a novel range of
  fixed points   has become available,
with no perturbative counterparts  in a Veneziano limit  (Fig.~\ref{pComparison2}).  We  thus conclude that the non-perturbative ``phase space" of fixed points is large, and substantially larger then the  one accessible at weak coupling.
 Further, all conformal fixed points  (Fig.~\ref{pConformalWindow}) are in accord with the $a$-theorem, bounds on central charges, Seiberg duality, and unitarity.  We also found a manifold  of Leigh-Strassler  models
arising at  the sign-flip-controlling boundary of the conformal window,  where theories  display a line of IR fixed points generated by an exactly marginal gauge interaction (Fig.~\ref{pSchematicLine}).
It is understood that similar UV conformal windows,  bounded by  Leigh-Strassler manifolds, arise naturally  in other semi-simple gauge theories  coupled to matter.

From the viewpoint of particle physics and model building,
it is  quite promising that fixed points persist   at low field multiplicities (Tab.~\ref{tsolutions}),  including  with Standard Model-like gauge groups.
Next natural  questions to ask  are whether  fixed points also arise in MSSM extensions  \cite{Hiller:2022hgt} or in  non-supersymmetric setting at strong coupling.
From the viewpoint of conformal field theory, it will   be interesting to classify asymptotically safe SUSY theories more systematically, and to extract CFT data or structure coefficients directly from the RG fixed points.
 \vspace{.5em}\step\vspace{.5em}

\centerline{\bf
 Acknowledgements}
The work of DL is supported by the Science and Technology Research Council (STFC) under the Consolidated Grant [ST/T00102X/1] .

\bibliographystyle{mystyle}

redpajama_set_name: RedPajamaArXiv


text:
Q: Force HTTPS with mod_rewrite, including proxied SSL I've got a server getting some traffic from an SSL terminating load balancer- in which case it comes in as HTTP over port 80 with a http_x_forwarded_proto = "https"
I want a mod_rewrite rule that only allows direct HTTPS traffic or forwarded HTTPS traffic.
I have this so far:
RewriteEngine On
RewriteCond %{HTTPS} !=on
RewriteCond %{HTTP:http_x_forwarded_proto} != https
RewriteCond %{HTTP:http_x_forwarded_proto} != HTTPS
RewriteRule ^ https://%{HTTP_HOST}%{REQUEST_URI} [L,R=301]

but I'm getting

RewriteCond: bad flag delimiters

error.
What do I need to correct to get this working, and is this the best approach?

A: The problem was the whitespace after the "!=":
Working version:
RewriteEngine On
RewriteCond %{HTTPS} !=on
RewriteCond %{HTTP:X-Forwarded-Proto} !https [NC]
RewriteRule ^ https://%{HTTP_HOST}%{REQUEST_URI} [L,R=301]

tricky...

A: If you have load balancer always use SSL when communicating with the server, you need to omit the first check as it will always be true.  (If you are offloading SSL at the load balancer, the first line is always true unless someone manages to directly hit your server using SSL in which case it would be false and not try to redirect since X-Forwarded-Proto would be missing.)
The code I'm using since we're always communicating over SSL between the ELB and the webserver:
RewriteEngine On
RewriteCond %{HTTP:X-Forwarded-Proto} !https [NC]
RewriteRule ^ https://%{HTTP_HOST}%{REQUEST_URI} [L,R=301]

redpajama_set_name: RedPajamaStackExchange


On behalf of Mahaffey & Associates, LLC posted in Estate Planning on Thursday, July 7, 2016.
Each and every Ohio family has a unique set of needs when it comes to creating an estate plan. Passing down accumulated wealth to one's children is a common estate planning theme. However, there are cases in which all of a couple's children are already doing well for themselves, and where the focus can turn to the next generation. Retirement trusts are an excellent option for families who are looking for ways to pass wealth down to grandchildren.
On behalf of Mahaffey & Associates, LLC posted in Estate Planning on Friday, June 10, 2016.
There are a great many Ohio families in which the spouses intend to leave all of their assets to each other in the event of death. In many cases, both parties feel that they have worked hard to amass their wealth and that the fruits of that labor belong solely to their partner. Unfortunately, many are unaware of the best way to ensure that outcome, and they make the mistake of assuming that marriage is sufficient estate planning all on its own.


When it comes to those couples out there who do not have any kids yet, these couples might be really sad because they can not have their very own children and if you are a couple who are suffering with these things, you have really come to the right place today. There are wives out there that are really trying to have children of their own but they just can not have them and if you are someone who is like this, we have some help coming your way. There are actually a lot of things that you can do in order to get to have a child of your own. There are those people out there who would adopt children so that they can get to be mothers and fathers of this child. If you do not want to adopt a child because you really want to have your very own child, you can opt for surrogacy. There is a lot to be said about surrogacy and if you are not familiar with it, you might be really interested to know more about this service and how it can help you.
Surrogacy is actually when a mother wants to have a child and therefore she gives her fertilized eggs to another woman to keep in her womb until it grows up into a baby. Maybe you have always wanted your very own child and if you do not like to adopt children, you should really go and get a surrogate mother to help you have your own child. When it comes to surrogacy, you can really get to have your very own child that you can really call your very own and this is something that is really great indeed. There are a lot of women out there who are surrogate mothers and if you would also like to become one, you should really go and try out and see if you are qualified to become a surrogate mother yourself.
If you would like to become a surrogate mother, the first thing that you should do is to decide if you really want to be one. After you have made your mind to become one, you should go and see if you meet the requirements to become a surrogate mother. After you have decided that you really want to become a surrogate mother, you should go and see if you meet the requirements of becoming a surrogate mother. You can then start looking for those couples who will need assistance with a baby.


text:
Watch live: Trump leaves White House to face uncertain future at Mar-a-Lago
Ask Amy: Friends with benefits might have a love match
Chicago Bears Q&A: What is a realistic possibility at quarterback? Why did Ryan Pace keep his job? Could the defense switch to a 4-3?
Chicago entering next phase of Illinois' COVID-19 vaccine program on Monday, as officials predict indoor dining could resume within days
Outgoing President Trump pardons Lil Wayne, Kodak Black and Roc Nation CEO Desiree Perez
Column: Can Joe Biden hold off the left?
City sticker artwork shows gang signs?
By By Hal Dardick and John Byrne
Tribune reporters |
The winning city sticker design. City Clerk Susana Mendoza said Tuesday she's investigating whether it contains gang signs.
City Clerk Susana Mendoza said Tuesday she's investigating whether gang signs are part of the winning design for Chicago's next city sticker, just days before the decals are scheduled to be printed.
The sticker, meant to honor city firefighters, paramedics and police, was designed by a 15-year-old boy who attends a school for troubled youth.
The artwork shows four hands lifted skyward toward symbols of the three professions — above a heart containing a carefully rendered city skyline against the city flag.
A blogger who writes about police issues identified the position of those hands as symbols often flashed by members of a notorious Chicago street gang. Concerned citizens began calling the clerk's office Tuesday after those allegations hit the web, the clerk's office said.
"Every artist has a back story and oftentimes artwork is controversial," Mendoza said in a statement. "However, the artwork on Chicago's city stickers should not be controversial.
"In a design that is meant to honor the city's first responders, I am very sensitive to this issue. I grew up in a neighborhood filled with crime and gang violence and I come from a law-enforcement family," Mendoza added.
Exactly what's on the city stickers is a touchy subject, in part because they will be the first to carry the names of Mendoza and Mayor Rahm Emanuel.
Jody Weis, a former police superintendent who is now president of the Chicago Crime Commission that puts out a handbook on city gangs, said he initially saw the imagery on a Blackberry and didn't think it looked like a gang sign.
But a short time later, when he blew up the picture on an iPad, he could see it much clearer and he changed his mind.
"It's very, very close to a gang sign,'' Weis told the Tribune when reached by phone Tuesday night. "It's not exactly, but it's very close,'' Weis said. "It's too close to be a coincidence.''
Weis explained that the left and right hands shown in the artwork are the same.
"Every finger configuration is identical,'' he said.
Weis said he believes authorities should take a closer look at the artwork.
"I don't know what was in the boy's heart, but somebody needs to find that out,'' Weis said.
"For a city sticker, it would spark controversy,'' Weis said. "You don't want that kind of controversy on a city sticker.''
The new city sticker was picked after a round of judging and another round of voting. In the first round, judges invited by the clerk's office crowd into City Council chambers and rank their favorites from among hundreds of designs submitted by high school students.
The top 10 then were posted at suntimes.com, and Mendoza said more than 18,000 votes were cast.
The winning design was announced last week at a news conference. The teen said his mother and his school, Lawrence Hall Youth Services, were helping him improve his academic performance. He said he was honoring firefighters because they rescued him when he was 4 and had lit his clothes on fire with a candle.
hdardick@tribune.com
jebyrne@tribune.com
Tribune reporter Rosemary Sobol contributed
Chicago Mayor
Jody Weis
Susana Mendoza
University of Illinois tuition hike, delayed because of COVID-19, will kick in next year, as trustees consider room and board cost increase
$11 billion in damage from last summer's derecho: 'Historically it produced the most damage in the least amount of time.'

redpajama_set_name: RedPajamaCommonCrawl


We're delighted to welcome the Waggon Inn in Northumberland as the latest addition to our ever-growing Nightstop scheme.
The former coaching inn has a cosy bar and separate restaurant; food is served every day. It's free to stay overnight at the pub if a meal is purchased.
Facilities available at the location include hardstandings and fresh water; toilets facilities are also available when the pub is open. Dogs are welcome to the site.
For further details about the location, visit the pub's website by clicking here. To find out if there's space for you to stay over, call 01661 881 872 between 11am-11pm.
Remember, all of our Nightstops are CLs that are fully licenced by our scheme partners, the Motor Caravanners' Club; so they're safe and legal to use. However, you don't need to be a member of any club to use them: they're open to everyone.
For full details of all our Nightstop locations, email us by clicking here.


A close friend called to tell me a man knocked on her door to sell her on repaving her driveway. In the process, he requested she invite him in to discuss it further and go over different options. The man was persistent and if my friend was anyone else, he may have gotten in. However, she is savvier than that and reminded him that her German Shepherd would not appreciate anyone coming in the house.
Call them con men, grifters, scammers, or thieves. Or simply call them liars. Lying is what they do best. Face to face, via email or over the phone they lie through their teeth. They do it casually and with such conviction that we have no reason not to believe them.
These people will stand in your doorway and, in some cases, keep you talking until you buy something or persist till they get into your home. Remember, whatever you tell them can be used against you.
For example, if they act as a home alarm salesman and find out you don't have an alarm, they may break into your house. If you tell them who your home alarm is with, they may call you at a later date posing as that alarm company and request "updated credit card numbers".
This "request" is best resolved by not answering any questions at all, or telling the person at the front door (while you speak to them through the locked door) you are not interested. No matter what, never give them Social Security or credit card numbers, or tell them whether or not you have a home alarm.
The key is to stop being so nice and SAY NO as quickly as possible and always do it through a locked door.
Robert Siciliano personal security expert to Home Security Source discussing scammers and thieves on The Big Idea with Donnie Deutsch.


If you like pretty movies with confusing plots and very little payoff, run — don't walk — to go see "Warcraft." Everyone else, save your ticket dollars for the summer months.
That's not promising for the adaptation's June 10 wide release. At least the CGI sounds solid.
"Imagine 'Battlefield Earth' without the verve, or the unintentional comedy, and you've got 'Warcraft,'" our resident critic adds.
Empire's critic hands "Warcraft" the same poor rating, saying the story "just about" coheres — that's a compliment, relatively speaking.
"It's unfortunate that, like the ugly green baby that becomes central to its plot, the result is a hard thing for anyone but its creators to love," Olly Richards writes.
GQ's Helen O'Hara believes the film may have actually worked better with the lengthier opportunity afforded by television.
O'Hara does praise director Duncan Jones for staying true to the PC game, but that doesn't mean it makes a good movie.
Warcraft managed three of five stars over at Total Film, but despite hitting that 60 percent threshold, the review is still counted as "Rotten" on Rotten Tomatoes, for some reason.
Perhaps this is why: "Jones embraces Warcraft's world with laudable commitment: but when it comes to charging it with life, sheer bulk gets the better of him," Kevin Harley pens.
"Warcraft" was back at two of five stars for The Independent and critic Jack Shepherd.
Even the former "World of Warcraft" gamer stated that it's "almost impossible" to keep up with all of the characters and subplots. That's not a good sign for the layman, or the box office.
Will 'X-Men: Apocalypse' Be Hurt by Its Mixed Reviews?


Custodian Electrical prepares for 2014.
Custodian Electrical would like to thank all our customers and friends for supporting us in 2013 and look forward to a better 2014.
Normal service resumes on the 2nd January 2014 and we look forward to your calls and emails.
Custodian Electrical has been involved in some fantastic projects in 2013 and thank all other contractors as it has been very peaceful working with some fantastic tradesmen.


Casadei ballet flats cut from shimmering, glitter-coated fabric. Pointed-toe profile. Metallic piping at top line. Calfskin lining and leather sole.
In San Mauro Pascoli, Italy, in 1958, Quinto and Flora Casadei created their first collection of sandals for the summering tourists on the Riviera Romagnola. From there, the storied footwear label began its ascent to the summit of high couture. In the decades that followed its inception, Casadei shoes made their way into boutiques around the world, becoming one of modern fashion's most sought-after footwear labels. Today, led by Quinto and Flora's son Cesare, Casadei continues its success with collections that are never less than stunning.


Shavasana | Inner Mega Power -Personal Development.
Shavasana (Sanskrit: शवासन; IAST: śavāsana), or Corpse Pose is an asana in hatha yoga, often used for relaxation at the end of a session.
The name comes from the Sanskrit words shava (शव, Śava) meaning "corpse" and asana (आसन, Āsana) meaning "posture" or "seat".
Savasana is intended to rejuvenate the body, mind, and spirit. In Savasana, practitioners' breath deepens, and the stress of the day is released. The yogi forgets all other thoughts and surrenders any psychological effort. While in Savasana, yogis slip into blissful neutrality and reflect on the practice.
Comfort is essential in the asana; the slightest point of discomfort can be endlessly distracting. Shavasana is a good way to reduce stress and tension. On the other hand, yoga-nidra ("yogic sleep") meditation is often practiced in a lying position. Drowsiness or restlessness of the mind while in Shavasana may be counteracted by increasing the rate and depth of breathing. While in Shavasana, it is important to be in a neutral position. Although Shavasana ends a practice in relaxation, some practitioners might not be able to relax and let go in this position. If this is the case, bend the knees and move the feet hip-distance apart.
^ "Shavasana". Ashtanga Yoga. Retrieved 2011-04-11.
^ "The History Behind Savasana". Boston Yoga. 6 October 2012. Retrieved 7 March 2017.
^ Singh, G.; Singh, J. (2010). "Yoga Nidra: a deep mental relaxation approach". British Journal of Sports Medicine. 44 (44(S1)): i71–i72. doi:10.1136/bjsm.2010.078725.238.
^ "The Purpose of Corpse Pose". YogaJournal.com. Retrieved 16 January 2018.
^ Physical Education Class – XII. Rachna Sagar. p. 64. ISBN 978-81-8137-350-2. Retrieved 11 April 2011.
Iyengar, B. K. S. (1 October 2005). Illustrated Light On Yoga. HarperCollins. ISBN 978-81-7223-606-9.
Saraswati, Swami Janakananda (1 February 1992). Yoga, Tantra and Meditation in Daily Life. Weiser Books. ISBN 978-0-87728-768-1.
Saraswati, Swami Satyananda (1 August 2003). Asana Pranayama Mudra Bandha. Nesma Books India. ISBN 978-81-86336-14-4.
Saraswati, Swami Satyananda (January 2004). A Systematic Course in the Ancient Tantric Techniques of Yoga and Kriya. Nesma Books India. ISBN 978-81-85787-08-4.


The Hobie H-Rail Mounting Plate gives you the freedom to mount any accessory to the all new Hobie H-Rail.
The Mounting Plate is easy to drill and thru-bolt to, and does not require drilling or tools to attach to the bar itself. Once the clamp is tight, the plate does not slip, slide or twist, keeping your gear safe and sound.
NRS Compact Rescue Bag perfect throw bag for your kayak. Its not the strongest rope but totally fine for pulling your friend outta that nasty hole. Its made from high visability and bomber durable Cordura.
Werner Amigo is perfect for kids and smaller paddlers!


- SGS qualified (certificate available).
- Light weight, high strength, anti-corrosive.
- Weather resistant (UV, rain, typhoon and frost).
- Water proof and moisture proof.
- Billboard, exhibition booth decoration.
- Insulator, cushion material, tray.


When your child is applying for a private school, the to-do list can get pretty long. There are application forms that need to be completed, recommendation letters to collect, and sometimes interviews to attend. There are also admissions exams. The Secondary School Admissions Test, or SSAT, is required by many top-tier private schools. If your child is stressed about taking this test, you need to be proactive. At Varsity Tutors, we can help you. If your child is preparing for the SSAT, let our educational directors connect you with a tutor in Brookline.
Plenty of tutors can be found for general subject areas. However, an SSAT tutor needs to have a specialized awareness of what your child could face on the test. We can find you a tutor in Brookline, MA, who is as adept at SSAT format as they are at teaching SSAT content. Your child will be prepared for the math, reading, and verbal sections of the test. The tutor will also share testing strategies and study techniques that can be helpful in preparing for the exam.
Your child's tutor will be able to focus on any areas of the test that your student will find most challenging. If that's the math section, your student's tutor will practice word problems and review basic and complex math functions, depending on your child's grade and level. If language arts have always been a challenge, your child's tutor will target reading comprehension strategies and vocabulary study. Each tutoring session will be completely customized and designed with your child's learning style in mind. For assistance in connecting with an experienced SSAT tutor in Brookline contact one of our educational directors.
Another great benefit of tutoring is time management. Your child's tutor will train your student in how to work through questions and manage each section. Your time is also valuable to your tutor. All study sessions will be scheduled with your convenience in mind. Your child's tutor will be able to meet in-person or online, depending on the time and place that works best for you. To strive to ensure your child gets into his or her first-choice school, makes sure they can deliver a great score on the SSAT. Varsity Tutors can help you find a professional SSAT tutor in Brookline, MA, who can offer your child engaging study sessions to make the most use of their time. You might be happy you invested in your student, and the results will move your child closer to academic achievement.
"We continued work on math - specifically decimals and fractions - and reading, areas in which the student showed strong improvement. We briefly covered the essay. He practiced with a prompt using techniques we had discussed in the previous session, and the results did not warrant further exercises. His vocabulary was adequate for an applicant for the 7th grade. We then went over "test skills" for when answers are not apparent."
"We worked on her math homework for the week. We also played a game matching vocab words to their synonyms. She was really good at this! She's great at solving synonyms even when she can't put her finger on a definition. We also talked a lot about the application process for schools and her hopes and plans for high school."
"Today the student and I covered many areas - mostly in preparation for his test Saturday morning. First we worked on the essay portion of the SSAT and discussed strategies and timing. Then we went through each section and discussed strategies on test day. He has been working hard and taking a lot of practice tests. He has raised his percentile rank on his practice tests quite a bit in the last several months. He is feeling confident and well prepared for his test. I left him with two note cards with test taking notes on them so he can easily reference between now and test day. "
"The student was exceptionally focused and motivated today! He did great the whole time! We practiced synonyms and reading comprehension. We even timed how long it took him to answer the questions, and he is within the time limits! It was a really wonderful lesson, and I hope that he is proud of himself."
"In this session we continued with verbal skills, and it is apparent that the student has been studying his vocabulary words and continues to improve; I am very impressed! I had him take the verbal section of a practice test on his own and gave him 30 minutes to complete 60 questions, as if he was actually taking the SSAT, and his score improved significantly from last week. We also began working on reading comprehension. With further practice and vocabulary review, the student should do very well on the SSATs and I look forward to continue working with him!"


text:
Rajasthan Horror: Woman Abducted from Assam, Gang-Raped, Forced to Marry Accused in Karveer
In a shocking incident, a woman, native of Assam was sedated, abducted and brought to Rajasthan by two men and was forced to marry one of the accused. Her three-year-old daughter was also abducted with the woman. The victim reportedly managed to escape the trap and reach Kohlapur on Saturday night.
Jaipur, January 13: In a shocking incident, a woman, native of Assam was sedated, abducted and brought to Rajasthan by two men and was forced to marry one of the accused. Her three-year-old daughter was also abducted with the woman. The victim reportedly managed to escape the trap and reach Kohlapur on Saturday night. Rajasthan Horror: 8-Year-Old Girl Raped, Strangled to Death in Pratapgarh District.
As per the reports, the woman, who is six-months pregnant was abducted from Assam and taken to the Panchgaon village in Karveer district where she was forced to marry of one of the accused and stay with him. The victim claimed that the accused threatened to kill her three-year-old daughter if she refuses to give into their demands. Rajasthan Horror: Man Rapes Minor Sister-in-Law, Elopes With Her After She Becomes Pregnant.
She told police that the accused forced her to have physical relationships with other men as well. The victim claimed that a woman was also involved in abducting and trapping her, as reported by Times of India. Police has launched an investigation into the matter.
Magh Bihu 2021 Greetings and HD Images: WhatsApp Stickers, Telegram Messages, GIFs, Facebook Photos and SMS to Send Wishes of Bhogali Bihu
"The role of the woman suspect is also being investigated along with how the woman was brought to Kolhapur from Rajasthan. We are also investigating why the suspects came to Kolhapur," Inspector Sandip Kolekar told Times of India.
The police has filed a complaint and arrested accused named by the woman. They have been presented in a local court and sent into 5-day police custody for further interrogation.


As women move into the work force in greater numbers, a lot of them are finding that their professional and white-collar employment brings job satisfaction, but that it's tempered by other realities. One is that the new responsibilities probably haven't replaced those of the woman's once-traditional role as homemaker and mother. Those roles remain, and many women find themselves not only holding down a full-time salaried position, but also still doing the shopping, the child rearing, and the laundry. Women probably were on their feet a lot more than men before work, now that increasing numbers of them are working, their feet are taking even more punishment. The footwear that some feel obliged to wear in the work place may be playing a role. All this adds up to a need for women to pay more attention to foot care, to avoid subjecting themselves to unnecessary foot problems, particularly those that might be caused by improper footwear and hosiery.
The wearing of high-heeled shoes is a prime example of women inviting foot problems. Podiatrists see no value in high heels (generally defined as shoes with heels of more than two inches). We believe them to be biomechanically and orthopedically unsound, citing medical, postural, and safety faults of such heels. We know, for example, that high heels may contribute to knee and back problems, disabling injuries in falls, shortened calf muscles, and an awkward, unnatural walking pattern. In time, high heels may cause enough changes in the feet to impair their proper function. Most women admit high heels make their feet hurt, but they tolerate the discomfort in order to look taller, stylish, and more professional. In a recent Poll, 37 percent of the women surveyed said they would continue to wear high heels, even though they did not think them comfortable.
If women persist in wearing high heels, there are ways to relieve some of the abusive effects; they can limit the time they wear them, for example, alternating with good-quality shoes or flat shoes for part of the day. High heels that are too tight compound the abuse; it's good advice to buy shoes in the afternoon, since feet tend to increase in size later in the day.Women have other heel-size choices, fortunately. They don't have to endure pain at the expense of their foot health. The key is wearing the right shoe for the right activity and that means varying heel height, determining what heel is most suitable. For example, there are comfortable and attractive "walking" shoes (also called "comfort" or "performance" shoes) for women for work and social activities. The shoe manufacturers who have introduced them seek a marriage of fashion considerations and comfort, offering fashionable shoes with athletic shoe-derived construction, with reinforced heels and wider toe room. They are using space-age materials, like long-lasting memory cushioning that acts as a shock absorber. And the soles are more pliable. Several companies have also designed footwear for certain athletic activities, including aerobics, specifically for women.
Women who always wear nylon tights are also inviting foot problems. Nylon is occlusive, it doesn't breathe and the heat that it generates and traps causes excessive perspiration. A warm, damp area is an ideal place for the encouragement of fungal infections such as athlete's foot. (Tights are not the only factor that causes excessive perspiration; the dress boots and shoe boots many women wear in cold weather shouldn't be worn all day in an office.) Nylon, which tends to be abrasive, is even more so when it's damp; in a tight shoe that's already irritating; it offers little protection against blisters. Support hose, because they're so much tighter, can be the worst culprits of all.Inexpensive nylon tights can also cause forefoot problems, because the stretch mechanism constricts normal expansion of the foot when walking, and may pull the toes backward when the pantyhose ride up. The cramping and pressure of the hose can contribute to ingrown toenails and hammertoes. A better quality nylon will provide a better fit, and the better the fit, the less likely it is that cramping will occur. Women's feet have grown larger because of improved health care and nutrition. The one-size-fits-all stocking no longer is the universal answer, if it ever was. Attention to proper fit is essential.
Pregnant women need to observe good foot health to prevent pain and discomfort. Since the body undergoes changes and acquires a new weight-bearing stance, women should wear shoes with broad-based heels that provide support and absorb shock. Additional body weight also calls for more support, to prevent foot "breakdown". The expectant mother often experiences more than ordinary swelling of her feet and ankles, which can aggravate existing foot conditions and promote inflammation or irritation. Pregnancy also triggers the release of hormones, which enhance laxity in ligaments, which can contribute to foot strain.
Older women have more trouble with their feet than younger ones, for the simple reason that fat pads on the bottom of the feet tend to deteriorate in the aging process. They can alleviate some foot problems by wearing properly fitted, well-constructed shoes. Shoes which provide cushioning and have a soft, flexible upper that will conform to the shape of their feet. They also need leather shoes, which "breathe" and can reduce the possibility of skin irritation. Soles should be lightweight, with enough flexibility and shock-absorbing quality to provide solid footing and not be slippery. Low-heeled shoes provide greater stability, more protection for the feet, and greater comfort. Because older women often have circulatory problems, they have a special need to keep their feet warm in cold weather, to prevent frostbite.
Whether the sources are congenital problems, foot abuse, high heels, poorly fitting shoes, or other maltreatment of the feet, women are subject to a number of afflictions involving the feet (most of which can also occur to men).
inflammation of the Achilles tendon, the link between the calf muscle and heel bone. Those who wear high heels regularly can expect to acquire shortened tendons; switching to low heels for strenuous physical activity without appropriate warm-up exercises creates an ideal scenario for Achilles tendentious.
Misaligned big toe joints, which become swollen and tender. Bunions tend to be hereditary, however, shoes that are too narrow in the forefoot can aggravate the tendency to bunions.
A condition in which the toe is contracted in a claw-like position. Although the condition usually stems from muscle imbalance, it is often aggravated by ill-fitting shoes, socks, or hosiery that cramp the toes.
General pain in the ball of the foot; often caused by wearing high heels. Neuromas: enlarged, benign growths of nerves, most commonly between the third and fourth toes. They may stem, in part, from ill-fitting shoes, resulting in pain, burning, tingling or numbness between the toes and in the ball of the foot. Treatment includes orthotic devices and/or steroid injections, and sometimes surgery.
inflammation of the long band of tissue running from the heel to the ball of the foot, a main cause of rear-foot pain. This condition is sometimes caused by shoes that cramp the feet, especially in the arch area.
A bone enlargement at the back of the heel bone, in the area where the Achilles tendon attaches to the bone. The deformity generally is the result of faulty foot function causing increased motion of the heel bone against the shoe counter.


I support the principles of compassionate conservation, promoting consideration of animal welfare in a conservation context. As such, the well-being of animals I photograph takes precedence over the photo. We all have an impact on wildlife, however unintentional at times. But as a general rule and to the best of my ability, I work to reduce how intrusive my presence might be.
• I use the NANPA Ethics Guidelines and the Audubon Guide to Ethical Photography as models. I'm in permanent, continuing education on animal behavior and natural history, to better understand the wildlife I photograph.
• In macro photography of insects, all photos are taken as I find them, in their natural settings, no staging.
I am especially cautious about photographing young animals and nesting areas, and don't photograph nests often out of concern for interrupting feeding behavior or alerting predators area. There are situations, however — like certain Cliff Swallow colonies, heron rookeries, or Osprey nests — where with adequate distance, the birds are generally not disturbed by human presence. Again, I use my best judgment and try to get the pulse of other photographers and birders who also know the area and the local animals.
With wildlife, I shoot in natural light, no flash or artificial illumination. It's less intrusive to wild animals, and I enjoy the challenge of working with the nuances of available light. Because I appreciate photographs as emotive and story-telling devices, I see difficult light — darkness, fog, and high-contrast situations — as a call to adapt and find something new in the frame. If I use artificial light in any situation, I'll note it in the description.
The term anthropomomorphism is often used to deny animals their rightful entitlement to complex emotional and social behaviors. I don't believe in stripping an animal of his or her individuality because she belongs to a different species, and will grant nonhumans the benefit of the doubt.
I encounter frustrating human-wildlife interactions often enough to feel protective of the wildlife and birds I photograph. I feel an obligation to contribute to their well-being, and not detract from it. It has nothing to do with hoarding a choice photography spot. In fact, most places where I've photographed wildlife are quite open to the public and well-known by birders and photographers throughout the year.
Feel free to comment, disagree, to challenge. I'm a believer in the First Amendment. I come at my subjects from the perspective of an observer and an artist, not a scientist, so there's subjectivity inherent in my views.