text:
\section{Introduction}

The standard model contains two physical CP violating parameters: (1)
the perturbative CKM phase,
which originates in the misalignment of the eigenvectors of the Yukawa
matrices $y_u$ and
$y_d$~\cite{PhysRevLett.55.1039},
\begin{align}
  \delta_{CKM} = \arg \det \left[y_u y_u^\dagger , y_d y_d^\dagger \right],
\end{align}
and (2) the strong CP phase
\begin{align}
  \bar{\theta} = - \arg \det \left[e^{-i\theta}y_u y_d\right],
\end{align}
which originates from the combination
of the QCD $\theta$ angle
and the determinant of the Yukawas.
Although these two phases appear to be intimately related through
their connection to the Yukawa matrices, $\delta_{CKM}$ is observed to
be $\mathcal{O}(1)$, while current limits give $\bar{\theta}\lesssim
10^{-10}$ \cite{Pospelov:1999ha,Baker:2006ts,Afach:2015sja}. This is
the strong CP problem: how can such a small value of $\bar\theta$ be
explained when the quark sector appears to feel $\mathcal{O}(1)$ CP
violation?
In view of the strong connection of the flavor sector with the strong
CP problem, it is natural to explore its solutions in the context of
models which also generate the flavor structure in the standard
model~\cite{Harnik:2004su,Cheung:2007bu,Calibbi:2016hwq}.
We present such a mechanism in this work.

One appealing class of solutions to this problem are those that
contain a new anomalous $U(1)_{PQ}$ symmetry.
The most economical possibility is the ``massless up quark solution'',
where setting  $m_u=0$ at a scale above the QCD scale leads to a
$U(1)_{PQ}$ symmetry.
This is not a priori inconsistent with current algebra
since non-perturbative effects can
generate an effective up-quark
mass~\cite{Georgi:1981be,Choi:1988sy,Kaplan:1986ru,Banks:1994yg}
(see~\cite{Dine:2014dga} for a review).
In the simplest
extensions of the standard model, non-perturbative QCD effects are only relevant
at the scale $\Lambda_{QCD}\sim\ \rm{GeV}$, and the mechanism can therefore
remove any contributions to $\bar\theta$ generated above the scale
$\Lambda_{QCD}$.
Unfortunately, the massless up-quark solution is now
strongly disfavored by lattice results, which
find a non-zero $\overline {\rm MS}$ value~\cite{Aoki:2013ldr,Agashe:2014kda},
\begin{align}
m_u
&= \left. 2.3^{+0.7}_{-0.5} \right|_{\mu=2\ \rm{GeV}}
\,.
\end{align}
The significance with which this rules out $m_u=0$ solutions is more
difficult to quantify. Refs.~\cite{Dine:2014dga,Frison:2016rnq} have
recently pointed out some ambiguities and suggested further direct
lattice tests that can support this conclusion.

In this work we consider an extension of the massless up quark solution into models where large non-perturbative effects are generated by embedding QCD as the diagonal subgroup of a $SU(3)^N$ gauge group.  This mechanism for ``factoring" the Strong CP problem was first presented in Ref.~\cite{Agrawal:2017ksf}, where all of the quarks are charged under a single SU(3) factor, and the PQ symmetry is realized by a heavy axion in each sector. In this work, we instead give a flavorful embedding of the quarks in a $SU(3)\times SU(3) \times SU(3)$ gauge group, with each quark generation charged under a separate factor. Each factor contains an independent PQ symmetry implemented by a perturbatively massless quark instead of a heavy axion, and the observed non-vanishing Yukawa couplings are generated entirely by non-perturbative effects at a high scale $M$. These non-perturbative effects can be sizable because although the SM QCD coupling is weak at high scales $M\gg \Lambda_{QCD}$, each individual $SU(3)$ factor can easily be near strong coupling\footnote{We will generalize this model to include
  additional $SU(3)$ factors with no charged matter, making each
  factor more strongly
  coupled, so that the non-perturbative effects can
be made larger still.}.
Higher dimension operators generate
the quark mixing matrix upon the breaking to the diagonal group.
Below the scale $M$ the theory matches
to the standard model with no additional matter. Since in the standard
model $\bar\theta$ is very well sequestered from
$\delta_{CKM}$~\cite{Ellis:1978hq,
Shabalin:1978rs,Khriplovich:1985jr,Dugan:1984qf}, solving the strong
CP problem at the scale $M$ solves it at low energy as long as no new
sources of flavor or CP violation are introduced \cite{Dine:2015jga}.
While $\bar\theta$ is suppressed in this model at tree-level, a
non-vanishing radiative contribution is generated with a size directly
connected to the observed quark masses and CKM angles. Remarkably, the
model predicts $\bar\theta \sim 10^{-10}$, just below the sensitivity
of current EDM experiments and within reach of proposed next
generation neutron EDM \cite{Ito:2007xd,Tsentalovich:2014mfa} and
proton storage ring experiments \cite{Anastassopoulos:2015ura}.

Other models that can explain
$\bar\theta=0$ at tree level in the UV typically require large
discrete symmetries and extensions of the flavor structure, and do not
preserve the radiative sequestering of $\bar\theta$ present in the SM.
For example, in Nelson-Barr models
\cite{Barr:1979as,Nelson:1983zb,Barr:1984qx,BENTO199195}, the
radiative contributions $\Delta\bar\theta$ generally exclude the most
appealing models unless some allowed couplings have unexplained
suppressions or the symmetry structure of the SM is substantially
extended \cite{Dine:2015jga,Vecchi:2014hpa}.
There are also other mechanisms that introduce new non-perturbative PQ violating
effects at higher energies $M \gg \Lambda_{QCD}$ to solve the strong
CP problem.
Refs.~\cite{Rubakov:1997vp,Berezhiani:2000gh,
Hook:2014cda,Fukuda:2015ana,Dimopoulos:2016lvn}
consider models where the $\bar\theta$ of the SM is related by a $Z_2$
symmetry to a mirror copy of the standard model with
$\bar\theta'=\bar\theta$. Spontaneous $Z_2$ breaking
\cite{Blinov:2016kte} allows the states of the mirror sector to be
decoupled, and non-perturbative mirror $SU(3)'$ effects to become
strong at a scale $\Lambda_{QCD}' \gg \Lambda_{QCD}$ and
simultaneously relax $\bar\theta'$ and $\theta$ either with a
heavy-axion
\cite{Rubakov:1997vp,Berezhiani:2000gh,Hook:2014cda,Fukuda:2015ana,Dimopoulos:2016lvn}
or a heavy perturbatively massless quark \cite{Hook:2014cda}. These
theories are significantly constrained by the cosmology of the mirror
sector and new colored TeV-scale particles.
Another possibility is that the SM QCD itself becomes embedded in a
strongly coupled gauge group at high
energies--Refs.~\cite{Holdom:1982ex, Holdom:1985vx, Dine:1986bg,
Flynn:1987rs, Choi:1998ep} considered the possibility that extra
matter causes QCD to run back to strong coupling at a scale $M$ where
it is embedded in a larger gauge group, e.g. $SU(3+N)$. In general to
obtain sizable effects these models also require the addition of new
dynamics breaking the chiral symmetries, and contain new CP violating
phases which cause a misalignment between the non-perturbative
violations of the PQ symmetry at  $\Lambda_{QCD'}$ and
$\Lambda_{QCD}$, spoiling the solution to the strong CP problem
\cite{Dine:1986bg}.

\section{Massless Quark Solution in QCD: the baby version}
\label{sec:two-flavor}

We start from a simpler version of the standard model with only a
single generation of quarks -- the $SU(2)$ doublet $q=(u,d)$
and two singlets $u^c, d^c$ -- charged as in the
standard model.
We include an $SU(2)$ doublet Higgs $H$  and assume a UV cut-off $\Lambda_{UV}$.
We make use of an
anomalous $U(1)_{PQ}$ symmetry under which only $u^c$ transforms,
\begin{align}
u^c \rightarrow e^{i\alpha} u^c
\end{align}
which forbids an up Yukawa coupling at the
perturbative
level (more precisely, we assume that the dominant source of PQ breaking is from non-perturbative effects within the effective theory far below the scale $\Lambda_{UV}$). The relevant terms in the Lagrangian are
\begin{align}
\mathcal{L}_{\Lambda_{UV}}
&\supset
\frac{\alpha_s }{8\pi} \theta\,\tilde{G}_{\mu\nu}^aG^{a,\mu\nu}
+ y_d q H^\dagger  d^c
\end{align}
The $U(1)_{PQ}$ symmetry and a chiral rotation of $d^c$ can be used
to remove the topological phase $\theta$ and the phase of the
non-vanishing Yukawa coupling $y_d$. Therefore there is no physical CP
violating parameter, $\bar\theta=0$. This is effectively
the massless
up quark solution to the strong CP problem.

Non-perturbative $SU(3)$ effects violate the anomalous $U(1)_{PQ}$, so
non-perturbative effects suppressed as $\sim e^{-2\pi/\alpha_s}$  will
generate a non-vanishing effective $y_u$ coupling at energies below
$\Lambda_{UV}$. In the weak coupling limit, the dilute instanton gas
approximation captures the leading non-perturbative effects, and the
instantons can be integrated out to generate an effective Lagrangian
for the fermions \cite{tHooft:1976snw,Andrei:1978xg,Choi:1988sy}. For
$SU(3)$ with two flavors of quarks, both four-fermion and bilinear
terms are generated from single-instanton effects,
\begin{align}
  \mathcal{L}_{inst}
  &=
  \int_{\rho=\Lambda_{UV}^{-1}}^{\rho=\Lambda_{IR}^{-1}}
  \frac{d\rho}{\rho} e^{i\theta}
  D\left[\alpha_s(1/\rho)\right]
  \left.\left(
  -c_0 y_d^*  q H u^c
  + c_0 (2\pi^2)  \rho^2
  (u^\alpha d^c_\alpha d^\beta u^c_\beta
  - d^\alpha d^c_\alpha u^\alpha u^c_\alpha)
  \right)
  \right|_{\mu=\rho^{-1}}
  \label{eq:Linst2}
\end{align}
where $\alpha,\beta$ are QCD indices and the dimensionless instanton
density is
\begin{align}
D[\alpha]
  &=
  D_0 \left(\frac{2\pi}{\alpha}\right)^6
  e^{-\frac{2\pi}{\alpha}}
\end{align}
which features the
non-perturbative exponential suppression factor at weak coupling. The
analytic constants are $D_0 \approx
0.02$ and $c_0\approx 1.79$~\cite{Choi:1988sy}. The couplings in the integrand are evaluated at
the scale $\rho^{-1}$  (higher order corrections can be found in
Ref.~\cite{Dine:2014dga}). Higher dimension operators are suppressed
by further powers of $D[\alpha]$ , and  $D[\alpha]\sim 1$ signals the
breakdown of the dilute instanton gas approximation.

The effect of instantons on the Yukawa couplings can be conveniently
described as a non-perturbative contribution to the running of the
Yukawa couplings \cite{Choi:1988sy},
\begin{align}
  \frac{d}{d \ln \mu}
  \begin{pmatrix}
    y_u & 0 \\
    0 & y_d
  \end{pmatrix}
  &=
  -c_0 D[\alpha(\mu)] e^{i\theta}
  \begin{pmatrix}
    y_d^* & 0 \\
    0 & y_u^*
  \end{pmatrix}
  \label{eq:yrun}
\end{align}
Recall that the perturbative contributions to the
running of Yukawas are multiplicative, and are negligible here.
Now that non-perturbative effects are included,  $y_u\neq 0$ is
generated and the PQ-symmetry appears to be violated perturbatively in
the low energy effective Lagrangian. However,  the physical CP angle
$\bar\theta$ remains vanishing: the non-perturbatively generated $y_u$
has just the right phase to allow the $\theta$ angle and the phase in
$y_d$ to be simultaneously rotated away, as is clear from
\cref{eq:Linst2}.

Two-flavor QCD is asymptotically free and the instanton density grows
in the IR. If $SU(3)$ is Higgsed at the scale $M$, the
instanton contribution to $y_u$ is cut-off and dominated by instantons
of size comparable to the Higgsing, $\rho^{-1}\sim M$ (we will discuss
the nature of the Higgsing sector in the following section). Using the
one-loop running of the gauge coupling $d\alpha^{-1}=\frac{b}{4\pi} d
\ln{\mu}$, with $b=29/3$ for 2-flavor QCD, the  linear solution to the
running \cref{eq:yrun} gives
\begin{align}
  \frac{|y_u|}{|y_d|}
  &=-\frac{2 c_0 D_0}{b}\int_{2\pi /\alpha(\Lambda_{UV})}^{2\pi/ \alpha(M)}
  \left(\frac{2\pi}{\alpha}\right)^6
  e^{-\frac{2\pi}{\alpha}}
  d(2\pi/ \alpha) \approx \frac{2 c_0 D_0}{b}\Gamma(7, 2\pi \alpha(M))
  \label{eq:singleinstanton}
\end{align}
where we have assumed $\alpha^{-1}(\Lambda_{UV}) \ll 1$ for the last
equality, and $\Gamma(n,x)$ is the upper incomplete $\Gamma$-function.
\Cref{fig:yuydRatio} shows the ratio $|y_u|/|y_d|$ after
integrating out effects above $M$ as a function of the QCD coupling at
the scale of Higgsing, $\alpha(M)$.  As $\frac{|y_u|}{|y_d|}$ approaches $\sim 1$,
multiple-instanton effects captured by higher order solutions to
\cref{eq:yrun} become important, and the ratio asymptotes to $|y_u|/|y_d|=1$.  For $\alpha(M)
\sim 0.4-0.8$ an $\mathcal{O}(1)$ ratio can be generated as required by the observed light quark masses. In this regime the dilute instanton gas approximation is only a
qualitative picture of the non-perturbative QCD effects, but strongly
suggests that they are $\mathcal{O}(1)$ and that a viable ratio $|y_u|/|y_d|$ can be realized before the theory enters the chiral-symmetry breaking phase which would be expected to occur at $\alpha(M) \gtrsim 0.7-1$ \cite{Roberts:1994dr,Appelquist:1997gq}.  As the theory flows to weak coupling at scales above M, the PQ violating effects are rapidly suppressed. For example for
$\alpha \approx 0.1$, as in the SM near the weak scale, the
non-perturbative  contribution to $y_u$ is  $|y_u|/|y_d|\lesssim
10^{-16}$.

\begin{figure}
\centering
\includegraphics[width=0.65\textwidth]{yuyd.pdf}
\caption{The ratio of Yukawa couplings generated by instanton effects
  as a function of $\alpha=g^2/4\pi$ evaluated at the scale
  $M$ in the 2-flavor model assuming a vanishing
  perturbative value for one of the couplings. The solid curve cuts
  uses the full solution to \cref{eq:yrun}, cutting off the running at
  $\mu=M$. The dashed curves cut off the instanton
  effects at $\mu = M/2, 2M$, and are shown
  as a rough estimate of the theoretical uncertainty. The dotted line
  is the single-instanton approximation \cref{eq:singleinstanton}.
The horizontal dashed lines are, from top to bottom, the experimental
values of $y_u/y_d$, $y_s/y_c$, and $y_b/y_t$.
}
\label{fig:yuydRatio}
\end{figure}
This simple 2-flavor example shows that instanton effects can generate
large non-perturbative contributions to a perturbatively vanishing
Yukawa coupling. In fact such effects are known to be important near the
scale of QCD confinement, $\Lambda_{QCD}$,
in the standard model, as reviewed in~\cite{Dine:2014dga}.
However,  as mentioned
above, lattice results strongly disfavor a massless up quark solution
to the strong CP problem in the SM.

The suppression of this effect in the SM is partly due to the fact
that the strange quark is also relevant at $\Lambda_{QCD}$, and
instanton contributions to $m_u$ are further suppressed by $m_s$. In
fact, 2+1 flavor lattice QCD results fully include all instanton
configurations and can be interpreted as a  calculation of the 2nd
order term in the Chiral Lagrangian giving an effective up-quark mass
proportional to $m_d^* m_s^* / \Lambda_{QCD}$ -- these results suggest
that the size of the desired non-perturbative effect is only $\sim
10$--$40\%$ of the experimentally required value \cite{Dine:2014dga}.

So, although qualitatively non-perturbative effects in the SM near the
scale $\Lambda_{QCD}$ are nearly the right size to allow $m_u=0$
solution to the strong CP problem, quantitatively the possibility is
strongly disfavored by precision lattice results.  In the following
section we will describe an extension to the SM in which
non-perturbative effects can become important again at a high energy
scale $M\gg\Lambda_{QCD}$, and these additional contributions allow a
solution to the strong CP problem reminiscent of the massless up quark
solution.

\section{Massless Quark Solution in QCD: the real thing}
\label{sec:ThreeGen}
Going beyond the illustrative two-flavor example, there are two
challenges to generating a large non-perturbative contribution to the
Yukawa couplings at a new scale $M \gg \Lambda_{QCD}$. The first is
that QCD must be embedded in a strongly coupled theory at the scale
$M$ so that non-perturbative effects are important, but must match to
the weak coupling of QCD in the standard model at high energies, e.g. $\alpha_s(1000 \ \rm{TeV}) \approx 0.05$. The second challenge is that at high energies in QCD, all
three generations of quarks are relevant, leading to further Yukawa
suppressions of high energy contributions from instantons at small
sizes $\rho^{-1} \gg v$. For example, as illustrated in
\cref{fig:SixFlavorInstanton}, the high energy contributions to $y_u$
in the 3-generation SM are further suppressed as
\begin{align}
|y_u|/|y_d| \sim
\frac{|y_c y_s y_t y_b|}{(16\pi^2)^2}
\end{align}
because the explicit breaking
of each non-anomalous $U(1)_{PQ}$ by the Yukawa couplings must be felt
to generate $y_u \neq 0$.

\begin{figure}
\centering
\includegraphics[width=2in]{SixFlavorInstanton.pdf}
\caption{An instanton contribution to the up quark mass in the SM at high energies. All six quark flavors appear, and the non-vanishing contributions are proportional to the products of all the Yukawa couplings. For $M \gg v$, diagrams with the Higgs looped off are more important than Higgs vev insertions. }
\label{fig:SixFlavorInstanton}
\end{figure}

Both these challenges can be solved by embedding the standard model
$SU(3)_c$ into a $SU(3)_1 \times SU(3)_2 \times SU(3)_3$ product gauge
group above the scale $M$, as depicted in \cref{fig:threesite}. Each
generation of quarks is charged under a separate $SU(3)$ factor. The
theory will be Higgsed at the scale $M$ to the diagonal gauge group by
bifundamental scalar fields, as discussed in more detail in the following section. The unbroken diagonal $SU(3)_c$ group's coupling is
\begin{align}
  \frac{1}{\alpha_s}
&=
\frac{1}{\alpha_{s_1}}
+\frac{1}{\alpha_{s_2}}
+\frac{1}{\alpha_{s_3}}
\label{eq:weakcoupling}
\,,
\end{align}
allowing  to match to the weakly coupled SM QCD even when each
individual factor is more strongly coupled.

\begin{figure}
\centering
\includegraphics[width=3in]{ThreeSite.pdf}
\caption{The model is a 3-site $SU(3)_1 \times SU(3)_2 \times SU(3)_3$ theory, with one generation charged under each $SU(3)$ factor. The link field $\Sigma$ vevs break the gauge group down to the diagonal  SU(3) of the standard model. One quark in each generation obtains its mass from non-perturbative effects, making the $\bar\theta$ angle in each individual gauge factor unphysical. }
\label{fig:threesite}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=4in]{ProductGroupInstantons.pdf}
\caption{Schematically, high energy instantons act separately at scales above $M$ to generate one of the Yukawa couplings in each individual $SU(3)$ factor. }
\label{fig:manyFlavorInstanton}
\end{figure}

Since there are now three separate $SU(3)$ factors, there are now
three separate $\theta$ problems! Fortunately, all the $\theta$ angles
can be made unphysical if there is an independent anomalous
$U(1)_{PQ}$ symmetry in each sector.  The minimal realization of this
PQ symmetry involves a perturbatively massless quark in \emph{each}
sector. Since the non-perturbatively generated Yukawa couping is
always smaller than the unprotected Yukawa, a natural choice is to
choose PQ symmetries that enforce $y_u=0, y_s=0, y_b=0.$ Above the
scale $M$ of Higgsing, each site behaves as the two-flavor model of
~\cref{sec:two-flavor}. Schematically, the generation of the Yukawa
couplings is depicted in~\cref{fig:manyFlavorInstanton}.
From~\cref{fig:yuydRatio}, we can read off the size of the gauge
couplings at the scale $M$ that are necessary for the instantons in
each factor to generate the observed Yukawa ratios:
\begin{eqnarray}
y_u / y_d \approx 2/5
& \rightarrow & \alpha_{s1}(M) \approx 0.45-0.85
\nonumber \\
y_s / y_c \approx 1/12
& \rightarrow & \alpha_{s2}(M) \approx 0.36-0.6
\nonumber \\
y_b / y_t \approx 1/35
& \rightarrow & \alpha_{s3}(M) \approx 0.33-0.55
\end{eqnarray}
\Cref{eq:weakcoupling} then gives the coupling of the unbroken
diagonal group at the matching scale  $\alpha_s(M) = 0.12-0.22$.
Flavor constraints will require us to match to the SM at a scale
$M\gtrsim 1000\ \rm{TeV}$ where $\alpha_s(1000\ \rm{TeV}) = 0.05$, so it appears
unlikely that this minimum ${SU(3)_1\times SU(3)_2 \times SU(3)_3}$
model is viable unless our dilute instanton calculation significantly
underestimates the size of non-perturbative effects.

One way to overcome this obstacle is to enlarge the product gauge
group to $SU(3)_1 \times SU(3)_2 \times SU(3)_3 \times SU(3)_X^N$,
where the extra gauge factors do not contain chiral matter and
therefore can remain more weakly coupled.  Removing the $\theta$ angle
in these extra factors will involve introducing a PQ symmetry at each
new site, as shown for example in~\cref{fig:foursite}. For
example, the $\theta$ parameter in the extra sites can be removed with
very heavy axion degrees of freedom as discussed in
Ref.~\cite{Agrawal:2017ksf}. Another simple viable possibility is to
add $N=3$ or $4$ sites, each with a colored vectorlike particle
$\Psi_X, \Psi_X^c$ and $M_\Psi =0$ perturbatively to realize a PQ
symmetry. Instantons in each factor generate a mass $M_\Psi \sim
D(\alpha(M)) M$. For $N=4$ and a scale $M=1000\ \rm{TeV}$, each extra site
needs a coupling $\alpha_{sX}(M) \approx 0.26-0.35$, giving masses
$M_\Psi \sim 1 \ \rm{TeV} - 100 \ \rm{TeV}$, while for $N>4$ we find $M_\Psi \sim
M$.  The possible presence of these light vector-like colored fermions
with masses $M_\Psi \ll M$ generated by non-perturbative effects could
be an interesting signature of this theory to study in further work,
but for the remainder of this work we assume these states decouple and
focus on the details of the $SU(3)_1 \times SU(3)_2 \times SU(3)_3$
theory.

Another alternative possibility to avoid enlarging the gauge group
with extra $SU(3)$ factors is to consider a model with PQ symmetries
ensuring $y_d=0$ instead of $y_s=0$, so that smaller non-perturbative
effects are required to generate the quark mass ratios. This
possibility is appealing but is in tension with constraints on
$\bar\theta$, as described in~\cref{app:altyukawa}.

\begin{figure}
\centering \includegraphics[width=4in]{FourSite.pdf}
\caption{The
  3-site $SU(3)_1 \times SU(3)_2 \times SU(3)_3$ theory of
  \cref{fig:threesite} extended to contain an extra site with a
  more weakly coupled $SU(3)_4$ factor. There is no chiral matter at
  this site, and the $ \theta_4$ angle is removed by an anomalous
  $U(1)_{PQ}$ symmetry of a single vector-like quark species
$\Psi,\Psi^c$.  While $M_\Psi=0$ perturbatively, non-perturbative
effects violating the PQ symmetry generate $M_\Psi \neq 0$. }
\label{fig:foursite}
\end{figure}

\subsection{The scalar sector}

The full description of the relevant particle content of the $SU(3)_1 \times SU(3)_2 \times SU(3)_3$ model is
given in \cref{tab:particles}.  There are several possibilities for the scalar
fields breaking the gauge group to the diagonal, here we take a simple
choice motivated by CKM mixings as described in the following
section.

\begin{table}
\centering
\begin{tabular}{l|rrr}
 & $SU(3)_i$ & $SU(3)_j$ & $(B-L)_i $ \\
 \hline
 $q_i$ 	 & $\square $ & 1 &$ \frac{1}{3}$ \\
 $u^c_i$ & $\bar\square $ &1& $ -\frac{1}{3}$\\
 $d^c_i$ & $\bar\square $ & 1&$-\frac{1}{3}$ \\
$\Sigma_{ij}$ 	& $\bar\square$	& $\square$& $\mp\frac13$ \\

\end{tabular}
\caption{Particle content and charges under the family $SU(3)$
factors and family $B-L$ symmetries  for a
simple 3-generation model ($i,j=1,2,3$).}
\label{tab:particles}
\end{table}

We assume that the scalar link fields $\Sigma_{12}$, $\Sigma_{23}$,
and $\Sigma_{31}$ get vevs $f_{12} \sim f_{23}\sim f_{31}$ to break
the gauge group down to the diagonal, with $M$ corresponding to the
scale of Higgsing $M\sim g f$ (only two link fields are necessary to
break the gauge group, but the simplest renormalizable flavor models
will involve three link fields). The renormalizable potential allowed
by the symmetries  leads to spontaneous breaking of the gauge group
without introducing any new CP phases or uneaten light Goldstone boson
degrees of freedom. A standard renormalizable Higgs-like potential
drives a vev for each field,
\begin{align}
V
&=
V_0(\Sigma_{12}) + V_0(\Sigma_{23}) + V_0(\Sigma_{31}) +
  (\gamma {\rm Tr}(\Sigma_{12}\Sigma_{23}\Sigma_{31}) + h.c.), \\
 V_0(\Sigma_{ij})
  &=

 -m_{\Sigma_{ij}}^2 {\rm Tr}({\Sigma_{ij}\Sigma_{ij}^\dagger})
+\frac{\lambda_{ij}}{2}[{\rm Tr}({\Sigma_{ij}\Sigma_{ij}^\dagger})]^2
+\frac{\kappa_{ij}}{2}{\rm Tr}(\Sigma_{ij}\Sigma_{ij}^\dagger\Sigma_{ij}\Sigma_{ij}^\dagger) {\rm~(no~ sums~on~i,j)}
\label{eq:ScalarPotential}
\end{align}
The couplings $\lambda_{ij}$, $\delta_{ij}$, and $\lambda_{ij}$ are independent real parameters for each field $\Sigma$. The phase of $\gamma$ can be removed by a field redefinition, and causes the vacuum to align with vevs $\langle \Sigma_{12} \rangle$ , $\langle
\Sigma_{23} \rangle$ , $\langle \Sigma_{31} \rangle$ that can all consistently be chosen to be real.
Taking $\gamma$ to be a small perturbation for simplicity, we find \cite{Bai:2010dj,Bai:2017zhj}
\begin{equation}
\langle \Sigma_{ij} \rangle = \frac{m_{\Sigma_{ij}}}{\sqrt{\kappa_{ij}+3\lambda_{ij}}} \mathbb{I}_3 \equiv \frac{f_{\Sigma_{ij}}}{2} \mathbb{I}_3.
\end{equation}
For simplicity assume all of the scales are comparable, $f_{12} \sim
f_{23} \sim f_{31} \sim f$, giving a common scale $M\sim g f$ cutting
off the instanton integrals in each $SU(3)$ factor.

 Renormalizable
cross-couplings of the form $\lambda' |\Sigma_{12}|^2|\Sigma_{23}|^2$,
$\lambda'' |\Sigma_{12}\Sigma_{23}|^2$, etc. are also allowed by the
symmetries, but do not  introduce any new CP phases and are
qualitatively unimportant as long they do not destabilize the vevs.
The symmetry breaking pattern at the level of the scalar potential so
far is $SU(3)^3 \times U(1)_{B-L}^3 \times U(1)_\Sigma \rightarrow
SU(3)\times U(1)_{B-L}$. The 16 colored Goldstone bosons are eaten by
the Higgs mechanism for the broken $SU(3)$ factors. The 8 remaining
colored pseudo-Goldstone bosons radiatively obtain masses at the scale
$\sim g^2 f^2$.   There remain three singlet Goldstones to lift. The
renormalizable term, $\gamma{\rm Tr}(\Sigma_{12}\Sigma_{23}\Sigma_{31}) + h.c.$
explicitly breaks the $U(1)_\Sigma$ symmetry, lifting one Goldstone.
Gauging two of the $U(1)_{B-L}$ factors can lift the remaining
Goldstones without introducing any new phases. Alternatively,
explicitly breaking the $U(1)_{B-L}$ factors with terms of the form
$\kappa' {\Sigma_{12}}^a_i{\Sigma_{12}}^b_j{\Sigma_{12}}^c_k
\epsilon_{abc}\epsilon^{ijk}$ would introduce new CP phases to the
theory, but in a controlled way for $\kappa'\ll f$.

\subsection{CKM and no $\bar\theta$ at tree level}
\label{sec:CKMhigherdim}

The model we have introduced so far generates the diagonal Yukawa
couplings and breaks the product gauge group down to the standard
model $SU(3)_c$, all while maintaining an accidental CP symmetry at
the renormalizable level. After integrating out the non-perturbative
effects near the scale $M$, the theory matches to the standard model
with non-vanishing diagonal Yukawa couplings for all of the quarks
and phases that preserve $\bar\theta=0$.
\begin{align}
  D^u
  &=
  \begin{pmatrix}
    r_1 e^{i\theta} Y^{d*}_{11} &  & 0 \\
    0   & Y^u_{22} & 0 \\
    0   & 0 &  Y^u_{33}
  \end{pmatrix}
  \\
  D^d
  &=
  \begin{pmatrix}
    Y^d_{11} & 0 & 0\\
    0 &  r_2 e^{i\theta_2} Y^{u*}_{22} &  0 \\
    0   & 0 &  r_3 e^{i\theta_3} Y^{u*}_{33}
  \end{pmatrix}
  \label{eq:Yukdiag}
\end{align}
The PQ symmetries and large
non-perturbative effects are crucial to the accidental CP symmetry,
since they allow the breaking of the quark chiral symmetries without
introducing extra CP violating parameters.

The next challenge is to introduce the CKM mixing without spoiling
this  protection.  When we introduce additional  off-diagonal Yukawa
couplings, the accidental CP symmetry can no longer survive, since the
observed CKM phase must be generated. However, $\bar\theta_{SM}$ will
still vanish at tree level and remain highly suppressed even at loop
level due to the residual approximate flavor symmetries.

Introducing quark-mixing between generations requires higher
dimensional operators involving the link fields, e.g.
\begin{align}
  \mathcal{L}_{d=5}
  &=
  \lambda^u_{ij} \left(q_i \frac{\Sigma_{ij}}{\Lambda_f} u^c_j\right)
  H \label{eq:yHDO}
\end{align}
generates the
effective Yukawa matrices
when the $\Sigma$ fields acquire vacuum expectation values. We can
write the off-diagonal Yukawa couplings below the scale of
Higgsing,
\begin{align}
O^{u,d}
&=
\frac{\lambda^{u,d}_{ij} f_{ij}}{\Lambda_f}.
\label{eq:YOffDiagonal}
\end{align}
Since the off-diagonal entries in the Yukawa matrices can be small,
the flavor scale $\Lambda_f \gg M$  is possible, with a
separation as large as  $\Lambda_f \lesssim 10^4 M$
consistent with unitarity and the size of the observed off-diagonal
Yukawa elements. However, a natural assumption that the couplings
$\lambda$  of the UV completion are comparable to the non-vanishing
diagonal Yukawa couplings would require e.g. $\Lambda_f \sim f$ to
generate the $\mathcal{O}(1)$ Cabibbo angle.

\begin{table}
\centering
\begin{tabular}{l|rrr}
 &  $PQ_1$ & $ PQ_2$ & $PQ_3$  \\ \hline\hline

 $q_1$ 	& 		$0$ & $0$ & $-1$ \\
 $u^c_1$&	$1$ & $0$ & $0$ \\
 $d^c_1$&	$0$ & $0$ & $1$\\
 \hline

  $q_2$  & $0$ & $0$ & $-1$ \\
 $u^c_2$ & $0$ & $0$ & $1$ \\
 $d^c_2$ & $0$ & $1$ & $0$ \\
 \hline

  $q_3$  & $0$ & $0$ & $0$ \\
 $u^c_3$ & $0$ & $0$ & $0$ \\
 $d^c_3$ & $0$ & $0$ & $1$

\end{tabular}
\caption{Particle charges under family $PQ$ symmetries, for a
simple 3-generation model with CKM mixings ($i,j=1,2,3$).}
\label{tab:particlesCKM}
\end{table}

For general off-diagonal couplings, it is no longer true that the
tree-level $\bar{\theta}$ vanishes after matching to the SM,
\begin{align}
  \bar{\theta}
&=
\arg\det\left[
  e^{-i(\theta_1+\theta_2+\theta_3)}
  (D^u + O^u)
  (D^d + O^d)
\right]
\\&=
\arg
\left(
\det\left[
  e^{-i(\theta_1+\theta_2+\theta_3)}
  D^u D^d
\right]
\det \left[(1+  (D^{u})^{-1} O^u)\right]
\det \left[(1+  (D^{d})^{-1} O^d)\right]
\right)
\,.
\end{align}
The first determinant factor is real, as shown above. For the
other two factors,
it is
simple to see that we must require that the off-diagonal matrices
$O^{u,d}$ can be put in a
strictly triangular form (up to $SU(3)$ rotations). We
would like the quarks to transform under
(possibly anomalous) $U(1)$ symmetries that perturbatively protect
this form, and in fact there are only two possible textures satisfying
these constraints and giving viable CKM mixings. The texture we will
focus on is:
\begin{align}
  Y^u
  &=
  \begin{pmatrix}
    0   & Y^u_{12} & 0 \\
    0   & Y^u_{22} & 0 \\
    0   & 0 &  Y^u_{33}
  \end{pmatrix}
  \\
  Y^d
  &=
  \begin{pmatrix}
    Y^d_{11} & 0 & Y^d_{13}\\
    Y^d_{21} & 0 & Y^d_{23} \\
    0   & 0 &  0
  \end{pmatrix},
  \label{eq:UVYuks}
\end{align}
and the assignment of PQ charges in~\cref{tab:particlesCKM} protects this
form of the Yukawa matrix. The other possible texture, described briefly in~\cref{app:altyukawa}, gives a less natural realization of the CKM structure.

The three anomalous $U(1)_{PQ}$ symmetries allow us to rotate away the
$\theta$ angle in each $SU(3)$ factor, and field redefinitions leave
only two remaining physical CP phases in the Yukawa matrix, which we
choose by convention to put in the $Y^d_{23}$ and $Y^d_{21}$ elements.
Including non-perturbative instanton effects and for the moment
ignoring all other radiative effects, below the scale $M$ the theory
matches to the SM with Yukawa matrices

\begin{align}
y^u
&= \begin{pmatrix}
r_1 e^{i\theta_1} {Y^d_{11}}^*    & Y^u_{12} & 0 \\
0   & Y^u_{22} & 0 \\
0   & 0 &  Y^u_{33}
\end{pmatrix}
\\
y^d  &= \begin{pmatrix}
 Y^d_{11} & 0 & Y^d_{13}\\
 Y^d_{21} & r_2 e^{i\theta_2} {Y^u_{22}}^* & Y^d_{23} \\
0   & 0 &  r_3 e^{i\theta_3} {Y^u_{33}}^*
\end{pmatrix}
\end{align}
We can check explicitly that $\bar\theta_{SM}=0$ at tree level,
\begin{align}
  \bar{\theta}
&=
-\arg\det\left[
  e^{-i(\theta_1+\theta_2+\theta_3)}
\begin{pmatrix}
r_1 e^{i\theta_1} {Y^d_{11}}^*    & Y^u_{12} & 0 \\
0   & Y^u_{22} & 0 \\
0   & 0 &  Y^u_{33}
\end{pmatrix}
\begin{pmatrix}
 Y^d_{11} & 0 & Y^d_{13}\\
 Y^d_{21} & r_2 e^{i\theta_2} {Y^u_{22}}^* & Y^d_{23} \\
0   & 0 &  r_3 e^{i\theta_3} {Y^u_{33}}^*
\end{pmatrix}
\right]
=
0
\,.
\end{align}
The real
coefficients $r_{1,2,3}$ parameterize the size of the instanton
suppression of PQ breaking in each $SU(3)$ factor. The couplings $Y$ can
now be determined from the CKM matrix and the observed SM fermion
masses. The only undetermined parameter is $Y^d_{21}/Y^d_{11}$, but we
will be motivated shortly to focus on the limit $Y^d_{21}\ll
Y^d_{11}$. Then to leading order in the small Yukawa ratios
$y_{u,d}/y_{c,s,t,b}$, $y_{s}/y_{b}$, and small CKM mixings
$|V_{31}|= 0.0089, |V_{32}|=0.041$ \cite{Agashe:2014kda}  we obtain
\begin{align}
Y^u_{33} &=y_t, Y^u_{22}= y_c \cos\theta_c, Y^u_{12}=y_c \sin\theta_c
\nonumber \\
Y^d_{11}&=y_d, Y^d_{13}= y_b |V_{31}| e^{i\delta_0}, Y^d_{23} = y_b |V_{32}|
\nonumber\\
r_1 &= \frac{y_u}{\cos\theta_c y_d}, r_2 = \frac{y_s}{\cos\theta_c
y_c}, r_3 = \frac{y_b}{y_t}
\,,
\end{align}
where we have made a field definition choice to put the CKM phase
entirely into $Y^d_{13}$ and $\theta_c$ is the Cabibbo angle. An
alternative solution with the same texture but flipping the role of
the strange and down quarks is discussed in \cref{app:altyukawa}.

Now that the CKM elements are introduced, the gauge basis in the
$SU(3)\times SU(3)\times SU(3)$ theory is no longer aligned with the
flavor basis, and four-fermion operators generated by gauge
interactions at the scale $M$ will introduces non-MFV contributions to
CP-preserving flavor observables. The dominant constraint is due to
the $\Delta C=2$
operator generated by exchange of the heavy broken SU(3) gauge bosons,
given in the quark mass basis as
\begin{align}
  \mathcal{O}_{\Delta C=2}
  \sim\frac{4\pi \alpha_{s_{1,2}}
\sin^2\theta_C}{M^2} (\bar c \gamma^\mu u)^2.
\end{align}
Constraints on the $D^0$ splitting generated by this operator give
$M\gtrsim 1000\ \rm{TeV}$~\cite{Isidori:2010kg}. The
leading $\Delta B = 2$ and $\Delta S =2$ operators are suppressed
respectively by $|V_{13,23}|^2$  and $|V_{12,23}y_d / y_s|^2$  and
give less stringent constraints.

\subsection{$\Delta\bar{\Theta}$ from thresholds }

With the two physical CP violating phases in the $Y^d_{23}$ and
$Y^d_{12}$ elements, it is clear that at leading order in the Yukawa
couplings, neither contributes to the low energy theta angle. However
higher order perturbative corrections to the non-perturbative effects
at $M$ can give a non-vanishing threshold correction to
$\bar\theta_{SM}$.

At energies below $M$, the additional breaking of the SM
flavor symmetry generated by the gauging of $SU(3)_1 \times SU(3)_2
\times SU(3)_3$  decouples and the theory is just the standard model,
where the flavor symmetries suppress the running of  $\bar\theta$ to
negligible effects starting at 7-loops
\cite{Ellis:1978hq,Dugan:1984qf}. At energies far above
$M$, the non-perturbative PQ violating effects are
exponentially suppressed by the weak coupling of the gauge groups, and
the PQ symmetry protects the form of the Yukawa matrices with
$\bar\theta=0$ manifest, \cref{eq:UVYuks}. Therefore the
dominant effect on $\bar\theta$ is a threshold effect at energies near
$M$, where the non-perturbative violation of the PQ
symmetries are still large and the extra breaking of the SM flavor
symmetries through the gauging of $SU(3)_1 \times SU(3)_2 \times
SU(3)_3$ has not decoupled.

The leading effects occur at third order in the Yukawa couplings,
schematically generated from diagrams of the form
of~\cref{fig:ThetaThreshold}.
\begin{figure}
  \centering
\includegraphics[width=4in]{DeltaThetaDiagram.pdf}
\caption{One of the leading diagrams generating a non-vanishing
  threshold correction to $\Delta\bar\theta$. The off-diagonal Yukawa
  couplings appear in order to introduce a CP phase, and the instanton
  violates the anomalous PQ symmetry protecting the UV form of the
  Yukawa
  couplings as in~\cref{eq:UVYuks}.}
\label{fig:ThetaThreshold}
\end{figure}
Roughly, these diagrams describe how the Yukawa elements closing the
instanton diagrams depend on the scale of the instanton -- there is a
mismatch of the phase between instantons at different scales because
of the perturbative running of the Yukawas. Taking the $\Sigma$ fields
as background fields, the 1-loop running of the effective Yukawa
couplings \cref{eq:YOffDiagonal} takes the same form as in the SM
\cite{Machacek:1983fi}, with the non-vanishing CP phases entering
through the terms
3rd order in the Yukawa couplings:

\begin{align}
\beta_{P}(Y_{u,d})
&= \frac{1}{16\pi^2}\frac{3}{2}(Y_{u,d}
Y_{u,d}^\dagger - Y_{d,u}Y_{d,u}^\dagger)Y_{u,d}
\label{eq:PertRunning}
\end{align}
Since the phases entering in the instantons no longer align exactly with the low energy perturbative values of the Yukawa couplings, there is no longer an exact cancellation in phase between the non-perturbatively generated eigenvalues and the perturbative eigenvalues of $Y_{u,d}$.

To obtain a parametric estimate of these effects, we iteratively solve
the RGE including the perturbative running~\cref{eq:PertRunning} and
non-perturbative running~\cref{eq:yrun} of the Yukawas, as described
in detail in \cref{app:yukawarge}. We ignore the effects of
perturbative gauge interactions and the propagation of the $\Sigma$
fields -- all effects that modify $\bar\theta$ must involve both an
instanton and a Yukawa loop, so these higher order effects can give at
most $\mathcal{O}(1)$ corrections to our estimate if these states are strongly
coupled. Finite effects not captured by the RGE are also expected to
be of comparable size. There are two leading contributions to
$\bar\theta$. The size of the first is fixed by the experimentally
determined elements of the Yukawa matrix,
\begin{align}
\Delta\bar\theta
&= \frac{3}{16 \pi^2} \frac{1}{b}
\tan\theta_C V_{31} V_{32}
y_b^2\sin(\delta_0)[f_I(\alpha_{s_1}) + f_I(\alpha_{s_2})]
\label{eq:deltatheta1}
\\
&\approx  10^{-10} \times
\left(\frac{f_I(\alpha_{s_1}) +
f_I(\alpha_{s_2})}{2}\right)
\end{align}
where
\begin{align}
f_I(\alpha)
&= \frac{\Gamma(8, 2\pi\alpha^{-1})}
{\Gamma(7, 2\pi\alpha^{-1})}-2\pi\alpha^{-1}
\approx 0.8 + 1.6\alpha
\label{eq:fI}
\end{align}
with this linear approximation holding well in the coupling region of
interest $\alpha \sim 0.2-1$.
The small size of $\Delta\bar\theta \sim 10^{-10}$ is due to the loop
suppression and the smallness of the off-diagonal Yukawa elements. The
form of $\Delta\bar\theta$ is consistent with the observation that
$Y_{13}$ and $Y_{23}$ must appear as a product, since the  physical
phase can be rotated from one term to the other. The suppression by a
factor of $1/b = 3/29$ arises because there is only a small range of
energies where instanton effects are important, controlled by how
rapidly the gauge coupling runs.

There is another contribution proportional to the undetermined Yukawa
element $Y_{21}$,
\begin{align}
\Delta \bar \theta'
&= -\frac{3}{16\pi^2}\frac{1}{b}
\left( \cos\theta_C \sin\theta_C y_c^2 {\rm Im} [Y^d_{21}]
[f_I(\alpha_{s_1}) + f_I(\alpha_{s_2})]
\right)
\nonumber \\&
\approx
-(4\times10^{-8}) \times \frac{{\rm Im}[Y^d_{21}]}{y_d}
\left(\frac{f_I(\alpha_{s_1}) + f_I(\alpha_{s_2})}{2}\right)
\label{eq:deltatheta2}
\end{align}
If $Y^{d}_{21}$ takes on a value $\sim y_d$  with $\mathcal{O}(1)$ phase, this
extra contribution is inconsistent with experimental limits. However,
spurion arguments show that  $|Y^d_{21}|\ll y_d$ can be naturally
obtained. Since $Y^d_{21}$ breaks a different set of flavor
symmetries, its natural size can  be as small as
\begin{align}
|Y^d_{21}|\gtrsim Y^d_{11}
\times {\rm Max}({Y^d_{13}}^* Y^d_{23}, {Y^u_{12}}^* Y^u_{22})
\approx y_d
\times {\rm Max}(y_b^2 V_{31} V_{32}, y_c^2 \sin\theta_C \cos\theta_C)
\approx y_d\times 10^{-5}
\end{align}
making $\Delta\bar\theta'$ subdominant.

We have checked these estimates numerically at the one-loop level.

\subsection{UV Sensitivity}
\label{sec:UVsensitivity}

It is useful to discuss the degree to which this mechanism is
insensitive to ultraviolet physics at some scale $\Lambda_{UV}$ where
CP may be violated in a sector strongly coupled to the standard model.
For CP violation to be communicated from this sector to $\bar\theta$,
the anomalous breaking of the PQ symmetry must be active. There are two
possible contributions: small instantons of scale $\Lambda_{UV}^{-1}$
interacting directly with the new UV physics, and the unsuppressed
instantons at the scale $M^{-1}$ interacting with the physics at
$\Lambda_{UV}$ through higher dimensional operators.

The contributions of small instantons of size $\Lambda_{UV}^{-1}$ is
suppressed by the exponentially small instanton density
$D(\Lambda_{UV}^{-1})$ as long as the individual $SU(3)$ factors have
run back to weak coupling. For example, suppose the physics at
$\Lambda_{UV}$ introduces an $\mathcal{O}(1)$ phase $\alpha$ in the
non-vanishing Yukawas, e.g. $y_d(\Lambda_{UV}) \approx e^{i\alpha}
y_d(M)$. Then instantons at the scale generate a
contribution to $y_u$ with a phase that will appear in $\bar\theta$,
\begin{align}
\frac{{\Delta y_u}_{UV}} {y_u}
&\sim
e^{i\alpha} \frac{D(\Lambda_{UV}^{-1})}{D(M^{-1})}
\end{align}
In a sector with two-flavors, this contribution is consistent with
$\Delta\bar\theta \lesssim 10^{-10}$ if $\Lambda_{UV} \gtrsim 100 M$.

The physics at the scale $\Lambda_{UV}$ can also generate higher
dimensional operators consistent with the PQ symmetries and other
approximate chiral symmetries that carry CP phases and can interact
with the unsuppressed instantons at the scale $M$ (such operators also
interact with instantons at the scale $\Lambda_{QCD}$ and generate a
shift in $\bar\theta$ even in the standard PQ axion or massless up quark solution
\cite{Hamzaoui:1998yu}, but here these effects are subdominant by a
factor $\Lambda_{QCD}^2/M^2$). The most dangerous
operators are momentum dependent contributions to the phase of the
perturbatively allowed diagonal Yukawas,
\begin{align}
  \mathcal{L}_{d=6}
  &\supset
  {Y_{11}^d}' H^\dagger Q_1 \frac{D_\mu^2}{\Lambda_{UV}^2} d^c_1
  +  {Y_{22}^u}' H Q_2 \frac{D_\mu^2}{\Lambda_{UV}^2} u^c_2
  + {Y_{33}^u}' H Q_3 \frac{D_\mu^2}{\Lambda_{UV}^2} u^c_3
  \label{eq:D6}
\end{align}
Combined with instanton insertions at $M$, these give
\begin{align}
\Delta\bar\theta
&\sim
{\rm Im}\left[\frac{Y'}{Y}\right]\frac{{M}^2}{\Lambda_{UV}^2}
\,.
\end{align}
When $Y' \sim Y$ and
the phases are uncorrelated, this requires $\Lambda_{UV} \gtrsim 10^5 M$
to avoid generating $\Delta \bar\theta$.

Another dangerous $d=6$ operator that can generate contributions to
$\bar\theta$ even in the absence of PQ breaking are mixed topological
terms, for example
\begin{align}
\mathcal{L}_{d=6}
&\supset \hat\theta_{12} G^{(1)\;b}_{\;a}\tilde{G}^{(2)\; i}_{j} \frac{ {\Sigma_{12}}_b^{\;j} {\Sigma_{12}^\dagger}^a_{\;i}}{\Lambda_{UV}^2}
\label{eq:D6mixedtopo}
\end{align}
gives a contribution $\Delta\bar\theta \approx \hat{\theta}_{12}
\frac{M^2}{\Lambda^2_{UV}}$, again requiring
$\Lambda_{UV} \gtrsim 10^5 M$ unless $\hat\theta_{12}$ is suppressed.

\section{A Flavor UV Completion}
\label{sec:UVFlavor}

The $d=5$ operators in \cref{eq:yHDO} generating the off-diagonal
Yukawa elements require a UV completion at the scale $\Lambda_f$.
Unitarity of the $d=5$
operator in \cref{eq:yHDO} generating the off-diagonal Yukawas
requires $\Lambda_f \lesssim 10^{-4} M$. Taking the effective action to $d=6$ introduces operators consistent
with the PQ symmetries that could allow the CP violation generating
$\delta_{CKM}$ to enter directly into $\Delta\bar\theta$, as
discussed in~\cref{sec:UVsensitivity}.

In this section we give an example of a simple UV completion in which
the higher dimension operators do not make large contributions to
$\Delta
\bar\theta$  and which can also explain the origin of the spurion
argument giving $|Y^d_{21}|\ll y_d$ . The model is extended to involve
a set of vector-like fermions $Q_3, \bar{Q}_3, U^c_1, \bar{U^c_1}$,
with charges under the gauge and PQ symmetries as given
in~\cref{tab:vectorlikeCKM}. Renormalizable mixings between heavy
states and the SM-like fields generates the higher dimensional
operators~\cref{eq:yHDO} after integrating out the vector-like
states.

\begin{table}
\centering
\begin{tabular}{l|lll}
 & \tiny ${SU(3)_1}_{(B-L)_1, PQ_1}$ &\tiny ${SU(3)_2}_{(B-L)_2, PQ_2}$ &\tiny ${SU(3)_3}_{(B-L)_3, PQ_3}$
 \\
 \hline\hline\rule{0pt}{4ex}
 $U^c_1$		& $\bar\square_{(-\frac{1}{3}, 0)}$ & - & $-_{(0,1)}$
 \\
 $\bar{U^c_1}$		& $\bar\square_{(-\frac{1}{3}, 0)}$ & - & $-_{(0,-1)}$   \\
 \hline\rule{0pt}{4ex}
  $Q_3$ 			& - & - & $\square_{(\frac{1}{3},-1)}$   \\
  $\bar{Q_3}$ 			& - & - & $\square_{(\frac{1}{3},1)}$   \\
\end{tabular}
\caption{Vector-like quark content allowing a simple UV completion of
the off-diagonal higher dimension CKM mixing operators.}
\label{tab:vectorlikeCKM}
\end{table}

The renormalizable terms in the Lagrangian consistent with the gauge
and PQ symmetries are
\begin{align}
  \mathcal{L}_{\rm UV}
  &=
  M_U U_1^c
  \bar{U^c_1} + M_Q Q_3 \bar{Q_3}
  \nonumber\\& \qquad
  + z^u_{11} H q_1 U_1^c
  + z^d_{33} H^\dagger Q_3 d_3^c
  \nonumber\\ & \qquad
  +x^U_{12} \Sigma_{12} u^c_2 \bar{U^c_1}
  +x^Q_{13} \Sigma_{13} q_1 \bar{Q_3}
  +  x^Q_{23} \Sigma_{23} q_2 \bar{Q_3}
  \nonumber\\& \qquad
  +{\hat{Y}}^d_{11} H^\dagger q_1 d_1^c
  +{\hat{Y}}^u_{22} H q_2 u_2^c
  + {\hat{Y}}^u_{33} H q_3 u_3^c
\end{align}
The field redefinition freedom leaves one physics CP violating phase
in this Lagrangian, which can be rotated between the parameters $M_U,
x^u_{12}, x^Q_{13}, x^Q_{23}, z^u_{11}, \hat{Y}^u_{22}$.  Taking
$M_{Q,U}\gg M $ and integrating out these states at tree level, we
obtain the effective theory of~\cref{sec:CKMhigherdim}, with
couplings to leading order in $\langle \Sigma \rangle / M$
\begin{align}
& Y^d_{11}={\hat Y}^d_{11}, \;\;Y^u_{22}={\hat Y}^u_{22}, \;\; Y^u_{33}={\hat Y}^u_{33} \\
& Y^u_{12}=\frac{x^U_{12}\langle \Sigma_{12} \rangle}{M_U} z^u_{11} \\
& Y^d_{13}=\frac{x^Q_{13}\langle \Sigma_{13} \rangle}{M_Q} z^d_{33}, \;\; Y^d_{23}=\frac{x^Q_{23}\langle \Sigma_{23} \rangle}{M_Q} z^d_{33}, \\
& Y^d_{21}= {\hat Y}^d_{11}  \frac{x^Q_{13}\langle\Sigma_{13}\rangle (x^Q_{23} \langle\Sigma_{23}\rangle)^\dagger}{|M_Q|^2}
\end{align}
By convention we can rotate the physical phase entirely into
$Y^d_{23}$ and $Y^d_{21}$ in the low energy Yukawa matrix, and to
leading order this corresponds to rotating the phase entirely into
$x_{23}^Q$ in the full theory.

The contribution to $\Delta\bar{\theta}$ due to $Y_{21}$
(\cref{eq:deltatheta2}) is suppressed by the mixing of $q_1$ and
$q_2$ with the vectorlike $Q_3$,
\begin{align}
\left|\frac{x^Q_{13}\langle\Sigma_{13}\rangle (x^Q_{23}
\langle\Sigma_{23}\rangle)^\dagger}{|M_Q|^2}\right|
&=
\left|\frac{Y^d_{13}{Y^d_{23}}}{{z^d_{33}}^2}^\dagger\right|
\approx
3 \times 10^{-7} \times \left(\frac{1}{|z^d_{33}|^2}\right),
\end{align}
giving
\begin{align}
\Delta\bar\theta'
&\approx
10^{-15} \times \left(\frac{1}{|z^d_{33}|^2}\right)
\end{align}
which requires the coupling $|z^d_{33}|\gtrsim 10^{-2}$ for this to be
subdominant.

A more dangerous contribution in this model comes from the $d=6$
operators of \cref{eq:D6} (the contributions from mixed topological
terms~\cref{eq:D6mixedtopo} are subdominant). There are
unsuppressed terms generated a tree level giving $Y'^u_{22} \sim
Y^u_{22}$ and $Y'^d_{11} \sim Y^d_{11}$, with scale $\Lambda_{UV} =
M_Q$. However, these operators do not contribute to $\Delta\bar\theta$
because the phases of $Y$ and this contribution to $Y'$ are aligned.
The leading contribution with a misaligned phase is
\begin{align}
{\Delta Y^u_{22}}' &= Y^u_{12}
\frac{x^Q_{13}\langle\Sigma_{13}\rangle (x^Q_{23}
\langle\Sigma_{23}\rangle)^\dagger}{|M_Q|^2}=Y^u_{12}\frac{Y^d_{13}{Y^d_{23}}^\dagger}{|z^d_{33}|^2}
\end{align}
This term is also suppressed by the mixing of $q_1$ and $q_2$ with
the vectorlike $Q_3$. For
$\langle\Sigma_{13}\rangle\sim\langle\Sigma_{23}\rangle \sim M$, the
contribution to the theta angle is
\begin{align}
\Delta\bar{\theta}_{D=6}\sim
\frac{{{\rm Im}[\Delta Y^u_{22}}']}{Y^u_{22}} \frac{{M}^2}{|M_Q|^2}
\sim \frac{Y^u_{12}}{Y^u_{22}}
|Y^d_{13}|^2|Y^d_{23}|^2\frac{g^2}{x^Q_{13}x^Q_{23}} \sin\delta_0
\approx
10^{-13}\times\left(\frac{\alpha_{s_{1,2,3}}}{0.5}\right)\left(\frac{1}{|x^Q_{13}x^Q_{23}
{z^d_{33}}^2| }\right)
\end{align}
As long as the marginal couplings
generating the $q_1$ and $q_2$  mixings are not too weakly coupled,
$x^Q_{13}, x^Q_{23}, z_{33} \gtrsim 0.2$, this contribution is
subdominant. This corresponds to a rough lower limit on the scale
$M_Q \gtrsim 100 M$.  Note that a hierarchy $M_Q \gg M_U \sim M$ can naturally explain the small third-generation quark mixings and $O(1)$ Cabibbo angle.

This flavor model has a similar structure to minimal Nelson-Barr
models \cite{Barr:1979as,Nelson:1983zb,Barr:1984qx}, which obtain CKM
mixings through vector-like quarks \cite{BENTO199195}, forbidding a
tree-level $\bar \theta$. However, in contrast to the present case
where the $U(1)$ symmetries are sufficient to protect the structure of
the theory, in  Nelson-Barr models discrete symmetries and additional
UV structure are required \cite{Vecchi:2014hpa, Dine:2015jga}. In both
cases, radiative contributions to $\bar\theta$ limit the allowed
parameter space, but in Nelson-Barr models these limits appear to
generically require unexplained suppressions of allowed couplings
\cite{Dine:2015jga}.

\section{Conclusions}

The solutions to the strong CP problem and the origins of the flavor
structure of the standard model may be intricately tied to each other.
In this work, we constructed a model where embedding QCD in a
$SU(3)^3$ gauge group with flavorful anomalous PQ symmetries can
naturally explain the non-observation of a neutron EDM, the smallness
of the third generation CKM mixing angles, and the relative
suppression of the down-like quark masses in the second and third
generation.
The theta angle in each $SU(3)$ factor can be set to zero using an
anomalous PQ symmetry. This symmetry is realized by forbidding a bare
mass for the lighter quark in each generation (i.e.\,$u,s,b$). Their
masses are generated through instantons, dominantly at the scale of
$SU(3)^3$ breaking, $M$, which can be far above the weak scale. The
instanton-generated mass terms have phases
that are naturally aligned with the theta angle, and hence do not
reintroduce a non-zero $\bar{\theta}$.

There is a non-zero $\bar{\theta}$ generated at the threshold $M$ through
loop corrections that involve both the instanton vertex as well as the
perturbative CKM phase.
In fact, in our model the smallness of the CKM mixing angles is
intimately tied to the smallness of $\bar\theta$ in this model, and
the observed CKM elements give a prediction $\bar\theta \sim 10^{-10}$
that can be probed at the next generation of neutron EDM
\cite{Ito:2007xd,Tsentalovich:2014mfa} and proton storage ring
experiments \cite{Anastassopoulos:2015ura}.  The solution to the
strong CP problem is in the spirit of the massless up quark solution,
and there are no axion-like states in the theory.
An interesting future direction would be to study models which
generate the full standard model flavor structure
while also implementing our mechanism to solve the strong CP problem.

We would like to thank Asimina Arvanitaki, Savas Dimopoulos, Bogdan
Dobrescu, Howard Georgi, Roni Harnik, Anson Hook, Junwu (Curly) Huang,
Gustavo Marques Tavares, Lisa Randall, Matt Reece and Raman Sundrum
for encouragement and helpful comments. This work was initiated at the
Aspen Center for Physics, which is supported by National Science
Foundation grant PHY-1066293.  PA is supported by the NSF grants
PHY-0855591 and PHY-1216270.  This manuscript has been authored by
Fermi Research Alliance, LLC under Contract No.  DE-AC02-07CH11359
with the U.S. Department of Energy, Office of Science, Office of High
Energy Physics. The United States Government retains and the
publisher, by accepting the article for publication, acknowledges that
the United States Government retains a non-exclusive, paid-up,
irrevocable, world-wide license to publish or reproduce the published
form of this manuscript, or allow others to do so, for United States
Government purposes.

redpajama_set_name: RedPajamaArXiv


text:
Peconic Land Trust, Inc.
aka PLT | Southampton, NY | www.peconiclandtrust.org
Mission: The Peconic Land Trust conserves Long Island's working farms, natural lands, and heritage for our communities now and in the future. Impact: The conservation of working farms, natural lands, and heritage is closely inter-connected to many benefits that are important to all us who know and love the area ~ fresh local produce, miles of hiking trails, clean and productive bays and wetlands, drinking water protection, strong local economy, healthy ecosystems, storm water protection and buffer against effects of climate change, free public garden, historic preservation, lower property taxes (because of undeveloped land), and continued enjoyment of this unique region for generations to come.
Footnote to audited Financial Statement: Total Assets primarily reflects the donated value of conservation restrictions (easements) since 1983 that are booked as permanently restricted net assets and are not available for operating expenses. Peconic Land Trust have been awarded accreditation by the Land Trust Accreditation Commission. This national honor is based on an extraordinary commitment to excellence, trust, and permanence in land conservation. The Trust has demonstrated sound finances, ethical conduct, responsible governance, and lasting stewardship.
Mr. John v.H. Halsey
Southampton, NY 11969 USA
Ms. Amanda Abraham
Southampton, 11968
Legal name of organization: Peconic Land Trust
Land Resources Conservation (C34)
Farmland Preservation (K25)
Environmental Quality, Protection, and Beautification N.E.C. (C99)
We work to conserve Long Island's working farms, natural lands, and heritage for our communities now and in the future.
Stewardship of Land and Conservation Easements
For the Peconic Land Trust, Stewardship is the perpetual care of the preserves and conservation easements held by the Trust. These protected properties are a legal and good faith responsibility that continues forever. All properties are routinely monitored and related records maintained.
Properties donated to the Trust in fee are called Preserves. A management plan is prepared for each of these properties. Some Preserves are passive and need minimal maintenance, while others require extensive measures to control soil erosion, restore wetlands, control public access, etc.
Conservation easements are a voluntary agreement between a landowner and a qualified conservation organization or municipality that protects the land from development, with the landowner retaining ownership and the rights to sell the property or pass it onto heirs. These properties are monitored at least annually to detect any changes and to ensure the property is being used in accordance with the terms of the restrictions placed on it. The commitment to monitor, defend, and enforce the easements is what ensures perpetual protection.
Our stewardship staff is available to assist landowners with a variety of land management activities including:
Field mowing
Planting of cover crops
Invasive vegetation removal
Trail creation and maintenance
The Trust can provide environmentally sensitive land care services to landowners – individuals, corporation or municipalities. As a result, interest in the Peconic Land Trust's Stewardship Program comes from many sources, including:
landowners who want to preserve the rural character of their property;
landowners who want to protect environmentally sensitive area(s) of their property;
landowners who find it difficult to manage their property;
corporations holding land as an investment;
developers seeking assistance in the management of open space areas within their plans; and
Municipalities that own natural areas or development rights but whose management capability is stretched by the maintenance of active recreational areas.
Our experienced staff is available to advise and assist landowners on long-term natural resource management strategies available, including establishing and implementing stewardship objectives that reflect the owner's goals and the unique elements of each property.
To learn more, visit https://peconiclandtrust.org/our-work/stewardship .
Land Protection/Conservation
Since 1983, the Trust has worked diligently with landowners, communities, municipalities, and partner organizations to protect 12,000 acres of land (as of 2019), conserving more working farms on Long Island than any other private conservation organization, and securing millions of dollars from the public and private sector for land protection. The Peconic Land Trust has worked with landowners to help plan and accomplish conservation projects that meet their goals and financial needs. We create partnerships with landowners, communities and all levels of government to develop win-win solutions that conserve land while respecting the rights of the individual and the interests of the public. To learn more, visit https://peconiclandtrust.org/our-work/conservation .
Education and Public Policy for Land Conservation
A. Education:
Build connections between all people, diverse communities, and the land:
Conduct educational programs about conservation, the role of the Trust, and the land we have protected.
Provide multiple opportunities for people to experience the land as an essential element of their lives.
Empower our staff, Board, and volunteers to be messengers of the Trust's mission.
Enhance the volunteer program in order to engage the community and to extend the work of the Trust.
Expand and strengthen coalitions with other organizations to further our mission.Define and promote public policy goals that further our mission and core values.
B. Public Policy:
Analyze public policy issues and formulate positions and strategies:
Educate and inform ourselves, our constituents, and the general public about these issues.
Meet regularly with public officials at all levels to further our public policy goals.
Build new coalitions of diverse interests to achieve consensus on public policy issues and strategies.
Farms for the Future Initiative
What is Farms for the Future? Long Island's farms and farmers are central to our economy, heritage, and the stewardship of valuable natural resources. Faced with intense development pressures, escalating real estate values, a decline in food production farming, and a lack of affordable farmland for both new and established farmers, the Peconic Land Trust is working to ensure that Long Island's farms and farming communities remain viable and productive now and in the future. "Farms for the Future" is a special initiative that the board and staff of the Trust have undertaken to explore these challenges in depth and problem-solve using pragmatic approaches. Since our founding in 1983, the Peconic Land Trust has preserved more working farms on Long Island than any other private conservation organization. Within the last decade, our focus has changed slightly -- land protection remains first and foremost, but protecting the land for food production farming has grown in importance. In 2008, the Trust introduced its Farms for the Future Initiative. Through a formalized program, the Trust actively seeks to not only protect the farmland -- particularly that which is threatened by development -- but to also grow the resource. Our initiative's primary objectives are to:
1. Explore and refine a new set of conservation tools and techniques that address the issues of farmland accessibility, affordability, and sustainability.
2. Buy, protect, restore, and resell farms to farmers
3. Develop and expand our Farmland Leasing Program
To learn more, visit https://peconiclandtrust.org/our-work/farms .
Land Trust Accreditation Commission 2019
Growing Farmers
Edward Krug
John Norbeck
Ann ffolliott
John v.H. Halsey
President, Peconic Land Trust
John Larkin
Nancy Gilbert
Harriet Dresher
Thomas Isles
Jennifer Pike
Marcia Bystryn
Carol Ahlers
Nora Catlin
James MacGregor
Has the board conducted a formal, written self-assessment of its performance within the past three years? Yes
Financial statements for 2017

redpajama_set_name: RedPajamaCommonCrawl


text:
Q: How to extract Strings from DataGrid regardless of what type contains in GWT I'am programming a generic DataGrid to Excel exporter. What I wanted to do is to extract an ArrayList which represents the rows, and for each row I have an array of Colomn. So basically,  I want to extract an ArrayList<ArrayList<String>>.
The DataGrid have some methods as dataGrid.getColumn(i) or dataGrid.getRowElement(i) but I couldn't find anything to get a String from a specific cell(row,col).
Any idea ?

A: you can use
dataGrid.getRowElement(i).getCells().getItem(j);

You would have to write a method to extract all the data and put it in your ArrayList>. Maybe something like that (please note this code was written without any IDE):
for(int i = 0; i < dataGrid.getRowCount(); i++){
  TableRowElement element = dataGrid.getRowElement(i);
  for(int j = 0; j < element.getCells().getLenght(); j++){
    list.add(element.getCells().getItem(j).toString();
  }
}

Hope this can help you

redpajama_set_name: RedPajamaStackExchange


Although exotic aquatic weeds have been reported to be present in Africa since the end of the nineteenth century (Tackholm and Drar, 1950), they started infesting massively African freshwater bodies during the early 1950s (Mitchell et al., 1990) and rapidly spread in many countries. The growth of these weeds is extremely fast and this allows them to develop huge infestations in areas where they had not been reported only a few years earlier. These plants invaded lakes, ponds, rivers, canals and agricultural fields, becoming noxious weeds. The damage to the environment and the economy is enormous, having a disrupting impact on agriculture, fisheries, production of electricity, transportation, health, means of sustenance, living conditions and social structure. Water Hyacinth, Water Lettuce and Water Fern pose the most serious problems.
Origin and Distribution. Water Hyacinth (Eichhornia crassipes (Martius) Solms-Laubach), which is native to the Amazon basin, Brazil (Penfound and Earle, 1948; DeLoach, 1976), became widespread throughout the world, also due to its attractive appearance. It is commercially available as an ornamental for ponds. Its spread started with its deliberate introduction into North America from Brazil, in the late nineteenth century, as an ornamental in ponds and subsequently escaped cultivation (DeLoach, 1976; Barrett and Forno, 1982). At present it occurs as a weed throughout tropical and subtropical regions of the world, including North and South America, Africa, Asia, Australia and New Zealand.
Morphology and Biology. This perennial herbaceous plant is a floating freshwater hydrophyte. It belongs to the Family Pontederiaceae and all the species in the Genus Eichhornia are aquatic. Water Hyacinth shows considerable variation in both leaf and flower form and colour, also depending on the age of the plant. The flowers are bluish purple, large and self-fertile. The seeds are produced in large numbers and are contained in capsules, each capsule containing up to 300 seeds (Manson and Manson, 1958). The seeds can remain viable for 5-20 years (Matthews et al., 1977). The plant can also reproduce vegetatively through the production of horizontal stolons.
Weed problem. The fast growth of Water Hyacinth allowed the plant to build huge populations in its ranges of introduction, developing dense mats on the surface of the water and becoming a major weed problem.
It is considered the worst aquatic weed in the world. The rapid increase and spread of the plant into new areas is due particularly to its vegetative reproduction, a single plant being able to develop very rapidly a significant infestation. Moving easily with water currents, winds or other accidental means, such as fishing nets and boats, the plant invaded rivers, canals, ponds, lakes, dams and other freshwater bodies. The main problems arising from the growth of Water Hyacinth in thick mats are (a) an enormous water loss through evapotranspiration, that alters the water balance of entire regions; (b) the impediment to water flow, that increases sedimentation, causing flooding and soil erosion; (c) the obstruction of navigation; (d) hampering fishing and dramatically reducing the catch and the source of food and income for local populations; (e) a drastic change in the physical and chemical properties of water and in the environment in the water bodies invaded, with detrimental effects on plants and animals; (f) the reduction of the activity of electrical power stations, jeopardizing the power supply of the country; and (g) a serious threaten to agricultural production, following the blockage of irrigation canals and drainage systems. The economy of the countries concerned was therefore seriously affected in many aspects. This weed represents an environmental problem as well and indirectly a public health problem, since it may create a microhabitat suitable for the breeding of many vectors of human diseases and for hosting poisonous snakes.
Water Hyacinth is the principal aquatic weed in Africa. It is a weed problem in many countries, especially in Egypt and East, West and South Africa. This plant appeared in Lake Kyoga in Uganda in the early 1980s and it also occurs in Lake Kwania and down the Kyoga Nile. In 1988 it was observed in Lake Naivasha in Kenya. However, the most disturbing development in the region during the 1990s was in Lake Victoria. In 1990 dense mats of this weed were found interfering with fishing on the Tanzanian shore and during the same year mats were recorded on the Ugandan shore and around the Sese Islands. In Zimbabwe the plant attained the status of noxious weed in the 1980s. It was first recorded in Lake Kariba in 1994 and by 1996 it infested over 200 ha. In the early 1990s Water Hyacinth was also found in the Pangani River in Tanzania and the lower reaches of the Kagera River in Rwanda. FAO has been conducting numerous projects for the control of this weed (see Chapter 3).
Origin and Distribution. The area of origin of Water Lettuce (Pistia stratiotes L.), is still speculative, being most probably South America (Cordo et al., 1981). The plant spread widely and at present occurs in all continents, except Europe and Antarctica. Probably the initial spread took place through ballast water in ships from South America. At present it is also commercially available as an ornamental for ponds. It is one of the most widely distributed water plants. In Africa it is ubiquist.
Morphology and Biology. This perennial freshwater hydrophyte is a herbaceous floating plant belonging to the Family Araceae. It consists of a rosette of pale green leaves, prominently veined and it resembles a small lettuce plant. Water Lettuce has velvety-hairy, erect leaf blades, a very short stem and long feathery roots suspended in water. The flowers are bisexual. The plant reproduces and spreads rapidly by means of stolons and seeds. The seeds are easily carried by water for long distances, since they float during the first two days after they reach maturity. The hairy cover of leaves traps air and is water repellent, preventing the plant from being submerged by heavy rains. Both Water Lettuce and Water Hyacinth can bioaccumulate heavy metals (Sridahar, 1986).
Weed problem. Water Lettuce is a major weed problem in the tropics, where its impact is similar to that of Water Hyacinth, both on the environment and on the economy of the countries concerned. However, it was not a weed problem in Africa until recently and its weed status appears to be due to the pollution of water bodies and the presence of organic wastes and residues of fertilisers. The plant develops dense mats and its noxious effects are similar to those caused by Water Hyacinth and include (a) causing an enormous water loss through evapotranspiration, with the consequent negative impact on the water balance in entire regions; (b) clogging rivers and canals and related problems; (c) interfering with the activity of hydroelectric power stations; (d) hampering fishing; (e) hampering navigation; (f) negatively affecting water control and use for agricultural purposes; (g) displacing of native species, because of the modification of environmental conditions: mainly due to oxygen depletion in water and sediments, increased sedimentation - caused by the roots - and excessive shading - caused by the leaves; and (h) indirectly affecting human health, since it provides a suitable breeding habitat for pests and vectors of diseases. Water Lettuce occupies habitats similar to those where Water Hyacinth occurs and is generally displaced by the latter, but, inversely, the control of Water Hyacinth may be followed by increased numbers of plants of Water Lettuce where mixed populations occur. Water Lettuce invaded inland waters of various African countries in the early 1980s. The weed problem is particularly serious in West Africa, especially in Niger and Volta River basins, where FAO has been conducting numerous projects (see Chapter 3).
Origin and Distribution. Also Water Fern (Salvinia molesta D.S. Mitchell) is native of South America, in coastal and inland regions of south-eastern Brazil. The plant was introduced to Sri Lanka in the 1930s and has rapidly spread since then, now occurring in tropical and subtropical regions worldwide (Holm et al., 1991). The species is commercially available for aquariums and ponds and this may likely have been the way it was initially introduced and may still contribute to its spread. In Africa it is a weed problem mainly in the inter-tropical zone.
Morphology and Biology. Water Fern is a free-floating fern belonging to the Family Salviniaceae that lives in freshwater systems. Stagnant or slowly moving waters are the habitats most favourable to its growth. It consists of a horizontal rhizome that floats just below the water surface and produces at each node three leaves, two of which, aerial, are light green, broadly elliptic, about 25 mm long, while the third leaf, submerged, is brownish, filamentous, feathery and root-like and about25 cm long. The plant does not have roots. The submerged leaf, which serves as roots by absorbing water and nutrients, bears the sporocarp, oval in shape, which is the structure normally producing the spores. Hairs growing orderly, in parallel rows, on the aerial leaves assure a water-repellent layer that allows the plant to float (Holm et al., 1991). Individual plants are up to 30 cm long. Their growth is extremely fast, allowing the population to double within about one week (Mitchell and Tur, 1975). Mature plants press each other into tight chains, forming thick mats on the water surface and can withstand periods of stress, such as low temperatures, or periods of drought, regrowing from dormant buds. Water Fern readily reproduces vegetatively, by fragmentation of the rhizome, small fragments allowing the development of new infestations. The African plants are sterile and more vigorous than other accessions (Holm et al., 1991).
Weed problem. Growing rapidly in dense mats, the plant invaded lakes and streams in its range of introduction, becoming a nuisance. Water Fern is also known as Kariba Weed, this common name originating from Lake Kariba on the Zambesi River in southern Africa, an area that was invaded by this weed. It has a heavy impact on the environment and the economy of the regions affected. It causes similar problems to those posed by Water Hyacinth and Water Lettuce, such as clogging canals, rivers and lakes; displacing native plants and animals; and interfering with the irrigation of many crops, navigation, fishing and the activity of electric power stations. In addition, it grows in rice paddies, where it poses serious problems. Water Fern invaded inland waters of various African countries in the early 1980s. The weed problem is particularly serious in West Africa, especially in Niger, Volta and Senegal River basins, where FAO has been conducting numerous projects (see Chapter 3).


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OECD Home FranceSkills Matter - Additional Results from the Survey of Adult Skills - en
Skills Matter
Additional Results from the Survey of Adult Skills
In the wake of the technological revolution that began in the last decades of the 20th century, labour-market demand for information-processing and other high-level cognitive and interpersonal skills have been growing substantially. Based on the results from the 33 countries and regions that participated in the 1st and 2nd round of the Survey of Adult Skills in 2011-12 and in 2014-15, this report describes adults' proficiency in three information-processing skills, and examines how proficiency is related to labour-market and social outcomes. It also places special emphasis on the results from the 3rd and final round of the first cycle of PIAAC in 2017-18, which included 6 countries (Ecuador, Hungary, Kazakhstan, Mexico, Peru and the United States). The Survey of Adult Skills, a product of the OECD Programme for the International Assessment of Adult Competencies (PIAAC), was designed to provide insights into the availability of some of these key skills in society and how they are used at work and at home. The first survey of its kind, it directly measures proficiency in three information-processing skills: literacy, numeracy and problem-solving in technology-rich environments.
Published on November 15, 2019Also available in: French
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Adults' proficiency in key information-processing skills
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Additionally, if you become a premium member you no longer have to wait to be eligible. Unverified Gamertag - This icon means that there is no gamertag on the roster. To update the gamertag, the player must link their gamertag /PSN .... According to the data on the general sales of public housing in Hong Kong, there are more than 2,600 new flats that will be completed by 2018. They are to be distributed through the HOS. More units will be up for selling. If you feel that you are eligible for a low-cost housing option, go ahead and apply for it.
Major League Gaming Why am I ineligible on the team roster?
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How to Become Eligible. Simply complete the membership request letter, print it on your company letterhead and have an officer of the organization sign it and return it to us.
Your gavelkind titles will also become de facto part of Byzantium. Furthermore, your son will have claims on them, allowing him to revoke the titles without angering other vassals. Furthermore, your son will have claims on them, allowing him to revoke the titles without angering other vassals.
All are eligible to take amrit; however, some b � asic rules do apply. One must be a lacto vegetarian, mustn't cut, trim, shave, or rely on any form of hair removal, must never tell lies and he or she shall fulfill the requirements of completing all daily prayers, etc.


Spring has arrived in Zurich as soon as the Böögg's head has exploded.
When in April the first flowers and trees start to bloom, the temperatures rise and the days become longer, the people of Zurich celebrate their traditional spring festival: Sechseläuten, or "Sächsilüüte" as it is known in local dialect.
Where this tradition originates, who plays the leading role, what the Zurich guilds have to do with it and what happens when, you can find out here.
Every April, the Zurich guilds celebrate Sechseläuten. But where does this custom originate?
Information about the Festival program and the route of the Sechseläuten parade.
Interesting facts and curiosities relating to the Sechseläuten tradition in Zurich.
The Böögg builder is particularly familiar with Zurich's highlights – here are his tips.
The tour explores the history of Zurich´s guilds, whose traditions continue to be celebrated today.
Right up until Sechseläuten, fans can grab a photo with the Böögg at the Tourist Information.


Online registration will begin Oct. 1 for the fiscal year 2016 Diversity Lottery, which provides 50,000 immigrant visas to individuals from countries with historically low immigration to the U.S.
Registration begins Wednesday, Oct. 1 at noon EDT (GMT -4) and ends Monday, Nov. 3 at noon EST (GMT -5). Applicants must register online at www.dvlottery.state.gov. The State Department is warning participants not to register during the final week, as last-minute demand may slow web registration.
Nationals of the following countries are not eligible because the number of immigrants to the U.S. exceeded 50,000 in the past five years: Bangladesh, Brazil, Canada, China (mainland-born), Colombia, Dominican Republic, Ecuador, El Salvador, Haiti, India, Jamaica, Mexico, Nigeria, Pakistan, Peru, Philippines, South Korea, United Kingdom (except Northern Ireland) and its dependent territories, and Vietnam. Individuals born in Hong Kong, Macau and Taiwan are eligible.
Registrants must meet two eligibility criteria: They must be born in an eligible country and meet educational or work requirements. Individuals who are not from an eligible country may qualify by claiming the country of their spouse or parents from an eligible country under certain circumstances. Applicants must have a high school degree or equivalent of two years of work experience in the past five years.
Applicants must complete an online form and submit a digital color photo fitting precise specifications for each family member. The State Department selects lottery winners through a random computerized selection process starting May 2, 2015 through June 30, 2016 and later. Lottery applicants can find out if they are selected only by entering their confirmation number on the official E-DV Entrant Status Check website. Lottery winners may apply for their green cards beginning Oct. 1, 2015.


text:
Starting Back
Posted: January 25, 2023 in Adult Contemporary Romance, Adult Fiction, Adult Romance, Contemporary, Contemporary Romance, Mature Fiction, Mature Romance, New, New Release, Second Chance Romance, Small Town Romance
Tags: Adult Fiction, Contemporary Romance, multi-cultural, New, New Release, Romance
Title: Starting Back
Author: Stephanie Rose
Publisher: That's What She Said Publishing
Tropes: Single Mom, Firefighter, Second Chance, He falls first, Small town, Hurt/comfort, multi-cultural
"This story is steamy, romantic, sweet and heartwarming. Everything you could want!" ★★★★★ Barbara A.
"… a sweet and sexy, small town, single mom romance with all the feels." ★★★★★ JudyAnnLovesBooks
My accidental solo vacation takes a turn for the less-depressing after a drink that leads to a date with a brawny, bedroom-eyed firefighter. I'm 42, divorced, with two daughters at home. I thought romance was in my rearview mirror until the mysteriously intense Leo makes me feel like I'm the only woman in the Florida Keys.
I know better than to get attached. But between midnight kisses and dirty talk under starry skies, our connection feels so real. So right.
I can't give up my entire life for a fling. Not even for a man that reminds me I'm not defined by a failed marriage and a car pool calendar. And especially not one who has no interest in a family of his own, let alone a package deal like me.
We were supposed to be temporary. Just like everything else in my life. So why can't I get Kristina out of my head? I'm the master of no strings, no attachments. But our time together haunts me after she leaves.
I'm not cut out to make a life with a woman. A woman with two kids? Forget it.
At least, that's what I thought until fate brings us together again. It turns out the move I made to be closer to family brings me to Kristina's hometown—and right in her path.
I can't still explain the overpowering pull between us, but I'm sure of who she is.
And I'm not letting her go this time.
Stephanie Rose is a badass New Yorker, a wife, a mother, and lover of all things chocolate. Most days you'll find her trying to avoid standing on discarded LEGO or deciding which book to read next. Her debut novel, Always You, released in 2015, and since then she's written several more—some of which will never see completion—and has ideas for hundreds to come.
Boss (Dick)
The Billionaire Fling


Formula 1's managing director of motorsport Ross Brawn believes Ferrari has no idea "what on earth happened" to its pace at the Australian Grand Prix after its strong showing during pre-season testing.
Ferrari had been fancied as the fastest team with its 2019 F1 car, the SF90, from pre-season testing at Circuit de Catalunya but throughout the opening round at Albert Park the Italian manufacturer remained off the pace of Mercedes.
During the race both Ferrari drivers dropped behind Red Bull Honda's Max Verstappen to see Sebastian Vettel finish fourth and Charles Leclerc in fifth, with the latter told to hold station behind his German teammate.
Both drivers struggled for pace during the race but in different stints and on different tyres, potentially confusing the cause of Ferrari's pace drop even more, and Brawn feels the reason for the team's struggles remains unclear to the squad itself.
"From the outside, it's hard to tell what went wrong and, listening to the drivers and engineers, it seems that even within the team they have not yet been able to identify what on earth happened," Brawn said. "Neither Vettel nor Leclerc were ever really competitive at any point over the weekend.
Brawn, who guided Ferrari to five straight F1 world title doubles between 2000-2004 with Michael Schumacher, feels the surprise performances were complicated further by the impressive pace from the Ferrari-powered Haas squad who led the F1 midfield battle and edged towards the factory squad. Haas driver Kevin Magnussen finished the Australian Grand Prix less than 20 seconds behind Leclerc in sixth place.
"Apart from lagging behind Mercedes, more surprising was the gap to the Haas team. The US squad uses the Ferrari Power Unit and had closed on the works team considerably since Barcelona testing," he said.
"Even if it's true that the Albert Park circuit is unusual and does not really reflect the true pecking order, it's also true that the same number of points is given out after the race than at any other track.


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Salary Report for Psychology Teachers in Greensboro-High Point
Home / Comprehensive Psychology Salary Reports 2023 / 2023 Psychology Teachers Salary Report / 2023 Salary Report for Psychology Teachers in North Carolina /
Rankings for Greensboro-High Point
Compared to North Carolina
FAQ for Greensboro-High Point Psychology Teachers
The number of Psychology Teachers currently employed in Greensboro-High Point is 80.
Psychology Teachers in Greensboro-High Point earn an average of $75,110 per year or $36.11 per hour.
The percentage of North Carolina Psychology Teachers who work in Greensboro-High Point is less than 1%.
The percentage of U.S. Psychology Teachers who work in Greensboro-High Point is less than 1%.
Ranks #3 in North Carolina for salary and #8 when adjusted for cost-of-living.
Ranks #3 in North Carolina for the number of psychologists employed.
Greensboro-High Point Psychology Teachers earn an average of $75,110 or $36.11 per hour which is 7% less than the North Carolina average salary of $80,960.
Greensboro-High Point
Greensboro-High Point Psychology Teachers earn an average of $75,110 or $36.11 per hour which is 15% less than the U.S. average salary of $88,390.
When adjusted for cost-of-living, Psychology Teachers in Greensboro-High Point earns an average annual salary of $83,086 which compares to the North Carolina average annual salary of $80,960. per year.
Cost-of-living in Greensboro-High Point is 10% less than the North Carolina average, which means Greensboro-High Point residents have an overall less cost for food, housing, utilities and transportation compared to North Carolina average.
North Carolina average salary is $80,960.
North Carolina cost-of-living adjusted salary is $88,153.
Greensboro-High Point average salary is $75,110.
Greensboro-High Point cost-of-living adjusted salary is $83,086.
Greensboro-High Point $75,110 90.40% $83,086
North Carolina $80,960 91.84% $88,153
In Greensboro-High Point, 50% of Psychology Teachers earn between $60,150 and $81,980 a year, depending on factors such as level of education, experience, location and place of employment.
About 10% of Psychology Teachers earn more than $104,270, which is a rate for highly skilled, experienced and educated Psychology Teachers.
Approximately 10% of Psychology Teachers earn under $47,470, a rate generally reserved for less experienced or entry level Psychology Teachers.
With an average annual salary of $75,110, Greensboro-High Point Psychology Teachers tend to earn less than Clinical and Counseling Psychologists ($101,350).
However, Psychology Teachers tend to earn more than Social Workers ($50,290), Substance Abuse Counselors ($52,590), and Health Care Social Workers ($63,710).
How Many Psychology Teachers Work in Greensboro-High Point?
How Much Do Psychology Teachers Earn in Greensboro-High Point?
Psychology Teachers earn an average of $75,110 per year or $36.11 per hour in Greensboro-High Point.
How Much do Psychology Teachers Earn when Adjusted for Cost-of-Living in Greensboro-High Point?
When adjusted for Cost of Living in Greensboro-High Point Psychology Teachers earn an average of $83,086 per year or $40 per hour.
Which Psychologists Are Paid the Highest Annaul Salary in Greensboro-High Point?
The types of psychologists that are paid the highest annual salary in Greensboro-High Point are Clinical and Counseling Psychologists($101,350), Psychology Teachers($75,110), Health Care Social Workers($63,710).
Additional Salary Reports for Psychology Teachers in North Carolina cities and Metros and U.S. States.
Official Data Sources for Psychology Teachers: Salary Data | Projected Employment | Cost of Living Data


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2021 "OsteoCHAMPS in a Box": A look at this year's virtual program
We talk to Mollie Mfodwo, M.S., manager of pre-college programs, about OsteoCHAMPS.
Q: Can you give a quick overview of OsteoCHAMPS?
A: OsteoCHAMPS was established in 2000, so it's been running for over 20 years. We've served 650 sophomore and junior high school students from across the nation, although the majority of the students are from Michigan.
The program was launched to help create a pathway to osteopathic medical school by exposing students from underserved backgrounds to osteopathic medicine and to open doors to learn about the osteopathic approach that may not have been afforded to them otherwise.
Over half of students who attend OsteoCHAMPS go on to participate in our undergraduate Osteopathic Medical Scholars Program, which continues to foster their interest and skill sets to get them ready for medical school, and over 90% of those OMSP students go on to matriculate into the College of Osteopathic Medicine.
We're unique in the way we run our pre-college program—it's longer than other med schools' programs, and brings together peers from across the nation. We also work with community partners and donors to fund scholarships for students to ensure the program is truly accessible.
Q: Tell us about the 2021 cohort.
A: The 2021 cohort consists of 60 students, the same as our in-person trainings. This year, all the students hail from Michigan. This year's group is very bright and engaged, which is reflective of our rigorous applicant review process—we get over 100 applications for 60 spots. All the students had their video on and asked insightful questions in every session.
Q: How many faculty, staff, and students are involved in putting on OsteoCHAMPS?
A: Over 50 faculty, staff, SpartanDO students, and OMSP undergraduate students are involved in the program. All sessions are taught by faculty or staff, and all volunteer their time. Med students drop in to different sessions to provide more insights, and OMSP students tutor the high school students throughout the week.
We also bring in as many diverse faculty and students as we can so all the participants have someone to connect to. It's important for students to see someone they can relate to and aspire to be. We talk about health disparities, bias, the importance of paying attention to patients, and other hard conversations.
Q: How did you engage students in this year's virtual format?
A: We built on the lessons we learned from running the program virtually for the first time last year due to the pandemic. The faculty pulled together to transition the sessions virtually. This year, we mailed students an "OsteoCHAMPS In A Box" kit, providing tools to facilitate their learning in every session. For the osteopathic manipulative medicine (OMM) training session, students received mini skulls used to showcase parts of our anatomy where OMM can be used to treat various illnesses. Students also palpated their own skin or a family member during the session. For clinical skills, they received future doctor kits that contained stethoscopes and other medical tools. Students were able to listen to their own heartbeat and work with medical students on other skills.
We also divided the students into six groups to work on research projects, led by DO/PhD student, Luca Kaiser. Students use google drive and google slides to collaborate, and at the end they'll present at a virtual ceremony. This gives them more insight into higher level research skills.
The students get additional tutoring and mentorship from an undergraduate OMSP tutor counselor throughout the week to increase their engagement. They're able to interact with faculty, staff, medical students, undergraduate students, program alumni, and each other to get a wide range of perspectives related to the field.
All the students this week have really enjoyed the program.
Q: What's unique about this year's program?
A: We shortened the program to avoid Zoom fatigue. The in-person program is eight days long with packed, 12-hour days. The virtual format runs for five hours per day (10 am – 3 pm), for five days (Monday through Friday). It's well-received so far.
The theme for this year is Mental Health and Wellness—it's pertinent to the experiences associated with the pandemic, and we address physician burnout, mental health conditions, and the impact of social media on mental health. The kits we mailed home included wellness items, such as an adult coloring book and bubbles, for wellness activities led by Dr. John Taylor and Dr. Miko Rose. We also showed students some wellness routines they can practice at home, such as a daily Tai Chi session led by an OMS-II student.
Q: How do you continue to engage students after the program closes?
A: Program participants are automatically enrolled into our Alumni Association, which includes graduates of all our pre-college programs. There's lots of benefits, and we get to engage with them again if they join OMSP in undergrad. We're also looking to expand our post-program engagement through monthly check-ins.
Q: What's next for OsteoCHAMPS?
A: We're planning to launch an OsteoCHAMPS in the Upper Peninsula next summer to make the program more accessible to rural students. We're partnering closely with War Memorial Hospital on that project.
Q: Any other messages you'd like to share?
A: We're always looking for new partners and to get the word out about the program.
Article posted on: July 16, 2021


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Q: Java: наследование контркуктора Объясните дураку(в основном работаю с php), почему при наследовании исключений, не наследуются все конструкторы.
Кейс №1
Есть класс расширяющий исключения(без конструктора)
package test.exceptions;
public class StatusException extends Exception {
    final public static String STATUS_NOT_EXIST = "Status: not exist";
}

Есть код выкидывающий исключение:
public static void test() throws StatusException {
    String message= StatusException.STATUS_NOT_EXIST;
    throw new StatusException(message);
}

При компиляции получаю ошибку о несоответствие сигнатур конструкторов
Error:(26, 15) java: constructor StatusException in class test.exceptions.StatusException cannot be applied to given types;
  required: no arguments
  found: java.lang.String
  reason: actual and formal argument lists differ in length

Кейс №2
Исключение с конструктором принимающим String
package test.exceptions;
public class StatusException extends Exception {
    final public static String STATUS_NOT_EXIST = "Status: not exist";

    public StatusException(String message) {
        super(message);
    }
}

Код выкидывающий исключение:
public static void test() throws StatusException {
    String message= StatusException.STATUS_NOT_EXIST;
    throw new StatusException();
}

Ошибка
Error:(24, 15) java: constructor StatusException in class test.exceptions.StatusException cannot be applied to given types;
  required: java.lang.String
  found: no arguments
  reason: actual and formal argument lists differ in length

A: Конструкторы не наследуются.
Касательно первого примера - вы должны явно создать конструктор с параметром:
public class StatusException extends Exception {

    final public static String STATUS_NOT_EXIST = "Status: not exist";

    public StatusException(String message) {
        super(message);
    }

}

Касательно второго примера.
Если явно не задан ни один конструктор, то по умолчанию создается пустой конструктор без параметров. То есть в первом примере у вас будет неявно создан такой конструктор. Если же явно определен хотя бы один конструктор, то конструктор без параметров следует так же явно указать. Следовательно, для второго примера нужно так:
public class StatusException extends Exception {
    final public static String STATUS_NOT_EXIST = "Status: not exist";

    public StatusException(String message) {
        super(message);
    }

    public StatusException() {
        //nothing or call super
    }
}


Helpful Links | John W. Tyrone M.D.
Cosmetic surgery should not be used as a weight loss program. Dr. Tyrone recommends patients reach their weight loss goals prior to surgery in order to have optimal surgical results. BMI can be used to indicate if you are overweight, obese, underweight or normal. A healthy BMI score is between 20 and 25. A score below 20 indicates that you may be underweight; a value above 25 indicates that you may be overweight.
Recommended sports bra to wear when you begin working out after surgery.
After surgery patients are anxious to start working out again. When Dr. Tyrone gives you the approval to start working out, you now must take care of your new breasts. Look for a proper support bra. Our patients have found this bra meets all of their needs.
Gainesville, Florida Skin Care and Cosmetic Enhancement practice.
Lots of useful information for patients considering Plastic Surgery.
The only board in the field of Plastic Surgery recognized by the American Board of Medical Specialties. This includes Plastic and Cosmetic Surgery of the breast and body, as well as Facial Plastic Surgery. Board certification is one of the most important distinctions that a doctor can receive.
Visit my colleague Dr. Joshua Greenwald, New York board-certified plastic surgeon specializing in facial plastic surgery, breast enhancement and body contouring for Westchester County, New York.
Visit my colleague Dr. Kent Hasen, Florida board-certified plastic surgeon specializing in facial plastic surgery, breast enhancement and body contouring for Naples and Fort Myers, Florida.
The official website of BOTOX® Cosmetic, the popular injectable that is approved for the temporary treatment of moderate to severe frown lines between the brows.


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Glyn Y Coed
Tel: 01766 522 870 or Email: julie@gychotel.co.uk
Snowdonia | Trains
The Ffestiniog Railway is a narrow gauge heritage railway, located in Gwynedd in Wales. It's one of the more popular attractions in the Snowdonia National Park.
The railway is about 21.7 km long and runs from the harbour at Porthmadog to the slate mining town of Blaenau Ffestiniog. The line travels through spectacular mountainous scenery and is single track with passing places. The track gauge is 1 ft 11+1?2 in (597 mm). The first part of the line runs along a 1 mile (1.6 km) long embankment called "the Cob", which is the dyke of the Traeth Mawr "polder". From 1949, various groups of rail enthusiasts attempted to revitalise the railway. Eventually, on 24 June 1954 a group of volunteers funded by Alan Pegler purchased the company to run it as a tourist attraction, and gradually restored the line to working order.
For more information see: http://www.ffestiniograilway.co.uk/
A Majestic Mountain Top Adventure to the Summit of Snowdon
Welcome to Snowdon Mountain Railway Majestic Mount Snowdon dominates the glorious, ancient landscape of North Wales. At 3,560ft (1,085m) it is a true mountain and a place of legend ? said to be the burial place of the giant ogre Rhita, vanquished by King Arthur. Some believe that Arthur?s Knights still slee… Continue reading at http://www.snowdonrailway.co.uk/the_journey.php
Trains don't just go along the flat, they can go up mountains too. taking one up Snowdon is highly recommended, Tickets must be booked in advanced.
Weather permitting trains will run every day from late-March to the first week of November. From March to Mid-June trains will terminate at either Clogwyn or Rocky Valley (subject to winter weather conditions). The Summit Station is the usual destination between Mid-June and the end of October. Trains may be operated by steam or diesel traction.
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