"It's my daughter's sweet 16, so we let her choose the family vacation, and she wanted to come back here," says the woman at one of the picnic tables during the Friday night beach party. When she finishes her dinner, her two girls pull her and their dad to the dance floor (dance sand?), where hundreds of kids, parents, and grandparents are boogeying and smiling at each other. Hey, what ever happened to cranky kids, family squabbles?
Cranky is in short supply at Beaches Turks & Caicos Resort Villages & Spa. This 614-room/suite all-inclusive manages to please kids, teens, parents, and grandparents, a not-so-minor miracle. Located on Providenciales' Grace Bay, a 12-mile strand of snow-white sand, this Sandals-operated resort includes three distinct "villages"—Caribbean, French and Italian—plus seaside villas, excellent food and libations, a long menu of events and recreational options for all ages, a spa, and superior children's programs, and all but the spa are included.
Lodgings feature kids' beds and range from doubles to 3-bedroom suites with butlers, and the suites in the Italian Village are standouts, with child-pleasing kids' rooms that have bunks and/or trundles, a TV, an Xbox, and a scaled-down toilet and sink. The mahogany four-poster in each master bedroom, the 42-inch TV, the well-stocked bar, and other amenities bespeak an attention to adults not often associated with family resorts.
Wharton says that Beaches Turks & Caicos' Luxury Included vacations initially strike some clients as expensive, but guests get a lot for their money, which is why the World Travel Awards has named this resort "The Caribbean's Leading Family All-Inclusive" 14 years in a row. The 16 dining options, for example, offer quality cuisine, whether it's continental or Caribbean, Southwestern or Italian. Some venues are casual while others request "resort evening wear"; there are buffets (Giuseppe's), adults-only restaurants (Sapodilla's), and child-oriented eateries (Mario's). Only one restaurant, Kimono's, requires reservations—and it's expanding to accommodate more diners. The ever-dangerous Cafe de Paris tempts guests all day with pastries that seem unnecessarily delicious for an all-inclusive, and the Sunday night chocolate buffet could serve as your primary selling point.


قام اليوم الخميس ملثمون بالاعتداء على خطيبة منسق اعتصام الكامور طارق الحداد، قائلين لها بانها رسالة للحداد كي يبتعد عن الاحتجاجات حسب ما افاد به مراسلنا بالجهة.
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text:
College football is bad football
David Marcus 8:00 PM on August 26, 2019
The fact that 100,000 thousand adults will file, full of booze, into Michigan's stadium to watch 19-year-olds play a pathetic version of a man's game should trouble us. The millions, maybe billions of dollars that adults spend betting on it should too. Aggrandizing a mediocre display of athletics does not accrue to the common good.
What's going on here is cultural lag. In the first half of the 20th century, college football was football. The NFL was a distant dream and teams like West Point and the University of Pennsylvania were dominant forces. But as the game gained popularity, a better variety emerged. Professional football came into being and created a game that makes college look like a Foosball tournament compared to the World Cup.
Look, I don't want to be the Grinch. I don't want to take joy out of anybody's life. If you like watching an inferior game played by kids, that's your right. But don't pretend it's some pure version of the game. It isn't. It's an inferior version of the game that is usually as dull as plain donuts. Yay! Let's all get excited about Alabama's 900-point victory over whatever tech!
Read More From David Marcus
Project Veritas: Pfizer isn't going to like this at all

redpajama_set_name: RedPajamaCommonCrawl


PRAGUE (AP) — Petra Kvitova says she's determined to play tennis again — no matter what that involves after a knife-wielding intruder attacked her inside her home and injured her playing hand.
With a big bandage covering her left hand and forearm, Kvitova said she was told by the surgeon who operated on her in a specialized clinic in the northern town of Vysoke nad Jizerou on Tuesday that her recovery is "looking good and without complications."
"I feel better day by day," she told reporters in Prague.
"Yesterday morning during a session (with) the doctor, I was able to move the fingers on my left hand which I think ... was the greatest Christmas present I could have wished for."
"I never received so many (supportive) messages in my life at a moment," she said. "I really appreciate that."
Martina Navratilova was among those who wished Kvitova "a speedy recovery."
"We can't wait to see you back on court again," the tennis great tweeted.
"I don't care about how much time it takes, three months, six months or a year," she said. "But that's for sure I want to return one day and I'll do all I can to make that happen. I had no doubt about my return to the tennis circuit for a second."
Kvitova didn't take questions but said she didn't want to look back and talk about Tuesday's incident. The attacker is still at large.


Households in the UK and elsewhere around the world are being asked by their respective governments to reduce carbon emissions but those same governments continue to back the top 200 companies that hold more carbon reserves than can ever be safely burned. According to a report by the BBC quoting University College London academics, it would be virtually impossible to meet goals in lowering carbon emissions if that oil, coal and gas were burned – EVER!
University College London warns that 30% of the world's oil, 55% of global gas reserves and 80% of coal reserves must not ever be burned if we are to even come close to meeting carbon emission reduction goals. Knowing that we already have reserves too great to safely use, why are governments continuing to invest in further exploration? It's like speaking with a forked tongue – don't use fossil fuels but put all your money into finding greater stores. Does this make sense?
The BBC notes that it was the mission of one lone man, a former investment manager, Mark Campanale, began wondering how much the world held in carbon reserves. Carbon reserves are simply a way of lumping all fossil fuel reserves into a single category. This includes gas, oil and of course, coal. Campanale was actually shocked that no one had yet tallied the unbelievable amount that was stockpiled across the globe.
The research that was commissioned by Campanale shows that there is absolutely no way to meet the goal of holding the rise in global temperature to under 2C if we were to consume that amount of fossil fuels. Referring to the above mentioned figures, it is astounding that so many of the 200 big energy firms are still seeking out new supplies when there is already so much stored. Now other interested institutions are calling for a halt to the mindless exploitation of resources. Amongst those are Harvard University and 350.org, a highly vocal pressure group.
With So Much in Store Why Do Prices Continue to Rise?
Consumers across the UK continue to look for the best deals on electricity because the cost continues to skyrocket. Producers claim that the reason tariffs continue skyrocketing is due to the cost of production underscored by higher fossil fuel prices. It appears that fossil fuels are being artificially priced as there is no need to keep them that high with so much in reserve. So then, why does the cost of energy continue to rise? Again, it must be artificial manipulation of the market – what else can it be?
According to some of the world's leading energy market analysts, those who are investing in carbon products might soon be in for a rude awakening. With increasing global pressure to our carbon footprint by producing electricity with renewable sources, there may come a day in the not so distant future where fossil fuels will become obsolete in the production of energy. Those same analysts are suggesting that investments in renewables is a far smarter choice for the long term.
What does this all mean to the average consumer who isn't looking to invest but rather just wants cost effective energy? Perhaps the most intelligent choice to make when looking for the best tariffs would be to find a supplier that has heavy investments in the renewables sector. It just might be that their prices will provide greater stability when the fossil fuel bubble bursts.


text:
\section{Introduction}
\label{sec:intro}

\subsection{SYM$_4$ and integrable spin chains}
The problem of computing the conformal dimensions of local,
gauge-invariant single-trace composite operators in planar
$\mathcal N \! = \! 4$ supersymmetric Yang-Mills theory
in (3+1)-dimensions, SYM$_4$, is integrable
\cite{Minahan:2002ve, Beisert:2003yb, Beisert:2005fw}.
At 1-loop level, the mixing matrix $\Gamma$ maps
to the Hamiltonian $\mathcal H_{PSU(2,2 | 4)}$ of an integrable
$PSU(2,2 | 4)$-symmetric spin chain with nearest-neighbor
interactions and periodic boundary conditions, such that
{\bf 1.} The eigenstates $\{\mathcal O\}$ of $\Gamma$ are in
one-to-one correspondence with the eigenstates
$\{ | \mathcal O \rangle \}$ of $\mathcal H_{PSU(2,2 | 4)}$, and
{\bf 2.} The eigenvalues $\{\gamma \}$ of $\Gamma$,
which are the anomalous dimensions of $\{\mathcal O\}$,
are equal to the eigenvalues $\{\mathcal E \}$ of
$\mathcal H_{PSU(2,2 | 4)}$
\footnote{\ In this note, $\mathcal O$ is a local gauge-invariant
single-trace composite operator, in SYM$_4$ or in QCD depending
on context, that is an eigenstate of the mixing matrix
$\Gamma$, with anomalous dimension $\gamma$.
For brevity, we will refer to $\mathcal O$ from now on simply
as {\it \rq a single-trace operator\rq}.
$| \mathcal O \rangle$ is the corresponding eigenstate of
the integrable spin chain Hamiltonian $\mathcal H$, whose
eigenvalue $\mathcal E$ $=$ $\gamma$. The notation $\{\mathcal O\}$
stands for sets of single-trace operators,
{\it etc.}}.
Since the eigenstates and eigenvalues of
$\mathcal H_{PSU(2,2 | 4)}$ can be computed using Bethe ansatz methods,
the problem is integrable
\footnote{\ The situation at higher loops is more complicated:
Spin chains with nearest-neighbor interaction are replaced
with spin chains with long range interactions, the algebraic
Bethe ansatz is replaced with an asymptotic Bethe ansatz, and
finite-size effects must be accounted for. In this note, we
restrict our attention to 1-loop level and nearest-neighbor
interacting spin chains.}.
For a recent review, see \cite{Beisert:2010jr} and references
therein.

\subsection{SYM$_4$ $SU(2)$-doublets and spin-$\frac{1}{2}$
chains}
SYM$_4$ contains a vector gauge field, four
chiral and four anti-chiral spin-$\frac{1}{2}$ fermions,
and six real scalars that can be expressed as three
complex scalars
$\{  X,   Y,   Z\}$ and their charge-conjugates
$\{\bar{X}, \bar{Y}, \bar{Z}\}$.
Any two complex scalars that are not charge
conjugates, such as $\{X, Z\}$ or $\{X, \bar{Z}\}$, mix only
amongst themselves to form an $SU(2)$-doublet and
an $SU(2)$-invariant scalar subsector of SYM$_4$.
In the planar limit at 1-loop level,
the single-trace operators $\{\mathcal O\}$, that are
composed of a single $SU(2)$ doublet, and that are eigenstates
of $\Gamma$, map to eigenstates $\{| \mathcal O \rangle\}$ of
the Hamiltonian $\mathcal H_{\frac{1}{2}}$ of a periodic XXX
spin-$\frac{1}{2}$ chain.

\subsection{QCD $SU(2)$-triplets and spin-1 chains}
In \cite{Ferretti:2004ba}, Ferretti, Heise and Zarembo noted that,
at 1-loop level, operators composed of self-dual components
$\{f_{+}, f_{0}, f_{-}\}$ of
the QCD field strength tensor mix only among themselves to
form an $SU(2)$-triplet. Using
that observation, as well as the fact that QCD with no matter
fields is conformally invariant (the beta function vanishes)
in the planar limit at 1-loop level, they showed that local
single-trace operators
$\{\mathcal O\}$ that are eigenstates of $\Gamma$ correspond to
eigenstates of the Hamiltonian $\mathcal H_1$ of an integrable XXX
spin-1 \cite{Zamolodchikov:1980ku}
chain. \footnote{\ All spin chains mentioned in this note will be
integrable (their $R$-matrices satisfy Yang-Baxter equations),
of XXX type (their $R$-matrices are parametrized by rational
functions in the rapidity variables), and satisfy periodic
boundary conditions, hence we need not repeat this from now
on.}
As in the spin-$\frac{1}{2}$ case,
the spin-1 chain eigenstates
and eigenvalues can be computed using Bethe ansatz methods
\cite{Kulish:1981gi, Kulish:1981bi, A.:1982zz, Babujian:1983ae}.

\subsection{SYM$_4$ structure constants}
Following
\cite{Lee:1998bxa, Okuyama:2004bd, Roiban:2004va, Alday:2005nd},
Escobedo, Gromov, Sever and Vieira
\cite{Escobedo:2010xs}
used the connection to spin-$\frac{1}{2}$ chains to obtain a sum
expression for the structure constants of 3-point functions of
single-trace operators $\{\mathcal O\}$ in $SU(2)$
scalar subsectors of SYM$_4$. They noted that the
three operators
$\mathcal O_i$, of lengths $L_i$, $i \in \{1, 2, 3\}$, could be chosen
to be non-BPS (their conformal dimensions are unprotected by
supersymmetry) and non-extremal ($L_i < L_j + L_k$, for any
choice of distinct $i$, $j$ and $k$).

In \cite{Foda:2011rr}, the sum expression of Escobedo {\it et al.}
was evaluated in determinant form. This was made possible by
the fact that, when expressed in spin chain terms, the essential
factor in the sum expression can be identified with (a special
case of) the scalar product of an eigenstate of $\mathcal H_{\frac{1}{2}}$
and a generic state (not an eigenstate of $\mathcal H_{\frac{1}{2}}$).

\subsection{QCD structure constants}
In this note, we extend the results of
\cite{Escobedo:2010xs, Foda:2011rr},
from SYM$_4$ and spin-$\frac{1}{2}$ chains to QCD and spin-1 chains,
to gain information about QCD operator product expansions, OPE's,
of the operators $\{\mathcal O\}$ of Ferretti {\it et al.}
\footnote{\ Our results are subject to the same restrictions as
in \cite{Ferretti:2004ba}, and are valid only in
the planar limit ($N_{c} \rightarrow \infty$
and $g \rightarrow 0$, with $\lambda=g^{2} N_{c}$ constant)
and at one-loop level,
so that the beta function vanishes, and the theory is conformally
invariant.}.

We show that
{\bf 1.} In the general case where all three operators
$\mathcal O_i$, $i \in \{1, 2, 3\}$ are non-BPS-like (all three states
map to eigenstates of $\mathcal H_1$ that are not spin-chain reference
states), the tree-level structure constants can be expressed in
a sum form that is similar to, but even less restricted than that
of Escobedo {\it et al.}
\footnote{\ The sum form of Escobedo {\it et al.} involves
a summation over all partitions of one set of rapidity
variables. The sum form that we obtain in the general
case of three non-BPS-like operators involves
summations over all partitions of three sets of rapidity
variables with constraints between them.}
{\bf 2.} In the special case where one operator, {\it e.g.}
$\mathcal O_3$, is BPS-like  (it maps to a spin-chain reference state),
the tree-level structure constants can be expressed in a determinant
form that is similar to that in \cite{Foda:2011rr}.

In other words, to express the tree-level structure constants in
determinant form, (at least) one of the three operators must be
BPS-like. In the following subsection, we outline why this is
the case. More details are given in Section
{\bf \ref{sec:structureconstants}}.

\subsection{SYM$_4$ structure constants that can be evaluated
as determinants}
The SYM$_4$ structure constants studied in
\cite{Escobedo:2010xs, Foda:2011rr}
involve four types of scalars, $\{X, Z, \bar X, \bar Z\}$.
The only non-vanishing Wick contractions (2-point functions)
are those between charge-conjugate pairs, that is
$\langle   X \bar{X} \rangle$,
$\langle \bar{X}   X \rangle$,
$\langle   Z \bar{Z} \rangle$, or
$\langle \bar{Z}   Z \rangle$.
Each operator $\mathcal O_i$, $i \in \{1, 2, 3\}$, consists
of two types of non-conjugate scalars, that is
$\{X,     Z\}$,
$\{X,   \bar{Z}\}$,
$\{\bar{X},   Z\}$, and
$\{\bar{X}, \bar{Z}\}$.

If
$\mathcal O_1$ is $\{ X, Z\}$-type
(a composite operator of scalars of type $\{ X, Z\}$), and
$\mathcal O_2$ is $\{\bar{X}, \bar{Z}\}$-type,
there are non-zero Wick contractions of both types,
$\langle X \bar{X} \rangle$ and
$\langle Z \bar{Z} \rangle$, between $\mathcal O_1$ and $\mathcal O_2$.
Now consider $\mathcal O_3$. There is no way to choose the
scalar content of $\mathcal O_3$ such that
{\bf 1.} It has non-zero Wick contractions of both types with $\mathcal O_1$,
{\bf 2.} It has non-zero Wick contractions of both types with $\mathcal O_2$,
and
{\bf 3.} The 3-point function is non-extremal, which requires
that $\mathcal O_3$ has non-zero Wick contractions with both $\mathcal O_1$
and $\mathcal O_2$.
The only way to have a non-extremal 3-point function is
to choose $\mathcal O_3$ to be
$\{\bar{X},   Z\}$-type or
$\{  X, \bar{Z}\}$-type.
Either way, the Wick contractions between
$\mathcal O_1$ and $\mathcal O_3$ will be of one type only,
and the Wick contractions between
$\mathcal O_2$ and $\mathcal O_3$ will also be of one type only,
different from that between $\mathcal O_1$ and $\mathcal O_2$.
These constraints simplify the structure constant
and allow one to evaluate the sum form
of Escobedo {\it et al.} in determinant form.

\subsection{QCD structure constants that can be evaluated
as determinants}
The QCD structure constants studied in this note
involve three types of scalars, $\{f_{+}, f_{0}, f_{-}\}$.
The non-vanishing Wick contractions are those between
spin-conjugate pairs, that is
$\langle f_{+} f_{-} \rangle$,
$\langle f_{-} f_{+} \rangle$,
and
$\langle f_{0} f_{0} \rangle$.

Since the action of the Bethe creation operators on the spin-1
reference states generates all three scalars, each operator
$\mathcal O_i$, $i \in \{1, 2, 3\}$, will consist of all three scalars.
Consequently, there are no constraints on the Wick contractions,
and the 3-point function of non-BPS operators is more complicated
than in the SYM$_4$ case
\footnote{\ In particular, while integrable spin-1 chains
are related to integrable spin-$\frac{1}{2}$ chains by fusion,
there is no way that one can use fusion to obtain a 3-point
function of non-BPS-like operators in the spin-1 case from the
corresponding spin-$\frac{1}{2}$ result.}.
This 3-point function between three non-BPS-like operators can
be expressed in sum form, as we will explain in the sequel, but
that sum form will be more complicated than that
in \cite{Escobedo:2010xs}, and less useful.

The aim of this note is to identify the structure constants
that can be evaluated in single determinant form using currently
available methods of integrability
\footnote{\ What we have in mind is Slavnov's determinant
expression for the scalar product of an eigenstate of
the Hamiltonian and a generic state. This determinant
expression is unique. It is conceivable that
determinant expressions for more general scalar products,
that will allow us to evaluate more general structure
constants, will eventually be found, but this is obviously
beyond the scope of this work.}.
Our result is that, in QCD and the spin-1 case, determinant
expressions for the structure constants require that one
operator is BPS-like. In other words, that it maps to a
spin chain reference state.

\subsection{Outline of contents}
In Section {\bf \ref{sec:ops}},
we review the construction of the single-trace
composite operators from the self-dual components of the field
strength tensor, the 1-loop mixing matrix, operator product
expansions, and the {\it \lq tailoring\rq} approach of Escobedo
{\it et al.} to the structure constants.
In Section {\bf \ref{sec:BA}},
we recall the algebraic Bethe ansatz solution for the eigenstates
and eigenvalues of the mixing matrix.
In Section {\bf \ref{sec:structureconstants}},
we present our results for the structure constants in terms of
solutions of the Bethe equations. Section {\bf \ref{sec:discussion}}
contains a brief discussion.
In Appendix {\bf \ref{sec:CBA}}, we recall the coordinate Bethe
ansatz and the $\mathcal F$-conjugation of \cite{Escobedo:2010xs}.
In Appendix {\bf \ref{sec:scalarproducts}}, we present the
scalar products that appear in the expression for the structure
constants.

\section{Composite operators, operator product
expansions and structure constants}\label{sec:ops}

\subsection{Self-dual field-strength components as an $SU(2)$-triplet}
Following \cite{Ferretti:2004ba}, we decompose the QCD Yang-Mills
field strength tensor
$F_{\mu\nu}=
\partial_{\mu} A_{\nu} -
\partial_{\nu} A_{\mu} + i g
\left[A_{\mu} \,, A_{\nu}\right]$
into self-dual, $f_{\alpha\beta}$, and
anti-self-dual, $\bar{f}_{\dot\alpha\dot\beta}$, components,

\begin{eqnarray}
F_{\mu\nu} = \sigma_{\mu\nu}^{\ \ \alpha\beta}f_{\alpha\beta}+
\bar{\sigma}_{\mu\nu}^{\ \ \dot\alpha\dot\beta}
\bar{f}_{\dot\alpha\dot\beta}\,,
\end{eqnarray}

\noindent where

\begin{multline}
\sigma_{\mu \nu} =
\frac{i}{4}
\sigma_{2}
\left(\sigma_{\mu}\bar{\sigma}_{\nu}
-
\sigma_{\nu}\bar{\sigma}_{\mu}\right),
\
\bar{\sigma}_{\mu\nu} =
- \frac{i}{4} \left(\bar{\sigma}_{\mu}\sigma_{\nu}
-\bar{\sigma}_{\nu}\sigma_{\mu}\right)\sigma_{2},
\
    {\sigma}_{\mu} = (1,  \vec \sigma),
\
\bar{\sigma}_{\mu} = (1, -\vec \sigma) \,.
\end{multline}

\noindent We further define
\begin{eqnarray}
f_{A} =
\left(
\sigma_{2} \sigma_{A}
\right)^{\alpha \beta}
f_{\alpha\beta}\,, \qquad
\bar{f}_{\dot A} =
\left(
\sigma_{\dot A}\sigma_{2}
\right)^{\dot \alpha \dot \beta}
\bar{f}_{\dot\alpha\dot\beta}\,,
\end{eqnarray}

\noindent where $A, \dot A = 1,2,3$.
The 2-point function of the field
strength tensor has the structure

\begin{eqnarray}
\label{FF2ptfunction}
\langle
F_{\mu \nu\    b}^{\ \ a}(x)
F_{\rho\sigma\ d}^{\ \ c}(0)
\rangle =
\phi(x)
\left( \eta_{\mu\rho} \eta_{\nu\sigma} - \eta_{\mu\sigma}
\eta_{\nu\rho} \right)\delta^{a}_{d}\delta^{c}_{b}\,,
\end{eqnarray}

\noindent where $a,b,c,d = 1, \ldots, N_{c}$ are color indices,
and $\phi(x)$ is a scalar function.  Hence,

\begin{eqnarray}
\langle f_{A\ b}^{\ a}(x) f_{B\ d}^{\  c}(0) \rangle
=
\phi(x) \delta_{AB}\delta^{a}_{d}\delta^{c}_{b}
\,, \qquad
\langle
f_{A\ b}^{\ a}(x) \bar{f}_{\dot B\ d}^{\  c}(0)
\rangle = 0 \,.
\label{f2pt}
\end{eqnarray}

\noindent Following \cite{Beisert:2004fv}, we write

\begin{multline}
f_{+} = f_{11} =\frac{1}{2} \left( f_{2} + i f_{1} \right),
\ \
f_{0} = \frac{1}{\sqrt{2}} \left( f_{12} + f_{21} \right) =
-\frac{i}{\sqrt{2}} f_{3},
\ \
f_{-} = f_{22} =\frac{1}{2} \left( f_{2} - i f_{1} \right) \,.
\end{multline}

\noindent From Equation ({\bf \ref{f2pt}}),
$\langle f_{\pm}(x) f_{\pm}(0) \rangle =
 \langle f_{\pm}(x) f_{0}  (0) \rangle = 0$,
and the only nonzero Wick contractions (2-point functions)
are
$\langle f_{\pm} f_{\mp}\rangle$ and
$\langle f_{0}   f_{0}  \rangle$.
$\{f_{+}, f_0, f_{-}\}$
is an $SU(2)$ triplet, and transforms in the spin-1
representation of $SU(2)$.

\subsection{Single-trace operators from the self-dual
components}
We focus on the single-trace operators
of length $\ensuremath{L}$ that are composed of self-dual components
only

\begin{eqnarray}
{\mathcal O}(x) = \operatorname{tr} \left\lgroup f_{A_{1}}(x) \cdots f_{A_{\ensuremath{L}}}(x) \right\rgroup \,.
\label{ops}
\end{eqnarray}

\noindent Following \cite{Ferretti:2004ba}, at 1-loop level, in the planar
limit, these operators mix only among themselves, as in Equation
({\bf \ref{f2pt}}), and their mixing matrix is given by

\begin{eqnarray}
\Gamma = \frac{\lambda}{48\pi^{2}}\sum_{l=1}^{\ensuremath{L}}
\left\lgroup
7 + 3 \vec S_{l} \cdot \vec S_{l+1}
- 3 (\vec S_{l} \cdot \vec S_{l+1})^{2}
\right\rgroup  \,,
\label{Hamiltonian}
\end{eqnarray}

\noindent where $\lambda=g^{2} N_{c}$, and
$\vec S_{l}$ are $SU(2)$ spin-1 generators,

\begin{equation}
\left\lgroup S^j f \right\rgroup_A = -i \epsilon_{j A B} f_{B} \,.
\end{equation}

\noindent Note that $\{ f_{+}, f_{0}, f_{-}\}$ are eigenstates
of $S^{3}$ with eigenvalues $\{+1, 0, -1\}$, respectively. Since
$\Gamma$ commutes with $\vec S^{2}$ (where
$\vec S = \sum_{l=1}^{\ensuremath{L}} \vec S_{l}$ is the total spin), and
$S^{3}$, all three operators can be diagonalized simultaneously.
An eigenstate of $\Gamma$ is an operator of definite conformal
dimension $\Delta = 2 \ensuremath{L} + \gamma$, where $\gamma$ is the
corresponding eigenvalue.

\subsection{Operator product expansion of single-trace
operators}

Following \cite{Escobedo:2010xs,Foda:2011rr}, we normalize
the operators of definite conformal dimension according to

\begin{eqnarray}
\langle {\mathcal O}_{i}(x_{i})\,  \bar {\mathcal O}_{j}(x_{j}) \rangle
\sim
\left\lgroup
{\mathcal N}_{i} {\mathcal N}_{j}
\right\rgroup^{\frac{1}{2}}
\frac{\delta_{ij}}{|x_{ij}|^{\Delta_{i}+\Delta_{j}}}
\end{eqnarray}

\noindent for $x_{ij} \equiv x_{i}-x_{j} \rightarrow 0$,
where ${\mathcal N}_{i}$ will be specified below in
Equation ({\bf \ref{Gaudin}}). The OPE of a pair of these
operators ${\mathcal O}_{1}(x)$ and ${\mathcal O}_{3}(x)$ is given by

\begin{eqnarray}
\qquad {\mathcal O}_1(x_1)\, {\mathcal O}_3(x_3) \sim
\sum_{{\mathcal O}_2}
\left\lgroup
\frac{{\mathcal N}_1 {\mathcal N}_3}{{\mathcal N}_2}
\right\rgroup^{\frac{1}{2}}
\frac{C_{132}}{|x_{13}|^{\Delta_1 + \Delta_3 - \Delta_2}}
\
{\mathcal O}_2 (x)
+ \ldots \,,
\quad
x = \frac{1}{2} (x_1 + x_3)\,,
\label{OPE}
\end{eqnarray}

\noindent for $x_{13} \rightarrow 0$, where the ellipsis denotes
subleading corrections involving conformal descendants of ${\mathcal O}_{2}$
\cite{Braun:2003rp}. The structure constants $C_{132}$ have
a perturbative expansion in $\lambda$,

\begin{eqnarray}
N_{c}\, C_{132} = c_{132}^{(0)} + \lambda c_{132}^{(1)}+ \ldots \,,
\end{eqnarray}

\noindent In this note, we focus on the leading (tree-level) contribution
$c_{132}^{(0)}$.

\subsection{{\it \lq Tailoring\rq} the structure constants}
\label{sec:tailoring}
Following \cite{Escobedo:2010xs}, we construct $c_{132}^{(0)}$ in
four steps.

\subsection*{Step 1} We map the length-$L_i$ single-trace operator
${\mathcal O}_{i}$ to an eigenstate
$| {\mathcal O}_{i} \rangle$ of a length-$L_{i}$ periodic spin-1 chain Hamiltonian
$\mathcal H_1$.

\subsection*{Step 2} We {\it \lq split\rq} the spin chains into left
and right subchains of lengths
\footnote{\ We restrict the discussion to the {\it \lq non-extremal\rq}
case where all $\ensuremath{L}_{i,l}, \ensuremath{L}_{i,r}>0$, for which there is no mixing
with double-trace operators \cite{Escobedo:2010xs}.}

\begin{equation}
\ensuremath{L}_{i,l} = \frac{1}{2}\left(\ensuremath{L}_{i}+\ensuremath{L}_{j} - \ensuremath{L}_{k}\right),
\quad
\ensuremath{L}_{i,r} = \frac{1}{2}\left(\ensuremath{L}_{i}+\ensuremath{L}_{k} - \ensuremath{L}_{j}\right),
\label{subchainlengths}
\end{equation}

\noindent respectively, with $(i,j,k)$ in cyclic order.
We perform a corresponding split of the states,

\begin{eqnarray}
| {\mathcal O}_{i}  \rangle = \sum_{a} | {\mathcal O}_{i_{a}}  \rangle_{l} \otimes |
{\mathcal O}_{i_{a}}  \rangle_{r}   \,,
\label{splitting}
\end{eqnarray}

\noindent where, roughly speaking, the sum is over all possible ways
of distributing the component fields into the left and right subchains.
(A more precise definition of this splitting, as well as a more accurate
version of Equation ({\bf \ref{splitting}}), will be given below after
introducing the Bethe ansatz.) Note that
$| {\mathcal O}_{i_{a}} \rangle_{l}$ and
$| {\mathcal O}_{i_{a}} \rangle_{r}$
are states of subchains with lengths $\ensuremath{L}_{i,l}$ and $\ensuremath{L}_{i,r}$,
respectively.

\subsection*{Step 3} We {\it \lq flip\rq} or ${\mathcal F}$-conjugate the right
kets into right bras

\begin{eqnarray}
| {\mathcal O}_{i}  \rangle = \sum_{a} | {\mathcal O}_{i_{a}}  \rangle_{l} \otimes |
{\mathcal O}_{i_{a}}  \rangle_{r}
\rightarrow
\sum_{a} | {\mathcal O}_{i_{a}}  \rangle_{l} \otimes\,
  {}_{r} \langle {\mathcal O}_{i_{a}}|   \,.
\label{flip}
\end{eqnarray}

\noindent Given a pair of elementary fields $A$ and $B$ that are
associated with the kets $|\Psi_{i}\rangle_{r}$ and
$|\Psi_{i+1} \rangle_{l}$, respectively, the flipped state
${}_{r}\langle \Psi_{i}|$ is defined such that

\begin{eqnarray}
\langle A B \rangle \sim {}_{r}\langle \Psi_{i}| \Psi_{i+1}\rangle_{l}
\label{prescription}
\,.
\end{eqnarray}

\noindent In view of the fact that the only non-zero 2-point functions
are between $f_{+}$ and $f_{-}$, and between two $f_{0}$ fields, the
prescription ({\bf \ref{prescription}}) implies that

\begin{eqnarray}
|f_{\pm}\rangle_{r} \rightarrow  {}_{r}\langle f_{\mp}|, \qquad
|f_{0  }\rangle_{r} \rightarrow  {}_{r}\langle f_{0  }|  \,.
\label{Fconjugation}
\end{eqnarray}

\noindent Our convention is that
$\langle f_{\pm}| f_{\pm}\rangle = 1$,
$\langle f_{ 0 }| f_{ 0 }\rangle = 1$,
while all other 2-point functions are zero.

\subsection*{Step 4} We construct the structure constants by taking scalar
products of bra and ket states
\cite{Escobedo:2010xs,Foda:2011rr}, to obtain

\begin{equation}
c_{132}^{(0)} = {\mathcal N}_{132}\sum_{a,b,c}
{}_{r}\langle {\mathcal O}_{2_{b}} | {\mathcal O}_{1_{a}}  \rangle_{l}\
{}_{r}\langle {\mathcal O}_{1_{a}} | {\mathcal O}_{3_{c}}  \rangle_{l}\
{}_{r}\langle {\mathcal O}_{3_{c}} | {\mathcal O}_{2_{b}}  \rangle_{l} \,,
\label{prelim}
\end{equation}

\noindent where

\begin{equation}
{\mathcal N}_{132} =
\left\lgroup
\frac{L_{1} L_{2} L_{3}}
{
\langle {\mathcal O}_{1}| {\mathcal O}_{1} \rangle
\langle {\mathcal O}_{2}| {\mathcal O}_{2} \rangle
\langle {\mathcal O}_{3}| {\mathcal O}_{3} \rangle}
\right\rgroup^{\frac{1}{2}
} \,.
\label{N123def}
\end{equation}

\noindent This is represented graphically in
Figure {\bf \ref{fig:fig1}}.  In order to further evaluate the
expression ({\bf \ref{prelim}}) for the structure constants, it
is necessary to have a more explicit construction of the states
with definite conformal dimensions. To this end, we now turn to
the Bethe ansatz.

\begin{figure}
\begin{centering}
\includegraphics[height=7cm]{fig1.pdf}
\par\end{centering}
\caption{A configuration of 3-point functions with contractions
among the self-dual Yang-Mills fields
(a solid line is $\langle f_{+}f_{-} \rangle$ and
a dotted line is $\langle f_{0}f_{0} \rangle$).
$\mathcal O_3$ is chosen to consist of $f_{+}$ fields only, so it maps
to a spin-chain reference state. This will be the case that can
be evaluated in determinant form.}
\label{fig:fig1}
\end{figure}

\section{Algebraic Bethe ansatz}
\label{sec:BA}

\subsection{Diagonalizing the Hamiltonian $\mathcal H_1$}
The 1-loop QCD mixing matrix $\Gamma$ ({\bf \ref{Hamiltonian}})
is identical
to the Hamiltonian $\mathcal H_1$ of an antiferromagnetic spin-1 chain,
with periodic boundary conditions, that is integrable
\cite{Zamolodchikov:1980ku},
and therefore can be diagonalized using the algebraic Bethe ansatz
\cite{Kulish:1981gi, Kulish:1981bi, A.:1982zz, Babujian:1983ae}.
The basic strategy to diagonalize $\mathcal H_1$ is to diagonalize
a transfer matrix
$t^{(\frac{1}{2})}(u)$ that is constructed from a monodromy matrix
with a 2-dimensional (that is, spin-$\frac{1}{2}$) auxiliary space.
Although $t^{(\frac{1}{2})}(u)$ does not generate $\mathcal H_1$
({\bf \ref{Hamiltonian}}), it is related by the fusion procedure
to another transfer matrix $t^{(1)}(u)$ that is constructed from
a monodromy matrix with a 3-dimensional (that is, spin-1) auxiliary
space and that contains $\mathcal H_1$
\cite{Kulish:1981bi,A.:1982zz, Babujian:1983ae}.
By diagonalizing $t^{(\frac{1}{2})}(u)$, we diagonalize
$t^{(1)}(u)$, $\mathcal H_1$ and $\Gamma$, all in one go.

\subsection{The $R$- and the monodromy matrices}
The transfer matrix $t^{(\frac{1}{2})}(u)$ can be constructed
using the $6\! \times \! 6$ R-matrix

\begin{multline}
R^{(\frac{1}{2},1)}(u,v) =
\\
\frac{1}{(u-v-\eta)}
\left\lgroup
\begin{array}{ccc|ccc}
    u-v+\eta  &              &                                  \\
              & u-v          &               &   \sqrt{2}\eta   \\
              &              &  u-v-\eta     & & \sqrt{2}\eta   \\
    \hline
              & \sqrt{2}\eta &               & u-v-\eta\\
              &              &  \sqrt{2}\eta &   &  u-v\\
              &              &               &   & & u-v+\eta
\end{array}
\right\rgroup \,,
\end{multline}

\noindent where eventually we shall set $\eta=i$. The matrix
elements that are zero are left empty.  We regard
$R^{(\frac{1}{2},1)}(u,v)$ as an operator acting on
${\mathcal C}^{2} \otimes {\mathcal C}^{3}$. This R-matrix can be obtained
by fusion \cite{Kulish:1981gi, Kulish:1981bi}
from $R^{(\frac{1}{2},\frac{1}{2})}(u,v) = u-v + \eta {\mathcal P}$,
where ${\mathcal P}$ is the permutation matrix on
${\mathcal C}^{2} \otimes {\mathcal C}^{2}$, together with a {\it \lq gauge\rq}
transformation that makes the matrix symmetric.

The (inhomogeneous) monodromy matrix is constructed from
the $R$-matrix as
\footnote{In the sequel, we use different brackets to indicate
the type of enclosed arguments. We write
$f( x, y) $ when neither $x$ nor $y$ is a set of variables,
$f\{x, y\}$ when both $x$ and $y$ are sets of variables,
and
$f[x, \{y\}]$ when $x$ is not a set of variables, but $y$ is.}

\begin{eqnarray}
T^{(\frac{1}{2})}_{0}[u; \{z\}_{\ensuremath{L}}] =
R_{01}^{(\frac{1}{2},1)}(u,z_{1}) \ldots
R_{0\ensuremath{L}}^{(\frac{1}{2},1)}(u,z_{\ensuremath{L}})  \,,
\label{monodromy}
\end{eqnarray}

\noindent where we have introduced the inhomogeneities
$\{z\}_{\ensuremath{L}}=\{z_{1}, \ldots, z_{\ensuremath{L}}\}$ for later convenience.
The auxiliary space (labeled 0) is 2-dimensional, while each of
the quantum spaces (labeled $1, \ldots, \ensuremath{L}$) are 3-dimensional
By tracing over the auxiliary space, we arrive at the (inhomogeneous)
transfer matrix

\begin{eqnarray}
t^{(\frac{1}{2})}[u; \{z\}_{\ensuremath{L}}] =
\operatorname{tr}_{0} T^{(\frac{1}{2})}_{0}[u; \{z\}_{\ensuremath{L}}] \,.
\end{eqnarray}

\noindent It has the commutativity property

\begin{eqnarray}
\left[
t^{(\frac{1}{2})}[u; \{z\}_{\ensuremath{L}}]  \,,
t^{(\frac{1}{2})}[v; \{z\}_{\ensuremath{L}}]
\right] = 0 \,,
\end{eqnarray}

\noindent by virtue of the fact that the R-matrix obeys the Yang-Baxter
equation.

\subsection{Constructing the eigenstates}
The eigenstates of this transfer matrix can be readily obtained by
algebraic Bethe ansatz: we define the operators $A, B, C, D$ by
\begin{eqnarray}
T^{(\frac{1}{2})}_{0}[u; \{z\}_{\ensuremath{L}}] =
\left\lgroup
\begin{array}{cc}
    A[u; \{z\}_{\ensuremath{L}}] & B[u; \{z\}_{\ensuremath{L}}] \\
    C[u; \{z\}_{\ensuremath{L}}] & D[u; \{z\}_{\ensuremath{L}}]
    \end{array}
\right\rgroup \,.
\label{ABCD}
\end{eqnarray}

\noindent We also introduce the reference states with all spins up or
all spins down,

\begin{eqnarray}
|0 \rangle_{\pm} = |f_{\pm} \rangle^{\otimes \ensuremath{L}} \equiv |f_{\pm}^{L}
\rangle \,.
\label{reference}
\end{eqnarray}

\noindent These states are eigenstate of both $A[u; \{z\}_{\ensuremath{L}}]$ and
$D[u; \{z\}_{\ensuremath{L}}]$,

\begin{multline}
A[u; \{z\}_{\ensuremath{L}}] |0 \rangle_{+} =
\left\lgroup
\prod_{l=1}^{\ensuremath{L}}\frac{u-z_{l}+\eta}{u-z_{l}-\eta}
\right\rgroup
|0 \rangle_{+} \,,
\quad
D[u; \{z\}_{\ensuremath{L}}] |0 \rangle_{+} = |0 \rangle_{+} \,,
\\
A[u; \{z\}_{\ensuremath{L}}] |0 \rangle_{-} =  |0 \rangle_{-} \,,
\quad \quad \quad \quad \quad
\quad \quad \quad
D[u; \{z\}_{\ensuremath{L}}] |0 \rangle_{-} =
\left\lgroup
\prod_{l=1}^{\ensuremath{L}}\frac{u-z_{l}+\eta}{u-z_{l}-\eta}
\right\rgroup
|0 \rangle_{-} \,.
\quad \quad
\label{ADeigenvals}
\end{multline}

\noindent We note that

\begin{eqnarray}
B[u; \{z\}_{\ensuremath{L}}]^{\dagger} = -
\left\lgroup
\prod_{l=1}^{\ensuremath{L}}\frac{u^{*} - z_{l}^{*}-\eta}{u^{*} -
z_{l}^{*}+\eta}
\right\rgroup
C[u^{*}; \{z^{*}\}_{\ensuremath{L}}] \,,
\label{Bdagger}
\end{eqnarray}

\noindent where we have used $\eta = i$, and $*$ denotes complex
conjugation. Choosing $|0 \rangle_{+}$ as the reference state,
one finds that the states

\begin{eqnarray}
| \{ u \}_{N} \rangle_{+} =
\left\lgroup
\prod_{j=1}^{N} B[u_{j}; \{z\}_{\ensuremath{L}}]
\right\rgroup
|0 \rangle_{+}
\label{Bethestates}
\end{eqnarray}

\noindent are eigenstates of the transfer matrix
$t^{(\frac{1}{2})}[u; \{z\}_{\ensuremath{L}}]$ provided that
$\{u\}_{N} =\{ u_{1}, \ldots,
u_{N}\}$ are distinct and satisfy the spin-1 Bethe equations

\begin{eqnarray}
\prod_{l=1}^{\ensuremath{L}}\frac{u_{j}-z_{l}+\eta}{u_{j}-z_{l}-\eta} =
\prod_{k=1 \atop k\ne j}^{N}
\frac{u_{j}-u_{k}+\eta}{u_{j}-u_{k}-\eta} \,.
\label{BAE}
\end{eqnarray}

\noindent In the homogeneous limit $z_{l}=0$, these states are
eigenstates of $\mathcal H_1$ ({\bf \ref{Hamiltonian}}) with eigenvalues
(anomalous dimensions) \cite{Ferretti:2004ba}

\begin{eqnarray}
\gamma = \frac{\lambda}{48\pi^{2}}
\left\lgroup
7 \ensuremath{L}-\sum_{k=1}^{N}
\frac{12}{u_{k}^{2}+1}
\right\rgroup \,.
\end{eqnarray}

\noindent The conformal dimensions are therefore given by
$\Delta = 2\ensuremath{L} + \gamma$. The Bethe eigenstates
({\bf \ref{Bethestates}})
are $SU(2)$ highest-weight states, with spin

\begin{eqnarray}
s=s^{3}=\ensuremath{L}-N \,,
\label{spin}
\end{eqnarray}

\noindent and therefore $N \le \ensuremath{L}$. If we choose
$|0 \rangle_{-}$ as the reference state, then the Bethe states
are given by

\begin{eqnarray}
| \{ u \}_{N} \rangle_{-} =
\left\lgroup
\prod_{j=1}^{N} C[u_{j}; \{z\}_{\ensuremath{L}}]
\right\rgroup
|0 \rangle_{-}
\label{Bethestates-} \,,
\end{eqnarray}

\noindent which are lowest-weight states, with $s=-s^{3}=\ensuremath{L}-N$,
so again $N \le \ensuremath{L}$.

In order to properly define the splitting of states
({\bf \ref{splitting}}), we follow \cite{Escobedo:2010xs}
and split the monodromy matrix ({\bf \ref{monodromy}}),

\begin{eqnarray}
T^{(\frac{1}{2})}_{0}[u; \{z\}_{\ensuremath{L}}] =
T^{(\frac{1}{2})}_{0, l}[u; \{z\}_{\ensuremath{L}_{l}}]\,
T^{(\frac{1}{2})}_{0, r}[u; \{z\}_{\ensuremath{L}_{r}}]\,,
\end{eqnarray}

\noindent where

\begin{eqnarray}
T^{(\frac{1}{2})}_{0, l}[u; \{z\}_{\ensuremath{L}_{l}}] &=&
R_{01}^{(\frac{1}{2},1)}(u,z_{1}) \ldots
R_{0\ensuremath{L}_{l}}^{(\frac{1}{2},1)}(u,z_{\ensuremath{L}_{l}})  \,, \nonumber \\
T^{(\frac{1}{2})}_{0, r}[u; \{z\}_{\ensuremath{L}_{r}}] &=&
R_{0,\ensuremath{L}_{l}+1}^{(\frac{1}{2},1)}(u,z_{\ensuremath{L}_{l}+1}) \ldots
R_{0\ensuremath{L}}^{(\frac{1}{2},1)}(u,z_{\ensuremath{L}})  \,,
\end{eqnarray}

\noindent and $\{z\}_{\ensuremath{L}_{l}} = \{z_{1}, \ldots, z_{\ensuremath{L}_{l}} \}$,
$\{z\}_{\ensuremath{L}_{r}} = \{z_{\ensuremath{L}_{l}+1}, \ldots, z_{\ensuremath{L}} \}$.
Correspondingly,

\begin{multline}
\left\lgroup
\begin{array}{cc}
    A[u; \{z\}_{\ensuremath{L}}] & B[u; \{z\}_{\ensuremath{L}}] \\
    C[u; \{z\}_{\ensuremath{L}}] & D[u; \{z\}_{\ensuremath{L}}]
    \end{array}
\right\rgroup =
\\
\left\lgroup
\begin{array}{cc}
    A_{l}[u; \{z\}_{\ensuremath{L}_{l}}] & B_{l}[u; \{z\}_{\ensuremath{L}_{l}}] \\
    C_{l}[u; \{z\}_{\ensuremath{L}_{l}}] & D_{l}[u; \{z\}_{\ensuremath{L}_{l}}]
    \end{array}
    \right\rgroup
\left\lgroup
\begin{array}{cc}
    A_{r}[u; \{z\}_{\ensuremath{L}_{r}}] & B_{r}[u; \{z\}_{\ensuremath{L}_{r}}] \\
    C_{r}[u; \{z\}_{\ensuremath{L}_{r}}] & D_{r}[u; \{z\}_{\ensuremath{L}_{r}}]
    \end{array}
    \right\rgroup  \,.
\end{multline}

\noindent In particular,

\begin{eqnarray}
B[u; \{z\}_{\ensuremath{L}}] &=&
A_{l}[u; \{z\}_{\ensuremath{L}_{l}}]\,
B_{r}[u; \{z\}_{\ensuremath{L}_{r}}] +
B_{l}[u; \{z\}_{\ensuremath{L}_{l}}]\,
D_{r}[u; \{z\}_{\ensuremath{L}_{r}}]  \,, \nonumber \\
C[u; \{z\}_{\ensuremath{L}}] &=&
C_{l}[u; \{z\}_{\ensuremath{L}_{l}}]\,
A_{r}[u; \{z\}_{\ensuremath{L}_{r}}] +
D_{l}[u; \{z\}_{\ensuremath{L}_{l}}]\,
C_{r}[u; \{z\}_{\ensuremath{L}_{r}}] \,.
\end{eqnarray}

\noindent The ${\mathcal F}$-conjugation ({\bf \ref{Fconjugation4}})
implies that

\begin{eqnarray}
B_{r}[u; \{z\}_{\ensuremath{L}_{r}}] | f_{+}^{L_{r}}
\rangle_{r} &\rightarrow& {}_{r}
\langle
f_{-}^{L_{r}}|
B_{r}[u; \{z\}_{\ensuremath{L}_{r}}] \,, \nonumber \\
C_{r}[u; \{z\}_{\ensuremath{L}_{r}}] | f_{-}^{L_{r}}
\rangle_{r} &\rightarrow&
{}_{r}\langle f_{+}^{L_{r}}|
C_{r}[u; \{z\}_{\ensuremath{L}_{r}}] \,.
\end{eqnarray}

\section{Evaluating the structure constants}
\label{sec:structureconstants}

\subsection{3-point functions with three non-BPS-like
operators in sum form}
We start with the general case where all three composite
operators $\mathcal O_i$, $i \in \{1, 2, 3\}$ are non-BPS-like
(they are not of highest or lowest conformal dimension),
so they map to Bethe eigenstates $| \mathcal O_i \rangle$ that
are not spin-chain reference states, and that can be
split into left and right parts as

\begin{multline}
|{\mathcal O}_{i} \rangle =
\\
\prod_{j=1}^{N_i}
\left\lgroup
A_l [u_{i, j}, \{z_{L_{i,l}}\}]
\
B_r [u_{i, j}, \{z_{L_{i,r}}\}] +
D_r [u_{i, j}, \{z_{L_{i,r}}\}]
\
B_l [u_{i, j}, \{z_{L_{i,l}}\}]
\right\rgroup
|f_{+}^{L_{i,l}}
\rangle_{l}
\otimes
|f_{+}^{L_{i,r}}\rangle_{r}
=
\\
\sum_{\alpha_i \cup {\bar\alpha_i} =
\{ u_i \}_{N_i}}
H_i \{ \alpha_i, \bar\alpha_i \}
| {\mathcal O}_{i,    \alpha_i} \rangle_l
\otimes
| {\mathcal O}_{i,\bar\alpha_i} \rangle_r
\label{Oigen}
\end{multline}

\noindent where

\begin{equation}
|{\mathcal O}_{i, \alpha_i} \rangle_l =
\left\lgroup
\prod_{j \in     \alpha_i} B_l [u_{i, j}, \{z\}_{L_{i,l}}]
\right\rgroup
| f_+^{L_{i, l}}
\rangle_{l},
\quad
|{\mathcal O}_{i, \bar\alpha_i} \rangle_r =
\left\lgroup
\prod_{j \in \bar\alpha_i} B_r [u_{i, j}, \{z\}_{L_{i,r}}]
\right\rgroup
| f_+^{L_{i, r}}
\rangle_{r},
\end{equation}

\noindent the coefficients
$H_i \{ \alpha_i, \bar\alpha_i \}$ are computed from Equation
({\bf \ref{ADeigenvals}}) to be

\begin{equation}
H_i \{ \alpha_i, \bar\alpha_i \} =
\prod_{u_{i, j} \in \alpha_i}
\prod_{z_k      \in \{z\}_{L_{i,l}}}
\frac{ u_{i, j} - z_k + \eta }{ u_{i, j} - z_k - \eta } \,,
\end{equation}

\noindent and $\{ u_i \}_{N_{i}}$ satisfy the Bethe
equations ({\bf \ref{BAE}}) with $L=L_{i}$.
Under ${\mathcal F}$-conjugation, this operator becomes

\begin{equation}
|{\mathcal O}_i \rangle
\to
\sum_{
\alpha_i \cup {\bar\alpha_i} = \{ u_i \}_{N_i}}
H_i \{\alpha_i, \bar\alpha_i\}
| {\mathcal O}_{i,\alpha_i} \rangle_l \otimes { }_r\langle
{\mathcal O}_{i,{\bar\alpha_i}}| \,,
\label{OiFconj}
\end{equation}

\noindent where

\begin{equation}
     { }_r\langle
{\mathcal O}_{i,{\bar\alpha_i}}| =
{}_r\langle f_{-}^{L_{i,r}}|
\left\lgroup
\prod_{j\in{\bar\alpha_i}} B_r[u_{i, j}, \{z\}_{L_{i,r}}]
\right\rgroup \,.
\end{equation}

Substituting the above expressions into Equation
({\bf \ref{prelim}}), we obtain the following sum expression for
the structure constant of the 3-point function with three
non-BPS-like operators

\begin{multline}
c_{132} =
\\
\lim_{z_l \rightarrow 0}
{\mathcal N}_{132}
\sum_{\alpha_i \cup {\bar\alpha_i} = \{ u_i \}_{N_i}}
\left\lgroup
\prod_{i=1}^{3} H_i \{ \alpha_i, \bar\alpha_i \}
\right\rgroup
{ }_r\langle {\mathcal O}_{2, \bar\alpha_2} | {\mathcal O}_{1, \alpha_1} \rangle_l
\
{ }_r\langle {\mathcal O}_{1, \bar\alpha_1} | {\mathcal O}_{3, \alpha_3} \rangle_l
\
{ }_r\langle {\mathcal O}_{3, \bar\alpha_3} | {\mathcal O}_{2, \alpha_2} \rangle_l
\,,
\label{general-product}
\end{multline}

\noindent
where each of the three factors of type
$
{ }_r\langle
{\mathcal O}_{i+1,{\bar\alpha_{i+1}}} | {\mathcal O}_{i,\alpha_i}
\rangle_l
$
in the summand is a generic scalar product as in Equation
({\bf \ref{korepin-sum-spin-1/2}}),
subject to the conditions in Equations ({\bf \ref{inhomospin1}}),
({\bf \ref{aspin1}}).

\subsection{The structure of the sum form in Equation
({\bf \ref{general-product}})}
The non-BPS-like operator $\mathcal O_i$, $i \in \{1, 2, 3\}$,
is composed of the operators
$\{f_{+},    f_{0},    f_{-}   \}$ with multiplicities
$\{n_{i, +}, n_{i, 0}, n_{i, -}\}$,
such that $n_{i, +} + n_{i, 0} + n_{i, -}$$=$$ L_i$
and $2n_{i,-} + n_{i,0}=N_{i}$.
Splitting $\mathcal O_i$ into
a  left-part $\mathcal O_{i, l}$ of length $L_{i, l}$, and
a right-part $\mathcal O_{i, r}$ of length $L_{i, r}$,
$L_{i, l} + L_{i, r} = L_i$, the operators $\{f_+, f_0, f_-\}$
can be on either part, such that

\begin{equation}
n_{i, +}^l + n_{i, +}^r = n_{i, +}, \qquad
n_{i, 0}^l + n_{i, 0}^r = n_{i, 0},  \qquad
n_{i, -}^l + n_{i, -}^r = n_{i, -},
\end{equation}

\noindent where $n_{i, +}^l$ is the number of $f_+$-operators
on the left-part of $|\mathcal O_i \rangle$, {\it etc.}

Let us consider one type of these operators, for example $f_+$,
to be a reference state operator, in the sense that if all
elementary operators in a single-trace operator $\mathcal O$ are
of type $f_+$, then $\mathcal O$ maps to a spin-chain reference state.
In that case, the other two operators, $f_0$ and $f_-$, become
{\it \lq excitations\rq}
\footnote{\ Either $f_+$ or $f_-$ can be chosen as a reference
state operator, as we will see in the sequel.}.
Since the total number of elementary operators in $\mathcal O_i$,
$i \in \{1, 2, 3\}$ is fixed, one can think of
single-trace operators that are {\it not eigenstates} of the
mixing matrix $\Gamma$, but whose weighted sum is a single-trace
$\mathcal O_i$ that is an eigenstate, as labeled by the positions
of the excitations in the trace.

The crucial point is that, while the lengths of the left- and
right-parts are fixed once and for all
\footnote{\ This follows from the fact that the lengths
$L_i$, $i \in \{1, 2, 3\}$ are fixed as initial conditions,
and the lengths of the left and right parts are fixed from
Equation ({\bf \ref{subchainlengths}}).},
the distribution of the excitations on the left and the right
parts of $\mathcal O_i$ is not fixed. This means that the sum in
Equation ({\bf \ref{general-product}}) is over all possible
distributions of excitations in $\mathcal O_i$, $i \in \{1, 2, 3\}$
over its left and right parts, subject to the conditions

\begin{equation}
n_{i, +} = n_{i+1, -}, \qquad
n_{i, -} = n_{i+1, +}, \qquad
n_{i, 0} = n_{i+1, 0}, \qquad
i + 3 \equiv i \,.
\end{equation}

In spin-chain terms, the action of the Bethe operators on
a reference state, that consists of one type of operators,
generates excitations of both types. Thus every state
$\mathcal O_i$, $i \in \{1, 2, 3\}$ that is not BPS-like will
consist of all three types $\{f_+, f_0, f_-\}$, and we
need to sum over all possible positions of
$\{f_+, f_0, f_-\}$ in $\mathcal O_i$. The result is that
{\bf 1.} The sum over partitions in
Equation ({\bf \ref{general-product}}) is computationally
non-trivial, particularly when the number of Bethe roots
involved is not small; and as mentioned above,
{\bf 2.} Each of the three factors of type
${ }_r\langle {\mathcal O}_{i+1,{\bar\alpha_{i+1}}} |
              {\mathcal O}_{i,       \alpha_{i  } } \rangle_l $
in the summand is a generic scalar product as in Equation
({\bf \ref{korepin-sum-spin-1/2}}),
subject to the conditions in Equation ({\bf \ref{inhomospin1}}).
This is a complicated expression.

To reduce the complexity of the sum form in Equation ({\bf
\ref{general-product}}) and obtain a computationally tractable
expression, which in our case is a determinant, we choose one of
the operators to be BPS-like so that it maps to a spin-chain reference
state.  We will choose $\mathcal O_3$ to be BPS-like.

\subsection{3-point functions with one BPS-like state in
determinant form}

Choosing ${\mathcal O}_{3}$ to consist of $f_{+}$-operators only,
the corresponding state is

\begin{eqnarray}
| {\mathcal O}_{3}  \rangle = |f_{+}^{\ensuremath{L}_{3}} \rangle =
|f_{+}^{\ensuremath{L}_{3,l}} \rangle_{l} \otimes
|f_{+}^{\ensuremath{L}_{3,r}} \rangle_{r} \rightarrow
|f_{+}^{\ensuremath{L}_{3,l}} \rangle_{l} \otimes {}_{r}\langle
f_{-}^{\ensuremath{L}_{3,r}}| \,,
\label{O3}
\end{eqnarray}

\noindent where $\ensuremath{L}_{3,l}$ and $\ensuremath{L}_{3,r}$ are given by
(\ref{subchainlengths}).
Evidently, since there is only one way to split this state,
no summation is necessary. We write the algebraic Bethe state for
operator ${\mathcal O}_{1}$ as in ({\bf \ref{Oigen}}) with $i=1$, and we define the
corresponding parameters $u_{j} \equiv u_{1,j}$,
which satisfy the Bethe equations
({\bf \ref{BAE}}) with $L = L_1$. Under $\mathcal F$-conjugation, this
state becomes ({\bf \ref{OiFconj}}) with $i=1$.
\noindent However, we write the state corresponding to the operator
${\mathcal O}_2$ instead as

\begin{eqnarray}
|{\mathcal O}_{2}\rangle\to\sum_{\beta \cup {\bar\beta} =
\{ v \}_{N_2}} H_2 \{\beta, \bar\beta \}
|{\mathcal O}_{2,\beta}\rangle_l
\otimes
{ }_r\langle {\mathcal O}_{2,{\bar\beta}}|\,,
\end{eqnarray}

\noindent with

\begin{eqnarray}
\qquad \quad  |{\mathcal O}_{2,\beta}\rangle_l =
\left\lgroup
\prod_{j\in\beta}C_l[v_j; \{z\}_{\ensuremath{L}_{2,l}}]
\right\rgroup
|f_{-}^{L_2, l} \rangle_l \,,
\qquad
{ }_r\langle {\mathcal O}_{2,{\bar\beta}}|={}_r\langle f_{+}^{L_{2,r}}|
\left\lgroup
\prod_{j\in{\bar\beta}} C_r[v_j; \{z\}_{\ensuremath{L}_{2,r}}]
\right\rgroup \,,
\end{eqnarray}

\noindent where $\{ v \}_{N_{2}}$ satisfy the Bethe equations
({\bf \ref{BAE}}) with $L=L_{2}$. Having chosen to construct the Bethe states
for ${\mathcal O}_1$ with the reference state $|0 \rangle_{+}$, it is necessary
to construct the Bethe states for ${\mathcal O}_2$ with the reference state
$|0 \rangle_{-}$.
We now insert these results into Equation ({\bf \ref{prelim}}) to get

\begin{multline}
c_{132} =
\lim_{z_l \rightarrow 0} {\mathcal N}_{132}
 \sum_{
      \beta  \cup { \bar\beta} = \{ v \}_{N_2}
\atop \alpha \cup {\bar\alpha} = \{ u \}_{N_1}
}
H_1 \{ \alpha, \bar\alpha \}\, H_2 \{ \beta, \bar\beta \} \
{ }_r\langle {\mathcal O}_{2, { \bar\beta}} | {\mathcal O}_{1,\alpha} \rangle_l \
{ }_r\langle {\mathcal O}_{1, {\bar\alpha}} |  f_{+}^{L_{3,l}} \rangle_l\
{ }_r\langle f_{-}^{L_{3, r}}        | {\mathcal O}_{2, \beta} \rangle_l
\\
=
\lim_{z_{l}\rightarrow 0}
{\mathcal N}_{132}\, H_2 \{ \varnothing, \{ v \}_{N_2} \}
\sum_{\alpha \cup {\bar\alpha} =  \{ u \}_{N_1} }
H_1 \{\alpha, \bar\alpha \} \
{ }_r\langle {\mathcal O}_2 |{\mathcal O}_{1, \alpha} \rangle_l \
{ }_r\langle {\mathcal O}_{1, \bar\alpha} | f_{+}^{L_{3, l}} \rangle_l \
{ }_r\langle f_{-}^{L_{3, r}}| f_{-}^{L_{2,l}} \rangle_l
\label{c132more} \,.
\end{multline}

\noindent In passing to the second line, we have made use of the
fact that the expression vanishes unless the set $\beta$ contains
no Bethe roots, and we defined

\begin{eqnarray}
{ }_r\langle{\mathcal O}_{2}| \equiv \langle f_{+}^{L_{2,r}}|
\left\lgroup
\prod_{j=1}^{N_2} C_r [v_j; \{z\}_{\ensuremath{L}_{2,r}}]
\right\rgroup \,.
\end{eqnarray}

\noindent With the help of Equation ({\bf \ref{ADeigenvals}}),
we see that

\begin{eqnarray}
H_2 \{\varnothing, \{ v \}_{N_2} \} =
\prod_{j=1}^{N_2}
\prod_{l=1}^{\ensuremath{L}_{2, l}} \frac{v_j -z_l +\eta} {v_j -z_l -\eta}
\end{eqnarray}

\noindent becomes equal to 1 in the homogeneous limit, $z_l = 0$,
by virtue of the zero-momentum constraint

\begin{eqnarray}
\prod_{j=1}^{N_{2}}\frac{v_{j}+\eta}{v_{j}-\eta} = 1\,,
\end{eqnarray}

\noindent which arises from the cyclicity of the trace in
${\mathcal O}_{2}$\ \footnote{
Note that this argument can be used only when all Bethe
roots of an original unsplit eigenstate belong to the same
part after splitting. This is the case for the eigenstate
$| \mathcal O_2\rangle$ in the 3-point function with one BPS-like
state.
In particular, the same argument cannot be used to simplify
the $H_i$ coefficients, $i \in \{1, 2, 3\}$,
in Equation ({\bf \ref{general-product}}). This is because
in the 3-point function with three non-BPS-like states,
each state is split into a right part and a left part,
and the Bethe roots can appear on either part. But neither
part satisfies cyclicity on its own and the zero-momentum
constraint cannot be used.}.
The remaining sum over partitions in Equation
({\bf \ref{c132more}}) can be performed by using
${ }_r\langle {\mathcal O}_{1,{\bar\alpha}}|f_{+}^{L_{3,l}}\rangle_l=
{ }_l\langle f_{-}^{L_{3,l}}|{\mathcal O}_{1,{\bar\alpha}}\rangle_r$.
Noting also that ${ }_r\langle
f_{-}^{L_{3,r}}|f_{-}^{L_{2,l}}\rangle_l=1$, we obtain

\begin{multline}
c_{132}= \lim_{z_{l}\rightarrow 0}
{\mathcal N}_{132}\,
\sum_{\alpha \cup {\bar\alpha} =\{ u \}_{N_{1}}}
H_1 \{\alpha, \bar\alpha \}\
{}_r\langle {\mathcal O}_{2}|{\mathcal O}_{1,\alpha}\rangle_l\
{ }_l\langle f_{-}^{L_{3,l}}|{\mathcal O}_{1,{\bar\alpha}}\rangle_r
\\
= \lim_{z_{l}\rightarrow 0} {\mathcal N}_{132}\,
\ { }_l\langle f_{-}^{L_{3,l}}|\otimes\ {}_r\langle
{\mathcal O}_{2}|{\mathcal O}_{1}\rangle.
\label{nopartition}
\end{multline}

We observe that this expression vanishes unless

\begin{eqnarray}
\ensuremath{L}_{2}-N_{2} = \ensuremath{L}_{1}+\ensuremath{L}_{3}-N_{1} \ge 0 \,.
\label{constraint}
\end{eqnarray}

\noindent Indeed, the factor
${}_r\langle {\mathcal O}_{2}|{\mathcal O}_{1,\alpha}\rangle_l$ in
the first line of Equation ({\bf \ref{nopartition}})
vanishes unless $|\alpha|$ (the number of Bethe roots in $\alpha$) is given
by $|\alpha| = N_{2}$. It follows that $|\bar{\alpha}| = N_{1}-N_{2}$.
Moreover, the two states in the factor
${ }_l\langle f_{-}^{L_{3,l}}|{\mathcal O}_{1,{\bar\alpha}}\rangle_r$
should have the same $S^{3}$ eigenvalue; hence,

\begin{eqnarray}
L_{1,r} - |\bar{\alpha}| = - L_{3,l} \,,
\end{eqnarray}

\noindent which then implies Equation ({\bf \ref{constraint}}).
The sum over ${\mathcal O}_{2}$ in Equation ({\bf \ref{OPE}}) can
therefore be understood as the sum over all $\ensuremath{L}_{2}$ and
$N_{2}$ satisfying the constraint ({\bf \ref{constraint}}).
The scalar product in the second line of
Equation ({\bf \ref{nopartition}}) is
a restricted Slavnov scalar product

\begin{eqnarray}
c_{132}^{(0)} = {\mathcal N}_{132}^{hom}\,
 S^{hom}(\{u\}_{N_{1}},
\{v\}_{N_{2}})\,,
\label{finalresult}
\end{eqnarray}

\noindent where $\{u\}_{N_{1}}, \{v\}_{N_{2}}$ are the Bethe
roots corresponding to operators ${\mathcal O}_{1}, {\mathcal O}_{2}$,
respectively. In Appendix {\bf \ref{sec:scalarproducts}}
we obtain an expression  ({\bf \ref{restSlavnov}}) for
the restricted Slavnov scalar product, which in
the homogeneous limit $z_{l}\rightarrow 0$ becomes

\begin{multline}
\label{homrestSlavnov}
S^{hom}(\{u\}_{N_{1}}, \{v\}_{N_{2}}) =
\\
\prod_{k=1}^{N_{2}}
\left\lgroup
\frac{v_{k}+\eta}{v_{k}-\eta}
\right\rgroup^{\left\lgroup \frac{2L_1 - N_1 + N_2}{2}\right\rgroup}
\prod_{j>k}^{N_{1}}\frac{1}{u_{j}-u_{k}}
\prod_{j>k}^{N_{2}}\frac{1}{v_{j}-v_{k}}
\prod_{k=1}^{N_{2}}\frac{1}{
\left\lgroup
(v_{k}-\eta) v_{k}
\right\rgroup^{(N_{1}-N_{2})/2}}
\\
\times
\operatorname{det}
\left\lgroup
\begin{array}{cc}
    {\mathcal M}_{ij} & 1 \le i \le N_{2}\,, \quad 1 \le j \le N_{1} \\
    \hline
    \Psi^{(i-1)}(u_{j},0) & 1 \le i \le (N_{1}-N_{2})/2 \,,
    \quad 1 \le j \le N_{1} \\
    \hline
    \Psi^{(i-1)}(u_{j}+\eta,0) & 1 \le i \le (N_{1}-N_{2})/2 \,,
    \quad 1 \le j \le N_{1}
    \end{array}
\right\rgroup \,,
\end{multline}

\noindent where

\begin{multline}
{\mathcal M}_{ij} = \frac{\eta}{(u_{j}-v_{i})}
\left\lgroup
\prod_{m=1 \atop m\ne j}^{N_{1}}(v_{i}-u_{m}-\eta) -
\left\lgroup
\frac{v_{i}-\eta}{v_{i}+\eta}
\right\rgroup^{\ensuremath{L}_{1}}
\prod_{m=1 \atop m \ne j}^{N_{1}}(v_{i}-u_{m}+\eta)
\right\rgroup \,,
\\
\Psi(u,z) = -\frac{1}{(u-z)(u-z-\eta)}
\prod_{j=1}^{N_{1}}(z-u_{j}) \,,
\quad
\Psi^{(j)}(u,z) =\frac{1}{j!}
\frac{\partial^{j}}{\partial z^{j}} \Psi(u,z) \,.
\end{multline}

Moreover, ${\mathcal N}_{132}$ in Equation ({\bf \ref{N123def}})
is given by

\begin{eqnarray}
{\mathcal N}_{132} =
\left\lgroup
\frac{\ensuremath{L}_{1}\ensuremath{L}_{2}\ensuremath{L}_{3}}{{\mathcal N}_{1}{\mathcal N}_{2}{\mathcal N}_{3}}
\right\rgroup^{\frac{1}{2}} \,,
\label{N123}
\end{eqnarray}

\noindent where ${\mathcal N}_{i}$ are given by
Equation ({\bf \ref{Gaudin}}). Indeed,

\begin{multline}
\langle {\mathcal O}_{1}|{\mathcal O}_{1}\rangle
=
{}_{+}\langle 0|\prod_{j=1}^{N_{1}} B[u_{j},\{z\}_{L_{1}}]^{\dagger}
\prod_{j=1}^{N_{1}} B[u_{j},\{z\}_{L_{1}}] |0\rangle_{+}
\\
=
\left\lgroup
\prod_{j=1}^{N_{1}}
\prod_{l=1}^{L_{1}}\frac{u_{j}^{*} - z_{l}^{*}-\eta}{u_{j}^{*} -
z_{l}^{*}+\eta}
\right\rgroup
{}_{+}\langle 0|\prod_{j=1}^{N_{1}} C[u_{j}^{*},\{z^{*}\}_{L_{1}}]
\prod_{j=1}^{N_{1}} B[u_{j},\{z\}_{L_{1}}] |0\rangle_{+} \,,
\end{multline}

\noindent where we have used Equation ({\bf \ref{Bdagger}}).
The prefactor becomes 1 in the homogeneous limit due to
the zero-momentum constraint. Furthermore, the set of all
Bethe roots $\{ u\}_{N_{1}}$ transforms into itself under
complex conjugation. Hence,

\begin{eqnarray}
\langle {\mathcal O}_{1}|{\mathcal O}_{1}\rangle^{hom} =
\lim_{z_{l}\rightarrow 0}
{}_{+}\langle 0|\prod_{j=1}^{N_{1}} C[u_{j},\{z\}_{L_{1}}]
\prod_{j=1}^{N_{1}} B[u_{j},\{z\}_{L_{1}}] |0\rangle_{+} =
{\mathcal N}_{1}^{hom} \,.
\end{eqnarray}

\noindent Similar considerations apply to
$\langle {\mathcal O}_{2}|{\mathcal O}_{2}\rangle$.
Finally, we note that ${\mathcal N}_{3}=1$.

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

We have obtained a determinant expression for the tree-level OPE
structure constants in planar QCD for operators of the type
({\bf \ref{ops}}), where one of them is BPS-like,
see Equation ({\bf \ref{O3}}).
Indeed, given $(\ensuremath{L}_{1}, N_{1})$
and $\ensuremath{L}_{3}$, the possible values of $(\ensuremath{L}_{2}, N_{2})$ are
determined by Equation ({\bf \ref{constraint}}); then the
corresponding Bethe equations ({\bf \ref{BAE}}) can be solved,
and the structure constants $c_{132}^{(0)}$ can be efficiently
computed using Equation ({\bf \ref{finalresult}}).

In the QCD literature, operators of the form ({\bf \ref{ops}})
would be classified as {\it \lq chiral odd\rq}. While chiral-odd
operators involving quark fields play an important role in certain
hadronic scattering processes \cite{Jaffe:1991ra}, the purely gluonic
chiral-odd operators that we have considered here (with no covariant
derivatives) do not seem to have direct relevance to QCD phenomenology.

It would be interesting to generalize this work to operators with
covariant derivatives, which are more relevant to phenomenology.
Such operators comprise the largest sector of QCD that is known to
be integrable at one loop
\cite{Beisert:2004fv, Braun:1998id, Braun:1999te, Belitsky:1999qh, Ahn:2007ky}.
Another challenge is to go to higher loops (see {\it e.g.}
\cite{Belitsky:2005bu, Gromov:2012vu}).

\section*{Acknowledgments}
We thank G Korchemsky for valuable comments and help with
references, and M Wheeler for discussions.
CA and RN are grateful for the warm hospitality extended to them
at the Nordita workshop
{\it \lq Exact Results in Gauge-String Dualities\rq},
where some of this work was performed. CA also thanks University
of Melbourne and University of Miami for hospitality.
This work was supported in part by the World Class University grant
R32-2009-000-10130-0 (CA), The Australian Research Council (OF),
and by the National Science Foundation
under Grant PHY-0854366 and a Cooper fellowship (RN).

redpajama_set_name: RedPajamaArXiv


The objective of this work package is to achieve a most complete understanding of the water jet/tooling-rock-interaction for a large variety of sedimentary and crystalline rocks, in order to introduce a drilling technology for EGS that is not limited only to one single geological setting but will be applicable all over Europe in low to high enthalpy reservoirs. The reduction of risks for an insufficient connection of the borehole to the reservoir will remain the major challenge for the geothermal industry worldwide. Thus, it is necessary a) to prove the existing water jet-technology for different rock characteristics in laboratory and field scale experiments, b) to detect and understand the gaps, c) to improve the design when possible, d) to investigate if additional information about the geological structure is obtainable while drilling and finally e) to iteratively develop an enhanced jetting technology for rock properties that the existing water jet-tooling is not applicable to.
Development of an enhanced - abrasives/mechanically assisted – water jet/drilling device for heterogeneous, very hard rocks.


The Uniting Church is unusual in a number of ways—one of them being its wholehearted embrace of women's ordination.
The Uniting Church has ordained women since union, as did its constituent denominations before that. We stick out among Australian churches in this regard; while other denominations ordain women, few Australian churches match the Uniting Church in their conviction that women as well as men are called to be ministers.
In this context, it's worth revisiting why the Uniting Church is willing to make this important stand. In 1990 the Uniting Church Assembly released the report, "Why Does the Uniting Church in Australia Ordain Women to the Ministry of the Word?" Fifteen years later it is still an illuminating read, and a thoroughly biblical case for making a break with what has been a very longstanding tradition in the Christian church.
Still, 37 years after union the church has only just elected its second female president-elect. Despite our firm conviction, there are still clearly challenges for women in this space.
On page six we explore Christian feminism and talk to some trailblazing women in the Uniting Church (and one author at the centre of this discussion in the United States).
Thanks to our former moderator Rev Kaye Ronalds for being such a good sport and posing as our "Rosie the minister" on the cover.