text:
\section{Introduction}

One of the major unsolved problems in nuclear physics is the so-called
$A_y$ puzzle in nucleon-deuteron (N$d$) scattering.
The nucleon analyzing power
$A_y$ is the difference in differential cross sections for scattering
of polarized nucleons~\cite{Seyler}:
\begin{equation}
A_y =
\frac{\frac{d\sigma}{d\Omega}\bigr|_{\uparrow}
- \frac{d\sigma}{d\Omega}\bigr|_{\downarrow}}{\frac{d\sigma}{d\Omega}\bigr|_{\uparrow}
+ \frac{d\sigma}{d\Omega}\bigr|_{\downarrow}} \,,
\label{Aydiff}
\end{equation}
where $\uparrow$ denotes the polarization normal to the reaction plane
(spanned by the center-of-mass momentum of the incident and scattered
nucleon). All modern nucleon-nucleon (NN) interactions lead to
practically the same results: They under predict $A_y$ by $30\%$
for laboratory energies $E_{\rm N} \lesssim 30 \, \text{MeV}$ (for a review
see~\cite{Gloeckle}), whereas the predicted $A_y$ is in very good
agreement for higher energies. The contributions of the existing
three-nucleon (3N) interactions to $A_y$ are small at low
energies~\cite{Gloeckle,Kievsky,chiral3NF}. A similar discrepancy
is found for the deuteron vector analyzing power
$i T_{11}$~\cite{Gloeckle}.

The NN contribution to $A_y$ is directly related to the $^3$P$_j$ phase
shifts~\cite{PwavesWitala}, but it is very unlikely that uncertainties
in these phases can resolve the puzzle~\cite{PwavesFriar}. For few $\, \text{MeV}$
energies, $A_y$ is maximal around a center-of-mass scattering angle
$\theta \approx 100^\circ$. This is the
location of the minimum of the differential
cross section so that small effects are amplified in $A_y$. As
a result, a number of small contributions to $A_y$ have been investigated.
For instance, magnetic moment interactions lead to a very small
contribution to $A_y$ near the maximum in $pd$ scattering, but are only
sizable at forward angles for $nd$~\cite{Stoks,KievskyMM}. Moreover, ad hoc
solutions have been proposed that range from introducing a phenomenological
3N spin-orbit force~\cite{KievskyLS} to including the effects of exchanging
one pion in the presence of a two-nucleon correlation~\cite{Canton}.

Recently, Fisher {\it et al.} have shown that the $A_y$ problem
increases from a $30\%$ discrepancy in N$d$ to
a $100\%$ puzzle in $p \, ^3$He~\cite{p3He}. Therefore, one
can expect that the problem becomes even more pronounced for
understanding heavier systems. In addition, the $A_y$ discrepancy
increases with the inclusion of the Coulomb interaction in the
$pd$ system~\cite{Coulomb1,Coulomb2,Coulomb3}.

In the three-body (or higher-body) system, not all pairs of
particles are simultaneously in the {\it two-body} center-of-mass
(cm) system, and therefore relativistic
corrections~\cite{rel1,rel2,rel3} have to be taken into account:
\begin{equation}
\delta V \sim \frac{Q^2}{m^2} \: V_{\rm NN} \,,
\end{equation}
where $Q^2$ includes at least one power of
the two-body cm momentum ${\bm P} = {\bm p_1}+{\bm p_2}$,
$m$ is the nucleon mass and $V_{\rm NN}$ denotes the NN interaction in the
cm frame (for $P=0$). The modern understanding is to consider
these corrections as 3N interactions, but using the formalism
developed in~\cite{rel1,rel2,rel3} it is straightforward to include
these effects to order $(Q/m)^2$ without any new parameters. The
naive expectation is that relativistic corrections are small
at low energies. This was confirmed for selected $nd$ observables
and for energies $E_n \geqslant 28 \, \text{MeV}$~\cite{rel_nd} (where
there is no $A_y$ problem).

In this work, we show that, in contrast to the naive expectation,
relativistic boost effects may be important at low energies.
This is due to spin-violating relativistic corrections, which
couple relative NN S-waves with the $^3$P$_j$ waves (combined
with a change of the two-body cm angular momentum). We find that
the interference with the large S-wave scattering lengths can
lead to resonant enhancements of $A_y$ at low energies. This
effect would explain why predictions for $A_y$ at $E_n \gtrsim
30 \, \text{MeV}$ agree well with experiment. We present a transparent
analytical calculation, based on using the plane-wave impulse
approximation, that explores the effect of relativistic corrections on
spin observables. The effects are small but significant and should be
combined with a complete solution of the Faddeev equations.

This paper is organized as follows. We begin in Section~\ref{Notation} with
a brief discussion of the relevant notation and scattering formalism.
In Section~\ref{RelCorr},
we classify all relativistic corrections to order $(Q/m)^2$
and their impact on the differential cross section and $A_y$.
We calculate analytically their effect on $A_y$ neglecting
distortions. The central findings of this paper are given in
Eq.~(\ref{dUsv}) and in Fig.~\ref{Ay_ndE3}. Our results and
Eq.~(\ref{dUsv}) are general and in a form that should be
implemented in future Faddeev calculations. The reader familiar with
the standard notation and 3N scattering can skip Section~\ref{Notation}.
In Section~\ref{Results}, the contribution to
$A_y$ is estimated using benchmarked $nd$~\cite{Kievsky,KievskyPS}
phase shifts and pionless effective field theory (EFT)
contact interactions for the relativistic corrections $\delta V$.
We conclude in Section~\ref{Concl}
that relativistic boost effects may be important for a precise
understanding of three-body spin observables.

\section{Notation and Scattering Formalism}
\label{Notation}

We follow the notation and conventions of Gl\"ockle {\it et
al.}~\cite{Gloeckle} and define the $nd$ scattering amplitude $M$
by
\begin{multline}
M_{m_j',m_n';m_j,m_n}({\bm q'},{\bm q}) \\[1mm]
= - \frac{2m}{3} \, (2\pi)^2 \,
\langle \phi_d , m_j' ; {\bm q'} , m_n' | U | \phi_d , m_j ;
{\bm q} , m_n \rangle \,,
\label{MU}
\end{multline}
where $m_j, m_n$ are the deuteron total angular momentum
and nucleon spin magnetic quantum numbers respectively, ${\bm q}, {\bm q'}$
are initial and final relative momenta of the nucleon in the
$nd$ cm system, and $U$ denotes the transition amplitude.
The relative momenta are on-shell
related to the neutron laboratory energy $E_n$ by $q =
|{\bm q}|=|{\bm q'}|= \sqrt{\frac{8}{9} \, m \, E_n}$, and the cm
scattering angle is $\cos\theta \equiv \widehat{\bm q} \cdot \widehat{\bm q'}$. Finally,
the plane-wave states are normalized as $\langle {\bm p'} | p \, l \,
m \rangle \equiv i^{-l} \, Y_{lm}\bigl(\widehat{\bm p'}\bigr) \, \delta(p'-p)/(p'p)$
with spherical harmonics $Y_{lm}(\widehat{\bm p})$, and thus
$\langle {\bm p} | {\bm p'} \rangle = \delta^{(3)}({\bm p}-{\bm p'})$ and $\langle {\bm p} | {\bm r} \rangle
= e^{-i {\bm p} \cdot {\bm r}}/(2 \pi)^{3/2}$.

In terms of the scattering amplitude, the spin-averaged differential
cross section $d\sigma/d\Omega$ is given by
\begin{align}
\frac{d\sigma}{d\Omega} &= \frac{1}{(2j+1)(2 s_{\rm N} +1)} \: {\rm Tr} \bigl(
M M^\dagger \bigr) \nonumber \\[1mm]
&= \frac{1}{6} \sum_{m_j',m_n',m_j,m_n}
\bigl| M_{m_j',m_n';m_j,m_n}({\bm q'},{\bm q}) \bigr|^2 \,,
\end{align}
where $j=1$ and $s_{\rm N}=1/2$ are the spin of the deuteron and nucleon
respectively. The nucleon analyzing power is defined by
\begin{equation}
A_i = \frac{{\rm Tr} \bigl( M \bm \sigma^i M^\dagger \bigr)}{{\rm Tr} \bigl( M M^\dagger \bigr)} \,,
\end{equation}
with Pauli matrices $\bm \sigma^i$ and standard conventions for the
coordinate system: $\widehat{\bf z} = \widehat{\bm q}$, $\widehat{\bf y} =
\widehat{\bm q} \times \widehat{\bm q'}/|\widehat{\bm q} \times \widehat{\bm q'}|$ and $\widehat{\bf x} =
\widehat{\bf y} \times \widehat{\bf z}$. This directly leads to
Eq.~(\ref{Aydiff}), if one chooses $\widehat{\bf y}$ as the spin
quantization axis. Finally, due to parity conservation, $A_x =
A_z = 0$~\cite{Seyler}. Using the Fourier transform of
an operator representation for the
deuteron wave function~\cite{BrownJackson}
\begin{align}
\widehat{\phi}_d({\bm p}) \equiv \widetilde{\phi}_d^0(p) + \widetilde{\phi}_d^2(p) \, \frac{S_{12}(\widehat{\bm p})}{\sqrt{8}}
\,, \\[1mm]
\langle {\bm p} , m_j' | \phi_d , m_j \rangle = \bigl\langle m_j' \bigl| \,
\widehat{\phi}_d({\bm p})  \, \bigr| m_j \bigr\rangle \,,
\end{align}
with tensor operator $S_{12}(\widehat{\bm p})$, the scattering amplitude
can be expressed in a convenient operator form
\begin{widetext}
\begin{equation}
M_{m_j',m_n';m_j,m_n}({\bm q'},{\bm q})
= - \frac{2m}{3} \, (2\pi)^2 \int d{\bm p'} \int d{\bm p} \:
\bigl\langle {\bm p'} , m_j' ; {\bm q'} , m_n' \bigl| \, \widehat{\phi}_d({\bm p'}) \, U \,
\widehat{\phi}_d({\bm p}) \, \bigr| {\bm p} , m_j ; {\bm q} , m_n \bigr\rangle \,.
\label{MUop}
\end{equation}
\end{widetext}
Finally, computing the analyzing power is simplified by
coupling the deuteron total angular momentum ${\bm j}$ with the nucleon
spin ${\bm s}_{\rm N}$ to a total spin ${\bm \Sigma} = {\bm j} +
{\bm s}_{\rm N}$. In this basis, the spin matrix elements of the
scattering amplitude are given by
\begin{widetext}
\begin{equation}
M_{\Sigma',m_\Sigma';\Sigma,m_\Sigma}({\bm q'},{\bm q}) = \sum_{m_j',m_j}
(1 \, m_j' \, 1/2 \, m_\Sigma'-m_j' | \Sigma' \, m_\Sigma' )
(1 \, m_j \, 1/2 \, m_\Sigma-m_j | \Sigma \, m_\Sigma )
M_{m_j',m_\Sigma'-m_j';m_j,m_\Sigma-m_j}({\bm q'},{\bm q}) \,.
\end{equation}
\end{widetext}

We use benchmarked $nd$ partial waves for the scattering
amplitude without relativistic corrections, so we briefly discuss
the partial wave expansion. The states with
good total spin $\Sigma$ read $| p \, (ls) j ; {\bm q} \, (j \, 1/2) \Sigma
\, m_\Sigma \rangle$, where $s=1$ is the spin of the deuteron and the
nucleon motion can also be expanded in angular momenta $| q \,
\lambda \, m_\lambda \rangle$. In these states the $nd$ scattering
amplitude is given by
\begin{widetext}
\begin{multline}
M_{\Sigma',m_\Sigma';\Sigma,m_\Sigma}({\bm q'},{\bm q}) = - \frac{2m}{3} \, (2\pi)^2
\sum_{\lambda',m_\lambda',\lambda,m_\lambda} i^{\lambda-\lambda'} \, Y_{\lambda',m_\lambda'}(\widehat{\bm q'})
\, Y^*_{\lambda,m_\lambda}(\widehat{\bm q}) \\[1mm]
\times \sum_{l,l'} \int p'^2 dp' \, \widetilde{\phi}_d^{l'}(p')
\int p^2 dp \, \widetilde{\phi}_d^l(p) \,
\bigl\langle p' \, (l' 1) 1 ; q' \, \lambda' \, m_\lambda' \, (1 , 1/2)
\Sigma' \, m_\Sigma' \bigl| \, U \, \bigr| p \, (l 1) 1 ; q \, \lambda \, m_\lambda \,
(1 , 1/2) \Sigma \, m_\Sigma \bigr\rangle \,.
\label{Mpw1}
\end{multline}
\end{widetext}
Next one couples the nucleon angular momentum with the total spin
to the total angular momentum ${\bm J} = {\bm \lambda} +
{\bm \Sigma}$, for which $U$ is diagonal in $J$ and independent
of $m_J$, thus $m_\lambda' + m_\Sigma' = m_\lambda + m_\Sigma$. With $\widehat{\bm q} =
\widehat{\bf z}$, we have $Y^*_{\lambda,m_\lambda}(\widehat{\bm q})=\delta_{m_\lambda,0} \,
\sqrt{\frac{2\lambda+1}{4\pi}}$, and consequently $m_J = m_\Sigma$
and $m_\lambda' = m_\Sigma - m_\Sigma'$. The second line in Eq.~(\ref{Mpw1})
in the coupled $(\lambda \Sigma') J \, m_J$ basis is independent of $m_J$
and can be decomposed as $(\delta_{\lambda',\lambda} \delta_{\Sigma',\Sigma}
- S_{\lambda',\Sigma';\lambda,\Sigma}^J)/(4 \pi i m q/3)$. With this at hand,
the partial wave decomposition reads
\begin{widetext}
\begin{align}
M_{\Sigma',m_\Sigma';\Sigma,m_\Sigma}({\bm q'},{\bm q}) &= \frac{i \sqrt{\pi}}{q}
\sum_{\lambda',\lambda,J} i^{\lambda-\lambda'} \sqrt{2\lambda+1} \:
Y_{\lambda',m_\Sigma-m_\Sigma'}(\widehat{\bm q'}) \nonumber \\[1mm]
&\times ( \lambda' \, m_\Sigma-m_\Sigma' \, \Sigma' \, m_\Sigma' | J \, m_\Sigma )
\, ( \lambda \, 0 \, \Sigma \, m_\Sigma | J \, m_\Sigma ) \,
\bigl( \delta_{\lambda',\lambda} \delta_{\Sigma',\Sigma}
- S_{\lambda',\Sigma';\lambda,\Sigma}^J \bigr) \,,
\label{Mpw2}
\end{align}
\end{widetext}
where $S_{\lambda',\Sigma';\lambda,\Sigma}^J$ is given in terms of the $nd$
phase shifts and mixing parameters~\cite{Seyler} (see also
Eqs.~(209)--(214) in Ref.~\cite{Gloeckle}).

\section{Relativistic corrections}
\label{RelCorr}

Boost corrections to the two-nucleon interaction depend on the
total momentum ${\bm P}$ of the pair and are obtained by
satisfying the commutation relations of the Poincar\'{e} group~\cite{rel1}.
To leading order in $(Q/m)^2$, the relativistic boost corrections are
given in momentum space by (for the corresponding coordinate space
expression, see Eq.~(1.7) in Ref.~\cite{rel3})
\begin{widetext}
\begin{align}
\delta v_{{\bm \sigma}_1,{\bm \sigma}_2}({\bm k'},{\bm k},{\bm P}) &= -\frac{P^2}{4m^2} \,
v_{{\bm \sigma}_1,{\bm \sigma}_2}({\bm k'},{\bm k})
+ \frac{i}{8 m^2} \, \bigl[ ({\bm \sigma}_1 - {\bm \sigma}_2) , v_{{\bm \sigma}_1,{\bm \sigma}_2}({\bm k'},{\bm k}) \bigr]
\times {\bm P} \cdot {\bm k} \nonumber \\[1mm]
&- \frac{i}{8 m^2} \, ({\bm \sigma}_1 - {\bm \sigma}_2) \times {\bm P} \cdot ({\bm k} - {\bm k'}) \,
v_{{\bm \sigma}_1,{\bm \sigma}_2}({\bm k'},{\bm k})
- \frac{1}{8 m^2} \, \bigl( {\bm P} \cdot ({\bm k} - {\bm k'}) \bigr) \, {\bm P} \cdot
\nabla_{{\bm k}-{\bm k'}} \, v_{{\bm \sigma}_1,{\bm \sigma}_2}({\bm k'},{\bm k}) \,,
\label{dVdirect}
\end{align}
\end{widetext}
where $v_{{\bm \sigma}_1,{\bm \sigma}_2}({\bm k'},{\bm k},{\bm P})$ is the direct NN interaction in
the cm system, with initial and final relative momenta ${\bm k} = ({\bm p}_1 -
{\bm p}_2)/2$ and ${\bm k'} = ({\bm p'_1} - {\bm p'_2})/2$, and Eq.~(\ref{dVdirect}) accounts
only for the direct term of the boost correction.
As explained in Ref.~\cite{rel3} the Poincar\'{e} group commutation
relations do not have a unique solution. The operator $\delta v$ can
have an additional term of the form
\begin{equation}
\delta v' = -i \, \bigl[ \chi , H_0 + v \bigr] \,,
\label{extra}
\end{equation}
where $\chi$ is a translationally invariant function and
$H_0$ is the non-interacting Hamiltonian. One must pay attention
to this term when studying scattering processes.

Some of the operators in Eq.~(\ref{dVdirect}) can be obtained from purely
classical considerations~\cite{rel3}. The first term arises from treating
the potential as a contribution to the nucleon mass and then
expanding the relativistic energy operator. The final term of
Eq.~(\ref{dVdirect}) is due to the effects of Lorentz contraction.
The third term results from Thomas precession in which objects with
spin precess when they accelerate, since rotations do not commute
with boosts. The commutator term of Eq.~(\ref{dVdirect})
does not have an analog in classical mechanics.

Our procedure will be to calculate the leading relativistic corrections
$\delta M$ in the basis $| {\bm p} , m_j ; {\bm q} , m_n \rangle$ of Eq.~(\ref{MUop})
by accounting for the change in $U$, $\delta U$ caused by $\delta v$
of Eq.~(\ref{dVdirect}). In this exploratory study we use the
plane-wave impulse approximation, which treats one of the nucleons
as a spectator. Taking the matrix element of $\delta U$
within plane-wave neutron-deuteron states yields $\delta M$. The use of
the plane-wave impulse
approximation enables us to analytically study the effect of
relativistic boost corrections and make a first assessment of their
importance. A full Faddeev calculation including distortions will
eventually be needed to make a complete assessment.
Our present use of the plane-wave impulse approximation has an additional
advantage: The matrix element of the term $\delta v'$ of \eq{extra},
taken between on-shell elastic scattering states vanishes.

There are three contributions to $\delta U$ arising from the
three pairs in the $nd$ system:
\begin{equation}
\delta U = \delta V_{12} + \delta V_{13} + \delta V_{23} \,.
\end{equation}

Since relativistic corrections are of order $(Q/m)^2$, we keep
only the central parts in $v_{{\bm \sigma}_1,{\bm \sigma}_2}({\bm k'},{\bm k})$. Non-central
interactions start at ${\mathcal O}\bigl((Q/m_\pi)^2\bigr)$ in pionless effective
field theory, which is relevant for the energies of interest.
Therefore, the spin structure is limited to
\begin{equation}
v_{{\bm \sigma}_1,{\bm \sigma}_2} = v_{\openone} + v_{\rm spin} \, {\bm \sigma}_1 \cdot {\bm \sigma}_2 \,.
\end{equation}
Furthermore we can neglect the last term in
Eq.~(\ref{dVdirect}) of ${\mathcal O}\bigl((Q/m)^2 (Q/m_\pi)^2\bigr)$.

The next step is to include the exchange term. We need to compute
\begin{multline}
\delta V_{{\bm \sigma}_1,{\bm \sigma}_2}({\bm k'},{\bm k},{\bm P}) = \\[1mm]
\delta v_{{\bm \sigma}_1,{\bm \sigma}_2}({\bm k'},{\bm k},{\bm P}) - P_{\bm \sigma} P_{\bm \tau} \:
\delta v_{{\bm \sigma}_1,{\bm \sigma}_2}(-{\bm k'},{\bm k},{\bm P}) \,,
\end{multline}
where the spin (isospin) exchange operator is  $P_{\bm \sigma}$
($P_{\bm \tau}$) and $V_{{\bm \sigma}_1,{\bm \sigma}_2}({\bm k'},{\bm k}) = v_{{\bm \sigma}_1,{\bm \sigma}_2}({\bm k'},{\bm k})
- P_{\bm \sigma} P_{\bm \tau} \: v_{{\bm \sigma}_1,{\bm \sigma}_2}(-{\bm k'},{\bm k})$ denotes
the antisymmetrized interaction. Writing the commutator term of
Eq.~(\ref{dVdirect}) explicitly and using the property that
$\{ ({\bm \sigma}_1 - {\bm \sigma}_2) , P_{\bm \sigma} \} = 0$ leads directly to the
result:
\begin{widetext}
\begin{equation}
\delta V_{{\bm \sigma}_1,{\bm \sigma}_2}({\bm k'},{\bm k},{\bm P}) = -\frac{P^2}{4m^2} \,
V_{{\bm \sigma}_1,{\bm \sigma}_2}({\bm k'},{\bm k})
- \frac{i}{8 m^2} \, V_{{\bm \sigma}_1,{\bm \sigma}_2}({\bm k'},{\bm k}) \, ({\bm \sigma}_1 - {\bm \sigma}_2)
\times {\bm P} \cdot {\bm k}
+ \frac{i}{8 m^2} \, ({\bm \sigma}_1 - {\bm \sigma}_2) \times {\bm P} \cdot {\bm k'} \,
V_{{\bm \sigma}_1,{\bm \sigma}_2}({\bm k'},{\bm k}) \,.
\label{dVdirplusex}
\end{equation}
\end{widetext}
In antisymmetrized states, relativistic corrections thus have the
form of $V$~(boost corrections in)$-$(boost corrections out)~$V$.

It is evident that the first ($P^2$) term in Eq.~(\ref{dVdirplusex})
will lead to a relativistic correction to the $nd$ scattering amplitude
that is a scalar in spin and of general structure:
\begin{equation}
\delta M_{P^2} \sim \openone \: \frac{Q^2}{m^2} \quad {\rm and} \quad
{\bm S} \cdot {\bm \sigma}_3 \: \frac{Q^2}{m^2} \,,
\end{equation}
where ${\bm S} = ( {\bm \sigma}_1 + {\bm \sigma}_2 )/2$ is the deuteron spin and
the nucleon spin operator is given by ${\bm s}_{\rm N} = {\bm \sigma}_3/2$.
In the following we will show that the spin-violating (sv)
relativistic corrections (the last two terms in
Eq.~(\ref{dVdirplusex})) lead to terms of the form
\begin{equation}
\delta M_{\rm sv} \sim {\bm S}^y \: \frac{Q^2}{m^2} \quad
{\rm and} \quad {\bm \sigma}_3^y \: \frac{Q^2}{m^2} \,.
\end{equation}
The leading contributions to the differential cross section and
to $A_y$ are from the interference of $\delta M$ of ${\mathcal O}\bigl(
(Q/m)^2 \bigr)$ with the leading $nd$ scattering amplitude at
low energies. Similar to the above considerations for the two-nucleon
interaction, the leading operators in $M$ are given by the central
part:
\begin{equation}
M_{{\bm S},{\bm \sigma}_3} = M_{\openone} + M_{\rm spin} \, {\bm S} \cdot {\bm \sigma}_3 +
{\mathcal O}\bigl((Q/m_\pi)^2\bigr) \,.
\end{equation}
We can now evaluate the relativistic corrections to the nucleon analyzing
power $\delta A_y$ and to the differential cross section
$\delta (d\sigma/d\Omega)$:
\begin{align}
\delta A_y &= \frac{{\rm Tr} \bigl( \delta M \, \bm \sigma^y M^\dagger + M \bm \sigma^y \,
\delta M^\dagger \bigr)}{{\rm Tr} \bigl( M M^\dagger \bigr)} \nonumber \\[1mm]
&- A_y \: \frac{{\rm Tr} \bigl( \delta M \, M^\dagger + M \, \delta M^\dagger \bigr)}
{{\rm Tr} \bigl( M M^\dagger \bigr)} \,,
\label{dAygen} \\[2mm]
\delta \frac{d\sigma}{d\Omega} &= \frac{1}{6} \:
{\rm Tr} \bigl( \delta M \, M^\dagger + M \, \delta M^\dagger \bigr) \,.
\label{dcrossgen}
\end{align}
Since $A_y$ is small, we can neglect the second term in
Eq.~(\ref{dAygen}). For the leading contributions, it then follows
that only $\delta M_{\rm sv}$ contributes to $\delta A_y$,
\begin{equation}
\delta A_y = \frac{{\rm Tr} \bigl( \delta M_{\rm sv} \, \bm \sigma^y M^\dagger + M \bm \sigma^y \,
\delta M^\dagger_{\rm sv} \bigr)}{{\rm Tr} \bigl( M M^\dagger \bigr)} +
{\mathcal O}\biggl( \frac{Q^4}{m^2 m_\pi^2} \biggr) \,,
\end{equation}
and only $\delta M_{P^2}$ contributes to $\delta (d\sigma/d\Omega)$.
A straightforward calculation of the spin-violating relativistic
corrections arising from $V_{\openone}({\bm k'},{\bm k}) + V_{\rm spin}({\bm k'},{\bm k}) \,
{\bm \sigma}_1 \cdot {\bm \sigma}_2 \equiv (1-P_{\bm \sigma} P_{\bm \tau} P_{\bm k})
(v_{\openone} + v_{\rm spin} \, {\bm \sigma}_1 \cdot {\bm \sigma}_2)$ yields
\begin{widetext}
\begin{equation}
\delta V^{\rm sv}_{{\bm \sigma}_1,{\bm \sigma}_2}({\bm k'},{\bm k},{\bm P}) =
- \frac{i}{8 m^2} \, ({\bm \sigma}_1 - {\bm \sigma}_2) \times {\bm P} \cdot ({\bm k}-{\bm k'})
\, \bigl( V_{\openone}({\bm k'},{\bm k}) -
V_{\rm spin}({\bm k'},{\bm k}) \bigr)
+ \frac{1}{4 m^2} \, ({\bm \sigma}_1 \times {\bm \sigma}_2) \times {\bm P} \cdot
({\bm k} + {\bm k'}) \, V_{\rm spin}({\bm k'},{\bm k}) \,.
\label{dVsv}
\end{equation}
\end{widetext}
The resulting spin-violating interactions connect the two-body
$^3$P$_j$ waves (which are crucial for $A_y$) with the two-body
S-waves. The S-waves are resonant at low energies with large
scattering lengths, $a_0 \equiv a_{^1{\rm S}_0} = - 23.768 \pm 0.006 \, \text{fm}$
and $a_1 \equiv a_{^3{\rm S}_1} = 5.420 \pm 0.001 \, \text{fm}$~\cite{as},
and therefore
the interference with the $^3$P$_j$ waves can lead to a resonant
enhancement of these relativistic corrections at low-energies.
For higher energies, the S-wave phase shifts decrease, so
the effect of the spin-violating interactions decreases.

Including isospin and restricting two-nucleon interactions to
S-waves, the central part of the
antisymmetrized two-body interaction can be written as
\begin{multline}
V_{i,3} = \frac{1}{8} \bigl[ V_0 \, (1-{\bm \sigma}_i \cdot {\bm \sigma}_3)
(3+ {\bm \tau}_i \cdot {\bm \tau}_3) \\[1mm]
+ V_1 \, (3+{\bm \sigma}_i \cdot {\bm \sigma}_3) (1- {\bm \tau}_i
\cdot {\bm \tau}_3)\bigr] \,,
\end{multline}
where $i=1,2$ and ${\bm \tau}_{i,3}$ denote Pauli matrices that operate in
isospin space and $V_{0,1}$ are projections on $s=0,1$ states.
The operator ${\bm \tau}_i \cdot {\bm \tau}_3$ vanishes in $nd$
states, and thus we have
\begin{equation}
V_{\openone} = \frac{3}{8} \, (V_0 + V_1) \quad {\rm and} \quad
V_{\rm spin} = \frac{1}{8} \, (V_1 -3 V_0) \,.
\end{equation}

We use leading-order $(Q/m_\pi)^0$ pionless EFT contact
interactions~\cite{ksw}, where the operators $V_0$ and
$V_1$ are momentum independent:
\begin{equation}
V_i = \frac{C_i}{2 \pi^2 m} \quad {\rm with} \quad
C_i = \frac{1}{\frac{1}{a_i} - \mu} \,,
\label{Bornc}
\end{equation}
for $i=0,1$. Here, $\mu$ is the renormalization
scale in dimensional regularization with power-divergence
subtraction scheme~\cite{ksw}. A similar expression is
obtained for a momentum-cutoff regularization.
The operator of \eq{dVsv} therefore has the form of a
spin-violating operator (${\bm \sigma}_1 - {\bm \sigma}_2$) dotted into a
momentum vector that induces transitions between spin triplet
(singlet)/relative S-wave and spin singlet (triplet)/relative
P-wave states. The momentum vector ${\bm k}$ (${\bm k'}$) in \eq{dVsv}
explicitly projects on incoming (outgoing) P-wave states.
The change in the orbital angular momentum is compensated
with a corresponding change of the two-body cm angular momentum,
so that the total angular momentum is preserved.

At low energies we can take the P-wave states to be plane waves,
but we include iterated S-wave interactions $V_i$ in the initial
state (for ${\bm k'}$) and final state (for ${\bm k}$). This leads to
replacing $C_i$ by
\begin{equation}
C_i \to \frac{C_i}{1 + C_i \, (\mu +i \sqrt{m \, E_{\rm rel}} )}
= \frac{1}{\frac{1}{a_i} + i \sqrt{m \, E_{\rm rel}}} \,,
\label{newc}
\end{equation}
where $E_{\rm rel}$ is the relative energy (in the two-body cm of system).
As a result of these initial and final state interactions, we
find that the operators $V_i$ that enter in \eq{dVsv} are
independent of the renormalization scale $\mu$.

\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.7,clip=]{diags.eps}
\caption{Contributions of the spin-violating interactions to the
relativistic corrections $\delta U_{\rm sv}$
and our conventions for the Jacobi momenta. In Born approximation,
the low-energy coefficients are given by the $C_i$ of \eq{Bornc}, and
in the plane-wave impulse approximation we use the $C_i$ of \eq{newc}.}
\label{diags}
\end{center}
\end{figure}

Next we calculate the contributions of the spin-violating relativistic
corrections to $\delta A_y$ for the $nd$ system (where the
Coulomb interaction does not operate). We neglect distortions
that would involve the nucleon treated as a spectator.
We work in the
three-body cm system ${\bm p}_1 + {\bm p}_2 + {\bm p}_3 = 0$, where nucleons $1$, $2$
constitute the deuteron, and nucleon $3$ is the free neutron.
Since the matrix element of $\delta V^{\rm sv}_{12}$ vanishes
when evaluated in the deuteron eigenstate, we need only to evaluate
two contributions to the spin-violating collision operator
$\delta U_{\rm sv} = \delta V^{\rm sv}_{13} + \delta V^{\rm sv}_{23}$,
shown diagrammatically in Fig.~\ref{diags}. We employ
Jacobi momenta and the incoming nucleon momenta are expressed as
${\bm p}_1 = {\bm p} - {\bm q}/2$, ${\bm p}_2 = -{\bm p} -{\bm q}/2$ and ${\bm p}_3 = {\bm q}$ (with
primed momentum labels for the outgoing nucleons). The second
contribution $\delta V^{\rm sv}_{23}$ can be obtained from
$\delta V^{\rm sv}_{13}$ by replacing ${\bm \sigma}_1 \to {\bm \sigma}_2$,
${\bm p} \to - {\bm p}$ and ${\bm p'} \to - {\bm p'}$. Since the deuteron is even
in momentum ($l=0,2$), we can change variables in Eq.~(\ref{MUop})
back to $-{\bm p} \to {\bm p}$ and $-{\bm p'} \to {\bm p'}$. Consequently, the
contribution of $\delta V^{\rm sv}_{23}$ to $\delta M_{\rm sv}$
is identical to the contribution of $\delta V^{\rm sv}_{13}$
after replacing ${\bm \sigma}_1 \to {\bm \sigma}_2$ in the latter.

Inserting this into Eq.~(\ref{dVsv}) (with $1,2 \to 1,3$ and $1,2
\to 2,3$), we obtain for the total leading $(Q/m)^2$ relativistic
corrections relevant for $\delta A_y$
\begin{equation}
\delta U^{\rm sv}_{{\bm S},{\bm \sigma}_3}({\bm p},{\bm q'},{\bm q}) =
\delta^{(3)}\bigl({\bm p'} - ({\bm p} + \Delta) \bigr) \,
\delta \widetilde{U}^{\rm sv}_{{\bm S},{\bm \sigma}_3}({\bm p},{\bm q'},{\bm q}) \,,
\end{equation}
where the delta function accounts for the conservation of the
two-body cm momentum, the momentum transfer is
$\Delta \equiv ({\bm q}-{\bm q'})/2$, and we have
\begin{widetext}
\begin{equation}
\delta \widetilde{U}^{\rm sv}_{{\bm S},{\bm \sigma}_3}({\bm p},{\bm q'},{\bm q})
= \biggl[ - \frac{i}{4 m^2} \, ({\bm S} - {\bm \sigma}_3) \times ({\bm p}+\frac{{\bm q}}{2})
\cdot ({\bm q'}-{\bm q}) \, \frac{3 V_0 + V_1}{4}
- \frac{1}{2 m^2} \, ({\bm S} \times {\bm \sigma}_3) \times ({\bm p}+\frac{{\bm q}}{2})  \cdot
({\bm q} + {\bm q'}) \, \frac{V_1 -3 V_0}{8} \biggr] \,.
\label{dUsv}
\end{equation}
\end{widetext}
Here the relative momentum arguments
of $V_{0,1}$ are ${\bm k} = \frac{{\bm p}}{2} - \frac{3{\bm q}}{4}$ and ${\bm k'} =
\frac{{\bm p}}{2}+\frac{{\bm q}}{4} - {\bm q'}$.
The result of Eq.~(\ref{dUsv}) is  general and useful as input to
Faddeev calculations, in which the term is dressed by the effects of
initial and final state strong interactions.

Using \eq{dUsv} in Eq.~(\ref{MUop}), we obtain our final expression
for the relevant change in the scattering amplitude:
\begin{widetext}
\begin{equation}
\delta M^{\rm sv}_{m_j',m_n';m_j,m_n}({\bm q'},{\bm q})
= - \frac{2m}{3} \, (2\pi)^2 \int d{\bm p} \:
\bigl\langle m_j' , m_n' \bigl| \, \widehat{\phi}_d({\bm p} + \Delta) \,
\delta \widetilde{U}^{\rm sv}_{{\bm S},{\bm \sigma}_3}({\bm p},{\bm q'},{\bm q}) \,
\widehat{\phi}_d({\bm p}) \, \bigr| m_j , m_n \bigr\rangle \,.
\label{MUsv}
\end{equation}
\end{widetext}
Next, we estimate the impact of these spin-violating boost
corrections on $A_y$ based on pionless EFT contact interactions
for $V_{0,1}$ and using benchmarked $nd$ phase shifts from
Kievsky {\it et al.}~\cite{Kievsky,KievskyPS} for $M$. This has the
advantage that $\delta A_y$ can be evaluated analytically and the
physics is transparent.

\section{Results}
\label{Results}

We can transform variables in Eq.~(\ref{MUsv})
from ${\bm p} \to {\bm p} - \Delta/2$. For momentum-independent interactions
$V_{0,1}$, terms linear in ${\bm p}$ in $ \widetilde{U}^{\rm sv}$ integrate
to zero after this variable transformation. Therefore, we can replace
${\bm p} + {\bm q}/2$ by $- \Delta/2 + {\bm q}/2 = ({\bm q} + {\bm q'})/4$ in Eq.~(\ref{dUsv}),
and as a result the ${\bm S} \times {\bm \sigma}_3$ term vanishes.
Note that a relatively small quantity $({\bm q} + {\bm q'})/4$ determines the change
in the computed $A_y$. Furthermore, we simplify the integral by
approximating $E_{\rm rel}$ of \eq{newc} by zero. The relative energy
is very low, $E_{\rm rel} = E_{\rm d} + 2 E_n/3 - 3 ({\bm p} + \frac{{\bm q}}{2})^2/(4m)$
(with deuteron binding energy $E_{\rm d} = -2.22 \, \text{MeV}$), and
we have $E_{\rm rel} < 0$ for the energy of interest ($E_n = 3 \, \text{MeV}$).
So the imaginary part vanishes. Since we do not include effective
range corrections, we further neglect the energy dependence
of the real part of \eq{newc}. It is necessary to reexamine this
treatment within the framework of a Faddeev calculation that we
shall advocate below. We thus find
\begin{widetext}
\begin{equation}
\delta M^{\rm sv}_{m_j',m_n';m_j,m_n}({\bm q'},{\bm q})
= \frac{i}{24 m^2} \, (3 C_0 + C_1 ) \int d{\bm p} \:
\bigl\langle m_j' , m_n' \bigl| \, \widehat{\phi}_d({\bm p} + \Delta/2) \,
\bigl[ ({\bm S} - {\bm \sigma}_3) \cdot {\bm q} \times {\bm q'} \bigr]
\, \widehat{\phi}_d({\bm p}-\Delta/2) \, \bigr| m_j , m_n \bigr\rangle \,.
\end{equation}
\end{widetext}
Since ${\bm S}$ commutes with the tensor operator $S_{12}(\widehat{\bm p})$, we
can move the operator $[ ({\bm S} - {\bm \sigma}_3) \cdot {\bm q} \times {\bm q'} ]$ to the
right of $\widehat{\phi}_d({\bm p}-\Delta/2)$ and insert a one operator in deuteron
spin space $\openone = \sum_{m_j''} | m_j'' \rangle \langle m_j'' |$. Using
\begin{multline}
\int d{\bm p} \: \langle m_j' | \, \widehat{\phi}_d({\bm p} + \Delta/2) \, \widehat{\phi}_d({\bm p}-\Delta/2)
\, | m_j'' \rangle \\[1mm]
= \delta_{m_j', m_j''} + {\mathcal O}(\Delta^2) \,,
\end{multline}
we can neglect the momentum dependence of the charge form factor,
as well as the magnetic and quadrupole form factors of the deuteron.
With ${\bm q} \times {\bm q'} = q^2 \sin\theta \: \widehat{\bf y}$, we
have for the spin-violating boost corrections in operator form
\begin{equation}
\delta M_{\rm sv} = i \, R \, ({\bm S} - {\bm \sigma}_3)^y = i \,
\frac{q^2 \sin\theta}{24 m^2} \, (3 C_0 + C_1) \, ({\bm S} - {\bm \sigma}_3)^y \,.
\end{equation}
Here we have for convenience combined all factors into the coefficient
$R$. Combining our results with Eq.~(\ref{dAygen}) leads to
\begin{equation}
\delta A_y = i \, R \,
\frac{{\rm Tr} \bigl( {\bm S}^y \, {\bm \sigma}_3^y \, M^\dagger - M \, {\bm S}^y \, {\bm \sigma}_3^y + M -
M^\dagger \bigr)}{{\rm Tr} \bigl( M M^\dagger \bigr)} \,.
\label{dAyfinal}
\end{equation}
The necessary spin matrix element follows from the Wigner-Eckert
theorem:
\begin{equation}
\langle m_j' | \, {\bm S}^y \, | m_j \rangle = i \, \bigl[
( 1 \, m_j' \, 1 \, 1 | 1 \, m_j ) +
( 1 \, m_j' \, 1 \, -1 | 1 \, m_j ) \bigr] \,.
\end{equation}

We are now in the position to study the impact on $A_y$. For the
$nd$ scattering amplitude $M$ in Eq.~(\ref{dAyfinal}) we use the
phase shifts from Kievsky {\it et al.}~\cite{Kievsky,KievskyPS}.
These are based on the Argonne $v_{18}$ NN and the
Urbana 3N interaction for $J^P$ up to $7/2^+$ (from Table~2
in Ref.~\cite{Kievsky}) and on the Argonne $v_{14}$ NN interaction
for $7/2^-$ and $9/2 \leqslant J \leqslant 13/2$ (from Tables~I
and~II in Ref.~\cite{KievskyPS}). No parameters are adjusted.
As a check, we have reproduced the differential cross section
of Ref.~\cite{Kievsky}, which is in very good agreement with
experiment.

\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.36,clip=]{Ay_ndE3.eps}
\caption{(Color online) The $nd$ analyzing power $A_y$ for $E_n
= 3 \, \text{MeV}$ as a function of center-of-mass scattering angle $\theta$.
The dashed curve is based on $nd$ phase shifts obtained from NN
and 3N interactions~\cite{Kievsky,KievskyPS} (for details
see text). The solid curves include our results for the boost
corrections without distortion. The data is taken from
McAninch {\it et al.}~\cite{McAninch}.}
\label{Ay_ndE3}
\end{center}
\end{figure}

The effect of the spin-violating boost corrections on $A_y$ is shown
in Fig.~\ref{Ay_ndE3} for $E_n = 3 \, \text{MeV}$ and in comparison to the data
from McAninch {\it et al.}~\cite{McAninch}. We see that the influence
of the spin-violating relativistic corrections is to increase
the computed value of $A_y(\theta)$ by about $10 \%$ at the peak.
This contribution is significant. It shows that relativistic effects
may be relevant even at very low energies due to resonant
enhancements. However, this effect alone is too small to solve the
$A_y$ puzzle. We therefore explore the effects of initial and
final state interactions with the nucleon that has been treated as a
spectator so far, and finally, we discuss the energy dependence of
these boost effects.

We have considered the effects of boosting the interaction between
nucleons 13 and 23, while treating the nucleon 2 and 1 as a spectator.
The total momentum of the boosted pair is effectively $({\bm q} +{\bm q'})/4$.
When for instance the projectile neutron interacts with nucleon 2
before interacting with nucleon 1, the total momentum of the 13
nucleon-pair will be increased due to the attractive interaction
between nucleons 2 and 3, and we expect our boost effect to be
enhanced. We explore the size of a 23 interaction with a schematic
square-well $^1$S$_0$ potential of Ref.~\cite{BrownJackson}, which
has a depth $V_0 = 13.4 \, \text{MeV}$ and range $R = 2.65 \, \text{fm}$. Using
conventional NN interactions, we estimate the probability to find
two nucleons in a deuteron closer than $R$ to be about $50 \%$,
so that a 23 interaction can be followed by a 13 interaction about
half of the time. If this occurs, the relative momentum inside
the well $\kappa$ is given by
\begin{equation}
\kappa^2 = \frac{3 q^2}{4} + m V_0 = m \, \biggl( \frac{2 E_n}{3} + V_0
\biggr) \,,
\end{equation}
so that $\kappa \approx 0.6 \, \text{fm}^{-1}$. These prescattering contributions
occur about half of the time, and thus the relevant average momentum
is $\approx 0.3 \, \text{fm}^{-1}$. This value is about three times larger than
$| {\bm q} + {\bm q'} |/4 \approx 0.1 \, \text{fm}^{-1}$. Therefore, we expect that distortion
effects will increase $A_y$ further. This simple estimate should be
taken only as an assessment that initial state interactions can make
a large contribution to the boost effects.

With increasing energy, the resonant enhancement of the spin-violating
relativistic corrections decreases. This is both due to the effective
range $r_i$ (and the decrease of the S-wave phase shifts with increasing
energy),
\begin{equation}
\frac{1}{a_i} \to \frac{1}{a_i} - \frac{r_i \, m \, E_{\rm rel}}{2} \,,
\end{equation}
and the impact of the imaginary part $i \sqrt{m \, E_{\rm rel}}$. A detailed
study of the energy dependence of these boost effects is beyond the
scope of this paper and will be left to a future
investigation~\cite{MS}.

\section{Summary and Future Steps}
\label{Concl}

In this paper, we have presented the first estimate of the effects
of relativistic boost corrections on the $nd$ analyzing power $A_y$.
We have focused on spin-violating relativistic corrections at order
$(Q/m)^2$, which can be important at low energies due to a resonant
enhancement from the large S-wave scattering lengths. Since boost
corrections depend on the two-body cm momentum, the modern viewpoint
is to consider their effects as 3N interactions. We have used the
formalism of Ref.~\cite{rel1,rel2,rel3}, where it is straightforward
to include relativistic corrections to order $(Q/m)^2$ without
any new parameters. The relevant spin-violating
contribution to the $nd$ transition
amplitude is given in \eq{dUsv}.

These corrections induce a $10 \%$ change in the computed value of
the $nd$ analyzing power $A_y$ for laboratory energy $E_n = 3 \, \text{MeV}$.
This is a small, but significant contribution of the sign necessary
to resolve the $A_y$ puzzle. Our result was estimated using the plane-wave
impulse approximation, which leads to a transparent analytical
calculation. The present study is clearly not complete. The
effects of initial and final state interactions allow for additional
contributions. For instance, the effects of $\delta V_{12}$ would
not vanish (as in the present calculation), if initial or final state
interactions excite the deuteron. Faddeev calculations that include
distortions are therefore needed to conclude on the $A_y$ puzzle. The
results presented here are mainly intended to stimulate the interest
of the few-body community to include relativistic corrections in their
complete solutions of the 3N problem.

For energies $E_n \gtrsim 30 \, \text{MeV}$, the predicted $A_y$ based on
microscopic NN and 3N interactions (without relativistic corrections)
is in very good agreement with experiment. Our present results are
not in contradiction to these findings, since the resonant
enhancement of our spin-violating boost corrections decreases with
energy. A detailed study of the energy dependence will be presented
in a future paper~\cite{MS}. In addition, future work will estimate
the scaling to larger systems and the impact on the $A_y$ puzzle
in $n \, ^3$H scattering~\cite{MS}.

\acknowledgments

We thank Dick Furnstahl,
Andreas Nogga and Rob Timmermans for useful discussions.
This work was supported in part by the US Department of Energy
under Grant No.~DE--FG02--97ER41014 and the Natural Sciences and
Engineering Research Council of Canada (NSERC). TRIUMF receives federal
funding via a contribution agreement through the National Research
Council of Canada.

redpajama_set_name: RedPajamaArXiv


text:
Q: Regex to get value between two colon excluding the colons I have a string like this:
something:POST:/some/path

Now I want to take the POST alone from the string. I did this by using this regex
:([a-zA-Z]+):

But this gives me a value along with colons. ie I get this:
:POST:

but I need this
POST

My code to match the same and replace it is as follows:
String ss = "something:POST:/some/path/";
Pattern pattern = Pattern.compile(":([a-zA-Z]+):");
Matcher matcher = pattern.matcher(ss);

if (matcher.find()) {
    System.out.println(matcher.group());
    ss = ss.replaceFirst(":([a-zA-Z]+):", "*");
}
System.out.println(ss);

EDIT:
I've decided to use the lookahead/lookbehind regex since I did not want to use replace with colons such as :*:. This is my final solution.
String s = "something:POST:/some/path/";
String regex = "(?<=:)[a-zA-Z]+(?=:)";

Matcher matcher = Pattern.compile(regex).matcher(s);

if (matcher.find()) {
    s = s.replaceFirst(matcher.group(), "*");
    System.out.println("replaced: " + s);
}
else {
    System.out.println("not replaced: " + s);
}

A: There are two approaches:

*

*Keep your Java code, and use lookahead/lookbehind (?<=:)[a-zA-Z]+(?=:), or

*Change your Java code to replace the result with ":*:"
Note: You may want to define a String constant for your regex, since you use it in different calls.

A: UPDATE
Looking at your update, you just need ReplaceFirst only:
String result = s.replaceFirst(":[a-zA-Z]+:", ":*:");

See the Java demo
When you use (?<=:)[a-zA-Z]+(?=:), the regex engine checks each location inside the string for a * before it, and once found, tries to match 1+ ASCII letters and then assert that there is a : after them. With :[A-Za-z]+:, the checking only starts after a regex engine found : character. Then, after matching :POST:, the replacement pattern replaces the whole match. It is totlally OK to hardcode colons in the replacement pattern since they are hardcoded in the regex pattern.
Original answer
You just need to access Group 1:
if (matcher.find()) {
    System.out.println(matcher.group(1));
}

See Java demo
Your :([a-zA-Z]+): regex contains a capturing group (see (....) subpattern). These groups are numbered automatically: the first one has an index of 1, the second has the index of 2, etc.
To replace it, use Matcher#appendReplacement():
String s = "something:POST:/some/path/";
StringBuffer result = new StringBuffer();
Matcher m = Pattern.compile(":([a-zA-Z]+):").matcher(s);
while (m.find()) {
    m.appendReplacement(result, ":*:");
}
m.appendTail(result);
System.out.println(result.toString());

See another demo

A: As pointed out, the reqex captured group can be used to replace.
The following code did it:
  String ss = "something:POST:/some/path/";
    Pattern pattern = Pattern.compile(":([a-zA-Z]+):");
    Matcher matcher = pattern.matcher(ss);

    if (matcher.find()) {
        ss = ss.replaceFirst(matcher.group(1), "*");
    }
    System.out.println(ss);

A: This is your solution:
regex = (:)([a-zA-Z]+)(:)
And code is:
String ss = "something:POST:/some/path/";

ss = ss.replaceFirst("(:)([a-zA-Z]+)(:)", "$1*$3");

ss now contains:

something:*:/some/path/

Which I believe is what you are looking for...

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text:
John Franklin (J. Frank) Bailey
in Obituaries
Thursday, June 19. 2014
Mr. John Franklin (J. Frank) Bailey, age 85 of Kinard, FL passed away Tuesday, June 17, 2014 at his home. J. Frank was born on February 15, 1929 in Frink, FL and lived in Kinard for most of his life.
He was a retired logger and a member of the Church of God Faith. J. Frank was preceded in death by his parents, James H. Bailey and Lura (Jones) Bailey, wife, Cynthia Delores Bailey, daughter, Margaret Ann Jenks, two brothers, Michael Wayne Bailey and Jimmy Bailey, grandson, Chase Bailey.
Survivors include,
5 sons, Scotty Bailey and his wife, Melissa of Kinard, FL
Phillip Anthony Bailey and his wife, Cathy of Kinard, FL
Frank Kevin Bailey and his wife Cindi of Sandy Creek Community
Tommy Wayne Bailey of Wewahitchka, FL
Tony Frank Bailey of Wewahitchka, FL
1 daughter, Shawn Pitts and her husband, Lonnie of Kinard, FL
6 sisters, Lois L. Blackburn of Blountstown, FL
Juanita McKay of Gulfport, MS
Hazel Herndon and her husband, Virgil of Clarksville, FL
Mary Edna Whitehead and her husband, Jim of Red Bay, FL
Betty Sue O'Conner of Rock Hill, FL
Yvonne Broome of Hattiesburg, MS
16 grandchildren and 6 great-grandchildren
Memorial services will be held Sunday, June 22, 2014 at 5:00 pm (CDT) at the Cypress Creek Baptist Church in Kinard with Reverend Daniel Merritt officiating. Memorialization will be cremation. All arrangements are under direction Marlon Peavy at Peavy Funeral Home in Blountstown, FL.
What is seven plus zero?

redpajama_set_name: RedPajamaCommonCrawl


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Down the Security Rabbithole, The Blog. Herein are thoughts, ideas, musings of my own making. I invite you to think freely, respond, or share. Together we move intellectual thought on our industry forward.
Let me be more precise -- I heavily dislike when a properly useful term is commandeered by the army of marketing people out there in the market space and promptly loses any real meaning. It makes me crazy, as it should make you, when terms devised to speak to some new method, utility, or technology becomes virtually meaningless when everyone uses it to mean everything and nothing all at once. Being in a highly dynamic technical field is hard enough without having to play thesaurus games with the marketing people. They always win anyway.
So when I see things like this post, "7 Security buzzwords that need to be put to rest" on one hand I'm happy someone is out there taking the over-marketing and over-hyping of good terms to task, but on the other hand I'm apprehensive and left wondering whether we've thrown the baby out with the bath-water.
"This is a term that has been knocked about in the industry for the last couple of years. It really amounts to little more than a glorified RSS feed once you peel back the covers for most offerings in the market place."
I'm unsure whether the author was going for irony or sarcasm, or has simply never seen a good Threat Intelligence feed before -- but this is just categorically wrong. Publishing this kind of thing is irresponsible, and does a disservice to the reading public who take these words for truth from a journalist.
Let's be honest, there are plenty of threat intelligence feeds that match that definition. I can think of a few I'd love to tell you about but non-disclosure agreements make that impractical. Then there are those that provide a tremendous amount of value when they are properly utilized at the proper point in time, by the proper resources.
Take for example a JSON-based feed of validated, known-bad IP addresses from one of the many providers of this type of data. I would hardly call this intelligence, but rather reputational data in the form of a feed. Sure, this is consumed much like you would an RSS feed of news -- except that the intent is typically for automated consumption by tools and technologies that requires very little human intervention.
Is the insinuation here that this type of thing has little value? I would agree that in the grand scheme of intelligence a list of known-bad IP addresses has a very short shelf-life and an complicated utility model which is necessarily more than a binary decision of "good vs. bad" -- but this does not completely destroy its utility to the security organization. Take for example a low-maturity organization who is understaffed, and relies heavily on network-based security devices to protect their assets. Incorporating a known-bad (IP reputation) feed into their currently deployed network security technologies may be more than a simple added layer of security. This may in fact be an evolution, but one that only a lower-level security organization can appreciate.
My point is, don't throw away the potential utility of something like a reputation feed without first considering the context within which it will be useful.
I don't know how to make this more clear. After spending a good portion of the last 4 months studying successful and operational security programs I can't imagine a scenario where a security program without the incorporation of threat intelligence is even viable. I'm sorry to report that without a threat-intelligence focused strategy, we're left deploying the same old predictable patterns of network security, antivirus/endpoint and other static defenses which our enemies are well attuned to and can avoid without putting much thought into it.
While I agree, the marketing organizations in the big vendors (and small, to be fair) have all but ruined the reputation of the phrase threat intelligence I dare you to run a successful security program without understanding your threats and adversaries, and be successful at efficient detection and response. Won't happen.
I guess I'm biased since I've spent so much time researching this topic that I'm now what you may consider a true believer. I can sleep well knowing that thorough (and ongoing) research into successful security programs which incorporate threat intelligence leads me to conclude that threat intelligence is essential to an effective and focused enterprise security program. I'm still not an expert, but at least I've seen it both succeed and fail and can tell the difference.
I get it, security people are experiencing fatigue from buzzwords and terms taken over by marketing people which makes our ears bleed every time someone starts making less than no sense. I get it, I really do. But let's not throw away the baby in the bathwater. Let's not dismiss something that has the potential to transform our security programs into something relevant to today's threats because we're sick of hearing talking heads mis-use and abuse the term.
Once you have these characteristics for your threat intelligence "feed" then you have significantly more than just an RSS feed. You have something that can act as a catalyst for your security program stuck in the 90's. Let's not let our pull to be snarky get the best of us, and throw away a perfectly legitimate term. Instead, let's take those who mis-use and abuse the term and point them out and call them out for their disservice to our mission.
Current weakness of Cyber Threat Intelligence is customer site specific threat indicators. Of the belief that once the CTI industry begins to provide customer specific context, that the fidelity of CTI will improve. Another weakness of CTI failure to adequately apply Graph Theory toward mapping relationships between CTI indicator attributes. Where attributes can be indicator node attributes, or relationship attributes. Uncovering relationships between attributes of formerly supposed disconnected clusters of CTI indicators. Successful application of Graph Theory in my opinion will begin to winnow out the leaders from the trailers within the CTI niche market space.
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Linden - Bleached Highlights
Linden is Joe McAlinden, formerly of Superstar, guest on Teenage Fanclub records and very formerly of The Boy Hairdressers.
Linden's debut album 'Bleached Highlights' has just been released on Edwyn Collins new label AED Records.
The album is already receiving some excellent reviews;
Album of the Week - Scotland On Sunday
"A beautiful record" 4/5 - Q Magazine
Album of the Week - Alexis Petridis The Guardian
"Beautifully assured" - The Scotsman
"This album is stunning" - Daily Record
Album of the Week - Vic Galloway BBC Radio Scotland
Album of the Week - Tom Morton BBC Radio Scotland
"Perfect pop moment" 4/5 - Sunday Express
"Glorious… A prime contender for Scottish Album of The Year" - The Sun
If I was to add something from this blog to the section above it would be;
"Immediate guitar pop with hooks and harmonies. Simple at times, always beautiful." everythingflowsglasgow
The first four tracks on this album are sublime, as good as anything you will hear all year, or any year. McAlinden has an effortlessly pure voice and he splices it perfectly with chiming and warm fuzzy guitars.
Opener 'Brown Bird Singing' has been released as a limited 7-inch via AED. Upon listening, it is easy to understand why it was chosen as the lead single, although there are a load of potential follow ups.
It is the sound of sunshine, beautifully produced by Edwyn Collins and Sebastian Lewsley. Check out the DIY video below.
The opening four tracks on the album grabbed me; immediate guitar pop with hooks and harmonies. Simple at times, always beautiful.
'I Just Wanna Be Here (When I'm Somewhere Else)' and the title track 'Bleached Highlights' flow beautifully, leaving me wonder if they were written around the same time.
'Thank You' has a chorus that just lifts the whole song;
I'm never gonna let you down girl, take my hand and welcome me back home
Hey, I'm always gonna be around girl, understand that you're my way back home
The CD (resplendent Jim Lambie print) comes with a lyric book and looking over the lyrics as I listen it is clearly evident that 'Bleached Highlights' is an out and out love album.
That is hammered home by 'If I Had Wings' which (on only a few listens) is a song that stands out, brilliant guitar work and McAlinden sounding particularly angelic and soulful. At only 2 minutes and 20 seconds long, you'll be playing it several times in a row.
I finally found you
My senses around you
People in love experience all kinds of emotions and 'Written' and 'Hear My Name' highlight insecurities, maintaining the flow of the album but just slowing it down ever so slightly.
'Something Wonderful' is acoustic based until the chorus with McAlinden sounding almost breathless as he is singing his hear out.
Somethings wonderful just got better
And I think of you every day
The album closes with 'Hide Away' with a melody that reminds me of something else but I just can't place it. However that is one of Linden's (and indeed McAlinden's) clear strengths - immediately accessible and understandable guitar pop with melodies and hooks. It is simple and pure, this is not an album that will take you a dozen listens to get into, you either like it and get it or it isn't for you.
I like it and I've got it!
Labels: AED Records, Edwyn Collins, guitar pop, harmonies, hooks, Jim Lambie, Joe McAlinden, Linden, Superstar, Teenage Fanclub, The Boy Hairdressers
18-years since Definitely Maybe by Oasis
Vigo Thieves - Forever
With Me Tonight - Keel Her
Introducing - The Velveteen Saints
Glasgow 2014 - Opening and closing ceremonies
Ace City Racers - Waiting
Under the Westway - Blur