text:
\section{Introduction}
The nearby radio galaxy M87, a giant elliptical near the center of the
Virgo cluster, hosts a striking optical jet extending 20\arcsec\, to
the northwest of a blazar-like core, as first observed by
\cite*{cur18}. This source
has been extensively observed in the radio, infrared, optical, and
X-rays
\citep[e.g.,][]{owen89_rad,sparks96_opt,per01_IR,walker08_vlbi,marshall2002_xray,perlman2005_xray}. At
a distance of only 16.7 Mpc \citep[81 pc/\arcsec,][]{bla09},
structures on the order of parsecs can easily be resolved using
high-resolution instruments such as the Hubble Space Telescope
(HST). The bright knots of emission spread along the 1.6 kpc
(projected) jet in optical and X-ray imaging correspond to features
seen in the radio, with a spectrum consistent with synchrotron
emission \citep{marshall2002_xray}.

Proper motion studies are integral to our understanding of
relativistic jets, as apparent velocities ($\beta_\textrm{app}$=v/$c$)
give lower limits to the bulk Lorentz factor and upper limits on the
angle to the line of sight, as well as detailed kinematics for
comparison to theoretical jet models. Almost all proper motion studies
of jetted AGN have used Very Long Baseline Interferometry (VLBI),
and in the case of M87 have revealed sub-relativistic speeds in small
components within a parsec of the core
\citep{reid89,junor95,kovalev07}. \emph{Superluminal} motions have
also been measured in the kpc-scale jet with both the VLA
\citep[][hereafter B95]{bir95}, and in the optical with HST
\citep[][hereafter B99]{bir99}. The latter study measured only the
inner 6\arcsec of the jet, and suffered from rather larger errors (on
the order of 0.3 - 1$c$). However, in comparison to radio
measurements, the optical jet emission traces higher energy electron
populations more closely associated with the sites of acceleration,
leading to more compact features which can be precisely measured.

In this letter we present the initial results of an archival study of
proper motions in the M87 jet with HST. The aim of this work is to
leverage 13.25 years of Hubble observations of M87 to vastly improve
on previous measurements of apparent velocities in the jet, reaching
accuracies better than 0.1$c$ in order to map the detailed velocity field of
the jet as it propagates from the core, including transverse motions
and accelerations.

\begin{figure*}
\centering
\includegraphics[width=6.5in]{Fig1.eps}
\caption{\emph{(Top Figure)} The entire jet, from a stack of ACS/WFC
  F606W images taken in 2006 with galaxy subtracted. All other images
  are taken from stacks of WFPC2 exposures in the F814W
  filter. \emph{(Lower Left)} Within knot D, D-East appears to be
  nearly stationary, while D-Middle appears to move at
  4.27$\pm$0.30$c$ along the jet. D-West is the only knot to show a
  strong deceleration in the jet-direction, apparent when comparing
  the 2001 and 2007 epochs. The new component noted with an arrow
  appears to move at $\sim$2.4$c$ (Vertical lines in all figures are
  to guide the eye). \emph{(Lower Middle)} Knot E appears complex,
  with a new extension to the East of the main knot (E-West) appearing
  in 2008.
  \emph{(Lower Right)} Knot F shows evidence of complex, even clumpy
  structure. F-Middle is moving at 0.36$\pm$0.14$c$ and appears
  to be fading over time. }
\label{fig:innerjet}
\end{figure*}

\section{Observations}
\subsection{Hubble Archival Data}

Because jet components are known to differ in size, intensity, and
apparent location with wavelength, we used archival images
only in a single filter, F814W (wide I band), which gives the longest
possible baseline of 13.25 years and a dense sampling in time. The
total dataset is comprised of nearly all the ACS/HRC, ACS/WFC, and
WFPC2 (PC chip) archival images in this filter. For ACS/HRC, we used
30 (48-50s) exposures spanning November 2002 to November 2006. For
ACS/WFC, we used 217 dithered exposures (all 360s) taken in 2006. For
WFPC2, we used 161 exposures of 160-600s, spanning 1995 to 2008;
several shorter exposures of 30-40s were not used due to poor
signal-to-noise ratio.

\subsection{Astrometry Methods}

The general method we have used is similar to that used (and described
in detail) in \cite*{and10} and \cite*{sohn12}. The reference
frame is based on the positions of hundreds of bright globular
clusters associated with the host galaxy which are easily detected in
the ACS/WFC images and are effectively stationary to proper motions
over the time of this study. As cluster positions do not change with
filter, the reference frame was built using 56 (500s) ACS/WFC
exposures of the field in the F606W filter, which has slightly better
resolution than F814W. The reference frame was created by first detecting the
positions of the globular clusters in each flat-fielded, CTE-corrected
ACS/WFC image\footnote{ACS/HRC and WFPC2 data were not used due to
  inferior in signal-to-noise quality to the longer ACS/WFC images,
  and to avoid complications of combining different instruments.}
using a PSF-peak-fitting routine, then applying the standard
(filter-specific) geometric correction to those positions, and then
finding the best linear transformation for each image which matches
the positions in the individual (geometrically-corrected) frames to a
master reference frame. This last step is done by a routine similar to
MultiDrizzle, but better optimized for astrometry. The process of
finding transformations for all frames is iterated so that the master
reference frame (super-sampled to a pixel scale of 0.025\arcsec/pixel)
effectively gives the average position of each reference source in a
geometrically-corrected frame.

The final reference system consists of positions and magnitudes for
1378 globular clusters within $\sim$100\arcsec of the M87 core
position. Among all clusters, the median one-dimensional RMS residual
relative to the mean position was 0.05 (reference-frame) pixels (1.25
mas), corresponding to a systematic astrometric accuracy ($\times$
1/$\sqrt 56$) of 0.17 mas; over the 13.25 year baseline this is
equivalent to 0.003$c$.

Astrometric solutions were then found for all the ACS/HRC , ACS/WFC,
and WFPC2 images in F814W, except that the linear transformation
solution between the individual (corrected) images and the reference
frame was found once rather than iterated. Typical numbers of globular
clusters used to match the frames were $\sim$200-500 for ACS/WFC,
$\sim$15-30 for WFPC2 images, and $\sim$15-25 for ACS/HRC;
corresponding systematic errors were $\sim$0.006, $\sim$0.03, and
$\sim$0.05 pixels, or 0.003$c$, 0.015$c$, 0.025$c$ over 13.25 years,
respectively.

\subsection{Measuring the Jet Knot Positions}
Because of the difficulty in measuring the diffuse/complex knot
structures particularly in the noisy HRC images and shorter WFPC2
images, we chose to first identify the peaks of interest using stacked
images. The image stacks include the 2006 ACS/WFC stack, four WFPC2
stacks (epochs 1995-1996, 1998-1999, 2001, and 2007-2008), and three
stacks of ACS/HRC exposures (November 2002 - November 2003, July 2004
- September 2005, November 2005 - November 2006). The host galaxy was
modeled using the ACS/WFC stacked image with the IRAF/STSDAS tasks
\emph{ellipse} and \emph{bmodel} and then subtracted. Seven jet
regions were cut out from these images, corresponding to knots
HST-1, D, E, F, I, A+B, and C (see Figure~\ref{fig:innerjet}).

A two-dimensional continuous functional representation of the image
cutouts was then created using the Cosine Transform function
(\emph{FourierDCT}) in \emph{Mathematica}, which allowed us to find
prominent peaks, as well as contours of constant flux level around
those peaks.  In general, the number of interesting peaks was chosen
by hand, and the contour line levels were at 50\% of the flux of the
peak, after a `background' level was subtracted, the latter estimated
as the minimum flux level between the particular peak and the next
nearest peak. The positions defining the contour were
`reverse-transformed' from master frame coordinates into each
distorted, galaxy-subtracted image so that an intensity-weighted
centroid position from the pixels within the contour could be
calculated.  These final positions were then transformed back into the
common reference frame, so that each peak of interest was measured in
every exposure, resulting in hundreds of position measurements spread
over the 13 year baseline.

\begin{figure}
\centering
\includegraphics[width=3.5in]{Fig2_Dwest.eps}
\caption{\emph{(Upper Panel)}: Distance from the core versus time for
  knot D-West. The blue points show the mean values for each time bin
  with errors estimated from nearby globular clusters. This knot shows
  a remarkable deceleration over time, from an initial speed of
  $\sim$2.7$c$ in 1996 to near zero by the end of the time
  series.\emph{(Lower Panel)} The rapid transverse velocity of knot
  D-West is apparent in this plot of y-coordinate versus time,
  corresponding to $-$0.59$c$.}
\label{fig:dwest}
\end{figure}

To measure the apparent motion of the jet components over time, we
define the x-direction as the line from the core through the center of
knot I (PA of 290$^\circ$), with y-direction perpendicular. Data were binned into 26 time
bins (depicted in Figure~\ref{fig:dwest}), and a linear model was fit using standard
weighted least squares.  The weights were taken to be the inverse of
the variance for each time bin, as measured from the 10 nearby
globular cluster reference sources (discussed below). For each feature
and in each direction, we also attempted to fit the data with a
quadratic to look for accelerations.  When the Analysis of Variance
(ANOVA) favored the quadratic over the linear model at a 95\% level or
greater, the quadratic fit is noted in Table~1 along with the linear
fit.  The apparent speed ($\beta_\mathrm{app}$) in each
direction in units of $c$ is also given; where the quadratic fit is
favored $\beta_\mathrm{app}$ corresponds to the speed at the mean date
of 2004.06.

\subsection{The Globular Clusters as Controls}

The method was tested using ten globular clusters near to the jet,
each of which was measured as described above, with contour levels
drawn at 50\% of the peak intensity. Each set of cluster positions was
binned into the 26 time bins spread between 1995 and 2008, and for
each bin the mean position for all ten clusters was subtracted from
the positions so that the variance in the estimates could be measured
from all clusters as a function of epoch for use in the weighted
linear least squares fit. The measured variance was essentially
constant across all epochs at $\sim$0.1 reference pixels. Linear fits
to the x- and y-direction positions versus time in all cases were
consistent with a slope of zero, with typical errors on the order of
0.003 pixels yr$^{-1}$ or 0.02$c$.

\section{Results}
Starting from the core, the first well-detected features in our
dataset include a small narrow extension from the core (out to
$\sim$0.5\arcsec) probably corresponding to knot L seen in VLBI
images, as well as the famous variable feature, HST-1. Both the core
and HST-1 are highly variable in flux and frequently saturated in our
images. Because special care must be taken with saturated images, an
analysis of the inner region of the M87 jet up to and including HST-1
will be addressed in a follow-up to this letter. For all the knots
described below, the WFPC2 image stacks (galaxy
subtracted) described above are shown in Figures~\ref{fig:innerjet}
and \ref{fig:outerjet}.\footnote{Movies of the jet and knots are also
  available in the online version of this paper.}.

\begin{figure*}
\centering
\includegraphics[width=6.7in]{Fig3_thin_with_vvectors.eps}
\caption{\emph{(Upper Left)} Knot I appears to fade and move backwards
  along the jet at 0.23$\pm$0.12$c$. \emph{(Lower Left}) Knot A is
  shown with a stretch to emphasize the fading of the knot over
  time. \emph{(Middle Panel)} The knot A/B complex shows remarkable
  variability, with both sub-relativistic and superluminal apparent
  motions. The white arrow is pointing to a `bar' which appears in the
  last epoch. \emph{(Right Panel)} Knot C shows speeds on the order of 0.5-1$c$. \emph{(Bottom Panel)} A depiction
  of velocities as vectors from their positions along the jet.}
\label{fig:outerjet}
\end{figure*}

\subsection{The Inner Jet: Knots D, E, F, and I}

Knot D is the most consistently bright feature after the knot A/B/C
Complex, and significant proper motions were measured in both previous
studies.\footnote{It is important to note that B95 used data from 1985
  through 1995, and so directly precedes the epoch of our dataset,
  while B99 overlaps with only the early part of it, spanning
  1995-1998. All $\beta_\mathrm{app}$ (including previous work) were
  computed with the conversion factor 0.264$c$ yr mas$^{-1}$.} As
suggested by the contours overlaid on Figure~\ref{fig:innerjet},
D-Middle is one of the fastest components with a speed of
4.27$\pm$0.30$c$ along the jet, while the brighter D-West shows
evidence of \emph{deceleration} radially, slowing to a near stop by
the final epoch in 2008, while maintaining one of the largest
transverse speeds of $-$0.59$\pm$0.05$c$ (see Figure~\ref{fig:dwest}).
These measurements are consistent with the results of B99
(5.26$\pm$0.92$c$ and 2.77$\pm$0.98$c$ for D-Middle and D-West,
respectively), where we compare with our estimated velocity of D-West
in 1996 of 2.73$\pm$0.40$c$ from the quadratic fit.

Previous results on knot D-East have been conflicting: B95 found that
it moved \emph{inward} along the jet at 0.23$\pm$0.12$c$ (possibly
consistent with being stationary), while B99 found a large outward
apparent velocity of 3.12$\pm$0.29$c$; our result of 0.28$\pm$0.05 is
more in line with B95. It is possible that there is a stationary
feature at D-East, through which components emerge (analogous to what
is seen in HST-1). In that case, the higher-resolution FOC was perhaps
tracking an emerging bright component, while over longer periods the
global feature at D-East is stationary.

As shown in the central lower panel of Figure~\ref{fig:innerjet}, the
main knot structure of knot E (E-West) is complex, with a moderately
superluminal speed of 1.91$\pm$0.14$c$, about half of that found in
B99 (4.07$\pm$0.85$c$).  In addition to the 50\% intensity contrast
contour line shown, a lower-intensity contour at 30\% has been drawn
around this area, to illustrate the morphological changes of this
feature over time. To the west there appears a bright extension in the
1995 epoch which advances westward faster than the brightest part, and
rapidly fades in the later epochs. Interestingly, a new component
appears to the East in 2008, suggesting again the possibility of a
favored location or stationary feature, which should be checked with
future observations.

Like knot E, the next feature in the jet, knot F, is diffuse, faint,
and apparently complex. We track two broad features, F-East and
F-Middle, as shown in the lower right of Figure 1. As B99 were unable
to measure proper motions for anything beyond knot E, the only
previous measurement gives 0.90$\pm$0.23$c$ in B95, for the entire
diffuse feature seen in the radio. This is slightly faster than the
value we measure for the most prominent component F-Middle
(0.36$\pm$0.14$c$), however, the epochs of observation are not
contemporaneous and slight deceleration is possible.

Finally, knot I has a slight negative velocity along the jet axis, of
$-$0.23$\pm$0.12$c$. The knot clearly fades steadily with time, as
shown in the upper left of Figure 3. This knot is the clearest case of
a possible stationary feature in the jet outside of the upstream component
HST-1d which has been extensively observed with VLBI
\citep[$\beta_\mathrm{app}<$0.25$c$;][]{cheung07}.

\subsection{The Outer Jet: Knots A, B, and C}

The outer part of the jet looks very different from the evenly spaced
knots discussed previously.  Knot A is the brightest feature in the
entire jet, excepting HST-1 during its remarkable flare of 2002 - 2008
\citep{har06}. It has been suggested that knot A is an oblique shock
which precipitates the break-up of the orderly structure of the jet,
thus resulting in the complex knot B region downstream, and the final
knot C feature which is considerably off the main jet axis
\citep{bic96}. Within the bright but extended knot A, we track two
features which appear on top of this more diffuse emission (see
Figure~\ref{fig:outerjet}). For A1, we find a relativistic velocity of
1.32$\pm$0.12$c$, however it is not clear that this feature is really
a single component, as the western side of it appears to move slowly
if at all. For A2, we measure a much slower radial velocity of
0.31$\pm$0.06$c$, consistent with the previous results of 0.4-0.6$c$
from B95.

Qualitatively, the A/B complex is surprisingly `active' with new
features brightening up and moving with high speeds relative to other
nearby components. A good example is pointed out by the arrow in the
last epoch of the central panel in Figure~\ref{fig:outerjet}, where a
`bar' has formed between B3 and B4. At the same time, within region B4
a separation is taking place, leaving a small knot behind to the
northeast. While B4 is probably a complex of at least 2-3 `subknots',
we used the inner contour (at 80\% median flux), giving us an apparent
speed of 1.66$\pm$0.11$c$ along the jet direction. B1,B2, and B3 as
well as the final knot C show significant apparent velocities in both
directions, on the order of 0.5-1.5$c$.

The final terminus of knot B is particularly interesting, as a new
distinct component (which we have labeled B5-n) appears in the
northern half in the 2001 epoch. Using data from 2001 - 2008 only (the
time when this component was detected), we measure an apparent
velocity of 2.16$\pm$0.35$c$ along the jet direction and
1.30$\pm$0.31$c$ in the transverse direction.  Qualitatively, the
northern end of B5 seems to continuously brighten while the whole
feature stretches up towards the northwest. In knot B generally there
is a repeating motif of two bright regions connected by a fainter
`bar'; this is apparent between knots B1 and B2, between B3 and the
remnant of B4, and the two halves of B5.

\begin{figure}
\centering
\includegraphics[width=3.2in]{Fig4_vel_envelope.eps}
\caption{Velocities along the jet (upper panel) and transverse to the
  jet (lower panel) as a function of distance from the core. Previous
  measurements shown for comparison taken from B95, B99,
  \citealt{cheung07}, and \citealt{kovalev07}.}
\label{fig:envelope}
\end{figure}

The most striking finding of the outer jet component velocities is
depicted in the lower panel of Figure~\ref{fig:outerjet}, where the
velocity vectors are plotted from the center of each knot.  There is a
conspicuous `tip-to-tail' alignment of almost all the vectors within
the outer knot A/B/C complex, strongly suggestive of a flattened view
of a helical motion which might result in such a `zig-zag' pattern,
though this should be checked with theoretical modeling.

\section{Discussion and Conclusions}
\label{sec:final}

We have presented new proper motion measurements for the knots in the
M87 optical jet, reaching accuracies better than 0.1$c$. The speeds
are largely in agreement with previous results where they exist (see
Figure~\ref{fig:envelope}); some discrepancies are likely due to the
higher resolution in the optical allowing us to track more compact
features than those measured in the radio, or changes such as the
observed deceleration of knot D-West. We also find evidence of
stationary features in knots D-East, I, and possibly E (if the new
extensions in 2008 is a recurrence at a preferred location), which
should be checked with future observations. Both D-Middle, and the
small component extending from D-East (labeled with an arrow in
Figure~\ref{fig:innerjet}) appear to steadily drop in intensity over
the 13 years of observation, a very similar behavior to the western
components of knot HST-1 (e.g., B99), and both HST-1 and knot D are
dominated by a transverse magnetic field as noted by
\citep{perlman99_pol}, which suggests a shock at these locations. It
has been suggested that at the eastern edge of HST-1 there is a
recollimation shock (\citealt{sta06,cheung07}; see also
\citealt{asada2012}), but the appearance of very different speeds all
along the jet might also be consistent with pairs of forward/reverse
fast-mode MHD shocks in a strongly magnetized relativistic flow
\citep{nakamura2010}.

In the outer jet (from knot A), we find apparent velocities that are still
superluminal and velocity vectors that appear to line up into a
helical/side-to-side pattern. However it is not clear that the
measured speeds can be identified with discrete moving components. It
is possible that if the jet of M87 has a helical structure which is
broken up at knot A, we are seeing the `unwinding' of the coil in knot
B. Thus the appearance of the bar noted with an arrow in 2008 could be
the result of a change in geometry as the coil of the jet unwinds,
bringing a length into our line of sight. Sideways oscillation of
components B1-B5 may presumably follow a helical path of underlying
magnetic fluxes \citep{owen89_rad,perlman99_pol} in three
dimensions. Thus, the knot A/B/C complex could be seen as a
consequence of the interplay between the slow-mode MHD shocks and
growing helical kink instability \citep{nakamura_2004_hk}.

\acknowledgments This project is part of the HSTPROMO collaboration
(www.stsci.edu/~marel/hstpromo.html), a set of HST projects aimed at
improving our dynamical understanding of stars, clusters, and galaxies
in the nearby Universe through measurement and interpretation of
proper motions. Support for this work was provided by NASA through a
grant for archival program 12635 from the Space Telescope Science
Institute (STScI), which is operated by the Association of
Universities for Research in Astronomy (AURA), Inc., under NASA
contract NAS5-26555. {\it Facilities:} \facility{HST (WFPC2, ACS)}.

\clearpage

\begin{deluxetable}{clcccccccccccc}

  \tabletypesize{\footnotesize}
\tablecolumns{14}
\tablewidth{0pc}
\tablehead{
\colhead{}        &  \colhead{}         & \colhead{} & \colhead{}                                  &   \multicolumn{2}{c}{Linear Fit}        & \colhead{} & \multicolumn{4}{c}{Quadratic Fit}                                            & \colhead{} & \multicolumn{2}{c}{Speed} \\
                                                                                \cline{5-6}                                          \cline{8-11}                                                                   \cline{13-14} \\
\colhead{Knot}    & \colhead{Subknot}   & \multicolumn{2}{c}{Distance\tablenotemark{$\dagger$}}    &   \colhead{$\mu_x$} & \colhead{$\mu_y$}  & \colhead{} & \colhead{$\mu_x$} & \colhead{$\xi_x$} & \colhead{$\mu_y$} & \colhead{$\xi_y$} & \colhead{} & \colhead{$\beta_x$} &  \colhead{$\beta_y$} \\
& & \arcsec & kpc & mas yr$^{-1}$ & mas yr$^{-1}$ & \colhead{} & mas yr$^{-1}$ & mas yr$^{-2}$ & mas yr$^{-1}$ & mas yr$^{-2}$ & \colhead{} & $c$ & $c$}
\startdata
D  & East     & \phantom{1}2.77 & 0.224 &    \phantom{$-$0}1.1$\pm$0.2  &            $-$0.2$\pm$0.1\phantom{0}   & &                           &                    &                              &                                 & &   \phantom{$-$}0.28$\pm$0.05 &            $-$0.05$\pm$0.03 \\
   & Middle\tablenotemark{a}   & \phantom{1}3.61 & 0.292 &    \phantom{$-$}16.2$\pm$1.1  &            $-$0.7$\pm$0.1\phantom{0}   & &                           &                    &              $-$1.3$\pm$0.2  &             $-$0.2$\pm$0.1      & &   \phantom{$-$}4.27$\pm$0.30 &            $-$0.35$\pm$0.05 \\
   & West     & \phantom{1}3.97 & 0.321 &    \phantom{$-$0}5.7$\pm$0.5  &            $-$2.2$\pm$0.2\phantom{0}   & & \phantom{$-$}3.9$\pm$0.5  & $-$0.80$\pm$0.2    &                              &                                 & &   \phantom{$-$}1.03$\pm$0.14 &            $-$0.59$\pm$0.05 \\[2ex]

E  & East     & \phantom{1}5.86 & 0.473 &    \phantom{$-$0}9.6$\pm$1.2  &            $-$1.8$\pm$0.3\phantom{0}   & &                           &                    &                              &                                 & &   \phantom{$-$}2.52$\pm$0.32 &            $-$0.48$\pm$0.08 \\
   & West     & \phantom{1}6.21 & 0.502 &    \phantom{$-$0}7.2$\pm$0.5  &  \phantom{$-$}0.0$\pm$0.2\phantom{0}   & &                           &                    &                              &                                 & &   \phantom{$-$}1.91$\pm$0.14 &            $-$0.01$\pm$0.05 \\[2ex]

F  & East     & \phantom{1}8.23 & 0.666 &    \phantom{$-$0}7.0$\pm$0.9  &            $-$1.3$\pm$0.4\phantom{0}   & &                           &                    &                              &                                 & &   \phantom{$-$}1.85$\pm$0.24 &            $-$0.35$\pm$0.12 \\
   & Middle   & \phantom{1}8.68 & 0.702 &    \phantom{$-$0}1.4$\pm$0.5  &  \phantom{$-$}0.7$\pm$0.5\phantom{0}   & &                           &                    &    \phantom{$-$}2.2$\pm$0.5  &   \phantom{$-$}0.6$\pm$0.2      & &   \phantom{$-$}0.36$\pm$0.14 &  \phantom{$-$}0.58$\pm$0.14 \\[2ex]

I  &          &           11.01 & 0.890 &    \phantom{0}$-$0.9$\pm$0.4  &            $-$0.1$\pm$0.2\phantom{0}   & &                           &                    &                              &                                 & &             $-$0.23$\pm$0.12 &            $-$0.02$\pm$0.07 \\[2ex]

A  & A1       &           12.28 & 0.993 &    \phantom{$-$0}5.0$\pm$0.4  &  \phantom{$-$}2.4$\pm$0.3\phantom{0}   & &                           &                    &    \phantom{$-$}3.1$\pm$0.3  &   \phantom{$-$}0.3$\pm$0.1      & &   \phantom{$-$}1.32$\pm$0.12 &  \phantom{$-$}0.82$\pm$0.09 \\
   & A2       &           12.60 & 1.019 &    \phantom{$-$0}1.2$\pm$0.2  &  \phantom{$-$}1.9$\pm$0.6\phantom{0}   & &                           &                    &                              &                                 & &   \phantom{$-$}0.31$\pm$0.06 &  \phantom{$-$}0.50$\pm$0.16 \\[2ex]

B  & B1       &      12.88      & 1.041 &    \phantom{$-$0}5.2$\pm$0.6  &            $-$0.6$\pm$0.9\phantom{0}   & &                           &                             &    \phantom{$-$}2.0$\pm$1.0  &   \phantom{$-$}1.1$\pm$0.3      & &   \phantom{$-$}1.37$\pm$0.16 &  \phantom{$-$}0.53$\pm$0.26 \\
   & B2       &      13.16      & 1.064 &    \phantom{$-$0}1.6$\pm$0.3  &            $-$1.4$\pm$0.2\phantom{0}   & &                           &                             &              $-$0.8$\pm$0.3  &   \phantom{$-$}0.2$\pm$0.1      & &   \phantom{$-$}0.42$\pm$0.07 &            $-$0.22$\pm$0.08 \\
   & B3       &      13.74      & 1.110 &    \phantom{$-$0}1.3$\pm$0.2  &  \phantom{$-$}1.2$\pm$0.1\phantom{0}   & &  \phantom{$-$}1.9$\pm$0.2 & \phantom{$-$}0.3$\pm$0.1    &    \phantom{$-$}1.6$\pm$0.2  &   \phantom{$-$}0.2$\pm$0.1      & &   \phantom{$-$}0.49$\pm$0.05 &  \phantom{$-$}0.42$\pm$0.04 \\
   & B4       &      14.29      & 1.155 &    \phantom{$-$0}6.3$\pm$0.4  &            $-$3.3$\pm$0.5\phantom{0}   & &                           &                             &              $-$1.7$\pm$0.3  &   \phantom{$-$}0.7$\pm$0.1      & &   \phantom{$-$}1.66$\pm$0.11 &            $-$0.45$\pm$0.08 \\
   & B5-s     &      15.18      & 1.227 &    \phantom{$-$0}7.2$\pm$0.3  &  \phantom{$-$}2.8$\pm$0.3\phantom{0}   & &                           &                             &    \phantom{$-$}1.8$\pm$0.3  &             $-$0.4$\pm$0.1      & &   \phantom{$-$}1.90$\pm$0.09 &  \phantom{$-$}0.48$\pm$0.07 \\
   & B5-n     &      15.18      & 1.227 &    \phantom{$-$0}8.2$\pm$1.3  &  \phantom{$-$}4.9$\pm$1.2\phantom{0}   & &                           &                             &                              &                                 & &   \phantom{$-$}2.16$\pm$0.35 &  \phantom{$-$}1.30$\pm$0.31 \\[2ex]

C  & C1       &      17.48      & 1.413 &    \phantom{$-$0}3.3$\pm$0.4  &            $-$0.6$\pm$0.3\phantom{0}   & &                           &                             &    \phantom{$-$}0.3$\pm$0.3  &   \phantom{$-$}0.4$\pm$0.1      & &   \phantom{$-$}0.86$\pm$0.10 &  \phantom{$-$}0.07$\pm$0.08 \\
   & C2       &      17.83      & 1.441 &    \phantom{$-$0}2.4$\pm$0.3  &            $-$1.7$\pm$1.2\phantom{0}   & &  \phantom{$-$}1.5$\pm$0.2 &        $-$0.4$\pm$0.1       &              $-$5.6$\pm$0.9  &             $-$1.6$\pm$0.3      & &   \phantom{$-$}0.41$\pm$0.06 &            $-$1.47$\pm$0.23 \\
   & C3       &      17.90      & 1.447 &    \phantom{$-$0}2.2$\pm$0.5  &  \phantom{$-$}0.7$\pm$0.6\phantom{0}   & &                           &                             &              $-$1.7$\pm$0.3  &             $-$1.0$\pm$0.1      & &   \phantom{$-$}0.57$\pm$0.13 &            $-$0.45$\pm$0.07 \\
   & C2+C3    &      17.83      & 1.441 &    \phantom{$-$0}3.2$\pm$0.4  &            $-$2.1$\pm$0.3\phantom{0}   & &                           &                             &                              &                                 & &   \phantom{$-$}0.83$\pm$0.12 &            $-$0.54$\pm$0.08 \\[2ex]

GC\tablenotemark{b} &   (single)&     & &       0.0$\pm$0.22           &             0.31$\pm$0.15\phantom{0}   & &                           &                             &                              &                                 & &   \phantom{$-$}0.00$\pm$0.06 &            0.08$\pm$0.04 \\
                    &   (mean)  &     & &       0.0$\pm$0.06           &             0.00$\pm$0.06\phantom{0}   & &                           &                             &                              &                                 & &   \phantom{$-$}0.00$\pm$0.02 &            0.00$\pm$0.02 \\
\enddata
\tablecomments{Quadratic fit given when preferred at $>$95\% level, where $\beta$ values in those cases correspond to the mean time 2004.06. No HRC data was used for knots D-Middle, F,A,B, and C due to poor signal-to-noise.}
\tablenotetext{$\dagger$}{Distance from the core as measured in the 1995 epoch WFPC2 stacked image.}
\tablenotetext{a}{No data after 2007 used in fit due to feature fading.}
\tablenotetext{b}{Examples of Globular Cluster (GC) fits, including a single faint GC near knot A and the ensemble average.}
\end{deluxetable}

\clearpage


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

\noindent
A key ingredient in determining whether supernovae evolve into
black-holes or neutron stars is the nuclear equation of state
(NEOS)~\cite{Brown:1993jz}.  The NEOS depends upon the degrees of
freedom at a given density, which in turn are determined by the masses
of the various hadrons in the medium and their interactions. One
possibility is that it is more energetically favorable for a
Bose-Einstein condensate of negatively charged mesons to form in the
hadronic matter.  Naively, one expects pion condensation to occur
before kaon condensation\footnote{ A second possibility is that it is
  energetically favorable to have $\Sigma^-$'s present in the nuclear
  material in addition to neutrons and protons. The in-medium mass of
  the $\Sigma^-$ is expected to be significantly less than in
  free-space due to its interactions with neutrons, but the precise
  mass-shift is uncertain due to the model dependence of existing
  theoretical calculations~\cite{Page:2006ud}. Experimentally, little
  is known about the $\Sigma^-$-neutron interaction and a precise
  Lattice QCD calculation of these interactions~\cite{Beane:2006gf}
  would greatly reduce the uncertainty in this particular contribution
  to the NEOS.}, however, the repulsive pion-nucleon interactions and
the attractive kaon-nucleon interaction lead to kaon condensation
being the prime candidate for softening the NEOS at moderate nuclear
densities.  In theoretical explorations of kaon condensation, chiral
perturbation theory ($\chi$PT) is used to describe the kaon-nucleon
interactions and the kaon self-interactions at leading order (LO) or
next-to-leading order (NLO) in the chiral
expansion~\cite{KaplanNelson}.  The energy of the hadronic plus
electronic system is minimized with respect to the vacuum expectation
values (the condensates) of the meson fields subject to the
constraints imposed by charge neutrality and the conservation of
baryon number.  In the case of non-interacting mesons, once the
chemical potential exceeds the rest mass of a given species of meson a
condensate will form with an infinite density of mesons.  However, in
the case of a meson with repulsive self-interactions, the number
density of mesons will remain finite and a smooth function of the
chemical potential.  Depending upon the many-body technique employed
to describe the system, a condensate of $K^-$'s may become
energetically favorable at baryon densities, $n_b$, as low as three
times nuclear matter density (for a nice discussion, see
ref.~\cite{Page:2006ud}).

In this work we perform the first Lattice QCD studies of kaon
condensation.  In particular, we calculate the correlation functions
of systems containing up to twelve ($n=12$) $K^-$'s on the coarse MILC
lattices with spatial extent $L\sim 2.5~{\rm fm}$ and $L\sim 3.5~{\rm
  fm}$, both with lattice spacing $b\sim 0.125~{\rm fm}$, and on the
fine MILC lattices with spatial extent $L\sim 2.5~{\rm fm}$ and
lattice spacing $b\sim 0.090~{\rm fm}$.  LO $\chi$PT is found to
describe the chemical potential, and its density dependence,
remarkably well over the densities explored.  This work is an
extension of our previous work on charged-pion
condensates~\cite{Beane:2007es,Detmold:2008fn} and, for completeness,
we update that analysis with the new ensembles studied herein.

\section{Condensation of
  Pseudo-Goldstone Bosons in $\chi$PT}
\label{sec:XPT}

At LO in $\chi$PT, the Lagrange density describing the low-energy
dynamics of the pseudo-Goldstone bosons associated with the
spontaneous breaking of the approximate $SU(3)_L\otimes SU(3)_R$
chiral symmetry of QCD has the form
\begin{eqnarray}
{\cal L} & = &
{f^2\over 8}\ {\rm  Tr}\left[\ \partial_\mu\Sigma\ \partial^\mu\Sigma^\dagger\ \right]
\ +\ \lambda\ {\rm  Tr}\left[\ m_q\ \left(\Sigma\ +\ \Sigma^\dagger\ \right)\
\right]
\ \ \ ,
\label{eq:lochipt}
\end{eqnarray}
where
$\Sigma$ is the exponential of the meson fields,
\begin{eqnarray}
\Sigma & = & e^{i 2 M/ f}
\ \ ,\ \
M\ =\
\left(
\begin{array}{ccc}
{1\over\sqrt{2}}\pi^0 + {1\over\sqrt{6}}\eta^0 & \pi^+ & K^+\\
\pi^- & -{1\over\sqrt{2}}\pi^0 + {1\over\sqrt{6}}\eta^0 &K^0\\
K^- & \overline{K}^0 & -{2\over\sqrt{6}}\eta^0
\end{array}
\right)
\ \ \ ,
\end{eqnarray}
and where $f$ is the chiral limit of the pion decay constant (in this
normalization convention the physical pion decay constant is
$f_\pi\sim 132~{\rm MeV}$).  At LO in the chiral expansion, the quark
mass matrix, $m_q$, can be written in terms of the meson masses,
\begin{eqnarray}
\lambda\ m_q & = & {f^2\over 8}
\left(
\begin{array}{ccc}
m_\pi^2 & 0 &  0\\
0 & m_\pi^2 & 0 \\
0&0& 2 m_K^2-m_\pi^2
\end{array}
\right)
\ \ \ .
\end{eqnarray}
The Lagrange density in eq.~(\ref{eq:lochipt}), in addition to
describing the free-field dynamics of the mesons, also contains
interactions containing arbitrary numbers of mesons.  At zero
momentum, the LO contributions to the interactions between three
charged pions or between three charged kaons are given by
\begin{eqnarray}
{\cal L} & = &
{2\over 3 f^4}
\left[\
 m_\pi^2 (\pi^+)^3(\pi^-)^3\ +\
m_K^2 (K^+)^3 (K^-)^3\
\right]
\ \ \ .
\end{eqnarray}
This contribution provides the naive dimensional analysis (NDA)
estimate of the size of the three-meson
interaction~\cite{Beane:2007qr,Detmold:2008gh} of $1/(m_\pi f^4)$ for
the $\pi^-\pi^-\pi^-$ interaction and $1/(m_K f^4)$ for the
$K^-K^-K^-$ interaction in the non-relativistic theory describing the
very-low momentum interactions between the mesons~\footnote{The
  relativistic and non-relativistic fields differ in their
  normalization by a factor of $\sqrt{m}$.}.

In the presence of a chemical potential for the third component of
isospin, $\mu_I$, and strangeness, $\mu_S$, the partial derivatives in
eq.~(\ref{eq:lochipt}) become covariant derivatives
\cite{Son:2000xc,Son:2000by,Kogut:2001id}
\begin{eqnarray}
\partial_\mu\Sigma \rightarrow D_\mu\Sigma = \partial_\mu\Sigma + i \delta_{\mu0} \mu_I
\left[\hat I_z , \Sigma\right]\ +\  i \delta_{\mu0} \mu_S \left[\hat S , \Sigma\right]
\ \ \ ,
\end{eqnarray}
where
\begin{eqnarray}
\hat I_z & = &
{1\over 2}
\left(
\begin{array}{ccc}
+1&0&0\\0&-1&0\\0&0&0
\end{array}
\right)
\ \ ,\ \
 \hat S\ =\
\left(
\begin{array}{ccc}
0&0&0\\0&0&0\\0&0&+1
\end{array}
\right)
\ \ \ .
\end{eqnarray}
In the situation where $\mu_{\pi^-}=-\mu_I > m_{\pi}$ and
$\mu_{K^-}=\mu_S-{1\over 2}\mu_I < m_{K}$, it is energetically
favorable to form a $\pi^-$ condensate but not a $K^-$ condensate, and
at LO in $\chi$PT the number density of $\pi^-$'s
is~\cite{Son:2000xc,Son:2000by}
\begin{eqnarray}
\rho_{\pi^-} & = &
{f^2_{\pi^-}\over 2} \ \left(\ \mu_{\pi^-}\ -\ {m_\pi^4\over \mu_{\pi^-}^3}
\ \right)
\ \ \ .
\label{eq:pionCOND}
\end{eqnarray}
The analysis leading to this result easily extended to
kaon-condensates \cite{Kogut:2001id}.  For $\mu_{\pi^-} < m_{\pi}$ and
$\mu_{K^-} > m_{K}$ there will be a condensate of $K^-$'s but not of
$\pi^-$'s~\footnote{Mixed states with different numbers of pions and
  kaons are the subject of ongoing work.}.  For chemical potentials in
this range, the number density of $K^-$'s as a function of the
chemical potential, $\mu_{K^-}$, is, at LO in $\chi$PT,
\begin{eqnarray}
\rho_{K^-} & = &
{f^2_{K^-}\over 2} \ \left(\ \mu_{K^-}\ -\ {m_K^4\over \mu_{K^-}^3}
\ \right)
\ \ \ .
\label{eq:kaonCOND}
\end{eqnarray}
In eqs.~(\ref{eq:pionCOND}) and (\ref{eq:kaonCOND}), we have used LO
$\chi$PT relations to express the result in terms of the physical
masses and decay constants.  It is this relation that we will compare
with Lattice QCD calculations.  The reason for ``testing'' the
relation in eq.~(\ref{eq:kaonCOND}) is that, given the relatively
large expansion parameters of $SU(3)_L\otimes SU(3)_R$ $\chi$PT, it is
natural to expect that there will be corrections to $\rho_{K^-}$, and
hence to the dependence of $\mu_{K^-}$ on $\rho_{K^-}$, at the level
of $m_K^2/\Lambda_\chi^2 \sim 25\%$, where $\Lambda_\chi\sim1$~GeV is
the scale of chiral symmetry breaking.

In order to assess the relevance of the present work, it is
instructive to compare the value of the $K^-$-condensates we calculate
to those found to be relevant to the dense nuclear matter in the
interior of neutron stars.  Decomposing the charged kaon fields into
their real and imaginary parts, $K^\pm = \left(K_1 \mp i
  K_2\right)/\sqrt{2}$, and denoting the condensates $\langle
K_1\rangle = v_1$ and $\langle K_2\rangle = v_2$ and further defining
$\langle K\rangle = \sqrt{v_1^2+v_2^2}/\sqrt{2}$, the condensate is
related to the chemical potential via
\begin{eqnarray}
{ \langle K\rangle \over f_K} & = & {1\over 2}\cos^{-1}\left({m_K^2\over
    \mu_{K^-}^2}\right)
\ \ \ .
\label{eq:kaonCONDvev}
\end{eqnarray}
Condensates of magnitude $\langle K\rangle/f_K \ \raisebox{-0.7ex}{$\stackrel{\textstyle <}{\sim}$ } 0.3$ are found
to be possible at the densities in the interior of neutron
stars~\cite{KaplanNelson}.  This corresponds to $\mu_{K^-}/m_K -1
\ \raisebox{-0.7ex}{$\stackrel{\textstyle <}{\sim}$ } 0.1$, which is contained in the region explored in the Lattice
QCD calculations described in this work.

\section{Multi-Meson Energies in a Finite Volume}
\label{sec:multi-meson-energies}

\noindent
In recent works \cite{Beane:2007qr,Detmold:2008gh,Tan:2007bg}, the
analytic volume dependence of the ground-state energy of $n$ identical
bosons in a periodic volume of side $L$ has been computed to ${\cal
  O}(L^{-7})$, extending the classic results of Bogoliubov \cite{Bog}
and Lee, Huang and Yang \cite{Lee:1957}.  The resulting shift in
ground-state energy of $n$ particles of mass $M$ due to their
interactions is \cite{Detmold:2008gh}
\begin{eqnarray}
 \Delta E_n &=&
  \frac{4\pi\, \overline{a}}{M\,L^3}\Choose{n}{2}\Bigg\{1
-\left(\frac{\overline{a}}{\pi\,L}\right){\cal I}
+\left(\frac{\overline{a}}{\pi\,L}\right)^2\left[{\cal I}^2+(2n-5){\cal J}\right]
\nonumber
\\&&\hspace*{2cm}
-
\left(\frac{\overline{a}}{\pi\,L}\right)^3\Big[{\cal I}^3 + (2 n-7)
  {\cal I}{\cal J} + \left(5 n^2-41 n+63\right){\cal K}\Big]
\nonumber
\\&&\hspace*{2cm}
+
\left(\frac{\overline{a}}{\pi\,L}\right)^4\Big[
{\cal I}^4 - 6 {\cal I}^2 {\cal J} + (4 + n - n^2){\cal J}^2
+ 4 (27-15 n + n^2) {\cal I} \ {\cal K}
\nonumber\\
&&\hspace*{4cm}
+(14 n^3-227 n^2+919 n-1043) {\cal L}\
\Big]
\Bigg\}
\nonumber\\
&&
+\ \Choose{n}{3}\left[\
{192 \ \overline{a}^5\over M\pi^3 L^7} \left( {\cal T}_0\ +\ {\cal T}_1\ n \right)
\ +\
{6\pi \overline{a}^3\over M^3 L^7}\ (n+3)\ {\cal I}\
\right]
\nonumber\\
&&
+\ \Choose{n}{3} \ {1\over L^6}\ \overline{\overline{\eta}}_3^L\
\ \ + \ {\cal O}\left(L^{-8}\right)
\ \ \ \ ,
\label{eq:energyshift}
\end{eqnarray}
where the parameter $\overline{a}$ is related to the scattering
length\footnote{In this work we use the Nuclear Physics sign
  convention for the scattering length, opposite to that used in
  Particle Physics.  In this convention, the $\pi^+\pi^+$ scattering
  length is positive, corresponding to a repulsive interaction.}, $a$,
and the effective range, $r$, by
\begin{eqnarray}
a
& = &
\overline{a}\ -\ {2\pi\over L^3} \overline{a}^3 r \left(\ 1 \ -\
  \left( {\overline{a}\over\pi L}\right)\ {\cal I} \right)\ \ .
\label{eq:aabar}
\end{eqnarray}
The geometric constants that enter into eq.~(\ref{eq:energyshift}) are
\begin{align}
  &{\cal I}\ =\ -8.9136329\,, &{\cal J}\ =\ 16.532316\,, \qquad\qquad
  {\cal K}\ = \ 8.4019240\,,
  \nonumber\\
  &{\cal L}\ = \ 6.9458079\,, &{\cal T}_0\ = -4116.2338\,,
  \qquad\qquad {\cal T}_1\ = \ 450.6392\,, &
\label{eq:sums}
\end{align}
and $^nC_m=n!/m!/(n-m)!$.  The three-body contribution to the
energy-shift given in eq.~(\ref{eq:energyshift}) is represented by the
parameter $\overline{\overline{\eta}}_3^L$, which is a combination of
the volume-dependent, renormalization group invariant quantity,
$\overline{\eta}_3^L$, and contributions from the two-body scattering
length and effective range,
\begin{eqnarray}
\overline{\overline{\eta}}_3^L & = & \overline{\eta}_3^L  \left(\ 1 \ -\
  6  \left({\overline{a}\over\pi L}\right)\ {\cal I} \right)
\ +\ {72\pi \overline{a}^4 r\over M L} \ {\cal I}
\ \ \ ,
\label{eq:eta3barbar}
\end{eqnarray}
where
\begin{align}
  \overline{\eta}_3^L & = \eta_3(\mu)\ +\ {64\pi a^4\over
    M}\left(3\sqrt{3}-4\pi\right)\ \log\left(\mu L\right)\ -\ {96
    a^4\over\pi^2 M} {\cal S}_{\rm MS} \ \ \ .
\label{eq:etathreebar}
\end{align}
The quantity $\eta_3(\mu)$ is the
renormalization scale dependent
coefficient of the three-$\pi^+$
interaction that appears in the effective Hamiltonian density
describing the system \cite{Detmold:2008gh}. The quantity ${\cal S}$ is renormalization
scheme dependent and its value in the minimal subtraction (MS)
scheme is ${\cal S}_{\rm MS}\ = \ -185.12506$.

For $n=2$, the last two terms in eq.~(\ref{eq:energyshift}) vanish and
the remaining terms constitute the small $\overline{a}/L$ expansion of the
exact eigenvalue equation derived by L\"uscher
\cite{Luscher:1986pf,Luscher:1990ux},
\begin{eqnarray}
p\cot\delta(p) \ =\ {1\over \pi L}\ {\bf
  S}\left(\,\left(\frac{p L}{2\pi}\right)^2\,\right)
\ \ ,
\label{eq:energies}
\end{eqnarray}
which is valid below the inelastic threshold, where
$p\cot\delta(p)$ is the real part of the inverse scattering amplitude.
The regulated three-dimensional sum is~\cite{Luscher:1986pf,Luscher:1990ux,Beane:2003da}
\begin{eqnarray}
{\bf S}\left(\, x \, \right)\ \equiv \ \sum_{\bf j}^{ |{\bf j}|<\Lambda}
{1\over |{\bf j}|^2-x}\ -\  {4 \pi \Lambda}
\ \ \  ,
\label{eq:Sdefined}
\end{eqnarray}
where the summation is over all triplets of integers ${\bf j}$ such
that $|{\bf j}| < \Lambda$ and the limit $\Lambda\rightarrow\infty$ is
implicit.

\section{Methodology and Details of the Lattice Calculation \label{sec:Method}}

\noindent
The computation in this paper uses the mixed-action lattice QCD scheme
developed by LHPC~\cite{Renner:2004ck,Edwards:2005kw} and described
fully in Ref.~\cite{Detmold:2008fn} where multi-pion systems are
investigated in detail.  The present calculations were performed
predominantly with the coarse,
asqtad-improved~\cite{Orginos:1999cr,Orginos:1998ue} MILC
configurations generated with rooted staggered sea
quarks~\cite{Bernard:2001av}\footnote{The results of this paper assume
  that the fourth-root trick recovers the correct continuum limit of
  QCD.} with a lattice spacing of $b\sim 0.125$~fm, and a spatial
extent of $L\sim 2.5$~fm.  These configurations were
HYP-smeared~\cite{Hasenfratz:2001hp,DeGrand:2002vu,DeGrand:2003in,Durr:2004as}
and used to generate domain-wall fermion
propagators~\cite{Kaplan:1992bt,Shamir:1992im,Shamir:1993zy,Shamir:1998ww,Furman:1994ky}.
The strange quark was held fixed near its physical value while the
degenerate light quarks were varied over a range of
masses\footnote{The mass of the valence quark was chosen to yield a
  pion mass that matches the pseudo-Goldstone staggered pion
  \cite{Detmold:2008fn}.}; see Table~\ref{tab:MILCcnfs} and
Refs.~\cite{Beane:2006mx,Beane:2006pt,Beane:2006fk,Beane:2006kx,Beane:2006gf}
for details.  Further calculations were performed using one fine MILC
ensemble, with $b\sim 0.090~{\rm fm}$ and $L\sim 2.5$~fm and on a
larger volume coarse MILC ensemble with $b\sim 0.125$~fm and $L\sim
3.5$~fm.  On the coarse, 2.5~fm MILC lattices, Dirichlet boundary
conditions were implemented to reduce the original time extent of 64
time-slices down to 32 and thus save in computational time.  While
this procedure leads to minimal degradation of baryon signals, it does
limit the number of time slices available for fitting meson
properties. For the fine MILC ensemble and the large volume coarse
MILC ensemble, anti-periodic boundary conditions in time were
implemented and all time slices are available for analysis.
\begin{table}[t]
 \caption{The parameters of the MILC gauge configurations and
   domain-wall quark propagators used in this work. The subscript $l$
   denotes light quark (up and down), and  $s$ denotes the strange
   quark. The superscript $dwf$ denotes the bare-quark mass for the
   domain-wall fermion propagator calculation. The last column is the
   number of configurations times the number of sources per
   configuration. For the ensembles labeled with ``P$\pm$A'', propagators
   that were periodic in the temporal direction were computed in
   addition to those with anti-periodic temporal boundary conditions
   ($b m_{res}$ was not computed for these sets but is expected to be
   very close to the result from the corresponding set of propagators
   with anti-periodic temporal boubdary conditions).}
\label{tab:MILCcnfs}
\begin{ruledtabular}
\begin{tabular}{ccccccc}
 Ensemble
&  $b m_l$ &  $b m_s$ & $b m^{dwf}_l$ & $ b m^{dwf}_s $ & $10^3 \times b
m_{res}$ & \# of propagators   \\
\hline
2064f21b676m007m050 &  0.007 & 0.050 & 0.0081 & 0.081  & 1.604 & 1038\ $\times$\ 24 \\
2064f21b676m010m050 &  0.010 & 0.050 & 0.0138 & 0.081  & 1.552 & 768\ $\times$\ 24 \\
2064f21b679m020m050 &  0.020 & 0.050 & 0.0313 & 0.081  & 1.239 & 486\ $\times$\ 24 \\
2064f21b681m030m050 &  0.030 & 0.050 & 0.0478 & 0.081  & 0.982 & 564\ $\times$\ 20 \\
\hline
2896f2b709m0062m031 & 0.0062 & 0.031 & 0.0080 & 0.0423 &
\ 0.380 \  & 1001\ $\times$\ 7 \\
2896f2b709m0062m031 P$\pm$A& 0.0062 & 0.031 & 0.0080 & 0.0423 &
\ --- \  & 1001\ $\times$\ (1+1) \\
\hline
2864f2b676m010m050 & 0.010 & 0.050 & 0.0138 & 0.081 &
\ 1.552 \  & 137 \ $\times$\ 8 \\
2864f2b676m010m050 P$\pm$A & 0.010 & 0.050 & 0.0138 & 0.081 &
\ --- \  & 274 \ $\times$\ (2+2) \\
\end{tabular}
\end{ruledtabular}
\end{table}
In order to extract precise measurements from the configurations,
propagators were computed from multiple sources displaced both
temporally and spatially on each configuration  The correlators were blocked
so that one averaged correlator per configuration was used in the
subsequent statistical analysis.  Further blocking over sets of
configurations did not change the resulting uncertainties associated
with each observable.

For the fine MILC ensemble and the large volume coarse MILC ensemble,
additional propagators were computed using periodic temporal boundary
conditions for a subset of the source points. Combining periodic and
anti-periodic (in time) propagators from the same source leads to a
propagator that is periodic in time over twice the temporal extent of
the lattice. These ensembles, labeled ``P$\pm$A'' in
Table~\ref{tab:MILCcnfs}, served to explore the effects of temporal
boundary conditions.\footnote{ For the correlators of systems
  containing multiple hadrons (which are extremely sensitive to
  truncation errors in arithmetic operations), we find that performing
  the P$\pm$A additions and subtractions in machine (double) precision
  leads to mild degradation of signals that worsens with increasing
  $n$. This indicates that differences in the iterative (conjugate
  gradient) propagator inversion for the two types of boundary
  conditions are becoming significant.  }

In order to determine the interaction energy in multi-meson systems,
both the single-meson, $C_{1}(t)$, and $n$-meson, $C_{n }(t)$,
correlation functions were computed, where $t$ is the Euclidean time
separation between the hadronic source and sink operators.  For
simplicity, meson correlation functions involving only zero-momentum
states were calculated.  The single-kaon correlation function is
\begin{equation}\label{eq:C_K}
        C_{K^-}(t) = \sum_\mathbf{x} \langle\, K^+(t,\mathbf{x})\
        K^-(0,\mathbf{0})\, \rangle\, ,
\end{equation}
where the sum over all spatial sites projects onto the zero-momentum
state, $\mathbf{p}=0$.  A correlation function which projects onto the
$n$ $K^-$ ground-state is
\begin{equation}\label{eq:C_nK}
        C_{n K^-}(t) =
                 \langle\,  \left( \ \sum_{\bf x } K^+(t,{\bf x})\ \right)^n \
                 \left(\ \phantom{\sum_{\bf x }} \hspace{-14pt} K^-(0,
                   {\bf 0})\ \right)^n \, \rangle\, ,
\end{equation}
which can be constructed from one light and one strange quark propagator.
In eqs.~\eqref{eq:C_K} and \eqref{eq:C_nK}, $K^+(t,\mathbf{x}) =
\bar{s}(t,\mathbf{x}) \gamma_5 u(t,\mathbf{x})$ is a Gaussian-smeared
interpolating field for the charged kaon.  In the relatively-large
spatial volumes used in the calculation, the interaction energy is a
small fraction of the total energy that is dominated by the
mass of the kaon.
To determine this energy, the ratio of correlation functions,
\begin{equation}\label{eq:C_nK_C_K}
        G_{n K^-}(t) \equiv \frac{C_{n K^-}(t)}{ \left[\ C_{K^-}(t)\ \right]^n
        }\ \longrightarrow \  \mathcal{A}_n \ e^{-\Delta E_{nK} t}\, ,
\end{equation}
was constructed, with the arrow denoting the large-time,
infinite-number-of-gauge-configurations limit (far from the boundary).
Due to the (anti-)periodic boundary-conditions imposed on the
propagators computed on the fine, and large-volume lattices, the $K^-$
correlation function becomes a single $\cosh$ function far from the
source, while the $n$-kaon correlation functions become sums of
multiple $\cosh$'s, leading to a non-trivial form for $G_{n K^-}(t)$;
further details are given in Appendix \ref{sec:effects-temp-bound}. As
an alternative method to calculating the interaction energy (and a
check of the systematics), a Jackknife analysis of the difference
between the energies extracted from the long-time behavior of the
multiple- and single-kaon correlation functions individually was
performed, finding results in agreement with those determined from
eq.~\eqref{eq:C_nK_C_K}.  The interaction energy is related to the
$n$-particle energy eigenvalues, $E_{n K}$, and the kaon mass,
\begin{equation}\label{eq:DeltaE}
        \Delta E_{n K}\  \equiv \ E_{n K} - n\ m_K\, .
\end{equation}
A comprehensive review of the lattice techniques we have used in this
work can be found in Refs.~\cite{Detmold:2008fn,NPLQCDreview}.

\begin{figure}[!ht]
  \centering
  \includegraphics[width=0.98\columnwidth]{figures/n12CorrDeltaEvsNSS007K.pdf}
  \caption{The energy-differences (in lattice units), $b\ \Delta E_{n K}$, for the
    multi-kaon systems with $m_\pi\sim 291~{\rm MeV}$.  The lighter
    (darker) shaded regions correspond to the statistical and
    systematic uncertainties combined in quadrature (statistical
    uncertainty) obtained from fitting the plateau regions.  The green
    line denotes the central value of the fit. }
  \label{fig:DeltaE007}
\end{figure}
To indicate the quality of the results of this calculation, in
fig.~\ref{fig:DeltaE007} we show the effective energy differences,
$\Delta E_{n K}$, as defined in eq.~(\ref{eq:DeltaE}), for the coarse
2.5 fm ensemble with $m_\pi\sim 291~{\rm MeV}$.  For each $n$ there is
clear, multi-time-slice, plateau from which to extract $\Delta E_{n
  K}$.  Figure ~\ref{fig:DeltaE007} shows that the systematic
uncertainties in the plateau region tend to become more significant as
the number of kaons in the system is increased, likely due to the fact
that the same propagators are being raised to high powers.  In the
extraction of $\Delta E_{nK}$ from the correlation functions, a
correlated $\chi^2$-fit is performed over the fitting interval.
Further, in the extractions of the two- and three- body scattering
parameters, correlations among the different $\Delta E_{nK}$ are
included by performing a coupled $n$- and $t$- correlated fit.
Systematic uncertainties are assigned to these extractions by varying
the ends of the fitting intervals.  It is interesting to note that the
uncertainty in the effective $n$-kaon energy grows with time. This is
expected when $m_\pi+m_\phi<2m_K$ (as is the case for the ensembles
studied here) based on the general arguments given in
Ref.~\cite{Lepage:1989hd}.

\section{Three-Meson  interactions}
\label{sec:threebody}

\noindent
The three-meson interactions extracted from our calculations, $m_{\pi}
f_{\pi}^4\ \overline{\overline{\eta}}^L_{3\pi^-}$ and $m_{K} f_{K}^4\
\overline{\overline{\eta}}^L_{3K^-}$, are shown in
Fig.~\ref{fig:threebodysummary}, in units of the NDA estimate,
$\overline{\overline{\eta}}^{L,(NDA)}_{3\pi^-} = 1/(m_\pi f_\pi^4)$
and $\overline{\overline{\eta}}^{L,(NDA)}_{3K^-} = 1/(m_K f_K^4)$.
Table~\ref{tab:results} contains the extracted values of the two-pion
and two-kaon scattering parameters and the three-pion and three-kaon
interactions for each ensemble.
\begin{figure}[!ht]
  \centering
  \includegraphics[width=0.45\columnwidth]{figures/mf4eta3barbarplot.pdf}
\qquad
  \includegraphics[width=0.45\columnwidth]{figures/mKfK4eta3Kbarbarplot.pdf}
  \caption{ The three-body interactions,  $m_{\pi} f_{\pi}^4\
    \overline{\overline{\eta}}^L_{3\pi^-}$ and $m_{K} f_{K}^4\
    \overline{\overline{\eta}}^L_{3K^-}$, defined in
    eq.~(\ref{eq:energyshift}), as a function of $m_\pi/f_\pi$ and
    $m_{K}/f_{K}$, respectively.  The statistical uncertainties and
    the statistical and systematic uncertainties combined in
    quadrature are shown as the inner and outer error-bounds,
    respectively.  The dashed vertical lines indicates the physical
    point.  }
  \label{fig:threebodysummary}
\end{figure}
The results of the calculation of the three-pion interaction on the
coarse MILC lattices have been presented previously, and the result
from the fine ensemble is consistent with the closest mass coarse
results, as can be seen in fig.~\ref{fig:threebodysummary}.  The
three-kaon interaction is found to be consistent with zero for all the
kaon masses that have been explored, but the uncertainties are
moderately large.  This result is consistent with the trend found in
multi-pion systems where there is a hint that the three-pion
interaction is decreasing with increasing pion mass.  As is the case
for the three-pion interaction~\cite{Detmold:2008fn}, it is not yet
possible to extract the RGI three-body interaction,
$\overline{\eta}_{3K}^L$ or the underlying parameter $\eta_{3K}(\mu)$
because of the current uncertainties in the calculation.  The
uncertainties of the three-body interactions measured on the large
volume ensemble are significantly larger than those shown in the
figure and we do not display them.

\begin{table}[t]
 \caption{Extracted values of the two-body and three-body interactions.
Note that  $\overline{\overline{\eta}}^L_{3\pi^-}$ and
   $\overline{\overline{\eta}}^L_{3K^-}$ depend on
   $L$, eq.~(\ref{eq:eta3barbar}) and eq.~(\ref{eq:etathreebar}) and are
   quoted at their physical volumes. The
   systematic uncertainties in the pionic quantities from the coarse
   $L\sim 2.5~{\rm fm}$  ensembles have been symmetrized about the central
value~\protect{\cite{Detmold:2008fn}}.
The   pion and kaon scattering lengths obtained on the  coarse
   $L\sim 2.5~{\rm fm}$  ensembles are in agreement with those of
   Ref.~\protect{\cite{Beane:2007uh,Beane:2007xs}}, but have  slightly smaller uncertainties
   stemming from increased statistics and from the $n$-correlated
   analysis performed here.
Quantities extracted from the correlators formed from the P$\pm$A propagators are
consistent with those from the individual propagators.
}
\label{tab:results}
\begin{ruledtabular}
\begin{tabular}{ccccc}
 Ensemble
&  $m_\pi\overline{a}_{\pi\pi}^{(I=2)}$ &
$m_\pi f_\pi^4\overline{\overline{\eta}}^L_{3\pi^-}$
&  $m_K\overline{a}_{KK}^{(I=1)}$ &
$m_K f_K^4 \overline{\overline{\eta}}^L_{3K^-}$ \\
\hline
2064f21b676m007m050 & 0.164(4)(8) & 0.8(1)(2) & 0.491(9)(13) & -0.1(2)(4) \\
2064f21b676m010m050 & 0.206(5)(6) & 1.0(1)(2) & 0.503(11)(19) & -0.1(2)(5) \\
2064f21b679m020m050 & 0.350(7)(10) & 0.9(1)(3) & 0.536(10)(19) & 0.3(2)(5)  \\
2064f21b681m030m050 & 0.476(10)(16) & 0.6(2)(4) & 0.584(12)(18) & 0.1(3)(5) \\
\hline
2896f2b709m0062m031 & 0.178(2)(8) & 0.64(3)(16) & 0.402(6)(18) & 0.3(1)(3)\\
\hline
2864f2b676m010m050 &  0.25(1)(6) & 0(1)(1) & 0.52(6)(17) & -5(3)(8) \\
\end{tabular}
\end{ruledtabular}
\end{table}

\section{The Equation of State and Chemical
  Potentials}
\label{sec:chemical}

\noindent
The energy of the $n$ $K^-$ system as a function of volume and of the
number of $K^-$'s is given explicitly in eq.~(\ref{eq:energyshift}) in
the large-volume expansion.  From the equation of state, the $K^-$
chemical potential is
\begin{eqnarray}
  \label{eq:Kplusdensity}
  \mu_{K^-} &=& \left.\frac{d E_{n K}}{d n}\right|_{V={\rm const}}
  \ \ \ ,
\end{eqnarray}
and can be constructed analytically from eq.~(\ref{eq:energyshift}) or
numerically from the results of the lattice calculations by using a
simple finite difference approximation.  The resulting ratios of the
$K^-$-chemical potentials to the kaon mass on each of the coarse
ensembles are shown in Fig.~\ref{fig:chemicalK} as a function of the
$K^-$-density, $\rho_{K^-}$ in units of (2.5 fm)$^3$.  For comparison
Fig.~\ref{fig:chemicalpi} shows the isospin chemical potential
computed from the $n$ $\pi^+$ systems, expanding upon the results of
Ref.~\cite{Detmold:2008fn} by including the large volume coarse
ensemble.  Finally, Fig.~\ref{fig:chemicalKfine} shows the kaon and
pion chemical potentials for the fine ensemble, and a comparison
between results obtained with anti-periodic boundary conditions and
those computed with the P$\pm$A propagators.  We note that for the
$12$-$K^-$ system, the number density is $\rho_{K^-}^{(12)}=12/L^3 =
0.77~{\rm fm}^{-3}$ on the ensembles with $L\sim 2.5~{\rm fm}$.  The
solid curves in Figs.~\ref{fig:chemicalK}, \ref{fig:chemicalpi}
and~\ref{fig:chemicalKfine} correspond to the prediction at ${\cal
  O}(L^{-7})$ from eq.~(\ref{eq:energyshift}) using the extracted
two-body and three-body scattering parameters, and differ
insignificantly from that at ${\cal O}(L^{-6})$. The dotted curves
result from setting the three-body interactions to zero.
\begin{figure}[!t]
\vskip0.5in
\centering
\includegraphics[width=1.0\textwidth]{figures/correlatedmuSvsnplotwithSonwith35.pdf}
\caption{\label{fig:chemicalK}
  The $K^-$ chemical potential, $\mu_{K^-}$, as a function of the
  density of $K^-$'s on the coarse MILC lattices.  The rectangular
  regions are the results of the lattice calculation where a
  finite-difference has been used to construct the derivative with
  respect to the number of $K^-$'s.  The lighter (green) rectangles
  are obtained from the lattices with spatial extent $L\sim 2.5~{\rm
    fm}$, while the darker (red) rectangles are obtained from the
  lattices with $L\sim 3.5~{\rm fm}$.  The dashed curves correspond to
  tree-level $\chi$PT.  The darker solid curve corresponds to the
  analytic expression for the energy of the ground state in the large
  volume expansion, eq.~(\protect\ref{eq:energyshift}), using the fit
  values for $\overline{a}_{KK}^{(I=1)}$ and $\overline{\overline{\eta}}_{3K^-}^L$ . The lighter dotted curve shows the
  analytic forms with the fit value of $\overline{a}_{KK}^{(I=1)}$ but with $\overline{\overline{\eta}}_{3K^-}^L$ set to
  zero.  }
\end{figure}
\begin{figure}[!t]
\vskip0.5in
\centering
\includegraphics[width=1.0\textwidth]{figures/correlatedmuIvsnplotwithSonwith35.pdf}
\caption{\label{fig:chemicalpi}
  The isospin-chemical potential, $\mu_{I}$, as a function of the
  density of $\pi^-$'s on the coarse MILC lattices.  The rectangular
  regions are the results of the lattice calculation where a
  finite-difference has been used to construct the derivative with
  respect to the number of $\pi^-$'s.  The lighter (green) rectangles
  are obtained from the lattices with spatial extent $L\sim 2.5~{\rm
    fm}$, while the darker (red) rectangles are obtained from the
  lattices with $L\sim 3.5~{\rm fm}$.  The dashed curves correspond to
  tree-level $\chi$PT.  The darker solid curve corresponds to the
  analytic expression for the energy of the ground state in the large
  volume expansion, eq.~(\protect\ref{eq:energyshift}), using the fit
  values for $\overline{a}_{\pi\pi}^{(I=2)}$ and $\overline{\overline{\eta}}_{3\pi^-}^L$ . The lighter dotted curve shows
  the analytic forms with the fit value of $\overline{a}_{\pi\pi}^{(I=2)}$ but with $\overline{\overline{\eta}}_{3\pi^-}^L$
  set to zero.  }
\end{figure}
\begin{figure}[!t]
\vskip0.5in
\centering
\includegraphics[width=0.457\textwidth]{figures/SS0062AandPpAmuKplotwithSon.pdf}
\qquad
\includegraphics[width=0.45\textwidth]{figures/SS0062AandPpAmuplotwithSon.pdf}
\caption{\label{fig:chemicalKfine}
  The $K^-$ chemical potential, $\mu_{K^-}$, as a function of the
  density of $K^-$'s on the fine MILC lattice ensemble (left panel)
  and the isospin chemical potential, $\mu_I$ as a function of the
  density of $\pi^+$'s on the same ensemble (right panel).  The
  rectangular regions show the results of the lattice calculation
  where a finite-difference has been used to construct the derivative
  with respect to the density of mesons.  The lighter (blue)
  rectangles are obtained from purely anti-periodic temporal boundary
  conditions on the quark propagators, while the darker (purple)
  rectangles are obtained from the P$\pm$A propagators.  The dashed
  curves correspond to LO $\chi$PT.  The darker solid curve
  corresponds to the analytic expressions for the energy of the ground
  state in the large volume expansion,
  eq.~(\protect\ref{eq:energyshift}), using the fit values of the
  two-body and three-body interactions. The lighter dotted curve shows
  the analytic forms with the fit value of the two-body interaction
  but with three-body interactions set to zero.  }
\end{figure}

The results of the lattice QCD calculation are consistent within
uncertainties with the prediction of tree-level $SU(3)_L\otimes
SU(3)_R$ $\chi$PT which are shown as the dashed lines in
Figs.~\ref{fig:chemicalK}, \ref{fig:chemicalpi}
and~\ref{fig:chemicalKfine}.  Given that one expects the tree-level
result to be accurate at the $\sim 25\%$-level, this is quite a
remarkable result.  The contributions from the higher-order
counterterms and loop diagrams have conspired to produce a shift from
tree-level that is much less than one expects from NDA.

In comparing these lattice QCD calculations of the chemical potential
to those calculated in $\chi$PT in Section \ref{sec:XPT}, we note that
they are performed in finite volumes and at non-vanishing lattice
spacings.  These approximations modify the chemical potential expected
in the lattice calculation from chiral perturbation theory at most at
NLO in the chiral expansion.  The fact that LO $\chi$PT is in good
agreement with our numerical results, on both the larger and smaller
volume coarse ensembles and the fine ensemble, suggests that finite
volume and lattice spacing effects are small. For $m_\pi\sim352$~MeV,
the two calculations at different volumes agree within their
uncertainties, also supporting this claim.

\section{Conclusions}

\noindent
Kaon condensation may play an important role in the evolution of
supernovae~\cite{Brown:1993jz}.  The theoretical analysis of the
condensation mechanism presently relies upon chiral perturbation
theory to determine both the interactions of kaons with the baryonic
matter and the kaon self-interactions.  We have performed the first
Lattice QCD investigation of the charged kaon condensate, isolating
part of the mechanism that may be in action in the interior of neutron
stars.  In particular, we have explored the $K^-$ chemical potential
as a function of density.  We have also updated our previous analysis
of the charged pion condensate.  Surprisingly, LO $\chi$PT is found to
reproduce the results of the Lattice QCD calculations to within the
uncertainties of the calculation.  This is significantly better than
the $\sim 25\%$-level expected from dimensional analysis.  However,
this is consistent with the findings of other lattice calculations
that higher orders in chiral perturbation theory make contributions
that are much smaller than naively expected, e.g.
Ref.~\cite{NPLQCDreview}, in this range of meson masses.

To move toward exploring the possibility of kaon condensation in a
baryonic background, lattice calculations of systems containing
multiple nucleons and multiple kaons will be required. Only one
dynamical lattice QCD calculation of the two-nucleon system has been
performed to date~\cite{Beane:2006mx}, and exploration of the
multi-nucleon-multi-kaon systems are left for the future.  In addition
to the increased exponential degradation of the signal-to-noise ratio
associated with baryonic correlation functions~\cite{Lepage:1989hd},
``disconnected diagrams'' (in which a quark is created and annihilated
at the source) will contribute to the multi-nucleon-multi-$K^-$
correlation functions, and therefore the computational resources
required to perform such calculations will be considerably larger than
currently available for the study of multi-hadron systems.

\section{Acknowledgments}

\noindent
We thank Silas Beane, Tom Luu, Assumpta Parre\~no and Aaron Torok for
contributing to the quark propagator calculations required for this
work.  We thank Michael Endres and Steve Sharpe for discussions, and
R.~Edwards and B.~Joo for help with the QDP++/Chroma programming
environment~\cite{Edwards:2004sx} with which the propagator
calculations discussed here were performed.  The computations for this
work were performed at Jefferson Lab, Fermilab, Centro Nacional de
Supercomputaci\'on (Barcelona, Spain), the University of Washington
and the National Energy Research Scientific Computing Center, which is
supported by the Office of Science of the U.S. Department of Energy
under Contract No. DE-AC02-05CH11231. This research was also supported
in part by the National Science Foundation through TeraGrid resources
provided by the National Center for Supercomputing Applications. We
are indebted to the MILC for use of their configurations.  The work of
MJS and WD was supported in part by the U.S.~Dept.~of Energy under
Grant No.~DE-FG03-97ER4014. The work of KO was supported in part by
the U.S.~Dept.~of Energy contract No.~DE-AC05-06OR23177 (JSA) and by
the Jeffress Memorial Trust, grant J-813, DOE OJI grant
DE-FG02-07ER41527 and DOE grant DE-FG02-04ER41302. The work of AW-L
was supported in part by DOE OJI grant DE-FG02-07ER41527 and DOE grant
DE-FG02-93ER40762.

redpajama_set_name: RedPajamaArXiv


text:
\section{Introduction}

The canonical seesaw model with three right-handed neutrinos \cite{seesaw} is one of the most natural and the simplest extensions of the Standard Model (SM) to account for tiny but nonzero neutrino masses. As a bonus, the seesaw model also provides an elegant explanation for the matter-antimatter asymmetry of the Universe via the leptogenesis mechanism \cite{Fukugita:1986hr}. Since the mass scale of right-handed neutrinos is typically much higher than the electroweak scale, i.e.,  $M \gg \Lambda^{}_{\rm EW} \sim \mathcal{O} \left( 10^2 \right)$ GeV, one can integrate out heavy right-handed neutrinos and work in the corresponding effective field theory (EFT), i.e., the so-called seesaw EFT (SEFT) to explore low-energy consequences of heavy right-handed neutrinos. In this way, one can easily perform large logarithm resummations by means of the renormalization group equations (RGEs) in the EFT. The tree-level structure of SEFT derived by integrating out right-handed neutrinos at the tree level has been obtained in Ref. \cite{Broncano:2002rw}, and contains the unique Weinberg operator \cite{Weinberg:1979sa} and two dimension-six (dim-6) operators in the  Warsaw basis. The former one generates tiny neutrino masses, whereas the latter two modify the couplings of neutrinos with weak gauge bosons after gauge symmetry is spontaneously broken. However, if one wants to take into account one-loop effects in the seesaw model, the tree-level SEFT is not enough. In this case, the one-loop SEFT achieved by integrating out heavy right-handed neutrinos at the one-loop level is indispensable for consistently exploring one-loop consequences of right-handed neutrinos, e.g., radiative decays of charged leptons in the SEFT \cite{Zhang:2021tsq}. In this talk, we summarize the results of one-loop matching of the seesaw model, i.e., integrate out heavy right-handed neutrinos at the one-loop level, and achieve the complete one-loop structure of the SEFT \cite{Zhang:2021jdf}. Then, we show self-consistent calculations of radiative decays of charged leptons.

\section{Matching between ultraviolet (UV) theory and EFT}

The idea to match a given UV theory to the corresponding low-energy EFT is to equate the one-light-particle-irreducible (1LPI) effective action (i.e., $\Gamma^{}_{\rm L, UV}$) in the UV theory with the one-particle-irreducible (1PI) effective action (i.e., $\Gamma^{}_{\rm EFT}$) in the EFT at the matching scale $\mu$, i.e.,
\begin{eqnarray}
	\Gamma^{}_{\rm L, UV} \left[ \phi^{}_{\rm B}\right] = \Gamma^{}_{\rm EFT} \left[ \phi^{}_{\rm B} \right] \;,
\end{eqnarray}
in which $\phi^{}_{\rm B}$ denotes the light background field and both effective actions can be calculated by means of the background field method \cite{Abbott:1981ke}. By making use of the functional approach, one can easily derive the tree-level Lagrangian of the EFT, namely
\begin{eqnarray}\label{eq:tree-level}
	\mathcal{L}^{\rm tree}_{\rm EFT} \left[ \phi^{}_{\rm B} \right] = \mathcal{L}^{}_{\rm UV} \left[ \hat{\Phi}_{\rm c} \left[ \phi^{}_{\rm B} \right] , \phi^{}_{\rm B} \right] \;,
\end{eqnarray}
where $\hat{\Phi}^{}_{\rm c} \left[ \phi^{}_{\rm B} \right]$ is the localized solution of the classical equation of motion for the heavy field $\Phi^{}_{\rm B}$,
\begin{eqnarray}
\left. 	\frac{\delta \mathcal{L}_{\rm UV} \left[ \Phi, \phi \right] }{\delta \Phi} \right|^{}_{\Phi = \Phi^{}_{\rm c} \left[ \phi^{}_{\rm B} \right], \phi = \phi^{}_{\rm B}} = 0 \;.
\end{eqnarray}
The one-loop Lagrangian of the EFT can be calculated via \cite{Cohen:2020fcu}
\begin{eqnarray}\label{eq:one-loop}
	\int {\rm d}^d x \mathcal{L}^{\rm 1-loop}_{\rm EFT} \left[ \phi^{}_{\rm B} \right] = \left. \frac{{\rm i}}{2} {\rm STr} \ln \left( - \bm{K } \right)  \right|^{}_{\rm hard} - \left. \frac{{\rm i}}{2} \sum^\infty_{n=1} \frac{1}{n} {\rm STr} \left[ \left( \bm{ K^{-1} X} \right)^n \right] \right|^{}_{\rm hard} \;,
\end{eqnarray}
where $\bm{K}$ and $\bm{X}$ are the inverse-propagator and interaction matrices, respectively, and can be extracted from the UV Lagrangian via
\begin{eqnarray}
	\left. \delta^2 \mathcal{L}^{}_{\rm UV} \right|^{}_{\Phi = \hat{\Phi}^{}_{\rm c} \left[ \phi^{}_{\rm B} \right]} = \left. 2\mathcal{L}^{}_{\rm UV} \left[ \varphi + \delta \varphi \right] \right|^{}_{\Phi = \hat{\Phi}^{}_{\rm c} \left[ \phi^{}_{\rm B} \right]} \supset \delta \overline{\varphi}^{}_i \left( \bm{K}^{}_i \delta^{}_{ij} - \bm{X}^{}_{ij}  \right) \delta \varphi^{}_j
\end{eqnarray}
with $\varphi$ being field multiplet and containing fields and their conjugates. Then, one may calculate supertraces in Eq. \eqref{eq:one-loop} with the help of the Mathematica package {\sf SuperTracer} \cite{Fuentes-Martin:2020udw} or {\sf STrEAM} \cite{Cohen:2020qvb} based on the covariant derivative expansion (CDE) method \cite{Gaillard:1985uh}.

\section{Complete one-loop structure of the SEFT}

\begin{table}[t!]
	\centering
	\renewcommand\arraystretch{1.5}
	\resizebox{\textwidth}{!}{
		\begin{tabular}{c|c|c|c|c|c}
			\hline\hline
			\multicolumn{2}{c}{$X^2H^2$} & \multicolumn{2}{|c|}{$\psi^2DH^2$} & \multicolumn{2}{c}{Four-quark}
			\\
			\hline
			$\mathcal{O}^{}_{HB}$ & $B^{}_{\mu\nu} B^{\mu\nu} H^\dagger H$ & $\mathcal{O}^{(1)\alpha\beta}_{HQ}$ &  $\left( \BLQi \gamma^\mu \LQi \right) \left( H^\dagger {\rm i}~ \mbox{$\raisebox{2mm}{\boldmath ${}^\leftrightarrow$}\hspace{-4mm}D^{}_\mu$} H \right)$ & $\mathcal{O}^{(1)\alpha\beta\gamma\lambda}_{qu}$ & $\left( \BLQi \gamma^\mu \LQi \right) \left( \BRUi[\gamma] \gamma^{}_\mu \RUi[\lambda] \right)$
			\\
			$\mathcal{O}^{}_{HW}$ & $W^I_{\mu\nu} W^{I\mu\nu} H^\dagger H$ & $\mathcal{O}^{(3)\alpha\beta}_{Hq} $ & $ \left( \BLQi \gamma^\mu \tau^I \LQi \right) \left(H^\dagger {\rm i}~ \mbox{$\raisebox{2mm}{\boldmath ${}^\leftrightarrow$}\hspace{-4mm}D^I_\mu$} H \right)$ & $\mathcal{O}^{(8)\alpha\beta\gamma\lambda}_{qu}$ & $\left(\BLQi \gamma^\mu T^A \LQi \right) \left( \BRUi[\gamma] \gamma^{}_\mu T^A \RUi[\lambda] \right)$
			\\
			$\mathcal{O}^{}_{HWB}$ & $W^I_{\mu\nu} B^{\mu\nu} \left( H^\dagger \tau^I H \right)$ & $\mathcal{O}^{\alpha\beta}_{Hu}$ & $\left( \BRUi \gamma^\mu \RUi \right) \left(H^\dagger {\rm i}~ \mbox{$\raisebox{2mm}{\boldmath ${}^\leftrightarrow$}\hspace{-4mm}D^{}_\mu$} H \right)$ & $ \mathcal{O}^{(1)\alpha\beta\gamma\lambda}_{qd}$ & $ \left( \BLQi \gamma^\mu \LQi \right) \left( \BRDi[\gamma] \gamma^{}_\mu \RDi[\lambda] \right)$
			\\
			\cline{1-2}
			\multicolumn{2}{c|}{$H^4D^2$} & $\mathcal{O}^{\alpha\beta}_{Hd}$ & $\left( \BRDi \gamma^\mu \RDi \right) \left( H^\dagger {\rm i} ~\mbox{$\raisebox{2mm}{\boldmath ${}^\leftrightarrow$}\hspace{-4mm}D^{}_\mu$} H \right)$ & $\mathcal{O}^{(8)\alpha\beta\gamma\lambda}_{qd}$ & $\left( \BLQi \gamma^\mu T^A \LQi \right) \left( \BRDi[\gamma] \gamma^{}_\mu T^A \RDi[\lambda] \right)$
			\\
			\cline{1-2}
			$\mathcal{O}^{}_{H\Box}$ & $\left(H^\dagger H\right) \Box \left( H^\dagger H \right)$ & $\mathcal{O}^{(1)\alpha\beta}_{H\ell}$ & $\left( \BLelli \gamma^\mu \Lelli \right) \left( H^\dagger {\rm i} ~\mbox{$\raisebox{2mm}{\boldmath ${}^\leftrightarrow$}\hspace{-4mm}D^{}_\mu$} H \right)$ & $\mathcal{O}^{(1)\alpha\beta\gamma\lambda}_{quqd}$ & $ \left( \overline{Q^a_{\alpha{\rm L}}} \RUi \right) \epsilon^{ab} \left( \overline{Q^b_{\gamma{\rm L}}} \RDi[\lambda] \right) $
			\\
			\cline{5-6}
			$\mathcal{O}^{}_{HD}$ & $\left(H^\dagger D^{}_\mu H \right)^\ast \left(H^\dagger D^\mu H \right)$ & $\mathcal{O}^{(3)\alpha\beta}_{H\ell}$ & $ \left( \BLelli \gamma^\mu \tau^I \Lelli \right) \left( H^\dagger {\rm i} ~\mbox{$\raisebox{2mm}{\boldmath ${}^\leftrightarrow$}\hspace{-4mm}D^I_\mu$} H \right) $ & \multicolumn{2}{c}{Four-lepton}
			\\
			\cline{1-2}\cline{5-6}
			\multicolumn{2}{c|}{$H^6$} & $\mathcal{O}^{\alpha\beta}_{He}$ & $\left(\BREi \gamma^\mu \REi \right) \left( H^\dagger {\rm i}~ \mbox{$\raisebox{2mm}{\boldmath ${}^\leftrightarrow$}\hspace{-4mm}D^{}_\mu$} H \right)$ & $\mathcal{O}^{\alpha\beta\gamma\lambda}_{\ell\ell}$ & $ \left( \BLelli \gamma^\mu \Lelli \right) \left( \BLelli[\gamma] \gamma^{}_\mu \Lelli[\lambda] \right) $
			\\
			\cline{1-4}
			$\mathcal{O}^{}_H$ & $\left(H^\dagger H \right)^3$ & \multicolumn{2}{c|}{$\psi^2 H^3$} & $\mathcal{O}^{\alpha\beta\gamma\lambda}_{\ell e}$ & $\left(\BLelli \gamma^\mu \Lelli \right) \left( \BREi[\gamma] \gamma^{}_\mu \REi[\lambda] \right) $
			\\
			\cline{1-4}
			\multicolumn{2}{c|}{$\psi^2XH$} & $\mathcal{O}^{\alpha\beta}_{uH}$ & $\left( \BLQi \widetilde{H} \RUi \right) \left( H^\dagger H \right) $ & &
			\\
			\cline{1-2}
			$\mathcal{O}^{\alpha\beta}_{eB}$ & $\left( \BLelli \sigma^{\mu\nu} \REi \right) H B^{}_{\mu\nu}$ & $\mathcal{O}^{\alpha\beta}_{dH}$ & $ \left( \BLQi H \RDi \right) \left( H^\dagger H \right) $ & &
			\\
			$\mathcal{O}^{\alpha\beta}_{eW}$ & $ \left( \BLelli \sigma^{\mu\nu} \REi \right) \tau^I H W^I_{\mu\nu} $ & $\mathcal{O}^{\alpha\beta}_{eH}$ & $ \left( \BLelli H \REi \right) \left( H^\dagger H \right)$ & &
			\\
			\hline
			\multicolumn{6}{c}{Semi-leptonic}
			\\
			\hline
			$\mathcal{O}^{(1)\alpha\beta\gamma\lambda}_{\ell q}$ & $ \left( \BLelli \gamma^\mu \Lelli \right) \left( \BLQi[\gamma] \gamma^{}_\mu \LQi[\lambda] \right)$ & $\mathcal{O}^{\alpha\beta\gamma\lambda}_{\ell u}$ & $\left( \BLelli \gamma^\mu \Lelli \right) \left( \BRUi[\gamma] \gamma^{}_\mu \RUi[\lambda] \right)$ & $\mathcal{O}^{\alpha\beta\gamma\lambda}_{\ell e d q}$ & $ \left( \BLelli \REi \right) \left( \BRDi[\gamma] \LQi[\lambda] \right) $
			\\
			$\mathcal{O}^{(3)\alpha\beta\gamma\lambda}_{\ell q}$ & $ \left( \BLelli \gamma^\mu \tau^I \Lelli \right) \left( \BLQi[\gamma] \gamma^{}_\mu \tau^I \LQi[\lambda] \right) $ & $\mathcal{O}^{\alpha\beta\gamma\lambda}_{\ell d}$ & $\left( \BLelli \gamma^\mu \Lelli \right) \left( \RDi[\gamma] \gamma^{}_\mu \RDi[\lambda] \right)$ & $\mathcal{O}^{(1)\alpha\beta\gamma\lambda}_{\ell e q u}$ & $\left( \overline{\ell^a_{\alpha \rm L}} \REi \right) \epsilon^{ab} \left( \overline{Q^b_{\gamma \rm L}} \RUi[\lambda] \right)$
			\\
			\hline\hline
	\end{tabular}}
	\vspace{-0.15cm}
	\caption{Dim-6 operators in the seesaw model at the one-loop level in the Warsaw basis, where the Hermitian conjugates of the operators in classes $\psi^2 X H$ and $\psi^2 H^3$ and four-fermion operators are not listed explicitly.}
	\label{tb:Wbasis}
	\vspace{-0.25cm}
\end{table}

With the help of Eqs. \eqref{eq:tree-level} and \eqref{eq:one-loop}, and the Lagrangian of the seesaw model, i.e.,
\begin{eqnarray}\label{eq:Lagrangian}
	\mathcal{L}^{}_{\rm UV} = \mathcal{L}^{}_{\rm SM} + \overline{N^{}_{\rm R}} {\rm i} \slashed{\partial} N^{}_{\rm R} - \left( \frac{1}{2} \overline{N^c_{\rm R}} MN^{}_{\rm R} + \overline{\ell^{}_{\rm L}} Y^{}_\nu \widetilde{H} N^{}_{\rm R} + {\rm h.c.} \right)
\end{eqnarray}
with $\mathcal{L}^{}_{\rm SM}$ being the SM Lagrangian, one can obtain the complete Lagrangian of the SEFT up to dim-6 at the one-loop level, namely,
\begin{eqnarray}
	\mathcal{L}^{}_{\rm SEFT} &=& \mathcal{L}^{}_{\rm SM} \left( m^2 \to m^2_{\rm eff}, \lambda \to \lambda^{}_{\rm eff}, Y^{}_l \to Y^{\rm eff}_l, Y^{}_{\rm u} \to Y^{\rm eff}_{\rm u}, Y^{}_{\rm d} \to Y^{\rm eff}_{\rm d}  \right)
	\nonumber
	\\
	&& + \left[ \frac{1}{2} \left( C^{(5)}_{\rm eff} \right)_{\alpha\beta} \mathcal{O}^{(5)}_{\alpha\beta} + {\rm h.c.} \right] + \frac{1}{4} \left( C^{(6)}_{\rm tree} \right)_{\alpha\beta} \left[ \mathcal{O}^{(1)\alpha\beta}_{H\ell} - \mathcal{O}^{(3)\alpha\beta}_{H\ell} \right] + \sum_i C^{}_i \mathcal{O}^{}_{i} \;,
\end{eqnarray}
where $\mathcal{O}^{}_i$ denote the dim-6 operators listed in Table~\ref{tb:Wbasis} including Hermitian conjugations of the non-Hermitian operators, while $C^{}_i$ refer to the one-loop contributions to corresponding Wilson coefficients. Explicit expressions of all the effective couplings and Wilson coefficients can be found in Ref. \cite{Zhang:2021jdf}. Here, we only show those for $\mathcal{O}^{}_{eB}$ and $\mathcal{O}^{}_{eW}$, i.e.,
\begin{eqnarray}
	C^{\alpha\beta}_{eB} = \frac{g^{}_1}{24 \left( 4\pi \right)^2} \left( Y^{}_\nu M^{-2} Y^\dagger_\nu Y^{}_l \right)^{}_{\alpha\beta} \;,\quad C^{\alpha\beta}_{eW} = \frac{5g^{}_2}{24\left( 4 \pi \right)^2} \left( Y^{}_\nu M^{-2} Y^\dagger_\nu Y^{}_l \right)^{}_{\alpha\beta} \;.
\end{eqnarray}
These two dim-6 operators directly contribute to radiative decays of charged leptons after spontaneous gauge symmetry breaking as follows \cite{Zhang:2021tsq}
\begin{eqnarray}\label{eq:rad}
	{\rm i} \mathcal{M} \left( l^{-}_\beta \to l^{-}_\alpha + \gamma \right) = \frac{{\rm i} e g^2_2}{6\left( 4\pi \right)^2 M^2_W} \left( RR^\dagger \right)^{}_{\alpha\beta} \left[ \epsilon^\ast_\mu \overline{u} \left( p^{}_2 \right) {\rm i} \sigma^{\mu\nu} q^{}_\nu \left( m^{}_\alpha P^{}_{\rm L} + m^{}_\beta P^{}_{\rm R} \right) u \left( p^{}_1 \right) \right]
\end{eqnarray}
where $R \equiv vY^{}_\nu M^{-1}/\sqrt{2}$ with $v\approx 246~{\rm GeV}$ and $m^{}_\alpha$ denotes the mass of $l^-_\alpha$. Together with one-loop contributions from the tree-level SEFT, Eq. \eqref{eq:rad} reproduces the results in the full theory \cite{Cheng:1980tp}.

\section{Summary}

We have carried out the complete one-loop matching of the seesaw model onto the corresponding SEFT, and obtained 31 independent dim-6 operators (barring flavor structures and Hermitian conjugates) in the Warsaw basis up to the one-loop level. Moreover, the SM couplings acquire threshold corrections in the SEFT, which are the very matching conditions for two-loop RGEs. The obtained one-loop structure of the SEFT is indispensable to consistent calculations of one-loop consequences of heavy right-handed neutrinos in the seesaw model.
\\[0.12cm]
{\it This work was supported by the National Natural Science Foundation of China under grants No. 12075254 and No. 11835013.}


text:
\section{Introduction}

It was mentioned more than once (see, e.g.\cite{Ku} ) that the use of the popular Bethe parametrization, suggested as early as in 1958 \cite{Be} , for the scattering amplitude with account of Coulomb exchanges
\begin{equation}
T_{C+N} (s;t) = T_{C} e^{i\alpha\varphi(s,t)}+ T_{N} (s;t)
\end{equation}
where
\begin{center}
$T_{C} = 8\pi s \alpha F^{2}(t)/t  $
\end{center}
(with $F(t)$ the electric form factor)
leads to the phase $ \Phi_{N}(s;t) $ of the strong interaction amplitude\footnote{$ T_{N}(s;t)\equiv T_{N} (s+i0, t), s \geq s_{el} $ .}  $ T_{N}(s;t)=\mid T_{N}(s;t)\mid \exp (i\Phi_{N}(s;t))$ which is independent on $ t $.

In phenomenological practice  this independence makes many things much easier and is often assumed (see e.g.\cite{We} ) and used to extract the ratio
\begin{center}
$ \rho (s)\doteq Re T_{N}(s;0) / Im T_{N} (s;0)$
\end{center}
from the experimental data.

\section{Analyticity}
In many cases, however, such a $ t $-independence is supposed to hold "for small values of $ t $" only.
Below we consider to which extent this reservation allows " freedom of choice" at arbitrary transfers.

To formalize the assumption in question, let us consider the quantity
  \begin{center}
 $ \rho (s,t) = Re T_{N}(s;t) / Im T_{N} (s;t) = \cot \Phi_{N}(s,t) $
 \end{center}
where $  T_{N}(s;t) $ is taken for simplicity the elastic scattering amplitude of identical strongly interacting scalars with $ s_{el}= 4m^{2} $.

Both $ Re T_{N}(s;t) $ and $ Im T_{N} (s;t) $ which are real and imaginary parts of the boundary value $ T_{N}(s+i0,t) $ of an analytic function $ T_{N}(s,t) $ at physical $ s $ are analytic in $ t $ in the Martin ellipse \cite{Ma}
\[\mid\ 4p^{2}+t\mid +\mid t\mid < 4p^{2}(1 + 4m^{2}/p^{2})^{1/2}\]
with focii at $ t=0 $ and $ t=-4p^{2} = s+4m^{2} $ and semi-major axis $ 2p^{2}(1+ 4m^{2}/p^{2})^{1/2} $.

Assume that $ \rho (s,t) $  ( and thus $ \Phi_{N}(s;t) $) is  $ t $-independent in the  interval $ (-\tau \leq t \leq 0] $
where $ \tau > 0 $ is arbitrarily small while $ s $ takes physical values in some interval on $ [4m^{2}, +\infty) $.
The following theorem then holds.

\textbf{Theorem I.}

\textit{ Let  $ \rho (s;t)$ is independent on $ t $ at $ t\in (-\tau, 0] $ and at  $ s \in [s_{1},s_{2}] \subset [4m^{2}, +\infty)  $. Then  $  \rho (s;t) $ is independent on $ t $ inside the Martin ellipse and, in particular, in the whole physical region
$ -4p^{2} \leq t \leq 0 $ with $ s \in \forall [s_{1},s_{2}] \subset [4m^{2}, +\infty)$.}

Proof.
The assumed $ t  $ - independence of $ \rho (s;t)\equiv \rho (s) $  at $ -\tau < t \leq 0 $ and physical $ s \in \forall [s_{1},s_{2}] \subset [4m^{2}, +\infty) $ implies the following relationship
\begin{equation}
Re T_{N}(s;t)\mid_{[-\tau \leq t \leq 0]}  =  \rho (s)\cdot Im T_{N} (s;t)\mid_{[-\tau \leq t \leq 0]}.
\end{equation}
The function
\[f(t)\doteq Re T_{N}(s;t) - \rho (s)\cdot Im T_{N} (s;t)\]
at fixed $ s $ as defined in the hypothesis of the theorem

1) is analytic inside the Martin ellipse and

2) vanishes on the interval $ (-\tau \leq t \leq 0] $ lying inside the analyticity

domain.

From the uniqueness theorem for analytic functions \cite{Pri} we conclude that
\[f(t)\equiv 0,\]
inside the Martin ellipse and $ s \in \forall [s_{1},s_{2}] \subset [4m^{2}, +\infty) $. This holds, in particular, at all physical values of $ t $ which are contained in this ellipse. So, $ \rho (s;t)\equiv \rho (s)  $, and hence the phase $ \Phi_{N}(s;t)$, does not depend on $ t $ at all physical $ t $:
\[\Phi_{N}(s;t) = \arctan (1/\rho(s))\] Q.E.D.

 Note that for the proof we could use even the narrower analyticity domain, the Lehmann ellipse.

\section{Unitarity}

The unitarity condition enables us to prove the following

\textbf{Theorem II}

\textit{If the phase $\Phi_{N}(s;t) $ of the physical scattering amplitude
\[T_{N}(s;t)=\mid T_{N}(s;t) \mid\exp (i\Phi_{N}(s;t))\] is independent on $ t $ at $ t\in (-\tau, 0] $, where a positive number $ \tau  $ is arbitrarily small and the energy region contains the values $s \in [s_{1},s_{2}]\subset [s_{inel}-\Delta s,s_{inel})$,where $ s_{inel}$ denotes the lowest inelastic threshold and $\Delta s \ll s_{inel} $, then $ T_{N}(s,t) \equiv 0 $ in all physical region of $ t $ and $ s $.}

Proof.
From the Theorem I  proved in the previous Section we deal with the amplitude
$ T_{N}(s,t) $ the phase of which is independent on $ t $,which leads to the following
relationship

\begin{equation}
 T_{N}(s;t)= T_{N} (s+i0,t)\vert _{s\in [s_{1},s_{2}]}= (1+i\omega(s))ReT_{N}(s;t).
 \end{equation}
Let us take use of the unitarity condition
\begin{equation}
a_{l}(s) = a_{l}(s)^{2} + r_{l}(s)^{2} + \eta_{l}(s)
\end{equation}
 for partial waves where
 \begin{equation}
 a_{l}(s) = \frac{1}{32\pi} \sqrt{1-\frac{4m^{2}}{s}} \int d\cos \theta P_{l}(\cos \theta)Im T_{N}(s,\cos \theta) )
 \end{equation}
 \begin{equation}
 r_{l}(s)= \frac{1}{32\pi} \sqrt{1-\frac{4m^{2}}{s}} \int d\cos \theta P_{l}(\cos \theta)Re T_{N}(s,\cos \theta)
 \end{equation}
 and $\eta_{l}(s)$ is the contribution of all inelastic channels,
 with
 $ t= -2p^{2}(1- cos \theta ) $.

 It follows from Eq.(3) that at $ s \in [s_{1},s_{2}]\subset [s_{inel}-\Delta s,s_{inel}) $  (when $ \eta_{l}(s) =0 $)
 we get the following "solution" of the unitarity equation ( $ a_{l}(s) = a_{l}^{2}(s) + r_{l}^{2}(s))$ :

  \begin{equation}
 r_{l}= \omega (s)/(1+\omega^{2}(s))= \rho(s)/(1+\rho^{2}(s)).
 \end{equation}
 From this we already see an evident disadvantage of this "solution": the partial wave amplitude does not decrease with the growth of angular momentum $ l $ as follows from the analyticity in $ t $ and polynomial boundedness in $ s $ \footnote{I am grateful to A. Samokhin for this observation.}  \cite{Ma}.
 The very form of the "solution" (7) implies that
 \begin{equation}
 T_{N}(s;t)= \frac{\omega (s)(1+i\omega (s))}{(1+\omega^{2}(s))}\frac{16\pi\sqrt{s}}{\sqrt{s-4m^{2}}}\sum_{l\geq 0}(2l+1)P_{l}(\cos \theta)=
 \end{equation}
 \[= \frac{\omega (s)(1+i\omega (s))}{(1+\omega^{2}(s))}\frac{32\pi\sqrt{s}}{\sqrt{s-4m^{2}}}\delta(1-\cos \theta).\]
 Thus we see that at $ \theta \neq 0 $ and $s \in [s_{1},s_{2}]\subset [s_{inel}-\Delta s,s_{inel}) $  the scattering amplitude vanishes.

 It follows, in particular, that on the interval $s \in [s_{1},s_{2}]\subset [s_{inel}-\Delta s,s_{inel}) $  the discontinuity of the scattering amplitude is zero. Now, even with a modest assumption that the scattering amplitude is analytic in $ s $ in the vicinity of the s-channel physical region
 $ [4m^{2},+ \infty) $ we conclude, again on the basis of the uniqueness property of analytical functions, that $ T_{N}(s,t) $ being vanishing on the segment $ [s_{1},s_{2}] $ should vanish in the whole analyticity domain, including the physical region.Q.E.D.
 \section{Conclusion and Discussion}
So, we see that - at first glance, not very restrictive - the condition of independence of the phase of the elastic scattering amplitude from the momentum transfer $ t $ in an arbitrarily small real neighborhood of the point $ t = 0 $ leads to a very significant result: the independence of the phase from momentum transfers \ textit {in the entire physical area} of the latter.

Accounting for the unitarity condition leads to the fact that in the region of purely elastic scattering (below inelastic thresholds) the phase of the amplitude along the momentum transfer \ textit {cannot be independent of $ t $} even on an arbitrarily small interval of transmitted momenta - otherwise the scattering amplitude turns out to be identically equal to zero.

The conclusion about the constancy of the phase as a function of the transmitted momenta given above does not use the fact that the real and imaginary parts of the scattering amplitude are not completely independent, because they are connected by dispersion relations due to analyticity  in energy . There are few strictly proved dispersion relations: they relate to pion-nucleon and kaon-nucleon scattering and for very limited momentum transfer values.

If you do not make too stringent requirements for rigor, then "at the level of heuristic considerations" and using dispersion relations, one can get the following result:

\textit {under the conditions of Theorem I and for "sufficiently high energies" the scattering amplitude is factorized:}

\begin {equation}
T_{N}(s;t)\mid_{s\rightarrow \infty} \Longrightarrow F(s))\cdot G (t).
\end {equation}

Such factorization is unacceptable, if only because it sharply contradicts the known experimental facts.
Thus, if we take into account the above as a sufficient basis, then the phase of the scattering amplitude \textit {cannot be independent of momentum transfers even at their arbitrarily small intervals.}

From the practical point of view  $ t $ -independence of the scattering phase can be implemented as "a weak $ t $-dependence "(at least at small $ t $) which can be implemented in the smallness of $ \frac{\partial\Phi_{N}}{\partial t} (s;0) $. At the  moment the only thing we know about the rate of $ t $ -dependence of the phase at small $ t $ is the relationship \cite{Kun}
\begin{equation}
\frac{\partial\Phi_{N}}{\partial t} (s;0) = \frac{1}{4}tg(\Phi_{N}(s;0))[\langle b^{2}\rangle_{tot}- 2B(s)]
\end{equation}
where $ B(s) $ is the forward slope of $ d\sigma/dt $ (observed) and $ \langle b^{2}\rangle_{tot} $
is the total transverse extent of the interaction region (unobserved .
With $ t $-independent phase we would have
$ \langle b^{2}\rangle_{tot} = 2B(s) $ (widely assumed).

 How weak such a rate can be without contradiction to basic principles is an open question at the moment.

 In Eq.(10) one can see the relationship of the scattering phase and its $ t $-dependence with the spatial characteristics, viz., \textit{the transverse range} $ \langle b^{2}\rangle_{tot} $ of the interaction region.

 There is a more direct relation if to consider also the \textit{longitudinal range.}
  The average longitudinal size of the interaction region in s.c.m.,  $ \langle \Delta x^{\ast}_{\parallel}\rangle $, is related to the phase of the scattering amplitude as follows \cite {Ptr}:
  \[\langle \Delta x^{\ast}_{\parallel}\rangle = 2\sqrt{s}\langle \frac{\partial \Phi_{N}(s,t)}{\partial t}\rangle = \frac{2\sqrt{s}}{\sigma_{el}}\int dt\frac{d\sigma_{el}}{dt}\frac{\partial \Phi_{N}(s,t)}{\partial t}. \]

  If the phase is independent of $ t $, then

  \[\langle \Delta x ^ {\ast}_{\parallel}\rangle = 0.\]

This, however, contradicts the long-known phenomenon of an increase in the longitudinal range of the interaction region with an increase of the collision energy \cite {Gri}.

 \section{Acknowledgements}
 I am indebted to Anatolii Samokhin for very fruitful discussions which helped to improve the formulation and  presentation of the results of this paper.

 This work is supported by the RFBR Grant 17-02-00120.