text:
\section{Introduction}
There are several widely known problems in the theoretical modeling of
\ion{Fe}{17} X-ray emission lines, making it difficult to use them as reliable
diagnostic tools for electron density, temperature, plasma opacity, and iron
abundances in X-ray astronomy. The first problem is related to the intensity
ratio of the $2p_{3/2}$-$3d_{5/2}$ transition at 15.26~{\AA}, generally
referred to as 3D, to the $2p_{1/2}$-$3d_{3/2}$ transition, or 3C. Most
theoretical predictions of this ratio are smaller than astrophysical
observations \citep{xu02, brinkman00, canizares00, mewe01, behar01} and
laboratory measurements \citep{brown98, brown01a, beiersdorfer01}. This
discrepancy has been used in the past to infer plasma opacity as the stronger
3C line is more susceptible to resonant scattering \citep{waljeski94,
  saba99}. However, this diagnostics depends critically on knowing the
optically thin limit of the 3D/3C ratio. The second problem concerns the
intensity ratio of the $2p$-$3s$ (hereafter $3s$) transitions, including 3F
(16.78~{\AA}), 3G
(17.05~{\AA}), and M2 (17.10~{\AA}), to the $2p$-$3d$ (hereafter $3d$)
transitions, including
3C, 3D, and a much weaker 3E (15.45~{\AA}). The astrophysical observations and
laboratory measurements are also significantly larger than most theoretical
values \citep{phillips99, beiersdorfer02, beiersdorfer04} for this
ratio. Finally, the
theoretical predictions of the intensity ratio of the two $2p$-$3s$
transitions, M2 to 3G, have also been shown to be significantly smaller than
astrophysical observations of low density plasmas \citep{ness05}. The M2/3G
ratio is density sensitive, and have been used to infer electron densities in
magnetic cataclysmic variables \citep{mauche01}. However, the inability of
reproducing the low density limit of this ratio in theory introduces
additional uncertainties in the density determination.

There have been claims that these modeling problems have been solved
with large scale close-coupling calculations of the electron collisional
excitation data of \citep{gchen02, gchen03}. The new theoretical values
of certain line ratios have been shown to agree with laboratory measurements
and a few astrophysical observations to within 10\%. However, the cross section
measurements of the 3C and 3D lines \citep{brown06} indicate that such
apparent agreements are rather fortuitous. The more recent work of
\citet{gchen07} and \citet{gchen08a} revised their earlier work by including
pseudo states and the $3d^2$ pair excitation configuration in the target
wavefunction expansion and obtained 15-20\% reductions in the 3C cross
sections compared with \citet{gchen03}, while those of 3D changed
little. \citet{gchen08b} further suggested that the theoretical radiative
recombination (RR) cross sections into the M-shell of \ion{Fe}{16} used for
normalization in \citet{brown06} are underestimated by 25\%, leading to
smaller measured 3C and 3D cross sections.
In this paper, we provide a
thorough assessment of the accuracies of existing atomic calculations on
\ion{Fe}{17} by comparing them with a large sample of \textit{Chandra}
observations of stellar coronae obtained with the high energy transmission
grating (HETG) spectrometer. We conclude that most of the existing predictions
do not offer satisfactory explanation of the observed line ratios. The recent
calculations of \citet{gchen07} and \citet{gchen08a} provide the best
agreement with the laboratory 3C and 3D cross sections. However, a detailed
comparison of these data with the \textit{Chandra} observations cannot be
done, as the full results of the calculations including all levels necessary
for collisional radiative modeling are not available. The major drawback of
\citet{gchen07} and \citet{gchen08a} is that it is not clear that the
accounting the target correlation effects has fully converged. In these
calculations, the largest improvement comes with the inclusion of $3d^2$ pair
excitation configuration. However, it is well know that such pair correlations are
notoriously slowly convergent. In this paper, we propose a
many-body perturbation correction procedure for the excitation cross section
calculations, taking all singly and doubly excited correlations into account,
but treating their effects on excitation cross sections approximately. We show
that the results agree well with \citet{gchen07} and
\citet{gchen08b} for the 3C and 3D cross sections, and all \ion{Fe}{17} line
ratios are brought into agreement with the \textit{Chandra} observations.

\section{Review of existing theoretical calculations}
Theoretical study of \ion{Fe}{17} X-ray lines and their applications in
astronomy have a long history. \citet{bely67} obtained the excitation cross
sections of $2p^5 3l$ configurations using the Coulomb-Born
approximation. \citet{loulergue73, loulergue75} developed collisional
radiative models of \ion{Fe}{17} and \ion{Ni}{19} with DW collision strengths,
and compared the resulting line intensities with solar
observations. \citet{smith85}, \citet{raymond86}, and \citet{goldstein89}
pointed out the
importance of resonant excitation and its role in determining the
temperature dependence of \ion{Fe}{17} line ratios. \citet{bhatia92} presented
a complete set of \ion{Fe}{17} line intensities obtained with the DW
approximation. \citet{cornille94} studied the \ion{Fe}{16} satellite transitions
that blend with \ion{Fe}{17} lines. \citet{mohan97} investigated the effects
of relativistic term coupling effects of the \ion{Fe}{17} collision strengths
using the Breit-Pauli $R$-matrix approximation.

\cite{beiersdorfer94} built a flat crystal spectrometer on the electron beam
ion trap (EBIT) of the Lawrence Livermore National Laboratory (LLNL), and
conducted the first laboratory measurement of \ion{Fe}{17} X-ray line
emission. \citet{brown98} carried out a systematic investigation of the
\ion{Fe}{17} lines with the same spectrometer and EBIT facility, and showed
that most previous theories significantly
underestimate the 3D/3C ratio. \citet{beiersdorfer02,beiersdorfer04}
showed that the $3s$/$3d$ ratio is also underestimated in theory by large
factors as compared with both EBIT and Tokamak measurements. Independent
laboratory measurements by \citet{laming00} with the National Institute of
Technology and Standards (NIST) EBIT obtained 3D/3C ratios that are consistent with
those from the LLNL EBIT, but reported $3s$/$3d$ ratios very different from
those measured at the LLNL EBIT and concluded that the $3s$/$3d$ ratios are
consistent with modern theoretical calculations. These laboratory work
prompted new theoretical studies to resolve these
discrepancies. \citet{bhatia03} improved the earlier work of \citet{bhatia92}
by including more configurations and computing collision strengths in a wider
energy range. \citet{doron02} investigated the effects of resonant
excitation, recombination of \ion{Fe}{18} and inner-shell ionization of
\ion{Fe}{16} as line formation processes using the Hebrew University Lawrence
Livermore Atomic Code (HULLAC), and concluded that such a multi-ion
approach increases the $3s$/$3d$ ratios, bringing them in better agreements
with laboratory measurements and astrophysical observations. \citet{gu03}
reached similar conclusions and also showed that recombination processes are
important for other iron ions. \citet{gchen02}, \citet{gchen03} and
\citet{gchen05} carried out
a large scale Breit-Pauli $R$-matrix calculation for electron collisional
excitation of \ion{Fe}{17}, and claimed that their results resolve the
discrepancies between theories, laboratory measurements, and astrophysical
observations. An independent close-coupling calculation with
the Dirac atomic $R$-matrix code \citep{loch06} confirms some of the findings
in \citet{gchen03}, but found
remaining discrepancies between theories and measurements. Most recently,
\citet{gchen07} and \citet{gchen08a} revised their earlier work by including
pseudo states and the $3d^2$ pair excitation configuration in the target
wavefunciton expansion, and obtained significantly lower 3C cross sections than
\citet{gchen03}.

In the present paper, we concentrate on these newer theoretical work. A
comparison of the \ion{Fe}{17} line ratios from these calculations are shown
in Figure \ref{fig:tr}, along with the predictions from the widely used
astrophysical
plasma emission code (APEC) of \citet{smith01}, which is based on DW collisional
data without resonances. All line ratios involved are calculated at the low
electron density limit of $\le 10^{10}$~cm$^{-3}$.
It is immediately clear from this comparison that the
predictions from APEC and \citet{bhatia03} are very different from those of
\citet{doron02}, \citet{gu03}, and \citet{loch06}. These differences can be
attributed to the lack of resonant excitation contributions in APEC and
\citet{bhatia03}. On the other hand, the results of \citet{doron02} and
\citet{gu03} are relatively close to each other both in the temperature
dependence of the line ratios and their magnitudes, which is not surprising
since the two calculations
employ essentially the same physics and approximations. The relatively large
difference in the 3E/3C ratio between \citet{doron02} and \citet{gu03} can be
attributed to the incomplete treatment of resonant excitation in
\citet{doron02}. The results of \citet{gu03} and \citet{loch06} are also in
very good agreement, although \citet{loch06} generally produce a steeper
temperature dependence of the line ratios, especially for the $3s/3d$ ratios. This
is due to the fact that \citet{loch06} do not include recombination
contributions to the line intensities. Recombination preferentially enhances
the $3s$ transitions and has larger contributions at higher temperatures due
to the higher fractional abundance of \ion{Fe}{18} relative to
\ion{Fe}{17}. One surprising fact in this comparison is the relatively large
differences between the 3F/3C ratios of \citet{gchen03}, 3s/3d ratios of
\citet{gchen05}, and the corresponding ratios of \citet{loch06}. In fact, the
ratios of \citet{gchen03} and \citet{gchen05}, which include resonant
excitation, are similar to the DW values without resonant
excitation. \citet{loch06} showed that the differences in the 3C and 3D cross
sections in the two $R$-matrix calculations are small. Therefore the large
differences we note in this comparison are due to the differences in the $3s$
intensities. The updated 3C and 3D cross sections of \citet{gchen07} give
larger 3D/3C ratios than all other calculations. Because the full results of
the new calculations of \citet{gchen07} are not available, we scaled the 3F/3C
and $3s$/$3d$ ratios of \citet{gchen03} and \citet{gchen05} to reflect the
updated 3C and 3D cross sections, but keeping $3s$ intensities
unchanged. These scaled ratios for 3F/3C become close to those of \citet{gu03}
and \citet{loch06}, but those for $3s$/$3d$ are still significantly smaller.

\section{\textit{Chandra} stellar observations of \ion{Fe}{17} spectra}
\label{sec:hetg}
In order to assess the accuracies of various calculations, we compare them
with an extensive set of stellar coronal observations with the HETG spectrometer
aboard \textit{Chandra} X-ray observatory. Since its launch, \textit{Chandra}
has observed a large sample of stellar coronal sources with the high resolution
HETG spectrometer. Table \ref{tab:src} lists the sources in our selection. The data
were retrieved from the \textit{Chandra} archive, and reprocessed with CIAO
version 3.4 and CALDB version 3.4. Some sources are observed multiple times,
and we combine the individual observations to improve the statistics. There
are 11 sources that do not provide decent counting statistics
individually, and we have also combined them to produce a single composite
observation referred to as ``Stack'' in Table \ref{tab:src}. Of the 24
targets (after grouping of multiple observations and weak sources) in our
sample, Capella provides the best statistical quality because it is an
calibration target, and observed many times.

We follow the differential emission measure (DEM) analysis method of
\citet{gu06} to characterize the coronal property of these sources. Using the
APEC database modified by the line emissivities of \citet{gu03} and wavelengths
of \citet{gu05}, we reconstruct the DEM distribution and elemental abundances
of the coronal plasma by jointly fitting the $\pm$1 orders of medium and high
energy grating (MEG and HEG) spectra. The resulting DEM and comparison of
theoretical and observed spectra in the 12--18~{\AA} region is shown in
Figures \ref{fig:dem} and \ref{fig:sp} for the composite observation of
Capella. The DEM is peaked near $10^{6.9}$~K, which is similar to the result of
\citet{gu06} using a single observation. To characterize the temperature
of the \ion{Fe}{17} emission regions, it is more appropriate to examine the
DEM weighted by the \ion{Fe}{17} emissivities, which is also shown in
Figure~\ref{fig:dem}. It indicates that the \ion{Fe}{17} emissivity weighted
DEM has a narrower profile. In fact, the characteristic
temperatures of the \ion{Fe}{17} emission regions are always in the range of
$10^{6.6}$--$10^{6.9}$~K for all sources, even though some of them have
significant high temperature tails beyond $10^7$~K. This is due to the
fact that \ion{Fe}{17} fractional abundance peaks near $10^{6.6}$~K in
collisional ionization equilibrium.

The spectral model shown in Figure~\ref{fig:sp} already provides a reasonable
fit to the observed spectrum. However, close inspection of the \ion{Fe}{17}
lines reveal significant discrepancies between them. Using the reconstructed
spectrum as a basis, we obtain a better fit to the \ion{Fe}{17} emission by
varying 3C, 3D, 3E, 3G, 3F, and M2 line intensities. \citet{brown01} and
\citet{brickhouse06} demonstrated that the 3D line blends perfectly with an
\ion{Fe}{16} satellite transition, and another nearby \ion{Fe}{16} transition
at 15.21~{\AA} can be used to remove the \ion{Fe}{16} contribution to the 3D
line. Many of the sources in our sample indicate the presence of the
15.21~{\AA} line, providing a measure of contamination to the 3D line. The
15.21~{\AA} line intensity is extracted from the measured spectra and
multiplied by a factor of 0.7, which is then subtracted from the apparent 3D
intensity. The branching ratio of 0.7 used here is determined with the Flexible
Atomic Code developed by the author, and is between the values 0.83 used by
\citet{brickhouse06} and 0.51 used by \citet{brown01}. However, because the
\ion{Fe}{16} contamination of the 3D line is relatively small, the correction
decreases the measured 3D/3C ratio only by $\sim$~10\%. The differences in the
branching ratios leads to 2--3~\% differences in the 3D/3C ratio, which are
smaller than the statistical errors for most sources. The derived line ratios
for the 24 sources are listed in Table~\ref{tab:ratio}. The \ion{Fe}{17}
emissivity weighted average temperature, $T_{\mbox{em}}$, of the coronae are
also shown. \citet{ness03} reported the ratios involving the 3C, 3D and 3F
lines for a large sample of coronal sources observed by \textit{Chandra} and
\textit{XMM-Newton}. The results obtained here are generally consistent with
those of \citet{ness03} for the set of common sources.

In comparing the measured and theoretical line ratios, the calculations should
ideally be performed for the appropriate DEM distributions. In practice, we
find that the ratios calculated with the DEM is very close to the ones
calculated at a single temperature of $T_{\mbox{em}}$. This is demonstrated in
Figure~\ref{fig:wr} for the 8 line ratios we investigate. In the rest of the
paper, we therefore use the isothermal line ratios calculated at
$T_{\mbox{em}}$ for comparison.

Figure~\ref{fig:mr} shows the comparison of measured and theoretical line
ratios in detail. Because the predictions of \citet{doron02} are generally
similar to those of \citet{gu03}, and those of \citet{bhatia03} are similar to
those of APEC, we omit those from Figure~\ref{fig:mr} to reduce
clutter. It is clear that none of the theories can explain all line
ratios satisfactorily. All theoretical ratios are below the measured ones,
except for 3D/3C, where the results of \citet{gchen07} agree with the
measurements, 3E/3C and 3F/3C, where the values of \citet{loch06},
\citet{gu03}, and the results of \citet{gchen03} scaled for the new 3C cross
sections of \citet{gchen07} agree with the measurements, and M2/3G, where the
values of APEC agree with the measurements. However, it should be pointed out
that the measured
3E/3C ratio have large statistical uncertainties. The APEC
ratios of M2/3C and 3G/3C are very different from the measurements, and the
agreement for M2/3G is therefore fortuitous. Despite the claim of excellent
agreements between their theoretical line ratios, laboratory measurements and
astrophysical observations in \citet{gchen02}, \citet{gchen03}, and
\citet{gchen05}, we find their theoretical values are significantly smaller
than the \textit{Chandra} measurements for the 3F/3C and $3s$/$3d$ ratios,
even more so than the calculations of \citet{loch06} and \citet{gu03}.
After scaling for the new lower 3C cross sections of \citet{gchen07}, the
$3s$/$3d$ ratios are still smaller than the \textit{Chandra} observations. Since
the data from \citet{gchen03} and subsequent work are not publicly available,
we are not able to calculate other line ratios for thermal plasmas for a more
detailed comparison.

\section{Distorted-wave with many-body perturbation correction}
The fact that the calculations of \citet{gu03} and \citet{loch06} agree with
each other very well except for the differences in the recombination
contributions indicates that the two theoretical methods, DW in \citet{gu03}
and $R$-matrix in \citet{loch06} are of comparable quality for
\ion{Fe}{17}. The differences between measured and calculated line ratios are
not due to deficiencies in the scattering approximations. The cross section
measurements for 3C and 3D lines of \citet{brown06} demonstrated that
theories tend to overestimate the 3C cross sections more than 3D. The general
trend in the comparison of theories and \textit{Chandra} measurements indeed
support this picture. If the theoretical 3C cross sections were to decrease
by $\sim$~20\%, the predictions of \citet{gu03} and \citet{loch06} would agree
with the observations much better. 3C is a dipole allowed transition, its
collisional excitation cross section is nearly proportional to the oscillator
strength, which is determined solely by atomic structure
calculations. Therefore we suspect that the root problem in the previous
theories of \ion{Fe}{17} line intensities is the incomplete treatment of electron
correlation effects in the target ion. This conclusion is supported by the
comparison of
various oscillator strengths calculations for 3C and 3D listed in Table~2 of
\citet{gchen03}. Of these calculations, the many-body perturbation theory
(MBPT) treatment of \citet{safronova01} is likely to be the most accurate in
describing electron correlation effects in \ion{Fe}{17}, which also yields
the smallest oscillator strength for the 3C transition. The recent
calculations of \citet{gchen07} and \citet{gchen08a} also showed that by
improving the target wavefunction expansion, the 3C cross sections are reduced
by 15-20\%. However, it is not clear that the target description in
\citet{gchen07} and \citet{gchen08a} has fully converged, given that the
largest change comes with the inclusion of the $3d^2$ pair excited
configuration, and such pair correlations are know to be notoriously slowly
convergent.

We have developed an independent MBPT method within the framework of FAC for
general open-shell ions, and have used it in calculating the wavelengths of
iron and nickel L-shell X-ray transitions \citep{gu05}. We recently extended
the method to also calculate radiative transition rates and oscillator
strengths. The 3C and 3D weighted oscillator strengths calculated in our MBPT
method are 2.17 and 0.62, respectively. This compares to 2.50 and 0.62
calculated by the DW approximation. These independent MBPT calculations
therefore suggest that the 3C cross sections are likely to be overestimated by
as much as $\sim$~10--20 \% in previous non-MBPT calculations. \citet{gchen07}
gave the weighted oscillator strengths of 3C and 3D to be 2.25 and 0.635,
respectively, which agree with our MBPT results very well. This indicates the
near convergence of their accounting of \ion{Fe}{17} target correlation effects.

However, the extension of the MBPT method to treat collisional excitation processes
is rather difficult beyond the plane-wave Born approximation. Here we propose
an approximate procedure for including MBPT corrections in the calculation of
DW excitation cross sections. \citet{barshalom88} showed that the DW collision
strengths can be factorized to contain angular and radial integrals as follows
\begin{equation}
\label{eq_cs}
\Omega_{01} = 2\sum_{k}\sum_{\alpha_0\alpha_1\atop\beta_0\beta_1}
Q^k(\alpha_0\alpha_1;\beta_0\beta_1)
<\psi_0||Z^k(\alpha_0,\alpha_1)||\psi_1>
<\psi_0||Z^k(\beta_0,\beta_1)||\psi_1>,
\end{equation}
where $\alpha$, $\beta$ denote bound orbitals making the transition, $Z^k$ are
angular factors, and $Q^k$ are radial integrals involving partial-wave sum
\begin{equation}
\label{eq_Qk}
Q^k(\alpha_0\alpha_1;\beta_0\beta_1) = \sum_{\kappa_0\kappa_1}[k]^{-1}
P^k(\kappa_0\kappa_1;\alpha_0\alpha_1)P^k(\kappa_0\kappa_1;\beta_0\beta_1),
\end{equation}
and
\begin{equation}
\label{eq_Pk}
P^k(\kappa_0\kappa_1;\alpha_0\alpha_1)=
X^k(\alpha_0\kappa_0;\alpha_1\kappa_1)+\sum_{t}
(-1)^{k+t}[k]\sixj{j_{\alpha_0}}{j_1}{t}{j_0}{j_{\alpha_1}}{k}
X^t(\alpha_0\kappa_0;\kappa_1\alpha_1),
\end{equation}
where $\kappa$ represents a continuum orbital and $X^k$ are the usual Slater
integrals. The first term in $P^k$ is the direct contribution, and the second
term is the exchange contribution. In the Coulomb-Bethe approximation, the
direct term can be written as
\begin{equation}
\label{eq_PkApprox}
P^k_{CB}(\kappa_0\kappa_1;\alpha_0\alpha_1) =
M_k(\alpha_0\alpha_1)R_k(\kappa_0\kappa_1),
\end{equation}
where
\begin{eqnarray}
M_k(\alpha_0\alpha_1) &=& <\alpha_0||C^k||\alpha_1>
\int (P_{\alpha_0}P_{\alpha_1}+Q_{\alpha_0}Q_{\alpha_1})r^kd r \nonumber\\*
R_k(\kappa_0\kappa_1) &=&
\int (P_{\kappa_0}P_{\kappa_1}+Q_{\kappa_0}Q_{\kappa_1})
\frac{1}{r^{k+1}}d r,
\end{eqnarray}
where $C^k$ is the normalized spherical harmonic, $P_{\alpha}$ and
$Q_{\alpha}$ are the large and small components of the Dirac radial
wavefunctions of orbital $\alpha$. Note that $M_k$ depends only on the target
orbitals and is proportional to the electric multipole integral of rank
$k$. This integral is the same as the one used in radiative transition rate
calculations. Therefore, the ratio of the MBPT corrected to the conventional
DW direct excitation contributions is the same as the ratio of the MBPT
corrected to uncorrected oscillator strenghts. We calculate such ratios from the
radiative transition rates with and without MBPT effects, and use them as
scaling factors to correct the direct excitation contributions to
$P^k$, and obtain the MBPT corrected collision strengths in the DW
approximation.

Following the procedure outlined above, and the three-ion model of
\citet{gu03}, we recalculate the \ion{Fe}{17} line ratios and compare them
with the \textit{Chandra} observations in Figure~\ref{fig:nr}. The results of
\citet{loch06} are shown again for easy comparison. It is obvious that our
MBPT corrections bring the calculated line ratios into much better agreement
with \textit{Chandra} observations. The M2/3C and M2/3G ratios seem to be still
slightly below the observed ones, but show significant improvements over the
existing theories.

Finally, we compare the MBPT corrected effective formation cross sections of
the 3C and 3D transitions with laboratory measurements of
\citet{brown06} in Figure~\ref{fig:cross}. Our effective cross sections
include contributions of resonant excitation and radiative cascades from
higher levels. The cross sections of \citet{gu03}, \citet{gchen03},
\citet{loch06}, and \citet{gchen07} are also shown for comparison. The
$R$-matrix calculations
include resonances, but do not taken into account radiative cascades. Clearly,
the MBPT corrections leave the 3D cross sections of \citet{gu03} almost
unchanged, but decrease the 3C cross sections by about 18\%. The cross
sections of \citet{gchen03} and \citet{loch06} are all higher than the present
MBPT corrected values for 3C. The cross sections of \citet{gchen07} are only
slightly lower than the present calculations by $\sim$5\%, which likely
represents the differences between different scattering
approximations. However, the present MBPT corrected cross sections for both 3C
and 3D are still about 15\% higher than the laboratory measurements overall,
while the
3D/3C ratios agree with each other very well. The energy dependence of the
measured cross sections shown in Figure~\ref{fig:cross} were obtained by
sweeping the electron beam energies and recording the energy dependence of the
3C and 3D line intensities. They are normalized to the RR emission into the
M-shell of \ion{Fe}{16} at a single energy point of 964~eV. There is
significant statistical fluctuation in the measured data, but the overall
discrepancies between the measured values and the present calculations are
less than 15\%, especially at the near threshold region below 1200~eV. The
discrepancies between the measured values and the results of \citet{gchen07}
are even smaller, at 10\% level. If the suggestion that the RR cross sections
used in \citet{brown06} are underestimated by 25\% \citep{gchen08b} is
correct, the new theoretical cross sections of 3C and 3D of the present paper
and those of \citet{gchen07} and \citet{gchen08a} would have become
smaller than the renormalized measurements.

Since our new \ion{Fe}{17} line ratios agree with \textit{Chandra}
observations, the lower experimental 3C and 3D cross sections would lead to
lower cross sections for all other lines as well, and to inaccurate iron abundance
determinations if these lines are used as main constraints. The cross sections
of \citet{brown06} were determined by normalizing the collisional excitation
lines to the RR emission onto the M-shell of
\ion{Fe}{16}. The overall normalization uncertainties were estimated to be on
the order of 10\%.  The differences between our new calculations and the
measurements are therefore not significantly larger than the experimental
uncertainties, especially if one takes into account that the more elaboate
R-Matrix scattering method seems to further lower the cross sections by
$\sim$5\%.

\section{Density diagnostics with the M2 line}
Due to the relatively slow radiative decay of the M2 transition, the M2 line
is collisionally quenched at electron densities above $10^{13}$~cm$^{-3}$. The
M2/3G ratio has
been used to infer electron densities in magnetic cataclysmic variables
\citep{mauche01} and stellar coronal sources \citep{ness05}. However,
\S\ref{sec:hetg} shows that the M2/3G ratios of all coronal sources with
densities known to be below $10^{12}$~cm$^{-3}$ are significantly larger than
previous theoretical predictions, which is identified as a major problem
in applying this diagnostic in \citet{ness05}. We note that \citet{ness05}
seem to indicate that the low density limit of M2/3G calculated with
\citet{gu03} data agree with the coronal observations. We have since confirmed
that it is due to some misinterpretation of the \citet{gu03} data in
calculating the M2/3G ratio, and the \citet{gu03} ratio of M2/3G is in fact
much lower than the observations. The present MBPT corrected M2/3G ratio, although
appears to be still slightly lower overall, is at least marginally consistent
with low density coronal observations.

In Figure~\ref{fig:emd}, we show the density dependence of the emissivities of
the six \ion{Fe}{17} lines at a temperature of $10^{6.7}$~K in comparison with
the predictions of \citet{loch06}. The publicly released data from
\citet{loch06} do not include forbidden transition rates. In order to derive
the density dependence of line emissivities, we augmented their dataset with
the forbidden transition rates from the present calculations. Two major
differences are seen between the present work and that of \citet{loch06},
namely, the smaller 3C (due to MBPT correction) and larger M2 (due to
recombination contributions) in the present calculation, which helps to
explain the observed coronal values. The density dependence of M2/3G ratios in
the present calculation and those of \citet{loch06} are shown in
Figure~\ref{fig:emr} for two electron temperatures, $10^{6.7}$ and
$10^{7.2}$~K. The differences between the present work and \citet{loch06} are
even more pronounced at higher temperatures due to the increasing importance
of recombination contributions to the M2 line intensity.

\section{conclusions}
We have conducted a systematic study of the \ion{Fe}{17} emission lines of
stellar coronal sources observed by the HETG spectrometer aboard
\textit{Chandra}. At least some of the line ratios involving the six principal
\ion{Fe}{17} lines are shown to disagree with most existing theories. The
calculations of \citet{doron02} and \citet{gu03}, which include resonant
excitation and recombination contributions in DW approximation provide as good
agreement with data as the more sophisticated close-coupling calculation of
 \citet{loch06} with the Dirac atomic $R$-matrix code. Similarly sophisticated,
Breit-Pauli $R$-matrix results of \citet{gchen02,gchen03,gchen05}, however, show
worse agreements with the \textit{Chandra} observations for the 3F/3C and
$3s$/$3d$ line ratios. The source of discrepancies in the two $R$-matrix
calculations are not clear. The most recent Breit-Pauli $R$-matrix results of
\citet{gchen07} gave the best agreement with laboratory measurements of 3C and
3D cross sections, but a systematic comparison with
\textit{Chandra} line ratios cannot be carried out, as the full results of
these calculations necessary for collisional radiative modeling are not
available. If we assume that the $3s$ line intensities of \citet{gchen07} are
the same as \citet{gchen03} and \citet{gchen05}, and only correct the ratios
for the updated 3C and 3D cross sections, the resulting 3D/3C and 3F/3C ratios
are in agreement with the \textit{Chandra} observations, but the $3s$/$3d$
ratios are still significantly lower than observed values.

Based on the MBPT calculations of oscillator strengths,
we suggest that the incomplete treatment of electron correlation effects of
the \ion{Fe}{17} target is the cause of overestimating the 3C cross
sections in most previous theories, as is also suggested by the recent
calculations of \citet{gchen07} and \citet{gchen08a}. We propose a simple,
MBPT based procedure to
incorporate such correlation effects in the calculation of electron
collisional excitation cross sections under DW approximation. The resulting
\ion{Fe}{17} line ratios are shown to agree with \textit{Chandra} observations
much better than previous predictions. The 3C and 3D cross sections of the
present work are only $\sim$5\% larger than the calculations of
\citet{gchen07}. Comparison of our MBPT corrected 3C
and 3D cross sections and the measurements of \citet{brown06} suggests
remaining discrepancies on the order of 15\%, although systematic
uncertainties of about 10\% may explain part of this difference, and the
simplied scattering approximation used here may be responsible for the
remaining discrepancies. We have shown that if the RR cross sections used for
normalization in \citet{brown06} are underestimated by 25\%, as suggested by
\citet{gchen08b}, the new theoretical cross sections of the present work and
those of \citet{gchen07} and \citet{gchen08a} would have become smaller than
the renormalized measurements.

We have presented the density dependence of the \ion{Fe}{17} emissivities and M2/3G
ratio, and compared them with previous calculations. The improved low density
limit of M2/3G ratio provides greater confidence in applying this density
diagnostics in astrophysical observations.

\acknowledgements
The authour would like to thank P. Beiersdorfer for extensive comments on the
original manuscript. Support for this work was provided by the National
Aeronautics and Space Administration through Chandra Award Number TM7-8001Z
issued by the Chandra
X-ray Observatory Center, which is operated by the Smithsonian Astrophysical
Observatory for and on behalf of the National Aeronautics Space Administration
under contract NAS8-03060. This work was performed under the auspices of the
U.S. Department of Energy by Lawrence Livermore National Laboratory under
Contract DE-AC52-07NA27344.


text:
\section{Introduction}
The central role the gauge potential plays in the theories of
fundamental interactions of nature always justifies new investigations
concerning its nature.
If we take into account gauge transformations it becomes quite apparent
that gauge potential is something special compared to any other particle
field.
It is not a field which transforms
itself like a tensor field. It is something more complicated than a
tensor field with more involved and deeper lying properties.

A gauge potential is carrier of a geometric structure in the sense of
the metric structure in the general relativity, but in a more general
sense.
Because of its complexities, its deepness and its importance, there
exist several formulations of it [1, 2]. In this paper, we would like to
present a new property of the gauge potential: the $TG$--equivariance of
a connection on a principal bundle $P(M,G)$ with $M$ the space-time
(basis manifold) and $G$ the stucture group. The $G$--equivariance
property of a connection is of course well known. Here we mean, as we
shall explain, the equivariance respectively to the bigger group, the
tangential group $TG$, of the structure group $G$. It is also important
to note that the $G$--equivariance property is shared by the usual tensor
fields, considered as vector-valued functions on a principal bundle
$P(M,G)$. But only the connection has in addition the $TG$--equivariance
property.

The motivation for this paper came from considerations within the
$G$--theory framework and applications of it in investigating the
structure of the space of reducible connections for Yang-Mills theories
[3], anomalies [4, 5] and recently within a non commutative geometrical
approach to the standard model [6]. The $G$--theory point of view
emphasizes particularly the action of a group and all what follows from
it on a  relevant space of physical objects.

Our starting point is as already mentioned a principal bundle $P(M,G)$
with a connection $\omega$ on it and the structure group $G$ acting by
definition freely on $P$ from the right. Going now to the tangential
space $TP$ and $TG$ of $P$ and $G$ respectively, it is important to
realize that $TG$ may be considered as a Lie group and that it acts
freely on $TP$. The multiplication law $TG \times TG \rightarrow TG$ and
the action $TP \times TG \rightarrow TP$, are to be defined below. This
leads to the fact that the tangential space $TP$ is also a principal
bundle with basis manifold $TM$ and structure group $TG$. We now
consider the space $TP/G$ as a fiber bundle (non-vector bundle) with
typical fibre the Lie algebra $\mbox{$\cal G$}$ of $G$. We show that $TP/G$ is
isomorphic to the associated fiber bundle of $TP : TP \times_{TG} \mbox{$\cal G$}$
where the action of $TG$ on $\mbox{$\cal G$}$ is deduced from the multiplication law
in $TG$. We give a new definition of a connection on $P$: a
connection on $P(M,G)$ is a linear section $\mbox{$\tilde{H}$}$ on the fiber bundle $TP
\times_{TG} \mbox{$\cal G$}$
\begin{eqnarray}\label{eqn1}
        \mbox{$\tilde{H}$} : TM &\rightarrow& TP \times_{TG} \mbox{$\cal G$} \quad.
\end{eqnarray}
It can be shown that $\omega$ (from the usual definition of a
connection) and $\mbox{$\tilde{H}$}$, as above, are equivalent. The section $\mbox{$\tilde{H}$}$ may
also be considered as a $TG$--equivariant and linear function $\mbox{$\bar{H}$}$ on
$TP$ with values on $\mbox{$\cal G$}$.
\begin{eqnarray}\label{eqn2}
        \mbox{$\bar{H}$} : TP &\rightarrow& \mbox{$\cal G$}\;,\nonumber\\
        X_p &\mapsto& \mbox{$\bar{H}$}(X_p)\quad\mbox{with}\quad
        \mbox{$\bar{H}$}(X_p \cdot \xi_g) = {\xi_g}^{-1} \cdot \mbox{$\bar{H}$}(X_p)\;,
\end{eqnarray}
and $X_p \in TP$, $\xi_g \in TG$. $X_p \cdot \xi_g$  represents the
action of $TG$ on $TP$ from the right and $\xi_g \cdot \eta$ with $\eta
\in \mbox{$\cal G$}$ the action of $TG$ on $\mbox{$\cal G$}$ (from the left). It follows
immediately that this property of $TG$--equivariance of $\mbox{$\bar{H}$}$ is also
adapted by the $\omega$. As we shall see, the following equation is
valid:
\begin{eqnarray}\label{eqn3}
        \omega(X_p) &=& -\mbox{$\bar{H}$}(X_p)\;.
\end{eqnarray}
The last topic we would like to discuss, are the gauge tranformations on
$\omega$ in the light of the above development. It is well known that a
gauge transformation $\Psi$ can be described by the equivariant function
$U$ on $P$ with values on $G$. So we have $\Psi(p) = p U(p)$ with
\begin{eqnarray*}
        U : P &\rightarrow& G
\end{eqnarray*}
and
\begin{eqnarray}\label{eqn4}
        U(p g) &=& g^{-1} U(p) g\;.
\end{eqnarray}
This makes the $G$--action point of view transparent and induces the
gauge transformation of the connection $\omega$ given by:
\begin{eqnarray}\label{eqn5}
        (\Psi^* \omega)(X_p) &=& \mbox{Ad}_{{U(p)}^{-1}}  \omega(X_p) +
        U^{-1} d U (X_p)\;.
\end{eqnarray}
This transformation may also be expressed by the equation
\begin{eqnarray}\label{eqn6}
        (\Psi^* \omega)(X_p) &=& -(TU X_p)^{-1} \cdot (-\omega(X_p))\;.
\end{eqnarray}
As we see, gauge transformations expressed pointwise are nothing else
but the $TG$--action on the values of connection. The non--linear term
in the transformation of a connection may now be explained by the
$TG$--action on $\mbox{$\cal G$}$. $TG$ is a group of affine type which acts non
linearly on $\mbox{$\cal G$}$.

In what follows we discuss in the next section the multiplication rule
in $TG$, the action of $TG$ on $TP$ and the space $TP$ as a principal
bundle over $TM$. In section three we present the new definition of a
connection and focus on its $TG$--equivariance property. In section four
we consider gauge transformations on a connection from the point of view
of the $TG$--action and we finally give a summary in section five.

In the physical literature we found no explicit treatment of the
$TG$--equi\-variance aspect of connections. The same is valid, to our
surprise, for the recent mathematical literature we examined. The only
place we found some, for us very inspiring, but short and compact
indications on $TG$--equivariance, is the very early paper [7]. Parts of
section two and three are implicitly or explicitly contained in [7] but
without connection to the gauge transformations which were the main
motivation for the present paper. For all these reasons we tried to be
quite detailed and explicit in our presentations.

\section{$TP$ as $TG$ principal bundle}
The action of the Lie--group $G$ in the principal bundle $P(M,G)$ is a
part of its definition. Therefore we expect the $G$--action to be also
relevant for further structures defined on $P$. This is the case i.e.
for the connection $\omega$ on $P$ and for the vector valued functions
on $P$ which correspond to particle fields in the space time $M$.

In this paper we would like to investigate the action of the group $TG$
which of course contains $G$ and in addition its Lie algebra $\mbox{$\cal G$} =
\mbox{Lie}(G)$. For this purpose we have first to determine the
multiplication law on $TG$ and then the action of $TG$ on $TP$. For the
reasons explained in the introduction, we shall proceed in a quite
detailed way. We start with the multiplication law in $TG$, although the
space $TG$ is isomorphic to the Cartesian product of $\mbox{$\cal G$}$ and $G$, the
relevant multiplication law is not the so expected trivial one, but it
makes $TG$ a semi--direct product of $\mbox{$\cal G$}$ and $G$:
\begin{eqnarray}\label{eqn7}
        TG \times TG    &\rightarrow&   TG\:,\nonumber\\
        (\xi_g, \eta_h) &\mapsto&       \xi_g \cdot \eta_h \mathrel{\hbox{$\mcolon \mkern0.75mu \mathord{=}$}} TR_h \xi_g
        + TL_g \eta_h\;.
\end{eqnarray}
The notation $R$ and $L$ is used to the right and left action of $G$ on
$G$. The inverse of $\xi_g$ is given by
\begin{eqnarray}\label{eqn8}
        {\xi_g}^{-1}    &=& - TL_{g^{-1}} TR_{g^{-1}} \xi_g\;.
\end{eqnarray}
If we denote with $e$ and $0_e$ the neutral elements in $G$ and $TG$ we
obtain by explicit calculation $\xi_g \cdot (\xi_g)^{-1} = 0_e$. We have
of course the injections:
\begin{eqnarray}\label{eqn9}
\begin{array}{rcl}
        G &\rightarrow& TG\\
        g &\mapsto& 0_g
\end{array}
        &\mbox{and}&
\begin{array}{rcl}
        \mbox{$\cal G$} &\rightarrow& TG\\
        \xi &\mapsto&   \xi_e
\end{array}\quad.
\end{eqnarray}
The verification of associativity is slightly involved and it is
therefore an advantage to use the isomorphism which contains the
semi--direct product splitting of $TG$ given by
\begin{eqnarray}\label{eqn10}
        \alpha : TG     &\rightarrow& \mbox{$\cal G$} \leftsemi G\;,\nonumber\\
        \xi_g   &\mapsto&  (\xi, 0_g) \mathrel{\hbox{$\mcolon \mkern0.75mu \mathord{=}$}} (TR_{g^{-1}} \xi_g, g)\;.
\end{eqnarray}
Denoting as usual $\mbox{Ad}_G = \mbox{Aut}(\mbox{$\cal G$})$ with
\begin{eqnarray}\label{eqn11}
        \mbox{Ad}_g \xi &=& TL_g TR_{g^{-1}} \xi = 0_g \xi 0_{g^{-1}}\;,
\end{eqnarray}
we have for the multiplication rule the multiplication
\begin{eqnarray}\label{eqn12}
        \xi_g \cdot \eta_h &\mapsto& (\xi,g) (\eta,h) = (\xi + \mbox{Ad}_g
        \eta, g h)\;.
\end{eqnarray}
At this point it should be noticed that of course all the above
structures are induced naturally by the $T$--functor and the canonical
isomorphism given in eq. (\ref{eqn10})

In what follows we need several times the action of $TG$ on $\mbox{$\cal G$}$. It
may be derived from the multiplication law above and the action of $TG$
on the $G$--orbit space $TG/G$ given by
\begin{eqnarray}\label{eqn13}
        TG/G &=& \{\xi_g G = TR_h \xi_g | h \in G\}
\end{eqnarray}
which we also denote as $\mbox{$\cal G$} = TG/G$ by misuse of notation. The rule
of addition in $TG/G$ may be defined by
\begin{eqnarray}\label{eqn14}
        \xi_g G + \eta_g G &\mathrel{\hbox{$\mcolon \mkern0.75mu \mathord{=}$}}& (\xi_g + \eta_g) G\:.
\end{eqnarray}
The $TG$--action on $\mbox{$\cal G$} = TG/G$ is given by\footnote{The restriction to
the subgroup $G$ of $TG$ leads as expected to the adjoined action of $G$
on $\cal G$:
\begin{eqnarray}
	G \times \cal G &\rightarrow& \cal G\nonumber\\
	(g, \eta ) &\mapsto& TR_{g^{-1}}(0_g \cdot \eta)
	= TR_{g^{-1}} TL_g \eta = \mbox{Ad}_g \eta\nonumber
\end{eqnarray}}
\begin{eqnarray}\label{eqn15}
        TG \times \mbox{$\cal G$} &\rightarrow& \mbox{$\cal G$}\;,\nonumber\\
        (\eta_h, \xi) &\mapsto&  \eta_h \cdot \xi = \mbox{Ad}_h \xi +
        TR_{h^{-1}} \eta_h\;.
\end{eqnarray}
The action of $TG$ on $TP$ is given canonically from the lifting of the
$G$--action on $P$ to the tangential space ($T$--functor). So we have in
an obvious notation from
\begin{eqnarray}\label{eqn16}
        R : P \times G &\rightarrow& P\;,\nonumber\\
        (p, g) &\mapsto& p g = R_g p = R_p g\:,
\end{eqnarray}
the $TG$--action on $TP$:
\begin{eqnarray}\label{eqn17}
        TP \times TG &\rightarrow& TP\;,\nonumber\\
        (X_p,\xi_g) &\mapsto& X_p \cdot \xi_g \mathrel{\hbox{$\mcolon \mkern0.75mu \mathord{=}$}} TR_g X_p + TR_p \xi_g\;.
\end{eqnarray}
In order to facilitate the reading, we give some further notations and some
examples. For the Maurer--Cartan form $C$ on $G$ given by
\begin{eqnarray}\label{eqn18}
        C(\xi_g) &=& TL_{g^{-1}} \xi_g\;.
\end{eqnarray}
we often use the expression $\xi(g) = C(\xi_g) \in \mbox{$\cal G$}$. We denote a
fundamental vector field on $P$ with $\hat{\xi}$ where we have $\xi \in
\mbox{$\cal G$}$ and  $\hat{\xi}_p = TR_p \xi$.

We now give some useful examples of the $TG$ action on $TP$.
\begin{eqnarray}\label{eqn19}
        X_p     \cdot   0_e     &=&     X_p\;,\nonumber\\
        X_p     \cdot   \xi     &=&     X_p +   \hat{\xi}_p\;,\nonumber\\
        X_p     \cdot   0_g     &=&     TR_g X_p\;,\nonumber\\
        0_p     \cdot   \xi     &=&     TR_p \xi \mathrel{\hbox{$\mathord{=} \mkern1mu \mcolon$}}  \hat{\xi}_p\;,
                                        \nonumber\\
        0_p     \cdot   \xi_g   &=&     TR_{p g} C(\xi_g) \mathrel{\hbox{$\mathord{=} \mkern1mu \mcolon$}} TR_{p g}
        \xi(g)\;.
\end{eqnarray}
After this preparation, everything is fixed and as a first result of the
$TG$--action we may formulate the follwing proposition:\\
{\it The space $TP$ is a principal bundle $TP(TM,TG)$ which structure
group $TG$ basis manifold $TM$ and canonical projection $T\pi : TP
\rightarrow TM$. $\pi$ is the canonical projetion in $P(M,G)$, $\pi : P
\rightarrow M$.}\\
This can be seen from the fact that
\begin{deflist}{(\romannumeral2)}
\item[\romannumeral1)] $TG$ acts freely on $TP$.
\item[\romannumeral2)] $T\pi$ commutes with the group action $TG$.
\end{deflist}
For i) we have to show that with $X_p \cdot \xi_g = X_p$,
\begin{eqnarray}\label{eqn20}
        \xi_g &=& 0_e
\end{eqnarray}
must be valid.\\
From
\begin{eqnarray}\label{eqn21}
        X_p \cdot \xi_g &=& TR_g X_p + TR_p \xi_g = X_p
\end{eqnarray}
we must first have $g = e$  so that using eq. (\ref{eqn19}), we obtain
\begin{eqnarray}\label{eqn22}
        X_p \cdot \xi_e &=& X_p +  \hat{\xi}_p =
        X_p\quad\mbox{and}\quad   \hat{\xi}_p
        = 0_p
\end{eqnarray}
which shows that
\begin{eqnarray}\label{eqn23}
        \xi = 0 \quad\mbox{and}\quad \xi_g = 0_e\;.
\end{eqnarray}
For ii) we have to show that
\begin{eqnarray}\label{eqn24}
        T\pi X_p &=& T\pi (X_p \cdot \xi_g)
\end{eqnarray}
is valid. Using eq. (\ref{eqn19}) and
\begin{eqnarray}\label{eqn25}
        X_p \cdot \xi_g &=& X_p \cdot 0_g + 0_p \cdot \xi_g
\end{eqnarray}
we obtain
\begin{eqnarray}\label{eqn26}
        X_p \cdot \xi_g &=& TR_g X_p + TR_{pg} \xi(g)\;.
\end{eqnarray}
$TR_{pg}  \xi(g)$ is vertical and
\begin{eqnarray}\label{eqn27}
        T\pi TR_g X_p &=& T\pi X_p\;,
\end{eqnarray}
so that ii) is indeed valid.

\section{The $TG$--equivariance of connection}
Having in mind the $G$--equivariance property of connections and the
$TG$--action on $TP$, as discussed in the previous section, we
study the space $TP/G$ and ask ourselves what its
connection with the connections on $P(M,G)$ is. This leads us to a new
definition of a connection which is of course, as we shall show,
equivalent to the usual one. The original definition of connections is
closely related to the $G$--action aspect whereas the definition in this
section will reveal the $TG$--action aspect and bring to light the most
characteristic property of connections: the $TG$--equivariance.

For that purpose we shall first discuss the $G$ orbit space
\begin{eqnarray}\label{eqn28}
	TP/G    &=& \{ X_p G = \{ TR_g X_p | g \in G\} | p \in P \}
\end{eqnarray}
and we shall show that it is isomorph to the $TP$ associated fibre
bundle with typical fiber the space $\mbox{$\cal G$}$:
\begin{eqnarray}\label{eqn29}
        TP/G     &\simeq&      TP \times_{TG} \mbox{$\cal G$}\;.
\end{eqnarray}
The $TG$--action on $\mbox{$\cal G$}$ is given in eq. (\ref{eqn15}). This action is
non-linear, therefore we do not expect $TP \times_{TG} \mbox{$\cal G$}$ to be a
vector bundle although it may be represented as $TP \times_{TG} \mbox{$\cal G$} =
TM \tilde{\times} \mbox{$\cal G$}$.

The isomorphism in eq. (\ref{eqn29}) follows from the fact that $TP$ is
a principal bundle and that $G$ is a subgroup of its structure group
$TG$:
\begin{eqnarray}\label{eqn31}
  TP/G   &\rightarrow&   TP \times_{TG} (TG/G)\;,\nonumber\\~
  [X_p \cdot \xi_g]_G  &\mapsto& [X_p, \xi_g G]_{TG}\;.
\end{eqnarray}
If we take $\xi_g = 0_e$, this may be seen more perspeciously:
\begin{eqnarray}\label{eqn32}
	X_p G  &\mapsto&   [X_p, 0_e G]_{TG}\;.
\end{eqnarray}
In the fibre bundle $TP/G$, we have to define an addition: for $\pi(p) =
\pi(q)$ and $p = q g$, we have:
\begin{eqnarray}\label{eqn33}
	X_p G + Y_q G &\mathrel{\hbox{$\mcolon \mkern0.75mu \mathord{=}$}}& X_p G + (TR_g Y_q) G = (X_p+TR_g Y_q) G\;.
\end{eqnarray}
In what follows we shall give three new definitions of a connection.
These are closely related to the horizontal lifting of the (original)
definition of connection:\\
{\it A connection on a principal bundle $P(M,G)$ is a linear section in
the associated fibre bundle $TP \times_{TG} \mbox{$\cal G$}$ ($\mbox{$\cal G$} = \mbox{Lie}(G)$).

The action of $TG$ on $\mbox{$\cal G$}$ is given in eq. (\ref{eqn15}). If we denote
by $\mbox{$\tilde{H}$}$ this section, we have:
\begin{eqnarray}\label{eqn34}
	\mbox{$\tilde{H}$} : TM &\rightarrow& TP/G\;,\nonumber\\
	   V_x &\mapsto&  \mbox{$\tilde{H}$}(V_x)\;.
\end{eqnarray}}
The existence of such a section leads to an equivalent but more abstract
definition:\\
{\it A connection on a principal bundle $P(M,G)$ is a reduction of the
structure  group $TG$ of the principal bundle $TP(TM,TG)$ to $G$.}

The most transparent definition we obtain if we consider the above section
$\mbox{$\tilde{H}$}$ in $TP/G$ as a $\mbox{$\cal G$}$ valued linear function $\mbox{$\bar{H}$}$ on $TP$:\\
{\it A connection on a principal bundle $P(M,G)$ is a linear
$TG$--equivariant function on $TP$ with values in $\mbox{$\cal G$} =
\mbox{Lie}(G)$. $\mbox{$\bar{H}$}$ is given by
\begin{eqnarray}\label{eqn35}
	\mbox{$\bar{H}$} : TP &\rightarrow& \mbox{$\cal G$}\;,\nonumber\\
	  X_p  &\mapsto&   \mbox{$\bar{H}$}(X_p)\qquad \mbox{with}\nonumber\\
	  \mbox{$\bar{H}$}(X_p \cdot \xi_g)  & = & (\xi_g)^{-1} \cdot \mbox{$\bar{H}$}(X_p)\;.
\end{eqnarray}}
At this point it is interesting to note that the function $\mbox{$\bar{H}$}$ is, as
we shall show, directly related to the original definition of a
connection $\omega$ (as a linear function on $TP$) whereas the section
$\mbox{$\tilde{H}$}$ is directly related to the horizontal lift $\mbox{$H^{\omega}$}$
which stems from the $\omega$. The relation between $\mbox{$\tilde{H}$}$ and $\mbox{$\bar{H}$}$ is
given by:
\begin{eqnarray}\label{eqn36}
	\mbox{$\tilde{H}$}(T\pi X_p) & = & [X_p, \mbox{$\bar{H}$}(X_p)]_{TG}\;.
\end{eqnarray}
The linearity of $\mbox{$\bar{H}$}$ is related as expected to the linearity of $\mbox{$\tilde{H}$}$:
\begin{eqnarray}\label{eqn37}
	[X_p,\xi]_{TG} + [Y_p,\eta]_{TG}
& = &   [X_p + \hat{\xi}_p,0]_{TG} + [Y_p + \hat{\eta}_p, 0]_{TG}\;,\nonumber\\
& = &   [X_p \cdot \xi, 0]_{TG} + [Y_p \cdot \eta, 0]_{TG}\;,\nonumber\\
& = &	(X_p + \hat{\xi}_p)G + (Y_p + \hat{\eta}_p)G\;,\nonumber\\
& = &	(X_p + \hat{\xi}_p + Y_p + \hat{\eta}_p)G\;,\nonumber\\
& = &   [X_p + Y_p +  \hat{\xi}_p + \hat{\eta}_p, 0]_{TG}\;,\nonumber\\
& = &   [X_p + Y_p, \xi + \eta]_{TG}
\end{eqnarray}
where we have used the isomorphism of eq. (\ref{eqn32}) and the
definition of the eq. (\ref{eqn33}).

This corresponds to
\begin{eqnarray}\label{eqn38}
	[X_p, \mbox{$\bar{H}$}(X_p)]_{TG} + [Y_p, \mbox{$\bar{H}$}(Y_p)]_{TG} &=&
	[X_p + Y_p , \mbox{$\bar{H}$}(X_p) + \mbox{$\bar{H}$}(Y_p)]_{TG}\;.
\end{eqnarray}
So we obain from the linearity of $\mbox{$\tilde{H}$}$
\begin{eqnarray}\label{eqn39}
	\mbox{$\tilde{H}$}(T\pi X_p) + \mbox{$\tilde{H}$}(T\pi Y_p) = \mbox{$\tilde{H}$}(T\pi (X_p + Y_p))
\end{eqnarray}
the linearity of $\mbox{$\bar{H}$}$:
\begin{eqnarray}\label{eqn40}
	[X_p,\mbox{$\bar{H}$}(X_p)]_{TG} + [Y_p, \mbox{$\bar{H}$}(X_p)]_{TG} &=&
	[X_p +Y_p, \mbox{$\bar{H}$}(X_p +Y_p)]_{TG}\nonumber\\
        \mbox{$\bar{H}$}(X_p + Y_p) &=&  \mbox{$\bar{H}$}(X_p) + \mbox{$\bar{H}$}(Y_p)\;.
\end{eqnarray}

For the relation between the horizontal lift $\mbox{$H^{\omega}$}$ and $\mbox{$\tilde{H}$}$, taking
$\pi(p) = x$ and $T\pi
X_p = V_x$, we have:
\begin{eqnarray}\label{eqn41}
	\mbox{$H^{\omega}$}_p :   T_xM &\rightarrow& T_pP\;,\nonumber\\
                  V_x  &\mapsto& \mbox{$H^{\omega}$}_p(V_x)\;.
\end{eqnarray}
Using the projection:
\begin{eqnarray}\label{eqn42}
        \mu :    TP    &\rightarrow& TP/G\;,\nonumber\\
                 X_p   &\mapsto& X_p G\;,
\end{eqnarray}
we have
\begin{eqnarray}\label{eqn43}
	\mbox{$\tilde{H}$}_x      &=& \mu \circ \mbox{$H^{\omega}$}_p\;.
\end{eqnarray}
The relation of $\mbox{$\bar{H}$}$ to connection $\omega$ is given by
\begin{eqnarray}\label{eqn44}
        \omega(X_p)  &=& -\mbox{$\bar{H}$}(X_p)\;.
\end{eqnarray}
All we have to show is that the so--defined $\omega = \omega(\mbox{$\bar{H}$})$ fulfils
the original axioms of a connection:
\begin{deflist}{(\romannumeral2)}
\item[\romannumeral1)] $\omega$ is $G$--equivariant.
\item[\romannumeral2)] the restriction of $\omega$ to the fundamental
	vector fields on $P$ corresponds, after the identification of
	the fundamental vector fields with the Lie algebra $\mbox{$\cal G$}$, to the
	identity.
\end{deflist}
The condition \romannumeral1) is clearly valid since the $\omega$ given
by $\mbox{$\bar{H}$}$ is $TG$--equivariant, $G$ is a subgroup of $TG$ and acts on
$\cal G$ like $\mbox{Ad}_G$.

The condition \romannumeral2) follows from the fact that for
$\hat{\xi}_p = TR_p \xi$ with $\xi \in \mbox{$\cal G$}$
\begin{eqnarray}\label{eqn45}
	\omega(\hat{\xi}_p) &=&
	-\mbox{$\bar{H}$}(\hat{\xi}_p)  =  -\mbox{$\bar{H}$}(0_p \cdot \xi_e)\;,\nonumber\\
  &=&   -{\xi_e}^{-1} \cdot \mbox{$\bar{H}$}(0_p)\;,\nonumber\\
  &=&   -\xi^{-1} \cdot 0\;,\nonumber\\
  &=&   -(-\xi) = \xi
\end{eqnarray}
where we have used the $TG$--equivariance of $\mbox{$\bar{H}$}$ (eq. (\ref{eqn35})) and
the linearity of $\mbox{$\bar{H}$}$ (eq. (\ref{eqn40})) to evaluate $\mbox{$\bar{H}$}(0_p)$.

The above considerations showed the equivalence of all the notions of
connection represented by $\omega$, $\mbox{$H^{\omega}$}$, $\mbox{$\tilde{H}$}$, $\mbox{$\bar{H}$}$ in every
combination and particularly in ($\omega$, $\mbox{$H^{\omega}$}$) with ($\mbox{$\bar{H}$}$, $\mbox{$\tilde{H}$}$).

At this point we give, for those who feel uneasy with the use of the
fibre bundle $(TP/G \rightarrow TM) = TP \times_{TG} \cal G$ in the
definition of a connection in eq. (\ref{eqn34}), a further definition in
terms of vector bundles. For that purpose we observe that $TP/G
\rightarrow M$ is a vector bundle with typical fiber $\Reell^m \times
\cal G$ and $m = \mbox{dim}(M)$. So, denoting this vector bundle by $E =
(TP/G \rightarrow M)$, we have $E = M \tilde{\times}(\Reell^m \times
\cal G)$.

This leads to the definition of a connection as a special vector bundle
homomorphism between $TM$ and $E$:\\
{\it A connection on a principal bundle $P(M,G)$ is a vector bundle
homomorphism $\mbox{$\hat{H}$}$:
\begin{eqnarray}
\mbox{$\hat{H}$}: TM &\rightarrow& E = (TP/G \rightarrow M)
\end{eqnarray}
with the restriction $\nu \circ \mbox{$\hat{H}$} = \mbox{id}_{TM}$ and $\nu = T\pi
\circ \mu^{-1}$ the projection $\nu : TP/G \rightarrow TM$. In the case
of a trivial $P(M,G)$, this definition corresponds to the graph of a
gauge potential $A$:
\begin{eqnarray}
A : TM &\rightarrow& \cal G
\end{eqnarray}
as it is used in particle physics phenomenology.}

\section{$TG$--equivariance and gauge transformations}
In this section we deal with the implications for gauge transformations
from the $TG$--action point of view. In this way we obtain a new
formulation for the well known expression for gauge transformations of
connections in terms of $TG$--elements instead of $G$--elements.

We consider a gauge transformation as an automorphism on $P(M,G)$ which
may be expressed equivalently by a function on $P$ with values on $G$
[8]:
\begin{eqnarray}\label{eqn46}
	\Psi : P &\rightarrow&  P\;,\nonumber\\
               p &\mapsto& \Psi(p)  = p U(p)
\end{eqnarray}
and
\begin{eqnarray*}
	U : P &\rightarrow&  G\;,\nonumber\\
            p &\mapsto&  U(p)
\end{eqnarray*}
with
\begin{eqnarray}\label{eqn47}
	U(p g)   &=&       g^{-1} U(p) g\;.
\end{eqnarray}
The automorphism $\Psi$ induces in the space of connections the
transformation $\Psi^*$ and we have for the connection $\omega$
pointwise the expression
\begin{eqnarray}\label{eqn48}
	(\Psi^* \omega)(X_p) = \mbox{Ad}_{U(p)^{-1}} \omega(X_p) +
	(U^{*} C)(X_p)
\end{eqnarray}
with $\xi_p \in TP$ and where $C$ is the Maurer--Cartan form on $G$:
\begin{eqnarray}\label{eqn49}
	C : TG &\rightarrow&  \mbox{$\cal G$}\;,\nonumber\\
        \xi_g   &\mapsto&  C(\xi_g) = TL_{g^{-1}} \xi_g\;.
\end{eqnarray}
We would like now to reexpress the right hand side of eq. (\ref{eqn48})
in terms of the $TG$--action on $TP$ and $\mbox{$\cal G$}$. For the action of $TG$
on $\mbox{$\cal G$}$ we obtain from
\begin{eqnarray*}
	\xi_g  \cdot \eta &=& \mbox{Ad}_g \eta + TR_{g^{-1}} \xi_g\;,
\end{eqnarray*}
given in eq. (\ref{eqn15}), the $(\xi_g)^{-1}$ action on $\eta$:
\begin{eqnarray}\label{eqn50}
	(\xi_g)^{-1} \cdot \eta  &=&
	\mbox{Ad}_{g^{-1}} \eta - TL_{g^{-1}} \xi_g
\end{eqnarray}
or
\begin{eqnarray}\label{eqn51}
	(\xi_g)^{-1} \cdot \eta &=& \mbox{Ad}_{g^{-1}} \eta - C(\xi_g)\;.
\end{eqnarray}
For the $\Psi^* \omega$, expressed in $\mbox{$\bar{H}$}$, we have pointwise:
\begin{eqnarray}\label{eqn52}
	(\Psi^* \omega)(X_p) &=& \omega(T\Psi X_p) = -\mbox{$\bar{H}$}(T\Psi X_p)\;.
\end{eqnarray}
Using the equivariance property of $\mbox{$\bar{H}$}$ (eq. (\ref{eqn35})) and eq.
(\ref{eqn51}), we obtain:
\begin{eqnarray}\label{eqn53}
        \mbox{$\bar{H}$}(T\Psi X_p) &=& \mbox{$\bar{H}$}(X_p \cdot TU X_p) =
        	(TU X_p)^{-1} \cdot  \mbox{$\bar{H}$}(X_p)
\end{eqnarray}
and
\begin{eqnarray}\label{eqn54}
        (TU X_p)^{-1} \cdot \mbox{$\bar{H}$}(X_p) &=&
        	\mbox{Ad}_{U(p)^{-1}} \mbox{$\bar{H}$}(X_p) - C(TU X_p)\;.
\end{eqnarray}
Eqs. (\ref{eqn52} - \ref{eqn54}) lead to
\begin{eqnarray}\label{eqn55}
	(\Psi^* \omega)(X_p) &=&
		-(TU X_p)^{-1} \cdot (-\omega(X_p))
\end{eqnarray}
or equivalently to
\begin{eqnarray}\label{eqn56}
	(\Psi^* \omega)(X_p) &=&
	\mbox{Ad}_{U(p)^{-1}} \omega(X_p) + U^* C(X_p)\;.
\end{eqnarray}

On the right hande side of eq. (\ref{eqn55}), the minus in front of
$\omega(X_p)$ cannot be eliminated because the $TG$--action is not
linear. Notice also that we obtained in this way eq. (\ref{eqn48}) in a
very straightforward manner.

\section{Summary}
The aim of this paper was to set some new lights on several aspects of
the notion of a connection on a principal bundle $P(M,G)$ and especially
on its gauge transformations. We expect this to be very useful in
physical applications.

We proceeded essentially by exploring the properties of the
$T$--functor, considering the action of the tangential group $TG$ on
$\cal G = \mbox{Lie}(G)$ and $TP$. It was perceived that $TP$ is also a
principal bundle $TP(TM,TG)$ and that the fibre bundle $TP/G \rightarrow
TM$ associated with it ($TP \times_{TG} \cal G$) is useful for a new
definition of a connection as a linear section on it.

This definition revealed the most characteristic property of a
connection, the $TG$--equivariance:
\begin{eqnarray}
\omega(X_p \cdot \xi_g) &=& - (\xi_g)^{-1} \cdot (- \omega(X_p))
\end{eqnarray}
where we have used the symbol $\;\cdot\;$ for the $TG$--action on $TP$
and $\cal G$.

The $TG$--equivariance property of the connection also gives the most
plausible explanation for the specific form the expression for gauge
transformations takes. In particular, the nonlinearity in those
transformations stems from the nonlinear action of the affine group
$TG$.
\section*{Acknowledgements}
We would like to thank S. Klaus and M. Kreck for very helpful
discussions and careful reading of the manuscript. We also thank R.
Coquereaux for an interesting discussion, and R. Buchert and F. Scheck
for useful comments.
\section*{References}
\begin{deflist}{8.\qquad}

\item[1.] S. Kobayaschi and K. Nomizu, Foundations of differential geometry,
          Volume 1, John Wiley \& Sons: New York, London, Sydney, 1963.

\item[2.] Y. Choquet--Bruhat, C. De Witt--Morette and M. Dillard--Bleik,
	  Analysis, Manifolds, and Physics, North--Holland: Amsterdam,
	  Oxford, New York, Tokyo, 1987.
\item[3.] --~ A. Heil, A. Kersch, N. A. Papadopoulos, B. Reifenh\"auser and
	  F. Scheck, Structure of the Space of Reducible Connections for
	  Yang--Mills Theories, Journal of Geometry and Physics: Vol, 7,
	  n. 4, 1990, 489.
	  --~ A. Heil, A. Kersch, N.A. Papadopoulos and B. Reifenh\"auser,
	  Abelian Charges in Nonabelian Yang--Mills Theory from the
	  Stratification of the Space of Gauge Potentials, Annals of
	  Physics, Vol. 217, No. 2 1992.
\item[4.] --~ A. Heil, A. Kersch, N.A. Papadopoulos, B. Reifenh\"auser,
	  F. Scheck and H. Vogel, Anomalies from the Point of View of
	  G--Theory, Journal of Geometry and Physics: Vol. 6, n. 2, 1989,
	  237.\\
          --~ A. Heil, A. Kersch, N.A. Papadopoulos, E. Reifenh\"auser
          and F. Scheck: Anomalies from Nonfree Action of the Gauge
          Group, Annals of Physics: 200, 1990, 206.
\item[5.] N.A. Papadopoulos, The Role of Stratification of Anomalies, in
	  J. Debrus and A.C. Hirshfeld (Eds.), Geometry and Theoretical
	  Physics, Springer-Verlag: Berlin, Heidelberg, 1991, 210.
\item[6.] R. H\"au{\ss}ling, N.A. Papadopoulos and F. Scheck,
	  Supersymmetry in the Standard Model of Electroweak
	  Interactions, Physics Letters B303, 1993, 265.
\item[7.] S. Kobayaschi, Theory of connection, Annali di Matematica, 43,
	  1957, 199.
\item[8.] M. G\"ockeler and T. Sch\"ucker, Differential Geometry, Gauge
	  Theories, and Gravity, Cambridge University Press: Cambridge,
	  1987.
\end{deflist}
\end{document}

redpajama_set_name: RedPajamaArXiv


text:
\section{Introduction}

In our understanding of the Standard Model the strong coupling $\gbar(\mu)$
plays an essential r\^ole as its parametric uncertainty is one of the dominant
sources of uncertainty in Higgs decays for instance. Using lattice gauge theory
we are in the fortunate position to study QCD for any number of dynamical
flavours $\nf$.  In contrast to any experimental data analysis in high energy
physics, we are able to employ non-perturbative definitions of the strong
coupling by means of any suitable lattice observable $O_{\rm lat}$, such
that~\cite{Flag:2013}
\begin{align}\label{eq:def-gen}
        \gbsq(\mu) &= \lim_{a\to 0} \cc_{O}(a,\mu)\cdot O_{\rm lat}(a,\mu)  \;.
\end{align}
The normalization factor $\cc_{O}(a,\mu)$ guarantees $\gbsq=g_0^2 + \Or(g_0^4)$
at leading order in lattice perturbation theory. Any lattice observable fulfilling
eq.~\eqref{eq:def-gen} defines a renormalization scheme with a different
scale-dependent renormalized coupling. As in QCD conformal symmetry of the
massless Lagrangian is broken on the quantum level, one can define a
renormalization group (RG) invariant, the $\Lambda$-parameter,
\begin{align}\label{eq:def-lambda}
   \Lambda &\equiv
            \mu\big[{b_0\gbsq(\mu)}\big]^{-{b_1}/({2b_0^2})}\,\ee^{-1/({2b_0\gbsq(\mu)})}
            \exp\bigg\{\!\!-\!\!\int_{0}^{\gbar^{\vphantom{A}}(\mu)}\!\dd g
            \bigg[\dfrac{1}{\beta(g)}+\dfrac{1}{b_0g^3} -\dfrac{b_1}{b_0^2g}\bigg]\bigg\} \;.
\end{align}
This definition holds for any $\mu$, and its value is trivially
scheme-dependent in the sense that the ratio $\Lambda_1/\Lambda_2$ for two
different schemes can be computed exactly.  It has become standard to quote
$\gbsq$ at the electroweak scale given by the $Z$-boson mass, $\mu=M_{Z}$, in
the intrinsically perturbative $\MSbar$ scheme.

A fully non-perturbative computation of the $\Lambda$-parameter
through lattice computations proceeds in the following way. One chooses an
appropriate definition of a renormalized coupling~\eqref{eq:def-gen} in a
finite-volume renormalization scheme which interlinks the energy scale and the
finite size of the system, $\mu\propto L^{-1}$. Starting at a high energy scale
where perturbation theory is applicable without doubt, say at $\mupt\sim
M_{Z}\sim 100\;\GeV$, one iteratively applies a finite-size rescaling technique
by computing the change of the renormalized coupling from a change of energy
scales (or $L$) by factors of $s=2$ for instance. After $N$ steps one
arrives at some hadronic scale $\muhad=\mupt/s^N\equiv \lmax^{-1}$, where a
connection to large volume (LV) lattice simulations can be established. This
allows to determine $\Lambda$ in terms of some experimentally known hadronic
observable, $\fhad$, used to set the overall energy scale in the LV
simulations, and thus in physical units. For $\Lmsbar$ this strategy decomposes
as follows:%
\footnote{To ease our discussion we work at fixed $\nf$, i.e.,
no quark-tresholds are taken into account.}
\begin{align}\label{eq:Lambda-comp}
        \Lmsbar &= \left[ \fhad \right]_{\rm exp} \times \left[\frac{\Lmsbar}{\Lambda} \right]_{\rm exact}  \times \frac{\Lambda}{\fhad}
    & &\text{with} &
    \frac{\Lambda}{\fhad} &= \frac{\lmax\Lambda}{\lmax\fhad}
    \;.
\end{align}
The total error on $\Lambda$ is composed of that from the LV scale setting
$(\lmax\fhad)$ and the determination of $\lmax\Lambda$ from the
non-perturbative running in the intermediate, finite-volume scheme. While for
$\nf=2$~\cite{Fritzsch:2012wq} the error of $\lmax\fhad\equiv L_1\fK=0.315(8)(2)$
contributed about $\tfrac{2}{5}$ to the total error of 6\% on ${\Lambda}/{\fhad}$,
it will become negligible in the near future due to new
developments~\cite{MattiaDB:Lat14}. The current world average(s) for
$\alpha^{(5)}_{\rm s}(M_{Z})$ are $0.1183(12)$ from PDG~\cite{Beringer:1900zz}
using experiments only, and $0.1184(12)$ from the average of present lattice
determinations~\cite{Flag:2013}. This 1\% error translates into an error of
about 6\% in $\Lmsbar^{(5)}$.
For our new estimate of $\Lambda^{(\nf=3)}/\fhad$, to be derived from LV lattices
within the current CLS effort~\cite{Bruno2014}, it is thus worthwhile to aim for
an accuracy of about 4\% or better.

\section{Why choose a new strategy?}

We have seen that by mainly controlling the accuracy of the non-perturbative RG
running, we can improve the determination of $\Lambda$. In the past, the
Schr\"odinger functional (SF) coupling $\gbsf$~\cite{Luscher:1992an} has been
the most useful definition of a finite-volume renormalization scheme compatible
with the strategy behind eq.~\eqref{eq:Lambda-comp}.
A more recent development that can be used along the same lines, is given by
one of the many running coupling definitions employing the Yang--Mills gradient
flow (GF)~\cite{Narayanan:2006rf}, $\gbgf$ in short. Initially introduced
in~\cite{Luscher:2010iy} it has been studied in a finite-volume setup
in~\cite{Fodor:2012td,Fritzsch:2013je,Ramos2014}.  Especially, the apparently
much better noise-to-signal ratio of $\gbgf$ raises hope to significantly
increase the accuracy in the RG running. To what extent this statement complies
with the renormalization group running covering two orders of magnitude in
energy scales, we will see below. First we start with a general comparison of
SF and GF couplings, both defined with Dirichlet boundary conditions in time.
A short summary is given in table~\ref{tab:compare}.
\begin{table}[t]
  \small
  \centering
  \begin{tabular}[]{llll}\toprule
    topic                                     & SF coupling                   & GF coupling                     & remark                      \\\midrule
    \sc Definition                            & $\gbsq_{\rm SF}(L) = k \langle\tfrac{\partial\Gamma}{\partial\eta}\rangle^{-1}_{\eta=0}$
                                              & $\gbsq_{\rm GF}(L) =   \langle t^2E\rangle /\mathcal{N}$        &                                 \\
    \sc SF boundary field                     & $\ne 0$                       & $=0$                            &                                 \\[0.4em]
    \sc PT matching $\sim$ 64\,GeV            & 2-loop                        & tree-level                      & GF @ higher energies            \\[0.4em]
    \sc typical \# meas.                      & O(100\,000)                   & O(1000)                         & $\Delta\gbsq_{\tiny\rm SF}\!\simeq\!\Delta\gbsq_{\rm GF}$ \\
    \hskip2em%
    ($\tau_{\rm int}$, $\mathcal{V}$, \ldots) &                               &                                 & $L\approx\!0.4\,\fm$ \\[0.4em]
    \sc cutoff effects                        & mild                          & rather large (so far)           &                                 \\
                                              & 2-loop improvement            & tree-level improvement          &                                 \\
                                              & $\Rightarrow {L}/{a}=6\ldots12$   & $\Rightarrow {L}/{a}=8\ldots16$      & controlled $a\to 0$             \\[0.4em]
    \sc $\Delta \gbsq/\gbsq$                  & $\sim \gbsq$                  & const.                          & for fixed \#meas.               \\
    \sc $\Rightarrow \Delta L/L$              & const.                        & $\sim \text{const}/\gbsq$       &                                 \\
    \bottomrule
  \end{tabular}
  \caption{General comparison of gradient flow and SF running coupling schemes.}
  \label{tab:compare}
\end{table}

The SF coupling, $\gbsf(L)$, is defined as the response to a variation of the
(QCD) action about a non-vanishing Yang--Mills background field imposed through
boundary conditions at Euclidean time $x_0=0,T$ $(T=L)$~\cite{Luscher:1993gh}.
As such it is entirely sensitive to the physical extent $L$. The coupling
$\gbgf(L)$ on the other hand is also sensitive to short distances since it is
given by the Yang--Mills energy density defined with vanishing background field
at finite gradient flow time $t=(cL)^2/8$ for some fixed constant $c\in
[0.25,0.5]$~\cite{Fritzsch:2013je}. While for both definitions the tree-level
normalization $\cc_{O}(a,\mu)$ is known for different lattice
actions~\cite{Luscher:1993gh,Fritzsch:2013je,Ramos2014}, cutoff effects are
mild for $\gbsf$ but large for $\gbgf$.%
\footnote{A consistent Symanzik improvement to reduce cutoff effects in
gradient flow observables and thus $\gbgf$ has been proposed
in~\cite{RamosSint:Lat14}.}
Additionally, for the SF coupling perturbative improvement is known up to
2-loop order but unknown for the GF coupling. Both facts together with
additional details such as the respective integrated autocorrelation time,
statistical variance and so on, influence the precision of the continuum limit
of the lattice step-scaling function $\Sigma$,
\begin{align}\label{eq:ssf}
        \sigma(u) &= \lim_{a\to 0} \Sigma(u,a/L) \;,
        &
        \Sigma(u,a/L) &= \left.\gbar^2(2L)\right|_{\gbar^2(L)=u,Lm=0} \;.
\end{align}
It has to be computed numerically at chosen values of
$u_{n}\equiv\gbar^2(L_{n})$, $n\in\{1,\ldots, N\}$ in order to cover the energy
range under consideration. From past experience we expect that lattice sizes
$L/a<12$ are sufficient to control the continuum extrapolation of
$\Sigsf(u,a/L)$. On the other hand lattices with $L/a>8$ are needed to achieve
an equivalent accuracy in the computation of $\Siggf(u,a/L)$.  At a fixed
volume with $L\approx 0.4\,\fm$, and choosing $c\equiv \sqrt{8t}/L = 0.3$ such
that $\gbsf^2(L)\simeq\gbgf^2(L)$, a numerical study has shown that in order to
achieve the same accuracy in both couplings one needs 100 times more
measurements of $\gbsf^2$.
\begin{figure}[t]
        \centering
        \includegraphics[width=0.50\textwidth]{plots/NewRunning_sketch}
        \hspace*{1em}
        \raisebox{3.5pt}{\includegraphics[width=0.45\textwidth]{plots/ssfNfdep}}
        \vskip-0.5em
        \caption{({\em Left}) Sketch of our proposed strategy to non-perturbatively
                              switch running coupling schemes at an intermediate
                              energy scale $\mu_{\rm swi}=1/L_{\rm swi}$.
                 ({\em Right}) Data of SF step-scaling functions for different
                              dynamical flavour content,
                              $\nf=0,2,3,4$~\cite{Luscher:1993gh,DellaMorte:2004bc,Tekin:2010mm},
                              and its leading asymptotic form $b_0+\Or(u)$ (dashed lines).
                }
        \label{fig:strategy}
\end{figure}
Finally, the two couplings show a very different leading scaling behaviour of
the relative error $R_{\gbsq}\equiv\Delta\gbsq/\gbsq$ which directly translates
via the RG into $R_{L}\equiv\Delta L/L = -\Delta\gbsq\big/2\gbar\beta(\gbar)$.
Considering only the dominant contribution, $\beta(\gbar)\simeq -b_0 \gbar^3 $,
this results into an error of $R_{L}= R_{\gbsq}\big/ 2b_0\gbsq$, with
$(2b_0)^{-1}\approx 9$ in three-flavour QCD. With a behaviour of $R_{\gbsq_{\rm
SF}}\sim\gbar^2$ and $R_{\gbsq_{\rm GF}}\sim\text{const}$, a relative scale
uncertainty of $\Delta L/L\sim\text{const}$ follows for the SF coupling and
$\Delta L/L\sim 1/\gbar^2$ for GF couplings at a fixed number of independent
measurements.

Every step in the finite-size scaling method contributes to the total
error budget of the RG running, giving as first rough estimate
\begin{align}\label{eq:budget}
        \left( \frac{\Delta[\lmax\Lambda]}{\lmax\Lambda} \right)^2
        &\simeq \sum_{n=1}^N \left( \frac{\Delta L}{L}\Big|_{u_{n}} \right)^2 \;,
        &
        u_{n} &= \sigma^{-1}(u_{n-1}) \;,
        &
        u_{0} &= \gbsq(\lmax) \;.
\end{align}
From this point of view it should be clear that at constant effort, the SF
coupling accumulates approximately the same error in each step while the GF
coupling definition shows an error that is steadily growing towards the high
energy regime. Of course, if computational costs are irrelevant this problem
could be solved by brute force calculations. So far we have discussed both
schemes under the aspect of statistical accuracy only, but there are more
points to be taken into account, for instance the matching to PT at high
energies. For $\gbsf$ the matching coefficients in
$\gbsq_{s}=\gbsq_{s^\prime}+\sum^{\ell}_{k=1}\chi^{(k)}\gbar^{2k+2}_{s^\prime}$
are known up to $\ell=2$ loop order~\cite{Bode:1999sm} while no coefficient is
known for $\gbgf$ so far. Even with a given 1-loop coefficient, $\gbgf$ still
has to be matched at higher energies to control this step equally well,
and thus increases the costs and statistical error in the RG running.
Additional points will be elucidated in more detail in a forthcoming
publication.

Having the global aspect of our strategy in mind we have to conclude that the
GF coupling scheme is advantageous at hadronic scales while the SF coupling
scheme still is to be favoured at high energy scales.

\section{Changing the standard strategy}

From the facts presented in the previous section the optimised way to improve
the computation of $\Lambda$ seems to be to combine both schemes. As depicted
in the left panel of Fig.~\ref{fig:strategy}, one computes the running of
the chosen gradient flow coupling at low energies up to an energy scale of
$\muswi=1/\lswi\equiv s^K/\lmax$:
\begin{align}\notag
        \ugf_{k} &= \gbgf^2(\lmax/s^k)  \;, \quad k\in\{0,1,\ldots,K\}\;,
        &
        \ugf_{0} &= \gbgf^2(\lmax) \;,  \\\label{eq:GFsteps}
        \siggf\left( \gbgf^2(L/s) \right) &= \gbgf^2(L) \;,
        &
        \ugf_{\rm swi} &= \gbgf^2(\lmax/s^K)\;.
\end{align}
At the scheme-switching scale $\lswi$ one has to set up additional simulations
defined through a line of constant physics, $(\gbgf^2(\lswi),\lswi
m)=(\ugf_{\rm swi},0)$, in order to compute the SF coupling via%
\begin{align}\label{eq:LCPswi}
        \usf_{\rm swi} &= \lim_{a\to 0} \Psi(u,a/L) \;,
        &
        \Psi(u,a/L) &= \left.\gbsf^2(L) \right|_{u=\ugf_{\rm swi},\lswi m=0 } \;,
\end{align}
as accurately as possible. From here on one continues in the new scheme
\begin{align}\notag
        \usf_{m} &= \gbsf^2(\lswi/s^m)  \;, \quad m\in\{0,1,\ldots,M\}\;,
        &
        \usf_{0} &= \gbsf^2(\lswi) \;,  \\\label{eq:SFsteps}
        \sigsf\left( \gbsf^2(L/s) \right) &= \gbsf^2(L) \;,
        &
        \usf_{\rm PT} &= \gbsf^2(\lswi/s^M)\;,
\end{align}
with $M=N-K$ if the same scale difference is to be covered as originally
anticipated by $\mupt^{-1}=\lmax/s^N$, and where ultimately the connection to
the perturbative running can be safely established.
We remark that we could have started our discussion also at high energies,
i.e., reversing the strategy and interchanging SF and GF in
eq.~\eqref{eq:LCPswi}. In fact, carrying out both procedures might help to
increase control over this non-perturbative scheme-switching step.

\section{Present status}

For various practical reasons we have started with the running of the SF
coupling from high to intermediate energies, reaching couplings as large as
$\usf=2$ within our new strategy. We are working in a massless renormalization
scheme with $\nf=3$ quark flavours and thus need to tune to vanishing PCAC
quark mass, $Lm=0$, for plaquette gauge action and non-perturbatively $\Or(a)$
improved Wilson quarks. This has been done for lattice sizes $L/a=4,\dots,16$
and bare gauge couplings $6 \le \beta\equiv 6/g_0^2  \le 9$. To minimise tuning
efforts in subsequent projects, we have done this more precisely than in the
past, allowing us to smoothly parameterise the critical line
$\mcrit(g_0^2,a/L)$ on the considered lattice sizes for $0\le g_0^2 \le 1$.
Employing the perturbative 2-loop formula for $\mcrit(g_0^2,a/L)$, we use the
fit ansatz
\begin{align}\label{eq:m_crit}
        a\mcrit(g_0^2,a/L) &= [a\mcrit(g_0^2,a/L)]_\text{2-lp} + k_1(a/L)\cdot g_0^6
                            + k_2(a/L)\cdot g_0^8 +   k_3(a/L)\cdot g_0^{10}
\end{align}
to describe our data, c.f. left panel of Fig.~\ref{fig:run_gSF}. The difference
between a direct fit for each $L/a$ and global fits with varying coefficient
functions $k_1,k_2,k_3$ serves as a quality criteria for our interpolating
function. We know that for the accuracy we are aiming at, any additional
uncertainty from a tuning to a line of constant physics becomes
\emph{negligible}, if $|Lm|<0.005$ is satisfied. Our final estimates fulfill
$|Lm|<0.001$.
\begin{figure}[t]
        \centering
        \includegraphics[width=0.45\textwidth]{plots/globfit_tuning}\hfill

        {\includegraphics[width=0.498\textwidth]{plots/ssf_redux}}
        \vskip-.75em
        \caption{({\em Left}) Data and interpolating functions for critical
                  line $m_{\rm crit}(g_0^2,a/L)$ along constant $L/a$.
                  ({\em Right}) Non-perturbative continuum step-scaling function
                  $\sigma_{\rm SF}(u)$ vs. perturbative evaluation.
                }
        \label{fig:run_gSF}
\end{figure}

To compute the SF step-scaling function $\sigsf(u)$ we aim for simulations with
$1.1\le \usf \le 2$, uniformly distributed in $1/\usf$. For $L/a\in\{4,6,8\}$
we use 8 different $\beta$ values but simulate only three for $L/a\in\{10,12\}$
in order to minimise computational costs in the finite-size rescaling step
$L\to 2L$.  Along the lines of~\cite{Appelquist_2009,Tekin:2010mm} we
interpolate our data for $\gbsf^2(\beta,L/a)$ and $\gbsf^2(\beta,2L/a)$. The
computation of the two-loop improved lattice step-scaling function
$\Sigsf^{(2)}(\usf,a/L)$~\cite{Bode:1999sm} thus requires $\gbsf^2(\beta,L/a)$
to be fixed at certain values of $\usf$ in line with~\eqref{eq:SFsteps}. The
continuum limit is taken individually at each value of $\usf_{m}$ with full
error propagation.
The coarsest lattice only serves the purpose of estimating the size of cutoff
effects but is never taken into account in the analysis. Furthermore, data for
$2L/a=20$ is still missing and cannot be included at present. The plot provided
in Fig.~\ref{fig:CL} thus shows a (preliminary) continuum limit of
$\Sigsf^{(2)}(u,a/L)$ using a local fit ansatz with linear scaling in $(a/L)^2$
towards the continuum.  This gives the largest error on $\sigsf(u)$ which will
be improved after studying appropriate global fit ans\"atze in more detail.
With the data at hand we use the standard interpolation
\begin{align}\label{eq:fit-sig}
     \sigma(u) &= u + s_0 u^2 + s_1 u^3 +s_2^{\rm fit} u^4 + s_3^{\rm fit} u^5 \;, &
           s_0 &= 2\ln(2) b_{0} \;,  &
           s_1 &= s_0^2 + 2 \ln(2)b_1 \;,
\end{align}
with parameters $s_2^{\rm fit},s_3^{\rm fit}$ and $s_0$, $s_1$ taken from
perturbation theory. In Fig.~\ref{fig:run_gSF} we plot the data points and
interpolating fit function of $\sigsf(u)$ together with their known
perturbative behaviour and remark that the important information lies in the
well-defined error of the former.  To look at our current results from a global
perspective, we compare it in Fig.~\ref{fig:strategy} (right panel) to previous
determinations of $\sigsf$ for $\nf=0,2,4$. For a better discrimination we plot
$[\sigsf(u)-u]/[2\ln(2)u^2]$ such that the axis intercept corresponds to $b_0$.
This figure provides a beautiful demonstration of our current understanding of
asymptotic freedom at the non-perturbative level.

\pagebreak
\section{Conclusions \& Outlook}
\begin{wrapfigure}{r}{0.5\textwidth}
  \vspace{-4em}
  \begin{center}
        \includegraphics[height=8cm]{plots/cextrap}
  \end{center}
  \vspace{-2em}
  \caption{Local (preliminary) continuum extrapolations of
           $\Sigsf^{(2)}(\usf,a/L)$ using $L/a\ge 6$ lattices.
          }
\label{fig:CL}
\end{wrapfigure}
We have presented an extension to the ALPHA-strategy towards future high
precision computations of the Lambda parameter. It still keeps all systematic
errors under control by following a traditional finite-size scaling approach to
non-perturbatively connect low- and high-energy regimes. The gain in precision
is achieved by exploiting complementary properties of the SF and gradient flow
coupling together with a non-perturbative scheme switch at intermediate
energies.  While the SF part of the computations has been finished we are now
working on the computation of the gradient flow step-scaling function and the
non-perturbative scheme-switching step.

\section*{Acknowledgments}
\small
This work is supported in part by the Deutsche Forschungsgemeinschaft under
SFB/TR~9.  S. Sint acknowledges support by SFI under grant 11/RFP/PHY3218.
M.D.B. is funded by the Irish Research Council, and is grateful for the
hospitality at DESY Zeuthen. We gratefully acknowledge the computer resources
provided by the John von Neumann Institute for Computing as well as at HLRN and
at DESY, Zeuthen.