\section{ Introduction}

The existence of dark matter (DM), as the invisible matter interacting by the
force of gravity, has been widely accepted by cosmological observations from
the experiments: Cosmic Microwave Background (CMB) \cite{eko09}, Galactic
Rotation Curves \cite{abo01}, Gravitational Lensing \cite{hho02} and Massive
Compact Halo Objects \cite{cal00}, and $etc$. About 83\% of the matter (around
23\% of the total energy density) in the universe is believed to be composed
of DM to account for the observations. However, the nature of DM is still
completely unknown despite decades of detection efforts. Many possible
explanations have been proposed. One of the alternative explanations from the
point of view of particle physics is that DM is composed of massive particles
and its interaction with ordinary matter is very weak.

Recently several DM models\footnote{Actually these models have been built to
explain the annual modulation signal from DAMA/NaI \cite{rbe03} and DAMA/LIBRA
\cite{rbe08} experiments with null results from other experiments (there is no
experimental evidence corroborating this signal yet). The scenario in Ref.
\cite{bfe10} is especially interesting, because the signal appears to be
electromagnetic energy deposit, not nuclear energy deposit, through the single
photon emission by the decay of the excited state.} (inelastic DM
\cite{bfe10,jing10}, asymmetric DM \cite{han10}, form factor DM \cite{bfel10})
with magnetic dipole interaction have been considered. DM in these models has
a few states and has no direct interaction with the photon. The candidate
particles could thus be stable and make up the invisible matter in our
universe. On the other hand, the direct interaction with photon through
magnetic dipole coupling has gotten some attention
\cite{heo09,mpo00,ksi04,vba11,wsc10,tba10,edn12} due to its plausibility. Most
of the works have concentrated on direct detections of DM. The main motivation
for the magnetic dipole interacting DM scenario is that the magnetic dipole
coupling can be sizable compared to other electromagnetic couplings, because
the magnetic dipole conserves the discrete symmetries like parity (P), time
reversal (T), and charge conjugation (C) or its combinations.

In this work, we consider cosmic ray signatures (indirect detections) of the
direct dipole-interacting DM with the shifted photon (hypercharge gauge boson
$B_{\mu}$). The cosmic ray signatures in the positron, antiproton and photon
channels are considered for the DM mass near the electroweak scale $(10-1000$
GeV$)$, essentially around 100 GeV. The dimension-5 operator which induces the
dipole interaction is $\overline{\psi}\sigma_{\mu\nu}\psi B^{\mu\nu}$, and it
may be expressed with photon and $Z$ boson in the standard model context since
the hypercharge gauge boson is a linear combination of photon and $Z$ boson
with Weinberg angle $\theta_{W}$. The relevant effective Lagrangian is given
b
\begin{equation}
\mathcal{L}_{\mathrm{eff}}=\frac{1}{2}\mu\overline{\psi}\sigma_{\mu\nu
\psi(F^{\mu\nu}-\tan\theta_{W}Z^{\mu\nu}),
\end{equation}
where $F^{\mu\nu}$ is the field strength for photon, $Z^{\mu\nu}$ for $Z$
boson and $\mu$ is the magnetic dipole moment. The DM annihilations therefore
produce the standard model particles via $\gamma,Z$ exchanges. The
annihilation processes were studied in Ref. \cite{heo09} in detail, and here
we take advantage of the results (annihilation rates or fractions). The
corresponding annihilation fractions are tabulated in Table I for our
benchmark mass, 100 GeV.

\begin{table}[t]
\caption{Annihilation fractions for each channel, in which $u$ denotes up type
quarks $(=u,c)$, $d$ down type quarks $(=d,s,b)$, $\nu$ neutrinos $(=\nu
_{e},\nu_{\mu},\nu_{\tau})$ and $e$ charged leptons $(=e,\mu,\tau)$. Five
fundamental channels are considered for dark matter mass of 100 GeV.
\label{table1
\begin{ruledtabular}
\begin{tabular}{cccccc}
$\text{Channel}$ & $u\bar{u}$  & $d\bar{d}$& $\nu \bar{\nu}$ & $e\bar{e}$ &$W^+W^-$ \\
\hline
$\text{Annihilation fraction (\%)}$ & $14.6$ & $7.4$&$3.2$ & $10.1$ & $8.3$ \\
\end{tabular}
\end{ruledtabular}\end{table}

\section{Cosmic ray signatures}

DM may annihilate at some point in Galaxy and produce the standard model
particles. These produced particles then propagate in the interstellar medium.
Antimatter particles and photons have been considered to be the subject of
indirect DM searches, because antimatter particles are rarely produced in
astrophysical process and gamma rays can transport freely without energy loss
or transmutation of the direction. They may thus provide important signatures
of DM in the Galaxy. Observations of such signals can reveal information on
the microscopic nature of DM.

The emissivity/energy (production rate or source for the signals)\footnote{If
DM is produced with a primordial asymmetry like baryons, there would be almost
no signals from DM-antiDM annihilations due to lack of anti-dark matters
(antiDMs). This source is for equal populations of DM and antiDM, and hence
our predicted signals will be upper limits of the predictions. Recently a
mechanism of DM-antiDM oscillations is suggested to re-equilibrate the
populations at late times \cite{stu12, mci11}.} at location $\mathbf{x}$ from
the Galactic center is obtained from the convolution over the various
annihilation channels $f$ of the annihilation rate $\left\langle \sigma
v\right\rangle _{f}$ with the differential yield (single particle
spectra)\footnote{We use PYTHIA \cite{tor07}, as implemented in DarkSUSY
\cite{pgo04} or MicrOMEGAs \cite{gbe10} program, to generate the differential
yields (injected particle spectra).} $\left(  dN^{f}/dT\right)  _{a}$ for the
final state particles $a$
\begin{equation}
Q_{a}\left(  \mathbf{x},T\right)  =\frac{1}{4}B\left\langle \sigma
v\right\rangle _{f}\left(  \frac{dN^{f}}{dT}\right)  _{a}\left(  \frac
{\rho(\mathbf{x})}{M}\right)  ^{2},
\end{equation}
where $M$ is the DM mass, $B$ is an overall boost (enhancement) factor and
$\rho(\mathbf{x})$ is the DM mass density at the location $\mathbf{x}$. The DM
mass density around the Galactic center (DM halo profile) is not known,
especially near center ($\leq100$ pc$)$. The theoretically motivated ones are
Navarro-Frenk White (NFW) \cite{jfn97}, Moore \cite{bmo98}, cored Isothermal
\cite{jnb80}, and recently Einasto profiles \cite{jei65,jfn08}. The kinetic
energy $T$ is often approximated to the total energy $E$, in the case when the
particles are energetic. Notice that the factor $\frac{1}{4}$ is different
from the one of self annihilating DM.

The DM density profile can be parameterized a
\begin{equation}
\rho(r)=\rho_{\odot}\left[  \frac{r_{\odot}}{r}\right]  ^{\gamma}\left[
\frac{1+(r_{\odot}/r_{s})^{\alpha}}{1+(r/r_{s})^{\alpha}}\right]  ^{\left(
\beta-\gamma\right)  /\alpha} ,
\end{equation}
where $\rho_{\odot}\simeq0.4$ GeV/cm$^{3}$ \cite{rca10} is the DM density in
the solar vicinity and $r_{\odot}=8.33$ kpc is the distance of the solar
system from the Galactic center. The profile parameters $\alpha,\beta
,\gamma,r_{s}$ are summarized in Table II. The Einasto profile is
$\rho_{\text{Einasto}}(r)$ =$\rho_{\odot}\exp\left[  -\left(  2/\alpha
)[(r/r_{s})^{\alpha}-1\right)  \right]  ]$ with $r_{s}=20$ kpc and
$\alpha=0.17$. As is well known, the NFW and Moore profiles exhibit a cusp at
the center of Galaxy.

\begin{table}[t]
\caption{The dark matter density parameters.
\begin{ruledtabular}
\begin{tabular}{cccccc}
$\text{Halo model}$ & $\alpha$  & $\beta$& $\gamma $ & $r_s$ \text {(kpc)}  \\
\hline
$\text{Navarro, Frenk, White}$ & $1$ & $3$&$1$ & $20$ \\
$\text{Moore}$ & $1.5$ & $3$&$1.5$ & $28$\\
$\text{cored Isothermal}$ & $2$ & $2$&$0$ & $5$  \\
\end{tabular}
\end{ruledtabular}\end{table}

The charged particles produced by the DM annihilation are predicted to come
from the halo near the Sun, not too far from the Sun at least, because they
may lose the energy while propagating through the Galactic halo. They are
deflected by the Galactic magnetic field, and this property has been described
by space diffusion \cite{fca02}. The charged particles suffer energy losses
from synchrotron radiation and inverse Compton scattering. The solar
modulation can also induce a certain amount of energy loss. Their energy
spectrum at the Earth, therefore, differs from the one produced at the source.
The equation that describes the evolution of the energy distribution for the
charged particles may be expressed a
\begin{align}
\frac{\partial}{\partial t}\left(  \frac{dn}{dT}\right)  _{a}-\nabla
\cdot\left(  K\left(  T\right)  \nabla\left(  \frac{dn}{dT}\right)
_{a}\right)  -  &  \frac{\partial}{\partial T}\left(  b\left(  T\right)
\left(  \frac{dn}{dT}\right)  _{a}\right)  +\frac{\partial}{\partial z}\left(
sign(z)V_{C}\left(  \frac{dn}{dT}\right)  _{a}\right) \nonumber\\
&  =Q_{a}\left(  \mathbf{x},T\right)  -2h\Gamma_{ann}\delta\left(  z\right)
\left(  \frac{dn}{dT}\right)  _{a},
\end{align}
where $dn/dT$ is the number density of particles per unit volume and energy.
The second term accounts for the space diffusion with the energy dependent
diffusion constant $K\left(  T\right)  =K_{0}\left(  T/\text{GeV}\right)  $.
The energy loss due to synchrotron radiation in the Galactic magnetic field
and inverse Compton scattering on CMB photons and on Galactic starlight is
described in the third term. The rate of energy loss is $b\left(  T\right)
=T^{2}/($GeV $\tau_{T})$, where $\tau_{T}=10^{16}$ s is the energy loss time.
The fourth term is the effect of convective wind. The last term accounts for
the annihilation of the produced matter(s) in the interstellar medium, H and
He atoms, with annihilation rate $\Gamma_{ann}$ in the disk of thickness
$2h\simeq0.2$ kpc, and hence it is provided as a negative source term. The
relevant coefficients were parameterized, and the established (transport)
parameters were estimated from the analysis of observed isotope ratios in
cosmic rays, primarily the boron to carbon (B/C) ratio \cite{fdo04}. Three
propagation models have been featured with the established parameters, and
these propagation models correspond to minimal (MIN), medium (MED) and maximum
(MAX) antiproton fluxes \cite{fdo04,tde08}.

The number density $dn/dT$ is obtained by solving Eq. (4) with the steady
state condition $\frac{\partial}{\partial t}\left(  \frac{dn}{dT}\right)
_{a}=0$ and boundary conditions in a two-zone model \cite{fdo01}, where the
region of diffusion of cosmic rays is represented by a thick disk of thickness
$2L\simeq5-20$ kpc and radius $R\simeq20$ kpc, and the thin Galactic disk lies
in the middle and has thickness $2h$, radius $R$. The boundary conditions are
such that the number density vanishes at $z=\pm L$ and at $r=R$.

\subsection{The positron Channels}

The energy spectrum of positrons is obtained by solving the diffusion
equation, keeping only contributions of space diffusion and energy losses
\begin{equation}
-K\left(  E\right)  \nabla^{2}\left(  \frac{dn}{dE}\right)  _{e^{+}
-\frac{\partial}{\partial E}\left(  b\left(  E\right)  \left(  \frac{dn
{dE}\right)  _{e^{+}}\right)  =Q_{e^{+}}\left(  \mathbf{x},E\right)  ,
\end{equation}
with the relevant parameters listed in Table III. This diffusion equation may
be solved by the Green function formalism or the Bessel-transform method, and
the solution results in the following form
\begin{equation}
\left(  \frac{dn}{dE}\right)  _{e^{+}}=\frac{1}{b\left(  E\right)  }\int
_{E}^{M}dE_{S}Q_{e^{+}}\left(  \mathbf{x}_{\odot},E_{S}\right)  I_{e^{+
}(E,E_{S}),
\end{equation}
where $\mathbf{x}_{\odot}$ is the location of the Sun from the Galactic
center. The function $I_{e^{+}}(E,E_{S})$ must fully encode the Galactic
astrophysics from the input energy $E_{S}$ to energy $E(\leq E_{S})$, and the
full expression of this function can be found in Ref. \cite{tde08}.

\begin{table}[t]
\caption{Typical diffusion parameters for positrons deduced from a variety of
cosmic ray data, which yield the minimum (MIN), median (MED) and maximal (MAX)
fluxes.
\begin{ruledtabular}
\begin{tabular}{ccccc}
$\text{Model}$ & $\delta$  & $K_0 \text{[kpc$^2$/Myr]}$& $L \text{(kpc)}$ \\
\hline
$\text{MIN}$ & $0.55$ & $0.00595$& $1$  \\
$\text{MED}$ & $0.70$ & $0.0112$& $4$ \\
$\text{MAX}$ & $0.46$ & $0.0765$& $15$ \\
\end{tabular}
\end{ruledtabular}\end{table}

The positron flux is then given b
\begin{equation}
\phi_{e^{+}}=B\frac{v_{e^{+}}}{4\pi}\left(  \frac{dn}{dE}\right)  _{e^{+
}=B\frac{v_{e^{+}}\left\langle \sigma v\right\rangle _{f}}{16\pi b\left(
E\right)  }\left(  \frac{\rho_{\odot}}{M}\right)  ^{2}\int_{E}^{M
dE_{S}\left(  \frac{dN^{f}}{dE}\right)  _{e^{+}}I_{e^{+}}(E,E_{S}),
\end{equation}
where $v_{e^{+}}$ is the velocity of the positron.

The enhancement may come from subhalo structure (dark clumps), and the
non-perturbative Sommerfeld effect because one of the force carriers is photon
in this scenario. The enhancement by the Sommerfeld effect can be calculated
from the original form \cite{aso31}. We can split the dipole operator in
energy and momentum dependent parts by the familiar Gordon decomposition
\begin{equation}
\mu\overline{v}(p^{\prime})\sigma^{\mu\nu}q_{\nu}u(p)=i\mu\overline
{v}(p^{\prime})(2M\gamma^{\mu}+(p^{\prime}-p)^{\mu})u(p).
\end{equation}
In the energy dependent part, we have the same type of coupling with the DM
mass dependence as for electric charge. The original Sommerfeld enhancement
factor i
\begin{equation}
S=\frac{\pi\alpha_{\mu}/v}{1-e^{-\pi\alpha_{\mu}/v}}\overset{\alpha_{\mu}\gg
v}{\sim}\frac{\pi\alpha_{\mu}}{v},
\end{equation}
where $v$ is the DM velocity $(\sim10^{-3})$. In the original form,
$\alpha_{\mu}$ is the electric fine structure constant, but $\alpha_{\mu
}=16\pi\mu^{2}M^{2}$ in this case. We do not find series of resonances because
the potential is not localized. We have the enhancement $S\simeq16$ for the DM
mass of 100 GeV and magnetic dipole $\mu\simeq0.1$ TeV$^{-1}$. According to
the recent work \cite{Trs09,Sga09}, DMs annihilating after recombination may
contribute to the CMB anisotropy spectrum, and the enhancement bound could be
set up. The CMB bound is $S<\left(  120/f\right)  \left(  M/\text{TeV}\right)
$ where the parameter $f$ indicates the average fraction of energy absorbed by
the gas and depends on final states. The bound is slightly over $17$ for
$e^{\pm}$ final state, $f\simeq0.7.$ Our enhancement factor lies very near the
estimated bound. We can also consider an enhancement from the metastable bound
state between DMs, called ``WIMPonium". This production process from DMs
annihilation is, however, kinematically forbidden, because the estimated
kinetic energy $(\sim Mv^{2})$ with velocity $v$ $\sim10^{-3}$ is too small to
incorporate the binding energy $(\sim M\alpha_{\mu}^{2})$.

The strength of magnetic dipole is, according to Ref. \cite{heo09}, almost
constant for the DM mass larger than 100 GeV, and thereby the enhancement
increases in the square of the DM mass. Due to this reason, the fluxes, Eq.
(7), must have the similar magnitude about the DM masses larger than 100 GeV
because the fluxes are not scaled by the DM mass. We notice that the fluxes
are scaled by the DM mass in most of the models, and large boost factors have
been required for larger DM mass because the fluxes decrease with the square
of the DM mass

\begin{figure}
[ptb]
\begin{center}
\includegraphics[
height=4.1641in,
width=6.1834in

{fig1.eps
\caption{Positron fraction from the annihilation of the fermionic dark matter
particle. The boost factors have been chosen to provide qualitatively good
fits to the data with dark matter mass 100 GeV. Shown are the background in
solid-line and the experimental datasets of the PAMELA \cite{oad09}, HEAT
\cite{swb97}, AMS-01 \cite{mag07}, Fermi LAT \cite{mac12}, and CAPRICES94
\cite{mbo00}.
\end{center}
\end{figure}

The positrons are affected by solar wind and lose energy while transporting in
the solar system. This effect leads to a shift in the energy distribution
between the interstellar spectrum (IS) and the spectrum at the top of the
atmosphere (TOA). This modulation is considered for the predicted fluxes by
the relation in force field approximation \cite{ljg68}
\begin{equation}
\phi_{e^{+}}^{\text{TOA}}(E)=\frac{E^{2}-m_{e^{+}}^{2}}{\left(  E+\left\vert
Z\right\vert e\Phi\right)  ^{2}-m_{e^{+}}^{2}}\phi_{e^{+}}^{\text{IS
}(E+\left\vert Z\right\vert e\Phi),
\end{equation}
where $\left\vert Z\right\vert $ is the magnitude of electric charge (1 in
this case), $e$ is the electric constant and $\Phi$ is the Fisk potential,
namely solar modulation parameter, which varies between 500 MV and 1.2 GV over
the eleven-year solar cycle. Since experiments are usually undertaken near
solar minimum activity, we choose $\Phi=600$ MV (the Fisk potential for the
PAMELA experiment) for our numerical analysis.

We show in Fig. 1 the predicted positron fractions for the boost factors,
$B=30,50,80$, with the computed background\footnote{Secondary positrons from
nuclear interactions of cosmic ray nuclei with interstellar gas have been
investigated in detail by the authors of Ref. \cite{ivm98}. Recently, it has
been suggested that the secondary positrons are increased by up to 60 per cent
at high energies above 100 GeV \cite{jla11}, based on analysis of the spectral
hardening in the cosmic ray proton and helium fluxes reported by the ATIC2
\cite{pan09} and CREAM \cite{ahn10} balloon experiments.} from Ref.
\cite{ivm98} and several experimental data sets. The predicted fractions have
almost no difference\footnote{If we only consider the fractions or fluxes of
signals excluding the background, there are sizable differences at the low
energies less than 10 GeV, especially in diffusion models} in the halo
profiles or the diffusion models. The predicted fractions exhibit a rather
sharp distribution at $E_{e^{+}}\simeq M$, since our candidate can directly
annihilate into electron and positron pair. The PAMELA \cite{oad09} have shown
a steep rise in the $10-100$ GeV range in their measurements and confirmed the
results of HEAT \cite{swb97} and AMS-01 \cite{mag07} experiments. Recently,
the steep rise has been extended to 200 GeV with three more data points over
100 GeV at the Fermi LAT \cite{mac12}. The predicted signals with the boost
factor $30-80$ nicely fit measurements of the PAMELA for the DM mass of around
100 GeV. An enhancement of about a factor of 16 comes from the Sommerfeld
effect, and the rest, an enhancement factor of $2-5,$ is expected to come from
subhalo structure (dark clumps). The existence of subhalos is a generic
prediction of the $\Lambda$CDM scenario of structure formation in the
Universe, and high resolution simulations \cite{jdi08} show that the large
scale structures form by continuous merging of smaller hallos which could be
in the form of subhalos. The contribution of subhalos to the flux could be
constrained from analysis of CMB data which do not rely on uncertain
assumptions of the DM distribution \cite{lpi11}. The subhalo boost factor has
been predicted to be ten \cite{lpi11} at most. Extracting the accurate
formalism of this boost factor is out of scope of this paper. Our subhalo
boost factors must be in the reasonable range.

We also have the difference between predictions and experimental measurements
or background at low energies ($\leq10$ GeV). It has been noticed that solar
modulation effect we consider has no charge-sign dependence and has to be
modified. This must be a future study.

\subsection{The antiproton channels}

The propagation of antiprotons is dominated by diffusion and the effect of the
Galactic wind. The energy spectrum of antiprotons is obtained by solving the
following diffusion equation:
\begin{equation}
\frac{\partial}{\partial z}\left(  sign(z)V_{C}\left(  \frac{dn}{dT}\right)
_{\overline{p}}\right)  -\nabla\cdot\left(  K\left(  T\right)  \nabla\left(
\frac{dn}{dT}\right)  _{\overline{p}}\right)  =Q_{a}\left(  \mathbf{x
,T\right)  -2h\Gamma_{ann}\delta\left(  z\right)  \left(  \frac{dn
{dT}\right)  _{\overline{p}},
\end{equation}
with the relevant parameters listed in Table IV. An important difference with
the positron case is that energy loss of antiprotons is negligible, because
antiprotons are more massive and hence it is absent in the diffusion equation (11).

\begin{table}[t]
\caption{Typical diffusion parameters for anti-protons deduced from a variety
of cosmic ray data, that yield the minimum (MIN), median (MED) and maximal
(MAX) fluxes.
\begin{ruledtabular}
\begin{tabular}{ccccc}
$\text{Model}$ & $\delta$  & $K_0 \text{[kpc$^2$/Myr]}$& $V_C \text{(km/s)}$ & $L \text{(kpc)}$ \\
\hline
$\text{MIN}$ & $0.85$ & $0.0016$&$13.5$ & $1$  \\
$\text{MED}$ & $0.70$ & $0.0112$&$12$ & $4$ \\
$\text{MAX}$ & $0.46$ & $0.0765$&$5$ & $15$ \\
\end{tabular}
\end{ruledtabular}\end{table}

The antiproton flux is then given b
\begin{equation}
\phi_{_{\overline{p}}}=B\frac{v_{_{\overline{p}}}}{4\pi}\left(  \frac{dn
{dT}\right)  _{\overline{p}}=B\frac{v_{\overline{p}}\left\langle \sigma
v\right\rangle _{f}}{16\pi}\left(  \frac{dN^{f}}{dT}\right)  _{\overline{p
}\left(  \frac{\rho_{\odot}}{M}\right)  ^{2}I_{\overline{p}}\left(  T\right)
,
\end{equation}
where the function $I_{\overline{p}}\left(  T\right)  $ encodes all the
astrophysics, and the full expression can be found in Refs. \cite{pch96,lbe99}

\begin{figure}
[ptb]
\begin{center}
\includegraphics[
height=4.1641in,
width=6.1834in

{fig2.eps
\caption{Antiproton over proton ratio as a function of kinetic energy. Shown
are the background in solid-line and the experimental data sets of the PAMELA
\cite{oad10}, BESS \cite{yas02}, CAPRICE98 \cite{mbo01} and IMAX \cite{jwm96}.

\end{center}
\end{figure}

Fig. 2 shows the predicted ratios of antiproton over proton with the computed
background \cite{aml05} and several experimental measurements. As in the case
of the positrons, the predictions have almost no difference in the halo
profiles, but they are sensitive to the propagation models. The predictions in
the MIN propagation model are selected, and they may be within the invisible
range for current detectors. However, this scenario is most likely ruled out
for other propagation models\footnote{To be strict, the MED propagation model
can also be viable because of the uncertainty of the transport parameters. The
parameters are established by the best fit of cosmic ray B/C data
\cite{fdo04}. However, the assigned values of transport parameters may differ
by one order of magnitude.}, MED and MAX, because the predicted fluxes in MED
or MAX propagation model are about ten or one hundred times larger than the
ones in the MIN propagation model.

\subsection{The gamma-ray channels}

The production of gamma-rays has been considered to be a very important
channel to search for the DM signals, since they travel in straight lines and
can travel greater distances without energy loss. For these reasons, they
contain spectral and directional information that can be well measured. The
gamma-ray flux from the DM annihilations at a given photon energy from a
direction that forms an angle $\psi$ between the direction of the Galactic
center and that of observation is accounted for by the line-of-sight (los)
integration method
\begin{equation}
\phi_{\gamma}=B\frac{\left\langle \sigma v\right\rangle _{f}}{16\pi M^{2
}\left(  \frac{dN^{f}}{dE}\right)  _{\gamma}\int_{\text{los}}\rho^{2
(r(s,\psi))ds,
\end{equation}
where $r=\sqrt{r_{\odot}^{2}+s^{2}-2r_{\odot}s\cos\psi}$ is the Galactocentric
distance. In terms of the galactic latitude $b$ and longitude $l$, one has
$\cos\psi=\cos b$ $\cos l$. The coordinate $s$ parameterizes the distance from
the Sun along the los.

This form can be reduced t
\begin{equation}
\phi_{\gamma}\simeq8.31\times10^{-11}\text{ }\left(  \text{cm}^{2
\cdot\text{sr}\cdot\text{s}\right)  ^{-1}\cdot\frac{B\left\langle \sigma
v\right\rangle _{f}}{10^{-26}\text{ cm}^{3}\text{s}^{-1}}\left(
\frac{100\text{ GeV}}{M}\right)  ^{2}\left(  \frac{dN^{f}}{dE}\right)
_{\gamma}\cdot\overline{J}_{\Delta\Omega}~,
\end{equation}
where $\overline{J}_{\Delta\Omega}$ is a dimensionless los integral averaged
over the solid angle $\Delta\Omega$ and is defined b
\begin{equation}
\overline{J}_{\Delta\Omega}=\frac{1}{\Delta\Omega}\int J\left(  \psi\right)
d\Omega~,
\end{equation}
wit
\begin{equation}
J\left(  \psi\right)  =\int\frac{\rho^{2}(r)}{\rho_{\odot}^{2}}\frac
{ds}{r_{\odot}}~.
\end{equation}

\begin{figure}
[ptb]
\begin{center}
\includegraphics[
height=1.2021in,
width=4.9692in

{fig3.eps
\caption{Feynman diagrams for annihilation of dark matter to one-photon or
two-photons. The hatched circles indicate the dipole couplings.
\end{center}
\end{figure}

In addition to the continuum emission, the direct DM annihilations produce
$\gamma\gamma$ and $\gamma Z$ final states\footnote{The production of the
single $\gamma$ or $Z$ final state is prohibited, because in this process it
is impossible to conserve energy and momentum together.} in this scenario, in
which Feynman diagrams are shown in Fig. 3. Such processes would yield the
very distinctive feature of monoenergetic gamma-ray lines (monochromatic
photons) with an energy $E_{\gamma}=M$ or $M\left(  1-m_{Z}^{2}/(2M)^{2
\right)  $. The full annihilation rates for the production of $\gamma\gamma$
are $\left\langle \sigma v\right\rangle _{\gamma\gamma}=\mu^{4}M^{2}/2\pi$ and
$\left\langle \sigma v\right\rangle _{\gamma Z}=\mu^{4}M^{2}\beta_{Z}^{2
\tan^{2}\theta_{W}/4\pi$ for $\gamma Z.$ The contributions for $\gamma\gamma$
and $\gamma Z$ final states are suppressed by the magnetic dipole $\mu^{4}$,
and the $\gamma Z$ final state has an additional suppression with Weinberg
angle $\tan^{2}\theta_{W}$.

We restrict our analysis to the possible signals from the Galactic halo for a
complementarity in the case of the positrons, and the same boost
factors\footnote{It has been predicted that the possible enhancement from
subhalo has an angular and/or energy dependence on the cosmic rays. The
enhancement can be different in each channel, but there are still no clear
experimental evidences for dark clumps. We select the same boost factors,
$B=30-80,$ for a reference. The main idea of the enhancement from subhalo
structure comes from $\left\langle \rho^{2}\right\rangle \geq\left\langle
\rho\right\rangle ^{2}$.} $B=30,50,80$ are selected. We compare our
predictions to experimental observations in two stringent cases. One is a
process that may contribute to the extragalactic gamma-ray background (EGB).
The other is, in the case that astrophysical sources account for the EGB in
the entire energy range, a process which satisfies the experimental exclusion
limit of the Fermi LAT \cite{aaa10}.

Fig. 4 shows the predicted gamma-ray spectra as a function of photon energy in
the region $0^{\circ}\leq\ell\leq360^{\circ},\left\vert b\right\vert
\geq10^{\circ}$. The spectra are superpositions of the continuum and
monoenergetic gamma-rays at the DM mass of 100 GeV. The spectra have almost no
difference in the halo profiles. The EGB from the Fermi LAT \cite{aaa10} is
given by
\begin{equation}
E_{\gamma}^{2}\phi_{\gamma}\simeq5.5\times10^{-7}\left(  \frac{E}{1\text{GeV
}\right)  ^{-0.41}\text{ }\left(  \text{cm}^{2}\cdot\text{sr}\cdot
\text{s}\right)  ^{-1}\text{GeV.
\end{equation}
The same type of background from the analysis of the EGRET measurements is
also described \cite{awst04} from the first analysis \cite{psr98}. The
predicted spectra are not exceeding the EGB, and we might have a signature if
it can be disentangled from astrophysical ones. In addition, we check if our
prediction can account for the EGRET anomaly\footnote{Although this anomaly is
likely caused by a systematic error of the effective detector sensitivity
calibration \cite{fws08}, we include this anomaly in our analysis for a
possible signal in case.} which is not confirmed at the Fermi LAT. Our
predictions are too soft to explain the observation of the EGB, and the more
enhancement would be needed

\begin{figure}
[ptb]
\begin{center}
\includegraphics[
trim=0.000000in 0.000000in 0.000000in -0.138416in,
height=3.5129in,
width=4.8179in

{fig4.eps
\caption{The gamma-ray spectrum from Galatic halo for dark matter mass of 100
GeV and the region, $0^{\circ}\leq\ell\leq360^{\circ},\left\vert b\right\vert
\geq10^{\circ}$. Shown are the experimental data sets from the Fermi LAT and
EGRET with their fitted spectral indices.
\end{center}
\end{figure}

The predicted fluxes have to be within the uncertainty of Fermi LAT data, in
the case that the EGB is accounted for by astrophysical sources in the entire
energy range. Fig. 5 shows the predicted fluxes with 90\%, 95\% and 99\% C.L.
experimental limits of the Fermi LAT, which are estimated from the data table
in Ref. \cite{aaa10}. The predictions are smaller than the exclusion limits,
and so satisfy the current experimental constraint

\begin{figure}
[ptb]
\begin{center}
\includegraphics[
trim=0.000000in 0.000000in 0.000000in -0.114324in,
height=3.5129in,
width=4.8179in

{fig5.eps
\caption{The gamma-ray flux from Galatic halo for dark matter mass of 100 GeV
and the target region $0^{\circ}\leq\ell\leq360^{\circ},\left\vert
b\right\vert \geq10^{\circ}$ with 90\%, 95\% and 99\% C.L. experimental limits
of the Fermi LAT.
\end{center}
\end{figure}

The Galactic center or the region close to it must be the most complex region
in the Galaxy due to many possible sources and the difficulty to model the
diffuse emission. Hence, it may be very difficult to disentangle possible DM
annihilation signals from the background fluxes. The monochromatic gamma-ray
lines appearing from DM annihilations could provide smoking-gun signatures for
these regions, because the line signals mostly cannot be mistaken for
astrophysical source.

Fig. 6 shows the predicted annihilation rate for $\gamma\gamma$ and $\gamma Z$
final states as a function of the DM mass. The Sommerfeld effect is included,
but not the subhalo structure. The strength of DM magnetic dipole is chosen to
satisfy the relic density from Ref. \cite{heo09}. The curves given for the
NFW, Einasto, and Isothermal DM distributions are 95\% C.L. upper limits of
Fermi LAT \cite{mack12} for the region of $\left\vert b\right\vert
\geq10^{\circ}$ plus a $20^{\circ}\times20^{\circ}$ square centered at the
Galactic center. The predicted annihilation rates are approximately
$4.0\times$ $10^{-29}$ cm$^{3}/$s at $E_{\gamma}\simeq100$ GeV for the
$\gamma\gamma$ final state and $4.0\times10^{-30}$ cm$^{3}/$s for the $\gamma
Z$ final state at $E_{\gamma}\simeq80$ GeV. Otherwise, the estimated upper
limits at the Fermi LAT are $10^{-27}-10^{-26}$ cm$^{3}/$s for both final
states, depending on the DM mass and the halo profiles. The predictions are
two or three orders smaller than the experimental upper limits. However, our
predictions can be enhanced if dense DM clumps are considered in regions close
to the Galactic center. Recently, the authors of Refs. \cite{cwe12,ete12}
pointed out the gamma-ray excess, $1-3\times10^{-27}$cm$^{3}$/s, around 130
GeV in the spectrum based on these measurements with 4.5 or 6$\sigma$
statistical significance. Our predictions with the subhalo boost factor of
about 100 can account for the gamma-ray excess

\begin{figure}
[ptb]
\begin{center}
\includegraphics[
trim=0.000000in 0.000000in -0.178258in 0.000000in,
height=4.1658in,
width=6.1834in

{fig6.eps
\caption{Annihilation rates with Sommerfeld enhancement factor $S$ for dark
matter annihilation to $\gamma\gamma$ or $\gamma Z$. Shown are the 95\% C.L.
upper limits of the Fermi LAT for the region of $\left\vert b\right\vert
\geq10^{\circ}$ plus a $20^{\circ}\times20^{\circ}$ square centered at the
Galactic center.
\end{center}
\end{figure}

One of the reasons we mostly get the low predictions for sharp peaks is due to
the relatively poor energy resolution. The current energy resolution of the
Fermi LAT is $11-13\%$ \cite{mack12} in the full width at half maximum. The
energy resolution could be as good as $1.5-2\%$ for a planned experiment,
AMS-02 \cite{ako10}, in which the upper limits of annihilation rate will be
$10^{-30}-10^{-29}$ cm$^{3}/$s. Our predicted signals are in the potential
probe at the AMS-02.

\section{Conclusion}

We considered cosmic ray signals in the positron, antiproton and photon
channels of dipole-interacting DM annihilation. The predicted signals in the
positron channel could nicely account for the excess of positron fraction from
Fermi LAT, PAMELA, HEAT and AMS-01 experiments for the DM mass larger than 100
GeV with a boost factor of $30-80$. An enhancement of about a factor of 16
could come from the Sommerfeld non-perturbative effect and the rest, an
enhancement factor of $2-5$, from subhalo structure (dark clumps). The
predicted signals have almost no dependence on the DM mass because of the
Sommerfeld effect. No excess of antiproton over proton ratio at the
experiments also gives a severe restriction for our scenario. This scenario
may be viable for MIN Galactic propagation model, but likely ruled out for
other propagation models, MED and MAX. The predicted signals from the Galactic
halo in the region $0^{\circ}\leq\ell\leq360^{\circ},\left\vert b\right\vert
\geq10^{\circ},$ and signals as the monoenergetic gamma-ray lines
(monochromatic photons) for the region ($\left\vert b\right\vert \geq
10^{\circ}$ plus a $20^{\circ}\times20^{\circ}$ square centered at the
Galactic center) close to the Galactic center were also considered. The
predicted signals from the Galactic halo must satisfy the current experimental
constraint, and the signals for the region near the Galactic center as
monoenergetic lines must be smaller than the experimental exclusion limits of
the Fermi LAT. The gamma-ray excess $1-3\times10^{-27}$cm$^{3}$/s around 130
GeV, pointed out by the authors of Refs. \cite{cwe12,ete12}, could be
accounted for in this scenario, with the subhalo boost factor of about 100.
Our predicted signals as monoenergetic lines for the region near Galactic
center are also in the potential probe at the planned experiment, AMS-02, with
the better experimental method.

\begin{acknowledgments}
The work was supported by the National Research Foundation of Korea (NRF)
grant funded by Korea government of the Ministry of Education, Science and
Technology (MEST) (No. 2011-0017430) and (No. 2011-0020333).
\end{acknowledgments}


\section{Introduction}
\label{intro}

\blfootnote{
    %

    %
    \hspace{-0.65cm}

    \textbf{\textit{*}} Equal contribution.\\
    This work is licensed under a Creative Commons Attribution 4.0 International Licence. Licence details: \url{http://creativecommons.org/licenses/by/4.0/}
    %

    %

    %

}

Definition Extraction from free text (DEFT) \cite{shirani2020semeval}  involves finding term definition pairs from free and semi-structured texts, especially those whose term-definition span crosses a sentence boundary and those which do not have a definition phrase. An example of a cross boundary sentence from the DEFT corpus is given below:
\vspace{0.3cm}
\begin{mdframed}
In doing so , $<$DEF$>$ monomers release water molecules as byproducts
$<$DEF$>$ (1). This type of reaction is known as $<$TERM$>$ dehydration
synthesis $<$TERM$>$ , which means “ $<$QUALIFIER$>$ to put together while losing water $<$QUALIFIER$>$ . ”(0)
\end{mdframed}
\vspace{0.3cm}
It can be observed that the definition (enclosed by DEF tags) and the corresponding terms (enclosed by TERM tags) are present in different sentences, thus increasing the difficulty of definition extraction. The task is thus relatively different and complex from the conventional definition extraction (DE) task in which a definition could be broken into the following sub-parts:

\begin{enumerate}
	\item The DEFINIENDUM field (DF) i.e., the word being defined.
	\item The DEFINITOR field (VF) i.e., the verb phrase used to introduce the definition.
	\item The DEFINIENS field (GF) i.e., genus phrase or the hypernym.
	\item The REST field (RF) i.e., additional clauses that help to distinguish the definiendum from its genus.
\end{enumerate}

and thus easily be captured by common verb phrases (DEFINITOR) like “means”, “refers to”, “is”, etc. These kind of conventional definitions could easily be tagged as follows:

\vspace{0.3cm}

\begin{mdframed}

$<$DF$>$Photosynthesis$<$/DF$>$ $<$VF$>$is$<$/VF$>$ $<$GF$>$the process $<$/GF$>$ $<$RF$>$by which green plants manufacture food. $<$/RF$>$
\end{mdframed}
\vspace{0.3cm}

In the example presented above, the definition (REST field) and the corresponding term (DEFINIENDUM field) are present in the same sentence. Moreover, the presence of DEFINITOR(s) in the text also eases the task of extracting such definitions.

\newpage

\noindent
The order of the tags in the WCL Corpus \cite{navigli2010annotated}, one of the conventional corpus for Definition Extraction task, is predefined viz. DEFINIENDUM, DEFINITOR, DEFINIENS, REST. On the other hand, there is no such predefined order for the DEFT corpus regarding the occurrence of BIO Tags. This absence of order makes it difficult for the models to identify the definitional sentences in the DEFT corpus and then associate the words present in the definitions with the appropriate tags.

Another observation is the variation in the pattern of the definitional sentences in the DEFT corpus due to the presence of the heterogeneous distribution of tags. Some tags like ‘O’,‘I-Term’,‘I-Definition’, etc. are very frequent whereas tags such as ‘Alias-Term’, ‘Secondary-Definition’ are rarely seen. This imbalance in dataset causes difficulty in finding a structure for the definition and hence makes the DEFT corpus relatively more complex when compared to conventional corpora like WCL.

Thus, given the complexity of the definitions present in the DEFT corpus, the whole pipeline for the extraction of meaningful term-definition pairs from free text can be restructured as presented in Figure \ref{fig:pipeline}.

\begin{figure}[H]
\center
\includegraphics[width=\textwidth]{pipe.eps}
\caption{Pipeline for Definition Extraction}
\label{fig:pipeline}
\end{figure}

\noindent
Since the corpus is already in the form of sentences, sentence segmentation is not required. For tagging tokens, authors of DEFT Corpus \cite{spala2019deft} have defined a new annotation scheme, a part of which has been touched upon in the introduction section. Moreover, it can be found in the corresponding paper.

In this paper, we present the approach used for the task ``DeftEval: Extracting term-definition pairs in free text" of SemEval 2020. The subtasks are as follows:

\begin{enumerate}
\item Subtask 1 - Sentence Classification
\item Subtask 2 - Sequence Labeling
\item Subtask 3 - Relation Classification

\end{enumerate}

Our presented methods for the sentence classification subtask (Subtask 1) revolve around the Transformer \cite{vaswani2017attention} based model, Bidirectional Encoder Representations from Transformers (BERT) \cite{devlin2018bert}. Experiments \footnote{Corresponding submissions have been made under the username of {\tt mler} on Codalab} have been done to improve the results and in the process we have come up with improvised architectures.

The rest of the paper is organized as follows: Related work with its limitations has been discussed in Section \ref{sec:relwork}, followed by a description of the data used in Section \ref{sec:data}. The proposed methods have been elaborated in Section \ref{sec:methods} \footnote{Source code available at \url{https://github.com/dsciitism/SemEval-2020-Task-6} }. Section \ref{sec:results} and \ref{sec:error} contains the results and error analysis respectively. Section \ref{sec:end} concludes the paper and also includes the possible future work.

\section{Related Work}
\label{sec:relwork}
The very first attempt at DE was by \newcite{kobylinski2008definition}. They used a “Balanced Random Forest classifier” to classify definitions so as to handle the imbalance of the dataset. A major contribution to DE was by \newcite{navigli2010learning}, by using a generalization of word lattices to model textual definitions in the form of Definiendum, Definitor, Definiens and Rest. However, this approach is unable to generalise definitions, especially the ones that defy the conventional semantics i.e. common patterns found in definitional sentences.

\noindent
Many existing works take advantage of linguistic features like syntactic dependencies. \newcite{jin2013mining} built DefMiner which is a supervised sequence labelling system using shallow parsing and dependency features. Training CRF (Conditional Random Fields) with lexical, terminological, and structural features extracted from data has also been tried by \newcite{anke2013towards}. The use of linguistic features lead to promising results and observations. \newcite{espinosa2014applying} took advantage of the subtrees in the dependency parsing of a sentence. Feature representation of sentences were created using these subtrees. \newcite{espinosa2015definition} incorporated both linguistic and semantic features of sentence for training classifiers. SensEmbed is used to reveal the semantic compactness of definition containing sentences.

With the advent of the deep learning era, the first attempt at DE using deep learning was by \newcite{li2016definition}. Feature representation of sentences were learned using Long Short Term Memory (LSTM) cells. Text preprocessing such as replacing some selected words with POS tags, were carried out thus obtaining brilliant results. \newcite{anke2018syntactically} developed Syntactically Aware Neural Networks by incorporating syntactic information (syntactic dependencies and dependency labels) along with the text sentences as input. With the help of pre-trained word embeddings, convolutional filters and Bi-LSTM cells, the model was capable of extracting both short range and long range dependencies from the text data.

In the most recent attempt at Definition Extraction, \newcite{veyseh2019joint} proposed a multi-task model to perform sentence classification and sequence labelling simultaneously. The model took advantage of the entire syntactic dependency tree rather than just dependencies, thus yielding state of the art results on the WCL dataset.

\section{Dataset}
\label{sec:data}

For the purpose of training, evaluation and testing, the DEFT corpus has been used. Not only is the corpus significantly larger than the previously available corpora in the field of Definition Extraction, but the dataset also contains definitions from complex, human-annotated data across a variety of topics and from both free (textbook) and semi-structured (legal document) language. Table \ref{dataset} shows the statistics of the major datasets available in the field of DE namely, WCL, W00 \cite{jin2013mining} and DEFT.

\begin{table}[htpb]
\centering
\begin{tabular}{|l|r|c|}
\hline \bf Dataset & \bf No. of positive annotations & \bf Size(in sentences) \\ \hline
WCL & 1,871 & 4,718 \\ \hline
W00 & 731 & 2,185 \\ \hline
DEFT & 11,004 & 23,746 \\ \hline
\end{tabular}
\caption{\label{dataset} Dataset statistics }
\end{table}

By complicated structure, we mean that the corpus does not contain simple sentences of the form “X is a Y” as is the case in most of the definitions present in the WCL dataset. Instead, roughly 50\% of term-definition pairs in the dataset appear across sentence boundaries or with an otherwise complex structure (e.g., containing secondary information, containing ambiguous references to previously stated terms or definitions) whereby the relationship between a term and definition requires more deduction than finding a definition verb phrase.

\section{Methods}
\label{sec:methods}

\subsection{Data Preprocessing}

Unlike conventional approaches for text classification (Tf-Idf, Bag of Words Model, etc.), deep learning approaches require minimal preprocessing. It is often believed that preprocessing leads to the loss of information. So, for preprocessing the we have only performed the following two steps:
\begin{enumerate}
\item Removal of leading line numbers: Some of the sentences in the data have a leading line number. It has been removed since it is irrelevant to the content of the sentence and acts as noise.

\item Addition of the subject token: It has been found in some cases of our experiments that adding the subject token i.e. the subject of the textbook from which the sentence to be classified is picked, helps improve the results.
\end{enumerate}

\subsection{Model Architectures}

Initially, we started off by using the approach of Syntactically Aware Neural Networks. However, owing to the complexity of the DEFT corpus, the model was unable to perform well. Hence, we shifted towards using larger pretrained models such as Transformers. We concentrate on the performance of BERT since it has achieved the State of The Art results in many NLP tasks with  limited  fine-tuning  on task-specific training data. Moreover, \newcite{10.1007/978-981-15-6318-8_13} have shown that BERT implicitly captures syntactic dependencies, which play an essential role in Definition Extraction.

We now describe the three approaches that we have implemented and submitted to the shared task.

\begin{enumerate}

\item {\bf BERT} : BERT, based on the Transformer architecture, consists of multi-attention heads which apply sequence-to-sequence transformation on the input text sequence.
BERT incorporates the following practices for training
(a) learn to predict a masked token using the left and right context of the text sequence (Masked Language Model)
(b) learn to predict whether two sentences occur in continuation or not (Next Sentence Prediction)

For our experiments, we use the BERT (base-cased) made publicly available by Huggingface \cite{wolf2019transformers}. It consists of 12 hidden layers in the encoder of the Transformer. The encoder outputs a feature vector of 768 dimensions. We select the cased model (pre-trained on cased English text) because lower casing the data leads to loss of information. The representation corresponding to the CLS token is fed through two feed-forward linear layers, thus giving two values corresponding to the logits of the two classes. We train the model for 5 epochs using AdamW \cite{DBLP:journals/corr/abs-1711-05101} optimizer with a learning rate of 2e-5.

\item {\bf BERT with fine tuned language model} : This model is the same as the first one, except we fine tune the Masked Language Model of BERT using the training data. This helps BERT to understand the context of the corpus in a better manner. Results show that fine tuning the language model of neural networks gives improved results \cite{howard2018universal}.

To fine tune the LM, we take help of Huggingface's \textit{run\_language\_modeling.py} utility\footnote{\url{https://github.com/huggingface/transformers/tree/master/examples/language-modeling}}. The other hyperparameter settings remain the same as the first model.

\begin{figure}[ht]
\center
\includegraphics[width=0.55\textwidth]{BertGCN1.eps}
\caption{Joint model of BERT and Text Level GCN}
\label{fig:hyb}
\end{figure}

\item {\bf Joint model of BERT and Text Level GCN} : The above models do not take into account linguistic features along with the text data. Researchers have obtained superior results by incorporating linguistic information into the models. However, creating such features requires extra effort and involves the use of dependency parsers, which may lead to error propagation (error in extracting linguistic information will lead to feeding of erroneous information to the model). Graph Convolutional Networks (GCN) \cite{kipf2016semi} have yielded superior performance on graph-structured data. Since text can also be represented as graph data (for example syntactic dependency trees), we use the feature representation obtained from a Text Level Graph Convolutional Network \cite{huang2019text} along with the feature representation obtained from BERT, to solve our task. As shown in Figure \ref{fig:hyb}, both the representations are concatenated and a linear layer is used to output the logits of the two classes.

\begin{figure}[ht]
\center
\includegraphics[width=0.35\textwidth]{TL_GCN.eps}
\caption{Inner Working of Text Level Graph Convolutional Networks}
\label{fig:gcn}
\end{figure}

In a GCN, data is represented as graph(s) and each node of the graph is represented by a set of attributes. A Text Level GCN represents a text sequence as a graph. Each token of the sentence represents a node and has its corresponding word vector as attributes. A token has an edge between its n-grams to the left and right of it (n is also referred as window size). The greater the value of n, greater is the range of dependencies that are captured. Figure \ref{fig:gcn} shows the corresponding edges for a token (T2) of a text sequence with 4 tokens, considering a window size of 2.

The weights of the edges are learned through training. The weighted representations (product of edge weight and word vector) is sent to its neighbours by each node. Each node gathers information from its neighbouring nodes, and uses it to update its own representation (i.e. attributes) and edge weights. The values of the edge weights represent the significance of the dependency between a token and one of its n-grams. Thus, the model understands the difference between strong and weak dependencies without any human intervention.

The language model of the BERT component of the model is not fine tuned. We use the publicly available pre-trained weights. The Text Level GCN component uses pre-trained GloVe word embeddings \cite{pennington2014glove} as node attributes. We use a window size of 5 in our experiments. The model is trained for 5 epochs using AdamW with learning rate 2e-5.

\end{enumerate}

\section{Results}
\label{sec:results}

We evaluate the performance of the three models on the validation set and the test set. The evaluation metric is F1 score on the positive class. The submissions of the test set were done using a 10-fold cross validation so as to increase the robustness of the results. The results are presented in Table \ref{results}.

\begin{table}[htbp]
\centering
\begin{tabular}{|l|l|l|}
\hline \bf Model & \bf Validation Set & \bf Test Set \\ \hline
BERT & 0.74 & 0.731 \\ \hline
BERT with fine tuned language model & 0.768 & \textbf{0.775} \\ \hline
Joint model of BERT \& GCN & \textbf{0.781} & 0.758 \\ \hline
\end{tabular}

\caption{\label{results} Comparison of results of the mentioned models }
\end{table}

The results show that fine tuning the language model of BERT leads to improved results. The fine tuned BERT achieves the greatest F1 score on the test set among the 3 models. The reason for such a performance boost can be attributed to the structure of the DEFT corpus. The corpus consists of long continuous segments from textbooks and legal documents. Since there is a lot of text in the same context, fine-tuning the language model helps BERT to get a better understanding of the domain of the corpus.

It is also evident from Table \ref{results} that the inclusion of Text Level GCN along with BERT is also beneficial. The Text Level GCN component of the joint model is capturing some extra information from the text data which is boosting the performance.

\section{Error Analysis}
\label{sec:error}

We analyze the predictions on the validation set. For the predictions of all the models, we calculate the binary cross entropy loss for each example and then sort the examples as per descending loss values. We examine the top commonly mis-classified examples by each of the 3 models (in Table \ref{table:error}).

\begin{table}[htbp]
\centering
\begin{tabular}{|p{12cm}| l|}
\hline \bf Sentence & \bf True Label \\ \hline
”United States v. Miller , 307 U.S. 174 ( 1939 ) . & 1 \\ \hline
Pathogens include bacteria , protists , fungi and other infectious organisms . & 1 \\ \hline
Toll goods are available to many people , and many people can make use of them , but only if they can pay the price . & 1 \\ \hline
\end{tabular}
\caption{\label{table:error} Common highly mis-classified sentences }
\end{table}

We observe that majority of these examples belong to the cross sentence definition scenario as mentioned in Section \ref{intro}. This implies that models are struggling to extract cross-sentence definitions. The models lack a context to the sentence to be classified, because of which it is unable to classify definitions that are covered in more than one sentence. To the human reader, the predictions of most of mis-classified sentences would appear correct. However, they are incorrect according to DEFT corpus.

An additional reason for the errors is the incorrect/ambiguous labelling of the dataset. For example, consider the sentence ‘”United States v. Miller , 307 U.S. 174 ( 1939 ) .' in table \ref{table:error}. The sentence is not a definition but due to the inverted comma (”) of the trailing sentence (which was a definition), this sentence was also labelled a definition. Similarly, the label for the sentence “Pathogens include bacteria , protists , fungi and other infectious organisms ." is debatable as it seems more of a description than a definition. Last but not the least, “Toll goods are available to many people , and many people can make use of them , but only if they can pay the price ." is clearly not a definition.

\section{Conclusion and Future Work}
\label{sec:end}

We study the performance of BERT on the DEFT corpus and tried to boost BERT with explicit linguistic information (in the form of dependencies) using a Text Level GCN model. For further improvements, we can use BERT with a fine-tuned language model as the component of the joint model. The models can also make use of a context in the form of the sentence preceding the sentence to be classified. The incorporation of a context can help models overcome the problem of misclassifying cross-sentence definitions.

\bibliographystyle{coling}


\section{Introduction}
\label{intro}

One of the most prospective non-perturbative approaches in
quantum field theory (QFT) is the functional (or exact)
renormalization group (FRG), which is based on the Wetterich
equation for the average effective action \cite{Wetterich,TimMorris}
(see the reviews
\cite{FRG-review,FRG-review-1,FRG-review-2,FRG-review-3}
and textbook \cite{Wipf} for an introduction to the subject).
The application of FRG to gauge theories was extensively
discussed, including in the recent work \cite{Wetterich-2018}.
The considerations in the last and many other papers are based
on the background field method, which enables one to maintain
the gauge invariance for the Yang-Mills (or gravitational) field
explicitly in the effective action. The background field method
is, in general, a useful formalism in the theory of gauge fields,
and that is why it attracted a very special attention recently,
see e.g. \cite{Barv,BLT-YM,FT,Lavrov}. The application of this
method to the average effective action has been done long ago
\cite{Reuter} (see also the recent work \cite{CODELLO}), but
in our opinion there are some important aspects of the problem
which should be explored in more details.

The main problem of FRG applied to the gauge theories is
that the dependence on the choice of the gauge fixing condition
does not disappear on-shell \cite{LSh}, as it is the case in
the usual perturbative QFT. As a result of the on-shell
gauge fixing dependence, the $S$-matrix of the theory is
not well defined, except at the fixed point, where the effective
average action coincides with the usual effective action. One can
expect that the renormalization group flow in the Yang-Mills theory
will also manifest a fundamental gauge dependence, and this
certainly shadows the interpretation of the results obtained
within the FRG approach in the gauge theories.

In order to better understand the situation with the gauge
symmetry at the quantum level and with the gauge dependence,
it is important to analyze the mentioned problems in the background
field method, that is the main purpose of the present communication.
In what follows we consider both gauge invariance and gauge fixing
dependence for the effective average action.

The paper is organized as follows. In Sec.~\ref{Sec2} we give a brief
description of the background field formalism in non-Abelian gauge
theories and the gauge independence of vacuum functional in this
method. In Sec.~\ref{Sec3} the background field symmetry is analyzed
within the  background field method - based functional renormalization group approach.
The regulator functions are dependent on the external (background)
fields but are chosen not to be invariant under gauge transformations
of external vector field.  In Sec.~\ref{Sec4} we present a solution
to the background field symmetry of background average effective action  with regulator functions
which are invariant under gauge transformations of the external
field. In Sec.~\ref{Sec5} the gauge dependence problem of the
background average effective action  is considered.  Finally, we discuss the results and draw
our conclusions in Sec.~\ref{Sec6}.

Our notations system mainly follows the DeWitt's book
\cite{DEWITT}. Also, the Grassmann parity of a quantity $\,A\,$
is denoted $\,\varepsilon(A)$.

\section{Background field formalism}
\label{Sec2}

We start by making a brief review of the background field formalism for
a gauge theory describing by an initial action $S_0(A)$ of fields $A=\{A^i\}$,
$\varepsilon(A^i)=\varepsilon_i$
 invariant under gauge transformations
\begin{eqnarray}
\label{YM}
S_{0,i}(A)R^i_{\alpha}(A)=0, \quad \delta
A^i=R^i_{\alpha}(A)\xi^{\alpha},
\end{eqnarray}
where $R^i_{\alpha}(A)$,
$\varepsilon(R^i_{\alpha}(A))=\varepsilon_i+\varepsilon_{\alpha}$
are the generators of gauge transformations, $\xi^{\alpha}$,
$\varepsilon(\xi^{\alpha})=\varepsilon_{\alpha}$ are
arbitrary functions. In general, a set of
fields $\,A^i=(A^{\alpha k},A^m)\,$ includes fields $A^{\alpha k}$
of the gauge sector and also fields $A^m$ of the matter sector
of a given theory. We
assume that the generators $ R^i_{\alpha}= R^i_{\alpha}(A)$
satisfy a closed algebra with structure coefficients
$F^\gamma_{\alpha\beta}$ that do not depend on the fields,
\begin{align}
\label{AlgebraAi}
R^i_{\alpha,j}R^j_{\beta} - (-1)^{\varepsilon_\alpha \varepsilon_\beta} R^i_{\beta ,j}R^j_{\alpha}  =
 - R^i_\gamma F^\gamma_{\alpha\beta},
\end{align}
where we denote the right functional derivative by $\delta_r X / \delta A^i = X_{,i}$. The structure coefficients
satisfy the symmetry properties
$\,F^\gamma_{\alpha\beta}=-(-1)^{\varepsilon_\alpha \varepsilon_\beta}F^\gamma_{\beta\alpha}$.
We assume as well that the generators are linear
operators in $A^i$, $R^i_{\alpha}(A)=t^i_{\alpha j}A^j+r^i_{\alpha}$.

We apply the background field method (BFM) \cite{DWITT,FADEEV,ABBOTT}
replacing the field $\,A^i\,$
by $\,A^{i} + \mathcal{B}^{i}\,$ in the classical action
$S_0(A)$,
\begin{eqnarray}
\n{BYM}
S_{0}(A) \,\,\longrightarrow \,\, S_{0}(A +\mathcal{B})\,.
\end{eqnarray}
Here
$\,\mathcal{B}^{i}$ are  external (background) vector fields
being not equal to zero only in the gauge sector.
The action $\,S_{0}(A +\mathcal{B})\,$ obeys the gauge invariance
in the form
\begin{eqnarray}
\n{BGI}
\delta  S_{0}(A +\mathcal{B}) = 0,
\qquad
\delta  A^{i} = R^i_{\alpha}(A+\mathcal{B})  \xi^{\alpha}\,.
\end{eqnarray}

Through the Faddeev-Popov quantization \cite{FP} the field configuration
space is extended to
\begin{eqnarray}
\n{Phi}
\phi^A = (A^i,B^{\alpha},C^{\alpha}, \bar{C}^{\alpha}),
\qquad
\varepsilon(\phi^A) = \varepsilon_A,
\end{eqnarray}
where $C^{\alpha}$, $\bar{C}^{\alpha}$ are the Faddeev-Popov ghost and antighost
fields, respectively, and $B^a$ is the auxiliary (Nakanishi-Lautrup) field.
The Grassmann parities distribution are the following
\begin{eqnarray}
\varepsilon(C^{\alpha}) = \varepsilon(\bar{C}^{\alpha}) =\varepsilon_{\alpha}+ 1,
\qquad
\varepsilon(B^{\alpha}) = \varepsilon_{\alpha}\,.
\end{eqnarray}
The corresponding Faddeev-Popov action $ S_{FP} (\phi, \mathcal{B})$
in the singular gauge fixing has the form~\cite{FP}
\begin{eqnarray}
\n{FP}
S_{FP} (\phi, \mathcal{B})
 =
 S_{0}(A +\mathcal{B})
+
S_{gh} (\phi, \mathcal{B})
+
S_{gf} (\phi, \mathcal{B})
\end{eqnarray}
where
\begin{align}
\n{Sgh}
S_{gh} (\phi, \mathcal{B})
&=
\bar{C}^{\alpha} \,
  \chi_{\alpha , i} (A, \mathcal{B}) \,
R^i_{\beta}(A+\mathcal{B}) C^{\beta},
\\
\n{Sgf}
S_{gf} (\phi, \mathcal{B})
&=
B^{\alpha}\chi_{\alpha}(A, \mathcal{B}).
\end{align}
In the last expression $\,\chi_{\alpha} (A, \mathcal{B})\,$ are functions
lifting the degeneracy for the  action $S_0(A+\mathcal{B})$. The standard
background field gauge condition in the BFM is linear in the quantum fields
\begin{eqnarray}
\n{GC}
 \chi_\alpha(A,\mathcal{B}) \,=\,F_{\alpha i}(\mathcal{B})A^i\,.
\end{eqnarray}

 The action \eqref{FP}
is invariant under the BRST symmetry \cite{BRS,T}
\begin{eqnarray}
\n{BRST}
\delta_{B} \phi^A = {s}^{A}(\phi, \mathcal{B}) \mu,
\qquad
\varepsilon\big({s}^{A}(\phi, \mathcal{B}) \big) = \varepsilon_{A} +1\,,
\end{eqnarray}
where
\begin{align}
\n{si}
s^{A}(\phi, \mathcal{B})
 = & \Big(
R^i_{\alpha}(A+\mathcal{B}) C^{\alpha},
\, 0, \,-\frac{1}{2} F^{\alpha}_{\beta\gamma} C^\gamma C^\beta (-1)^{\varepsilon_\beta },
(-1)^{\varepsilon_\alpha} B^\alpha
\Big)
\end{align}
and $\mu$ is a constant Grassmann parameter with $\,\varepsilon(\mu) = 1$.
One can write \eqref{si} as generator of BRST transformations,
\begin{eqnarray}
\hat{s}(\phi, \mathcal{B} ) =
 \frac{\overleftarrow{\delta}}{\delta \phi^{A}}
s^{A}(\phi, \mathcal{B} )
.
\label{shat}
\end{eqnarray}
Then, the action \eqref{FP} can be written in the form
\begin{eqnarray}
\n{FP+GF}
S_{FP} (\phi, \mathcal{B})
\,=\, S_{0} (A + \mathcal{B}) + \Psi(\phi, \mathcal{B} )
\, \hat{s}(\phi, \mathcal{B} )\,,
\end{eqnarray}
where
\begin{eqnarray}
\n{GFF}
\Psi(\phi, \mathcal{B} ) =
\bar{C}^{\alpha} \chi_{\alpha} (A, \mathcal{B}),
\end{eqnarray}
is the gauge fixing functional. The transformation (\ref{BRST})
is nilpotent, that means $\hat{s}^2 = 0$. Taking into
account that
$S_{0} (A + \mathcal{B}) \,\hat{s}(\phi, \mathcal{B} ) = 0$,
the BRST symmetry of $S_{FP}(\phi, \mathcal{B})$ follows
immediately
\begin{eqnarray}
\n{BRSTSFP}
S_{FP} (\phi, \mathcal{B}) \,\hat{s}(\phi, \mathcal{B} ) = 0
\, .
\end{eqnarray}

Due to the presence of external vector field $ \mathcal{B}^i$, the
Faddeev-Popov action obeys an additional local symmetry known
as the background field symmetry,
\begin{eqnarray}
\n{InSFP}
\delta_{\omega}  S_{FP}(\phi, \mathcal{B}) =0,
\end{eqnarray}
which is related to the background field transformations
\begin{eqnarray}
\n{TT}
&&\delta^{(c)}_{\omega} \mathcal{B}^i=
R^i_{\alpha}(\mathcal{B})\omega^{\alpha},
\nonumber
\\
&&\delta_\omega^{(q)} A^i
= \left[ R^i_\alpha (A+\mathcal{B})
- R^i_\alpha (\mathcal{B}) \right] \omega^\alpha ,
\nonumber
\\
&&\delta_\omega^{(q)} B^\alpha
= -F^{\alpha}_{\gamma\beta}  B^\beta \omega^\gamma,
\\
&&\delta_\omega^{(q)} C^\alpha = -F^{\alpha}_{\gamma\beta}  C^\beta \omega^\gamma
 (-1)^{\varepsilon_{\gamma}},
\nonumber
\\
&&\delta_\omega^{(q)} \bar{C}^\alpha
= -F^{\alpha}_{\gamma\beta}  \bar{C}^\beta \omega^\gamma  (-1)^{\varepsilon_{\gamma}}.
\nonumber
\end{eqnarray}
Here the subscript $(c)$ is used to indicate the background field
transformations in the sector of external (classical) fields while
the $(q)$ in the sector of quantum fields (integration variables in
functional integral for generating functional of Green functions).
The symbol $\delta_{\omega}$ means  the combined background
field  transformations $\,\delta_\omega  = \delta_\omega^{(c)} +\delta_\omega^{(q)} $.
Note that in deriving (\ref{InSFP}) the transformation rule for the
gauge fixing functions
(\ref{GC})
\begin{eqnarray}
\n{GFT}
\delta_\omega \chi_\alpha(\phi,\mathcal{B})
\,=\,  - \chi_\beta (\phi,\mathcal{B})  F^{\beta}_{\alpha\gamma} \omega^\gamma\, ,
\end{eqnarray}
under the background field transformations (\ref{TT}) is assumed.
It is useful to introduce the generator of the background field
transformation $\,{\hat{\cal R}}_{\omega}(\phi, \mathcal{B})$,
\begin{align}
{\hat{\cal R}}_{\omega}(\phi, \mathcal{B})
&=
\int dx \;\Big( \frac{\overleftarrow{\delta}}{\delta  \mathcal{B}^a_{\mu}}
\delta^{(c)}_{\omega} \mathcal{B}^a_{\mu}
+ \frac{\overleftarrow{\delta}}{\delta \phi^{i}}\delta_{\omega}^{(q)}\phi^i\Big)
\nonumber \\ &
=
{\hat{\cal R}}^{(c)}_\omega( \mathcal{B})+{\hat{\cal R}}^{(q)}_\omega(\phi),
\label{Rcq}
\end{align}
where $ \phi^j{\hat{\cal R}}^{(q)}_{\omega}(\phi) = {\hat{\cal R}}^j_{\omega}(\phi) $
and
\begin{eqnarray}
{\hat{\cal R}}^{j}_{\omega}(\phi)
=
\left(
{\cal R}^{(q)}_{\omega}(A)
,\, {\cal R}^{(q)}_{\omega}(B)
,\,{{\cal R}}^{(q)}_{\omega}(C)
,\,{{\cal R}}^{(q)}_{\omega}({\bar C}) \right).
\label{Rs}
\end{eqnarray}
Using the new notations (\ref{Rcq}), the background field invariance of the
Faddeev-Popov action (\ref{InSFP}) rewrites as
\begin{eqnarray}
\n{BFSSFP}
 S_{FP}(\phi, \mathcal{B}) \,{\hat{\cal R}}_{\omega}(\phi, \mathcal{B})=0.
\end{eqnarray}

The symmetries (\ref{BRSTSFP}) and (\ref{BFSSFP}) of the
Faddeev-Popov  action  lead to the two very important properties
at the quantum level. In order to reveal these consequences we
have to introduce the extended generating functional of Green functions
in the background field method in the form of functional integral
\begin{align}
\n{GFGF}
Z(J,\phi^*, \mathcal{B}) =
\int \mathcal{D} \phi  \,
\text{exp} \left\{\frac{i}{\hbar}
\left[ S_{FP} (\phi, \mathcal{B})
+ \phi^*(\phi {\hat s})+J \phi\right] \right\}
= \text{exp}\left\{\frac{i}{\hbar}
W (J, \phi^*,\mathcal{B}) \right\}\,,
\end{align}
where $W = W(J, \phi^*, \mathcal{B})$ is the extended generating functional of connected
Green functions and
\begin{eqnarray}
J_{A}
= \left(J_i,\, J^{B}_{\alpha}, \,\bar{J}_{\alpha},\, J_{\alpha} \right)
\end{eqnarray}
are the external sources to the
fields $\,\phi^A$ $\left( \varepsilon(J_{A}) = \varepsilon_{A} \right)$.
Furthermore, the new quantities (antifields)
$\phi^*_A$, with $\varepsilon(\phi^*_A) = \varepsilon_{A}+1$, are the sources
of the BRST transformations.

The introduction of antifields enable one to simplify the use of the BRST
symmetry at the quantum level. The next step is to introduce the extended
effective action $\Gamma=\Gamma(\Phi,\phi^*, \mathcal{B})$ through the Legendre
transformation of $W(J, \phi^*,\mathcal{B})$
\begin{eqnarray}
\Gamma(\Phi,\phi^*, \mathcal{B})
=W(J, \phi^*,\mathcal{B})-J\Phi,
\label{EA}
\end{eqnarray}
where
\begin{eqnarray}
\Phi^A = \frac{\delta_l W}{\delta J_A}
\quad
\mbox{and}
\quad
\frac{\delta_r \Gamma}{\delta \Phi^A} =-J_A.
\label{EA-ids}
\end{eqnarray}
From one hand, one can prove that the BRST symmetry~(\ref{BRSTSFP}) of $S_{FP}$
results in the Slavnov-Taylor identity \cite{Taylor,Slavnov}
\begin{eqnarray}
\n{STG}
\frac{\delta_r \Gamma}{\delta \Phi^A} \frac{\delta_l \Gamma}{\delta \phi^*_ A}
=0.
\end{eqnarray}
On the other hand, the background field symmetry (\ref{BFSSFP})
of  $S_{FP}$  leads to the symmetry of the effective action under
the background field transformations,
\begin{eqnarray}
\n{BSG}
\tilde{\Gamma}(\Phi, \mathcal{B})
{\hat{\cal R}}_{\omega}(\Phi, \mathcal{B})=0,
\quad
\tilde{\Gamma}(\Phi, \mathcal{B}) =\Gamma(\Phi,\phi^*=0, \mathcal{B}).
\end{eqnarray}

The fundamental object of the background field method is the background
effective action $\Gamma( \mathcal{B}) \equiv \tilde{\Gamma}(\Phi=0, \mathcal{B})$.
Thanks to the linearity of $ {\hat{\cal R}}_{\omega}(\Phi, \mathcal{B})$ with
respect to the mean fields $\Phi^ i$, from (\ref{BSG}) it follows
\begin{eqnarray}
\n{BSbG}
\delta^ {(c)}_{\omega}  \Gamma(\mathcal{B}) =0,
\qquad
\delta^{(c)}_{\omega} \mathcal{B}^i=
R^i_{\alpha}(\mathcal{B})\omega^{\alpha},
\end{eqnarray}
i.e. the background effective action is a gauge invariant functional of the
external field $\mathcal{B}^i$.

The last important feature of the Faddeev-Popov quantization is related to the universality of the $S$-matrix,
that is independent on the choice of the gauge fixing. According to
the well-known result \cite{KT}, the universality of the $S$-matrix
is equivalent to the gauge fixing independent vacuum functional.
In the background field formalism this functional
is defined starting from (\ref{GFGF}) as
\begin{align}
{Z}_{\Psi} (\mathcal{B}) = Z(\mathcal{B}, J=\phi^*=0)
= \int {\cal D} \phi\,\exp
\left\{\frac{i}{\hbar}\,S_{FP}(\phi, \mathcal{B})\right\}.
\label{VacFun}
\end{align}
Regardless this object depends on the background field,
it is constructed for a certain choice of gauge $\Psi (\phi, \mathcal{B})$.
However, it can be shown to be independent on this choice.
Without the presence of background field, the discussion
of this issue in usual QFT and in the FRG approach can
be found in Ref.~\cite{LSh}. Here we generalize it for the
background field method  case.

Taking an infinitesimal change of the gauge fixing functional,
$\,\Psi(\phi,\mathcal{B})\rightarrow \Psi(\phi,\mathcal{B})
+\delta\Psi(\phi,\mathcal{B})$, we get
\begin{align}
\label{WIZvd}
{Z}_{\Psi+\delta\Psi} (\mathcal{B}) = &
\int {\cal D}\phi\,
\exp \Big\{ \frac{i}{\hbar}\,\Big[
S_{FP}({\phi}, \mathcal{B})
+ \delta \Psi ({\phi}, \mathcal{B}) \hat{s} ({\phi}, \mathcal{B})
\Big]\Big\}
.
\end{align}
Then, after a change of variables in the form of BRST transformation
\eq{BRST} but with replacement of the constant parameter $\mu$ by the
functional
\begin{eqnarray}
\mu ({\phi}, \mathcal{B}) = \frac{i}{\hbar}\, \delta \Psi ({\phi}, \mathcal{B})
,
\end{eqnarray}
one can show that
\begin{eqnarray}
{ Z}_{\Psi + \delta\Psi} (\mathcal{B}) = { Z}_{\Psi} (\mathcal{B}),
\label{Inva-Vac}
\end{eqnarray}
which is the starting point for the proof of the gauge fixing
independence of the $S$-matrix \cite{KT,Tyutin}. In the next
sections we shall see how this and other features for the case of the Yang-Mills theory look in the framework of the FRG approach.

\section{Background average effective action }
\label{Sec3}

In this section we shall discuss the use of the BFM
 applied to the FRG, following the original publication on this
subject by Reuter and Wetterich \cite{Reuter} for the case of pure Yang-Mills theory
with the action
\begin{eqnarray}
\label{ActionYM}
S_{0} (A) = - \frac{1}{4} F^a_{\mu\nu}(A) F^a_{\mu\nu} (A),
\end{eqnarray}
where
$\,F^a_{\mu\nu}(A) = \partial_\mu A^a_\nu - \partial_\nu A^a_\mu
+ gf^{abc} A^b_\mu A^c_\nu\,$
is the field strength for the non-Abelian  vector field $A_\mu$ and $g$ is coupling
constant.
The correspondence
with the notations used in Sec.~2 reads
\begin{align}
&
A^i \rightarrow A^a_\mu ,
\qquad
\mathcal{B}^i \rightarrow \mathcal{B}^a_\mu ,
\qquad
F^{\alpha}_{\beta\gamma} \rightarrow f^{abc} ,
\nonumber \\ &
 R^i_\alpha(A) \rightarrow D^{a b}_\mu(A)
= \delta^{ab} \partial_\mu + gf^{acb} A^c_\mu.
\end{align}
Here the structure coefficients $f^{abc}$ of the gauge group are
constant. The action~\eqref{ActionYM} is invariant under the gauge
transformations defined by the generator $D^{ab}_\mu (A)$ with an
arbitrary gauge function $\omega^a$ with $\varepsilon(\omega^a) =0$. In the
Faddeev-Popov quantization, the Grassmann parity of the fields
$\,\phi^A = ( A^a_\mu, B^a, C^a, \bar{C}^a )$ is, respectively,
$\varepsilon_A = (0,0,1,1)$.

The background field formalism for Yang-Mills theory comprises
the definition of the background field transformation
\begin{eqnarray}
\label{YM-back}
&&
\delta_\omega^{(c)} \mathcal{B}^a_\mu = D^{ab}_\mu (\mathcal{B}) \omega^b ,
\qquad
\delta_\omega^{(q)} A^a_\mu \,=\,g f^{abc} A^b_\mu \omega^c ,
\nonumber
\\
&&
\delta_\omega^{(q)} B^a = gf^{abc}  B^b \omega^c ,
\qquad
\delta_\omega^{(q)} C^a = gf^{abc} C^b \omega^c ,
\qquad
\nonumber
\\
&&
\delta_\omega^{(q)} \bar{C}^a = gf^{abc} \bar{C}^b \omega^c  .
\mbox{\qquad}
\end{eqnarray}
Note that the generator of the
 transformation in the sector of fields $A^a_\mu$ reads
\begin{eqnarray}
D^{ab}_\mu (A + \mathcal{B}) - D^{ab}_\mu (\mathcal{B})
= gf^{acb} A^c_\mu ,
\end{eqnarray}
and thus all the quantum fields transform according the same rule.
The standard choice of the gauge-fixing function is
\begin{eqnarray}
\chi^a (A,\mathcal{B}) = D^{ab}_\mu (\mathcal{B}) A^b_{\mu}.
\end{eqnarray}
It leads to the tensor transformation rule for $\chi^a (A,\mathcal{B})$ under
the background field transformation,
\begin{eqnarray}
\delta_{\omega}\chi^a (A,\mathcal{B})=gf^{abc} \chi^b (A,\mathcal{B}) \omega^c  .
\end{eqnarray}

The main point
of the FRG approach is the introduction of the scale-dependent
regulator action $S_k (\phi,\mathcal{B})$, in the framework of
the background field method. Let us choose the regulator action
for the quantum fields $\,A^a_\mu\,$ and $\,C^a,\,\bar{C}^a$ in the form
\begin{align}
\n{SkB}
S_k (\phi,\mathcal{B})
= &
\frac12 A^{a}_{\mu}\,
R^{(1)\; ab}_{k \;\mu \nu}(D_T(\mathcal{B})) \,A^{b}_{\nu}
+
{\bar C}^a\,
R^{(2)\; ab}_{k}(D_S(\mathcal{B})) \,C^b
.
\end{align}
The regulator functions depend on the external field through the covariant derivatives of tensor $D_T$ and scalar  $D_S$
fields
\begin{align}
\n{AR}
({ D}_T {\cal (B)})_{\mu\nu}^{ab}
=& - \,\eta_{\mu\nu} (D^2)^{ab}
+ 2 g f^{acb} F^c_{\mu\nu} ({\cal B}),
\qquad
  (D^2)^{ab} =  D^{ac}_\rho ({\cal B}) D^{cb}_\rho ({\cal B}),
 \\
({ D}_S {\cal (B)})^{ab}
= & - \,(D^2)^{ab}  .
\end{align}
The form of these functions can be chosen e.g. as in \cite{Reuter},
\begin{eqnarray}
\n{R}
R_k(z) = Z_k \frac{ze^{-z/k^2}}{1-e^{-z/k^2}},
\end{eqnarray}
with $Z_k$ corresponding to the wave function renormalization.

Let us consider the variation of the regulator action \eq{SkB} under
the background field transformations \eqref{TT} in the first order
approximation, $R_k(z)=Z_k z$.  The first term in \eq{AR} can be rewritten
through integration by parts, as follows
\begin{eqnarray}
- \,A^a_\mu  \eta_{\mu\nu} (D^2)^{ab}
A^b_\nu
=
\chi^c_{\rho\mu} (A, {\cal B })\, \chi^c_{\rho\mu} (A, {\cal B }),
\end{eqnarray}
where
\begin{eqnarray}
\chi^a_{\rho\mu} (A, {\cal B })
\equiv D^{ab}_\rho {\cal (B) } A^b_\mu.
\end{eqnarray}
The transformation rule for $\,\chi^a_{\rho\mu} (A, {\cal B })\,$ under
the background field transformation is very close to \eq{GFT}. It has the form
\begin{eqnarray}
\delta_\omega \, \chi^a_{\rho\mu} (A, {\cal B }) =
g f^{acb} \chi^c_{\rho\mu} (A, {\cal B }) \omega^b.
\end{eqnarray}
As consequence, we find the first term invariance
\begin{align}
\delta_\omega ( - A^a_\mu  \eta_{\mu\nu} (D^2)^{ab} A^b_\nu )
&=
\delta_\omega (\chi^a_{\rho\mu} (A, {\cal B }) \chi^a_{\rho\mu} (A, {\cal B }))
\nonumber \\&=
2 g f^{acb} \chi^a_{\rho\mu}
(A, {\cal B }) \chi^c_{\rho\mu} (A, {\cal B }) \omega^b
\nonumber \\&
= 0\,.
\end{align}
Furthermore, taking into account that
\begin{eqnarray}
\delta^{(c)}_\omega F^a_{\mu\nu} ({\cal B})
= g f^{acb} F^c_{\mu\nu} ({\cal B}) \omega^b,
\end{eqnarray}
for the second term in \eq{AR}, we have
\begin{align}
 \delta_\omega \, \big(f^{acb} A^a_\mu F^c_{\mu\nu} ({\cal B}) A^b_\nu\big)
 = &
 gA^a_\mu A^b_\nu F^c_{\mu\nu}
\big(f^{ace} f^{ebd} +
+ f^{abe} f^{edc}
+ f^{ade} f^{ecb} \big) = 0,
\nonumber
\end{align}
because of the Jacobi identity. The invariance holds also
for the ghost regulator, as one can easily verify. In this approximation the scale-dependent
action $S_k (\phi,\mathcal{B})$ obeys the background field symmetry,
$\delta_{\omega}S_k (\phi,\mathcal{B})=0$.

The same consideration can be done for the terms of the higher
orders in $z$. Thus, we can ensure that the invariance is maintained
in all orders. With these results the action (\ref{SkB}) is invariant
under the background field transformations,
\begin{eqnarray}
\n{CISK}
\delta_\omega  S_k (\phi,\mathcal{B}) = 0.
\end{eqnarray}

The full action $S_{k \;FP} =S_{k \;FP} (\phi, \mathcal{B}) $
is constructed by the rule
\begin{eqnarray}
\n{FPSK}
S_{k \;FP} (\phi, \mathcal{B})
= S_{FP} (\phi, \mathcal{B}) + S_k (\phi,\mathcal{B})\,,
\end{eqnarray}
where $S_{FP} (\phi, \mathcal{B})$ is the Faddeev-Popov
action (\ref{FP}). Using the action (\ref{FPSK}), the generating
functional of Green function is given by the following functional
integral:\footnote{From here we adopt units in which $\hbar=1$.}
\begin{eqnarray}
\n{dZk}
Z_k(J,\mathcal{B})
 =
\int {\cal D} \phi \,
\exp\Big\{i[ S_{FP} (\phi, \mathcal{B})
+ S_k (\phi,\mathcal{B})
+J\phi]\Big\}
 = \exp\big\{iW_k(J,\mathcal{B})\big\},
\end{eqnarray}
where $W_k=W_k(J,\mathcal{B})$ is the generating functional
of connected Green functions. The main object of the FRG
approach in the background field method is the background average effective action
$\Gamma_k=\Gamma_k(\Phi, \mathcal{B} )$, defined through
the Legendre transform of $\,W_k$,
\begin{eqnarray}
&&
\Gamma_k(\Phi, \mathcal{B} )
\,=\,W_k(J,\mathcal{B})-J\Phi,
\end{eqnarray}
where
\begin{eqnarray}
\Phi^A = \frac{\delta_l W_k}{\delta J_A
\nonumber
\end{eqnarray}
and
\begin{eqnarray}
\frac{\delta_r \Gamma_k}{\delta \Phi^A}=-J_A.
\nonumber
\end{eqnarray}

The effective average action can be presented as a sum of
the regulator action of the mean field and the quantum correction,
\begin{eqnarray}
&&
\Gamma_k(\Phi, \mathcal{B} )
= S_k(\Phi, \mathcal{B})
 + {\bar \Gamma}_k(\Phi, \mathcal{B} ).
\label{EA-ca}
\end{eqnarray}
The functional $\bar{\Gamma}_k$ satisfies the flow equation, or
the Wetterich  equation \cite{Wetterich,Reuter},
\begin{eqnarray}
\n{FEG}
\partial_t {\bar \Gamma}_k(\Phi, \mathcal{B} )
=
\frac{i}{2}\,\mbox{sTr}\, \Bigg\{
\frac{\partial_t R_{k}( \mathcal{B})}
{\,\,
{\bar \Gamma}^{''}_k(\Phi, \mathcal{B})
\,+\,R_k( \mathcal{B})\,}\Bigg\}.
\end{eqnarray}
In (\ref{FEG}) $ \partial_t =k\frac{d}{d k} $ and the symbol $\,\mbox{sTr}\, $ means the
functional supertrace, this last is necessary due to the presence
of quantum fields $A^a_\mu\,$ and $\,C^a,\,\bar{C}^a$, with different
Grassmann parity. Another important notation is
\begin{eqnarray}
\Big({\bar \Gamma}^{''}_k(\Phi, \mathcal{B})\Big)_{AB}
=
\frac{\delta_l}{\delta\Phi^A}
\left(\frac{\delta_r {\bar\Gamma}_k(\Phi, \mathcal{B} )}{\delta\Phi^B}\right)
\end{eqnarray}
for the matrix of the second order functional derivatives with
respect to the mean fields $\Phi$.

As we have seen above, because of the invariance of the
scale-dependent regulator term (\ref{SkB}), the full action
(\ref{FPSK}) is invariant under the background field
transformations (\ref{InSFP}),
\begin{align}
\n{nISk}
\delta_{\omega}S_{k\;FP}(\phi,\mathcal{B})
=\delta_{\omega}S_k(\phi,\mathcal{B})
=
S_k(\phi,\mathcal{B}) {\hat{\cal R}}_{\omega}(\phi, \mathcal{B})=0.
\end{align}
At the quantum level (\ref{nISk}) provides the invariance of the
background
average effective action $\Gamma_k(\Phi, \mathcal{B} )$.
Indeed, variation of $Z_k(J,\mathcal{B})$ with respect to the
external field $\mathcal{B}^a_\mu$ reads
\begin{eqnarray}
\n{vZk}
\delta^{(c)}_{\omega}Z_k(J,\mathcal{B})=iJ_A
\;\!{\cal R}^A_{\omega}\left(\frac{\delta_l Z_k}{i\delta J}\right).
\end{eqnarray}
In terms of the functional $W_k(J,\mathcal{B})$ the relation
(\ref{vZk}) rewrites
\begin{eqnarray}
\n{vWk}
\delta^{(c)}_{\omega}W_k(J,\mathcal{B})=
J_A\;\!{\cal R}^A_{\omega}\left(\frac{\delta_l W_k}{\delta J}\right).
\end{eqnarray}
As a consequence of (\ref{vWk}), the background average
effective action is invariant under the background field
transformations,
\begin{eqnarray}
\n{vGk}
&&\delta_{\omega}\Gamma_k(\Phi,\mathcal{B})=0.
\end{eqnarray}
In terms of the functional $ {\bar \Gamma}_k(\Phi, \mathcal{B} )$
the relation (\ref{vGk}) becomes
\begin{eqnarray}
\n{vGka}
\delta_{\omega}{\bar \Gamma}_k(\Phi, \mathcal{B} )=0.
\end{eqnarray}
Thus, the background field symmetry is preserved for the background
average effective action $\,{\bar\Gamma}_k(\Phi, \mathcal{B} )$,
confirming the main statement of  the  paper \cite{Reuter}.

For the functional
${\bar \Gamma}_k( \mathcal{B})={\bar \Gamma}_k(\Phi=0, \mathcal{B})$,
the background field symmetry is preserved as well due to linearity
of the background field symmetry
\begin{eqnarray}
\n{vGkab}
\delta^ {(c)}_{\omega}{\bar \Gamma}_k( \mathcal{B})=0,
\end{eqnarray}
in agreement  with (\ref{BSbG}).  In particular this means that the
flow equation for~$\,{\bar \Gamma}_k( \mathcal{B})$,
\begin{eqnarray}
\partial_t {\bar \Gamma}_k(\mathcal{B} )
=
\frac{i}{2}\,\mbox{sTr}\, \Bigg\{
\frac{\partial_t R_{k}( \mathcal{B})}
{\,\,
{\bar \Gamma}^{''}_k(\Phi,\,\mathcal{B})\big|_{\Phi =0}
\,+\,R_k( \mathcal{B})\,}\Bigg\},
\end{eqnarray}
maintains the  background field symmetry.

\section{Background invariant regulator functions}
\label{Sec4}

The prove of invariance of $S_k$ under background field
transformations (\ref{CISK}) is based on the certain form
of the regulator functions and its arguments. In particular,
the regulator functions (\ref{R}) with argument (\ref{AR}) by itself are
not invariant under background field transformations
$\,\delta^{(c)}_{\omega} R^{(1)\; ab}_{k \;\mu \nu}(D_T(\mathcal{B}))\neq 0$,
$\delta^{(c)}_{\omega}R^{(2)\; ab}_{k}(D_S(\mathcal{B}))\neq 0$.
In this section we shall discuss the background field symmetry
of the background average effective action and formulate a possible
restriction on the regulator functions in the scale-dependent action
$S_k$ in the general settings that allow us to arrive at the
invariance of the background average effective action  under background field transformations.

Consider the scale-dependent regulator action
$S_k = S_k (\phi, \mathcal{B})$ in the background field
formalism, including the ghost sector,
\begin{align}
S_k (\phi, \mathcal{B}) =
\frac{1}{2}  A^{a}_{\mu}
R^{(1)\; ab}_{k \;\mu \nu} (\mathcal{B})
A^{b}_{\nu}
\,+\,
\bar{C}^{a}
R^{(2)\; ab}_{k}(\mathcal{B})
C^{b},
\end{align}
where  $R^{(1)\; ab}_{k \;\mu \nu}(\mathcal{B})\,$ and
$R^{(2)\; ab}_{k}(\mathcal{B})\,$  are the regulator functions.
We assume that they are local functions of external fields
$\mathcal{B}^a_{\mu}$ and their partial derivatives.
The full action has a standard FRG form
\begin{eqnarray}
\label{fullA}
S_{kFP} (\phi, \mathcal{B})
=
S_{FP} (\phi, \mathcal{B})+S_k (\phi, \mathcal{B}).
\end{eqnarray}
Due to the background field symmetry of the Faddeev-Popov
action (\ref{InSFP}), the full action (\ref{fullA}) will be invariant
under the background field transformations (\ref{TT}), if  the
scale-dependent regulator action $ S_k = S_k (\phi, \mathcal{B})$
satisfies the equation
\begin{eqnarray}
\n{ISK}
\delta_{\omega} S_k (\phi, \mathcal{B}) = 0.
\end{eqnarray}
Using the explicit form of the background field transformations
(\ref{TT}) the variation of $ S_k (\phi, \mathcal{B})$ reads
\begin{equation}
\begin{split}
\delta_{\omega} S_k (\phi, \mathcal{B})
= &
 \frac{1}{2} A^ a_{\mu}
\Big[g
\Big(f^ {adc}\omega^ d R_{k\;\mu\nu}^{(1)\; cb}(\mathcal{B})
- R_{k\; \mu\nu}^ {(1)\; ac} (\mathcal{B}) f^ {cdb} \omega^d
\Big)
+\delta^{(c)}_{\omega} R^{(1)\; ab}_{k \;\mu \nu} (\mathcal{B})
\Big] A^b_{\nu}
\\ &
\n{ISKa}
+
\int dx \, {\bar C}^a\Big[g
\Big(f^{adc} \omega^d
R^{(2)\; cb}_{k}(\mathcal{B})
- R^{(2)\;ac}_{k}(\mathcal{B}) f^{cdb} \omega^d
\Big)
+\delta^{(c)}_{\omega} R^{(2)\;ab}_{k}(\mathcal{B})\Big]
C^b.
\end{split}
\end{equation}
From Eq.~(\ref{ISKa}) follows that (\ref{ISK}) is satisfied if
\begin{align}
\n{R1}
&
g \Big( f^ {adc}\omega^d R_{k\;\mu\nu}^ {(1)\;cb}(\mathcal{B})
- R_{k\;\mu\nu}^{(1)\; ac}(\mathcal{B}) f^ {cdb} \omega^ d
\Big) + \delta^{(c)}_{\omega}R_{k\; \mu\nu}^ {(1)\;ab}(\mathcal{B})
=0,
\\
\n{R2}
&
g\left(f^{adc} \omega^d
R^{(2)\;cb}_{k}(\mathcal{B})
- R^{(2)\;ac}_{k }(\mathcal{B})
f^{cdb} \omega^d\right)
+\delta^{(c)}_{\omega} R^{(2)\;ab}_{k}(\mathcal{B})
=0.
\end{align}
Any solution of these equations provides the invariance of $S_k$
under background field transformations. Let us consider the case
when regulator functions are invariant under background
transformations of external field $\mathcal{B}^a_\mu$,
\begin{eqnarray}
\n{Rin}
\delta^{(c)}_{\omega}R_{k\; \mu\nu}^ {(1)\; ab}(\mathcal{B})
=0,
\qquad
\delta^{(c)}_{\omega} R^{(2)\;ab}_{k}(\mathcal{B})
=0.
\end{eqnarray}
Due to the arbitrariness in the choice of the functions
$\omega^{a}(x)$, from (\ref{R1}), (\ref{R2}) and
(\ref{Rin}) follow the relations
\begin{eqnarray}
\big[t^d,R_{k}^ {(1)\;\mu\nu}(\mathcal{B} ) \big]_{ab}=0,
\qquad
\big[t^d,R_{k}^ {(2)\;\mu\nu}(\mathcal{B}) \big]_{ab}=0,
\end{eqnarray}
for the generators $(t^a)_{bc} = f^{bac}$ of the Lie group.
Therefore, we see that the regulator functions commute with all the
generators of Lie group. Then, applying the Shur's lemma we find
\begin{align}
R_{k\;\mu\nu}^{(1)\;ab}(\mathcal{B}) =&
\delta^{ab} R_{k \;\mu\nu}^ {(1) }(D(\mathcal{B})),
\nonumber \\
R_{k}^{(2)\;ab}(\mathcal{B})
=& \delta^{ab} R_{k}^ {(2)}(D(\mathcal{B})),
\end{align}
where the quantities $R_{k\; \mu\nu}^ {(1)}(D(\mathcal{B}))$
and $R_{k}^{(2)}(D(\mathcal{B}) )$ are scalars with respect
to the background transformations of external field $\mathcal{B}^a_\mu$.
It means that the arguments of these quantities should be scalars
as well.  It is easy to construct an example of such kind of
a scalar argument, $D(\mathcal{B})=F^{a}_{\mu\nu}(\mathcal{B})
D^{ab}_{\mu}(\mathcal{B})\mathcal{B}^b_{\nu}$, where
$F^{a}_{\mu\nu}$ is defined in (\ref{ActionYM}).

So, in the case under consideration, the scale-dependent regulator
action has the form
\begin{align}
S_k (\phi,\mathcal{B})
= &
 \frac{1}{2}  A^{a}_{\mu}
R^{(1)}_{k \; \mu \nu}(D(\mathcal{B}) )
A^{a}_{\nu}
+ \bar{C}^{a}(x) R^{(2)}_{k}(D(\mathcal{B}) )
C^{a},
\end{align}
maintaining the background field symmetry $\delta_\omega S_k (\phi,\mathcal{B}) =0$.

\section{Gauge dependence of background average effective action }
\label{Sec5}

Here the problem of gauge dependence of background average effective action  will be discussed in general
setting of Sec. 2
The regulator action $S_k$ is invariant under the background
transformations (\ref{CISK}), but not under the
BRST transformations,
\begin{eqnarray}
S_k (\phi,\mathcal{B}) {\hat s}(\phi,\mathcal{B}) \neq 0.
\end{eqnarray}
Let us  discuss the implications of this fact for the
gauge dependence problem of the background average effective action.
Consider the extended generating functional of Green functions $Z_k(J,\phi^*, \mathcal{B})$, and the extended generating
functional of connected Green functions $\,W_k(J, \phi^*, \mathcal{B})$,
\begin{align}
\label{ZkWk}
Z_k(J,\phi^*, \mathcal{B})
= &
\int { \cal D} \phi\exp\{i[S_{FP}(\phi, \mathcal{B})
+ S_k(\phi, \mathcal{B})
+ J\phi
+\phi^*({\hat s}\phi)]\}
=
\exp\{iW_k(J,\phi^*, \mathcal{B})\},
\end{align}
As the first step we derive the modified Ward identity for the FRG
in the BFM which is a consequence
of the BRST invariance of the action $S_{FP}(\phi, \mathcal{B})$ (\ref{BRSTSFP}).
Making use the change of variables in the form of the BRST transformations
in the functional integral (\ref{ZkWk}),
$\phi^A\rightarrow \varphi^A(\phi)=\phi^A+({\hat s}\phi^A)\mu$, and taking into account
the triviality of the corresponding Jacobian if the conditions
\begin{eqnarray}
\label{nr}
(-1)^{\varepsilon_i}\frac{\delta_l R^i_{\alpha}}{\delta A^i}+
(-1)^{(\varepsilon_{\alpha}+1)}F^{\beta}_{\beta\alpha}=0
\end{eqnarray}
are satisfied (for detailed discussion of this point see \cite{GLSh}), we arrive at
the relation
\begin{align}
\nonumber
0 = &\int D\phi \;\big(J_A+S_{k,A}(\phi, \mathcal{B})\big)({\hat s}\phi^A)
\exp\{i[S_{FP}(\phi, \mathcal{B})
+ S_k(\phi, \mathcal{B})
+ J\phi+\phi^*({\hat s}\phi)]\}
\nonumber \\
\label{rel}
= &\left[J_A+S_{k,A}\left(\frac{\delta_l}{i\delta J}, \mathcal{B}\right) \right]
\int D\phi ({\hat s}\phi^A)\exp\{i[S_{FP}(\phi, \mathcal{B})
+ S_k(\phi, \mathcal{B})+ J\phi+\phi^*({\hat s}\phi)]\}.
\end{align}
From (\ref{rel}) it follows the modified Ward identity for the extended
generating functional of Green functions $Z_k(J,\phi^*,\mathcal{B})$
\begin{eqnarray}
\label{WIZk}
\left[J_A+S_{k,A}\left(\frac{\delta_l}{i\delta J}, \mathcal{B}\right) \right]
\frac{\delta_l}{\delta\phi^*_A}Z_k(J,\phi^*, \mathcal{B})=0.
\end{eqnarray}
This identity in terms of the extended generating
functional of connected Green functions
$\,W_k(J, \phi^*, \mathcal{B})$ reads
\begin{align}
\label{WIWk}
\left[J_A+S_{k,A}\left(\frac{\delta_l W_k}{\delta J}+
\frac{\delta_l}{i\delta J},\mathcal{B}\right) \right]
\frac{\delta_l}{\delta\phi^*_A}W_k(J,\phi^*, \mathcal{B})=0.
\end{align}
Introducing the generating functional of vertex functions
$\Gamma_k=\Gamma_k(\Phi,\phi^*, \mathcal{B})$ with the help
of Legendre transformation of $W_k=W_k(J,\phi^*, \mathcal{B})$
\begin{align}
&\Gamma_k(\Phi,\phi^*, \mathcal{B})=W_k(J,\phi^*, \mathcal{B})-J_A\Phi^A,
\qquad
\Phi^A=
\frac{\delta_l W_k}{\delta J_A},
 \\ &
\frac{\delta_r \Gamma_k}{\delta \Phi^A}=-J_A,\quad
\frac{\delta_l \Gamma_k}{\delta \phi^*_A}=
\frac{\delta_l W_k}{\delta \phi^*_A} \nonumber,
\end{align}
the modified Ward identity rewrites in the form
\begin{eqnarray}
\label{WIGk}
\frac{\delta_r \Gamma_k}{\delta \Phi^A}
\frac{\delta_l \Gamma_k}{\delta \phi^*_A}=
S_{k,A}({\hat\Phi}, \mathcal{B})\frac{\delta_l \Gamma_k}{\delta \phi^*_A},
\end{eqnarray}
where the notation
\begin{eqnarray}
\label{hatPhi}
{\hat\Phi}^A=\Phi^A+i\,(\Gamma_k^{''-1})^{AB}\,
\frac{\delta_l}{\delta\Phi^B},
\end{eqnarray}
has been used. The matrix \ $(\Gamma_k^{''-1})$ \ is inverse to the
matrix $\Gamma_k^{''}$, the last has elements
\begin{align}
&
(\Gamma_k^{''})_{AB} =
\frac{\delta_l}{\delta\Phi^A}
\Big(\frac{\delta_r\Gamma_k}{\delta\Phi^B}\Big)
\,,\quad \mbox{i.e.,} \nonumber \\&
\big(\Gamma_k^{''-1}\big)^{AC}\cdot
\big(\Gamma_k^{''}\big)_{CB}\,=\,\delta^A_{\,B}\,.
\end{align}

Now consider the variation of the extended generating functional of Green
functions under infinitesimal
variation of the gauge fixing functional,
$\,\Psi(\phi,\mathcal{B})\rightarrow \Psi(\phi,\mathcal{B})
+\delta\Psi(\phi,\mathcal{B})$. We find
\begin{align}
\label{varZk}
\delta Z_k(J, \phi^*,\mathcal{B})
= &  i\int D\phi \big(\delta\Psi(\phi,\mathcal{B}){\hat s}(\phi, \mathcal{B})\big)
\exp\{i[S_{FP}(\phi, \mathcal{B})
\nonumber \\ &
+ S_k(\phi, \mathcal{B})
+ J\phi+\phi^*({\hat s}\phi)]\}.
\end{align}
Now take into account that the functional integral of total variational
derivative is zero we have the relation
\begin{align}
0 = &  \int D\phi \frac{\delta_r}{\delta\phi^A}
\Big[ \big(\delta\Psi s^A\big)
\exp\{i[S_{FP}(\phi, \mathcal{B})
+ S_k(\phi, \mathcal{B})+
J\phi
+\phi^*({\hat s}\phi)]\}\Big]
\label{rel1}
\nonumber \\
= & \int D\phi\Big[i\delta\Psi s^A
\big(J_A+S_{k,A}\big)
+\big(\delta\Psi{\hat s}\big)\Big]
\exp\{i[S_{FP}(\phi, \mathcal{B})
+ S_k(\phi, \mathcal{B})
+ J\phi+\phi^*({\hat s}\phi)]\},
\end{align}
where the BRST invariance of $S_{FP}$ action, the nilpotency of BRST transformations
and the relations (\ref{nr}) have been used. From (\ref{rel1}) one has
\begin{align}
&
i\int D\phi \big(\delta\Psi(\phi,\mathcal{B}){\hat s}(\phi, \mathcal{B})\big)
\exp\{i[S_{FP}(\phi, \mathcal{B})
+ S_k(\phi, \mathcal{B})+
\nonumber \\ &
J\phi
+\phi^*({\hat s}\phi)]\}
=
\int D\phi \big(J_A+S_{k,A}\phi,\mathcal{B}\big)s^A(\phi,\mathcal{B})
\times
\nonumber \\ &
\times
\delta\Psi(\phi,\mathcal{B})
\exp\{i[S_{FP}(\phi, \mathcal{B})
+ S_k(\phi, \mathcal{B})
+ J\phi+\phi^*({\hat s}\phi)]\},
\end{align}
which allows to present the Eq. (\ref{varZk}) in the form closed with respect to
$Z_k(J, \phi^*,\mathcal{B})$,
\begin{align}
\label{varZk1}
\delta Z_k(J, \phi^*,\mathcal{B})
= &
- i \left[J_A+S_{k,A}\left(\frac{\delta_l}{i\delta J},\mathcal{B}
\right)\right]
\frac{\delta_l}{\delta\phi^*_A}
\delta\Psi\left(\frac{\delta_l}{i\delta J},\mathcal{B} \right)
Z_k(J,\phi^*,\mathcal{B}),
\end{align}
or, in terms of $W_k(J, \mathcal{B})$,
\begin{align}
\label{varWk1}
\delta W_k(J, \phi^*,\mathcal{B})
= &
-\, \left[J_A+S_{k,A}\left(\frac{\delta_l W_k}{\delta J}+
\frac{\delta_l}{i\delta J},\mathcal{B}\right)\right]
\,\frac{\delta_l}{\delta\phi^*_A}\,
\delta\Psi\left(\frac{\delta_l W_k}{\delta J}
+\frac{\delta_l}{i\delta J},\mathcal{B} \right).
\end{align}
In deriving (\ref{varWk1}) the modified Slavnov-Taylor identity (\ref{WIWk})
has been used.
The last equation can be rewritten for the background average effective action,
$\Gamma_k(\Phi,\phi^*,\mathcal{B})$, in the form
\begin{align}
\n{gGk}
\delta \Gamma_k(\Phi, \phi^*, \mathcal{B})
= &
\frac{\delta_r\Gamma_k}{\delta\Phi^A}
\frac{\delta_l}{\delta\phi^*_A}
\delta\Psi({\hat \Phi},\mathcal{B})
-
S_{k, A}({\hat \Phi},\mathcal{B})
\frac{\delta_l}{\delta\phi^*_A}\delta\Psi({\hat \Phi},\mathcal{B}),
\end{align}
where ${\hat \Phi}$ was introduced in (\ref{hatPhi}).
From Eq.~(\ref{gGk}) follows that
\begin{eqnarray}
\n{gGkn}
\delta \Gamma_k(\Phi,  \phi^*,\mathcal{B})
\Big|_{\frac{\delta \Gamma_k}{\delta\Phi}=0} \neq\ 0.
\end{eqnarray}
As result, the average effective action depends on gauge fixing even
on the equations of motion (on-shell) and the $S$-matrix
defined in the framework of the FRG approach is gauge dependent.

\section{Conclusions}
\label{Sec6}

We considered several aspects of background average effective action  in the FRG framework.
At the first place we confirmed the well-known classical result
of \cite{Reuter} concerning the background invariance of the
regulator actions and background average effective action  in the framework of the background field method for a wide class of regulator functions which include
(\ref{R}), but can be generalized to any other functions of
the arguments $z$. As a new technical result we formulated
general conditions of regulator actions being invariant with
respect to the purely background transformations.

The main motivation of this work was to check whether the
on-shell dependence of the average effective action \cite{LSh}
holds within the background field method formalism. The answer to this question
is given by the relation (\ref{gGkn}) and is strictly positive.
This output does not
contradict the recent works \cite{Wetterich-2018,CODELLO}
because in these publications the subject of study was the
gauge invariance of background average effective action,
and the question of gauge fixing dependence was not investigated.
From our viewpoint, the
on-shell gauge dependence of the average effective action
is a fundamental principal difficulty of the FRG approach
applied to the Yang-Mills theories. We have confirmed that
the situation does not improve in the background field method,
regardless of the different structure of lifting the degeneracy
of the classical action.

It is unclear whether one can achieve a reasonable physical
interpretation of the results obtained within the FRG formalism
applied to Yang-Mills theories, and therefore it makes sense
to discuss the possible ways out from this difficult situation.

Certainly the simplest way is to ignore the problem e.g. by
deciding that one special gauge fixing is ``physical'' or
``correct'', such that changing the gauge should be strictly
forbidden. As far as FRG provides valuable nonperturbative
results, the theoretically inconsistent formulation is the price
to pay for going beyond the well-defined perturbative framework.

Another possibility is to look for some observables that may
be gauge-fixing invariant. For instance, in the fixed point the
background average effective action  boils down to the standard QFT effective action and
then $S$-matrix, amplitudes and all related observables are
well-defined. Unfortunately, even in the vicinity of the fixed
point this is not true due to the relation (\ref{gGkn}). Since
the search of the nonperturbative fixed point is based on the
renormalization group flows and the last are supposed to be
gauge-fixing dependent, it is unclear how the fixed-point
invariance can be actually used.

Finally, there is an alternative formulation of the FRG in gauge
theories which is gauge-fixing independent, exactly as a conventional
perturbative QFT is \cite{LSh}. This scheme is technically more
difficult, since the regulator actions are constructed in a more
complicated way, that includes composite fields. At least by now,
the disadvantage of this approach is that there is no method to
perform practical calculations.

\section*{Acknowledgements}
\noindent
P.M.L. is grateful to the Departamento de F\'{\i}sica of the Federal
University of Juiz de Fora (MG, Brazil) for warm hospitality during
his long-term visit.  The work of P.M.L. is supported partially
by the Ministry of Education and Science of
the Russian Federation, grant 3.1386.2017 and by the RFBR grant
18-02-00153. This work of I.L.Sh. was partially supported by Conselho Nacional de
Desenvolvimento Cient\'{i}fico e Tecnol\'{o}gico - CNPq under the grant 303893/2014-1 and Funda\c{c}\~{a}o de Amparo \`a Pesquisa
de Minas Gerais - FAPEMIG under the project APQ-01205-16. E.A.R. is grateful to Coordena\c{c}\~ao de Aperfei\c{c}oamento de Pessoal
de N\'{\i}vel Superior - CAPES  for supporting his Ph.D. project.