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

Leptogenesis \cite{fukugita} (for reviews, see e.g.
  Refs.~\cite{buchmuller-origin,davidson-review})
is an appealing scenario which can
explain the baryon asymmetry of the Universe. It requires
the existence of sufficiently heavy
right-handed, or sterile neutrinos $ N _ i $ in addition
to the particles of the Standard Model. Their Yukawa couplings
with the Higgs and the Standard Model leptons have to contain $ CP $-violating
phases. When their interactions occur out of thermal
equilibrium, they can produce a lepton asymmetry $ L $, which, due to
$ B + L $ violating sphaleron processes, also leads to a
baryon asymmetry $ B $.

During leptogenesis most interactions are either much faster or
much slower than the expansion
of the Universe. Therefore most
physical quantities are either very close to their corresponding
thermal equilibrium values, or constant.
A few  quantities  evolve roughly at the same rate as the Hubble rate, and
can therefore deviate significantly from their equilibrium values.
Among them are the number density of the heavy neutrinos and
the $ B - L $ asymmetry, because they are only changed by the heavy neutrino
Yukawa interaction.

However,  not only  the number density of heavy neutrinos,
but also their momentum  distribution is out of equilibrium, because
the kinetic equilibration is also due to the Yukawa interactions.
Nevertheless, in the standard approach \cite{luty}  kinetic equilibrium is assumed.
In addition, Maxwell-Boltzmann statistics is used.
In order to account for the correct momentum
distribution Boltzmann equations for the
distribution function have been used in Refs.~\cite{basboll,garayoa,hahn},
together with quantum statistics,
and significant differences compared to the standard approach were reported.

In the standard approach both $ 1 \leftrightarrow  2 $ decays
and inverse decays of the right-handed neutrinos,
as well as $ 2 \leftrightarrow  2 $ scattering
processes are taken into account, and both were found to be important.
Recently the production rate of right-handed neutrinos
of mass $ M _ N $,
which is one ingredient of the rate equations describing leptogenesis,
were studied at next-to-leading order in the Standard Model
couplings~\cite{salvio,laine-nlo,laine-nlo-relativistic}.
It was pointed out that the $  2 \leftrightarrow  2 $
processes are only part of
the next-to-leading order contribution and that $ 1 \leftrightarrow 3
$ scattering and virtual corrections to the (inverse) decays
contribute at the same order.
If they are taken into account the large infrared contributions cancel
\cite{salvio}. Therefore a complete next-to-leading
order calculation would be important.
 Results for the production rate were obtained in the
limit $ T \ll M _ N $, together with $ T/M _ N $ corrections
in \cite{salvio,laine-nlo}, and
more recently also in the regime $ T \sim M _ N $  \cite{laine-nlo-relativistic}.

A measure for the interaction rate of right-handed neutrinos relative
to the Hubble rate $ H $ at the time when they become non-relativistic is
the washout factor
\begin{align}
   K \equiv \left . \frac{ \Gamma  _ 0} { H } \right | _ { T = M _ N }
   \label{K}
   ,
\end{align}
where $ \Gamma _ 0 $ is the total tree level
decay rate. In the so-called
strong washout regime $ K \gg 1 $  the right-handed neutrinos are close
to equilibrium  around $ T \sim M _ N $,  any pre-existing asymmetry is washed out,
and the final asymmetry is mainly created when
$ T $ has dropped below $ M _ N $.
At this point the right-handed
neutrinos are non-relativistic, since they are close to
equilibrium.
It turns out that the strong washout regime corresponds
to a mass range of the light neutrinos
which is favored by atmospheric and solar neutrino oscillation data
\cite{buchmuller-origin}.
Therefore the non-relativistic regime is of particular interest for
leptogenesis. A first principle treatment of leptogenesis, which
could allow for a complete next-to-leading order calculation can
be expected to be simpler in this regime than in the fully
relativistic case.

In this paper we obtain the rate equations for leptogenesis in the non-relativistic limit,
and we propose a systematic expansion to compute relativistic corrections.
This way one can  systematically control the accuracy of the non-relativistic
approximation, and one can obtain information about the momentum
distribution and thus about the deviation from kinetic equilibrium.
The rate equations are valid to all orders in the Standard Model couplings,
and a  complete  next-to-leading order calculation in these couplings
should be feasible. We take a first step in this direction by computing
the corrections for some coefficients in the rate equations, thereby consistently
including temperature dependent gauge and top Yukawa interaction in a
leptogenesis calculation for the first time.
We restrict ourselves to the case of hierarchical Majorana masses for the
right-handed neutrinos, and we only consider the leading order in the
right-handed neutrino Yukawa couplings.

This paper is organized as follows.
In Sect.~\ref{sec:non-relativistic} we
obtain the equations for leptogenesis  in the non-relativistic limit
and to all orders in the Standard Model couplings. In Sect.~\ref{sec:lo} we compute
the coefficients in these equations at leading order in the Standard Model couplings.
The conditions
for thermal (non-)equilibrium are discussed in Sect.~\ref{sec:neq}.
The first relativistic corrections to the equations of Sect.~\ref{sec:non-relativistic}
are obtained in Sect.~\ref{sec:relativistic}.  In Sect.~\ref{sec:radiative}
we use the results of \cite{laine-nlo} to compute some of the coefficients
at next-to-leading order in the Standard Model couplings.
Numerical results are shown in Sect.~\ref{sec:numerical}, and
the conclusions are in Sect.~\ref{sec:discussion}.

\section{Leptogenesis in the non-relativistic limit}
\label{sec:non-relativistic}

We assume that the mass of the right-handed neutrinos is hierarchical and that
we only need to take into account the lightest one which we denote by $ N $.
We work at leading order in the Yukawa couplings of the right-handed neutrinos.
We also assume that the interactions of the right-handed neutrinos are
much less frequent than gauge interactions. \footnote{\label{stopping}The rate for
large angle scattering via gauge interactions is of order
$  g ^ 4 T $ modulo logarithms, where $ g $ denotes a gauge
coupling. When a non-relativistic neutrino  with mass $ M _ N $ decays,
its decay products originally have energy of approximately
$ M _ N / 2 \gg T $. They lose
their energy within a stopping time of order $ t _ {\rm stop } $
with $ t _ {\rm stop } ^{ -1 }
\sim \sqrt{ T/M _ N  }\, g ^ 4 T  $
modulo logarithms \cite{baier,arnold-stopping}.}
Then on time scales which are relevant
to leptogenesis the Standard Model particles are in kinetic
equilibrium. Finally we assume that the Standard Model Yukawa interactions are either much
faster than the right-handed neutrino reaction rates so that they are in
thermal equilibrium, or that they are much slower. \footnote{If they
occur at roughly the same rate they both have to be included in the
equations for leptogenesis \cite{beneke-flavored}.}

We assume that only one lepton flavor is involved. This
is the case, for instance, at high temperature, when all Yukawa interactions
of the Standard Model leptons can be neglected. At lower
temperature the lepton Yukawa interactions become important,
and in general one has to deal with several asymmetries \cite{barbieri}.
However, for certain values of the Yukawa coupling matrix
one may still be approximately in a one-flavor regime.

We consider non-relativistic right-handed neutrinos with a characteristic
velocity $ v \ll 1 $. In a first approximation their motion can be
neglected.  Then the out-of-equilibrium state is fully specified by the
number densities $ n _ N $ and $ n _ { B - L } $. In a non-expanding
Universe the time derivative of these densities would only depend
on their deviations from their equilibrium values and on the temperature.
These considerations imply that the time
evolution of this system in an expanding Universe is governed by the equations
\begin{align}
   \left ( \frac{ d   } { d t }  + 3 H \right )  n _ N
       \, \, \, & = {} -
    \Gamma _ N  \, \, ( n _ N - n _ N ^ {\rm eq}  )
       +   \Gamma  _ { N, B-L } \, n _ { B - L }
   \label{dnNdt}
   ,
\\
   \left ( \frac{ d  } { d t }   + 3 H  \right )   n _ { B - L }
      & =

   \, \, \Gamma  _ {B-L, N }
   ( n _ N - n _ N ^ {\rm eq}  )
    -

    \Gamma  _ {B - L }
    \, n _ { B - L }
   \label{dnLdt}
   .
\end{align}
The  coefficients $ \Gamma  _ i $
only depend on the temperature of the ambient plasma.
$ \Gamma  _ N $ describes how quickly the number density of heavy neutrinos
approaches its equilibrium value, and
$  \Gamma  _ {B - L }  $ describes
the dissipation, or washout, of $ B - L $.
$ \Gamma  _ {B-L, N }  $
parametrizes how the deviation of $ n _ N $ from equilibrium
creates an asymmetry of $ B - L $. This reaction, as well as the one
described by
$ \Gamma  _ {N, B-L }  $ violate $ CP $.
We stress that these
equations are valid to all orders in the Standard Model couplings.
The only corrections are due to
relativistic effects of the motion of $ N $. \footnote{For large deviations
from equilibrium there could also be non-linear terms. These are
expected to be higher order in the $ N $-Yukawa couplings.}

For leptogenesis one is usually interested in an initial state
with vanishing $ n _ { B - L } $, which is then generated by
non-vanishing $ n _ N - n _ N ^{ \rm eq } $ as described
by Eq.~(\ref{dnLdt}). The coefficient $ \Gamma  _ { B-L, N } $
contains a small parameter $ \epsilon  $ (cf. Eq.~(\ref{epsilon}))
resulting in $ n _ { B-L } \ll n _ N - n _ N ^ {\rm eq }  $.
We expect the coefficient $ \Gamma  _ { N, B-L } $ to contain
a similarly small factor, so that the second term
on the right-hand-side of Eq.~(\ref{dnNdt}) can be safely neglected.

\section{Leading order coefficients}
\label{sec:lo}

We would like to determine the coefficients in Eqs.~(\ref{dnNdt}),
(\ref{dnLdt})
at leading order in the Standard Model couplings. We expect
that at this order
they can be obtained from a Boltzmann
equation containing the $1 \leftrightarrow 2 $ scattering processes.

We work at leading order in the Yukawa coupling
of right handed neutrinos, and  we therefore neglect processes
which change lepton number by two units via the
exchange between
Standard Model particles. The corresponding rates are suppressed
by an additional factor $ h ^ 2 $, where $ h $ is a generic
Yukawa coupling of the right handed neutrinos (see below).

Since leptons and Higgs bosons are in kinetic equilibrium,
their phase space distributions
can be parametrized by the temperature and by
chemical potentials. In the Boltzmann-equation
for $ f _ N ( t, p )$ with $ p
\equiv | \vec p | $ we can neglect the chemical potentials of Higgs and
leptons because they are of the order of the $ B - L $ asymmetry (recall that
$ n _ { B-L } \ll n _ N - n _ N ^ {\rm eq }  $). We use
Boltzmann statistics instead of Fermi- or Bose-statistics for leptons
and Higgs, because they have energies of order $ M _ N / 2 \gg T$ when they
produce an $ N $. Furthermore, we neglect the Pauli blocking
and Bose enhancement factors. The corresponding errors are suppressed
with $ \exp [ - M_N/ T   ]  $.  After performing the phase space integral
the Boltzmann equation becomes
\begin{align}
  \left (  \partial _ t - H p \partial _ p \right )  f _ N  =
    \frac{ M _ N \Gamma _ 0} { E _ N } \left ( e ^{ - E _ N / T } - f _ N \right  )
    \label{boltzmannN}
    ,
\end{align}
where $ \Gamma  _ 0 $ is the total tree-level decay rate of the
heavy neutrinos, and $ E
_ N = ( \vec p ^ 2 + M _ N ^ 2 ) ^{ -1/2 } $ is their energy.
We consider the Yukawa  interaction
\begin{align}  { \cal L } _ { N { \rm Yuk  } }
= h_{ij} \overline{N _i
                  }{\widetilde \varphi}^\dagger
   \ell_
   j}
   + \mbox{ h.c. }
\end{align}
with the left-handed Standard Model lepton doublets $ \ell _ j $ and the isospin
conjugate $ {\widetilde \varphi} \equiv i \sigma ^ 2 \varphi
^ \ast  $ of the Higgs doublet $ \varphi  $. Then
the decay rate of the lightest right-handed neutrino $ N \equiv N _ 1 $ is
\begin{align}
 \Gamma  _ 0 = \frac{ | h _ { 11 } | ^ 2 M _ N } { 8 \pi  }
 \label{Gamma0}
 ,
\end{align}
if $ \ell _ 1 $ is the lepton produced in the $ N $-decays (see Sect.~\ref{sec:non-relativistic}).

One can obtain an equation for $ n _ N = (2 s _ N + 1 ) ( 2 \pi ) ^{ -3 }
\int d ^ 3 p f _ N $, where $ s _ N = 1/2 $ is the spin of $ N $,
by integrating Eq.~(\ref{boltzmannN}) over $ \vec p $. However,
the loss term on the right-hand side would not only depend on the number density
$ n _ N $, but also on the momentum spectrum of the right-handed neutrinos.
In most papers on leptogenesis this problem is circumvented
by assuming that the $ N $'s are in kinetic equilibrium. Then the
deviation from thermal equilibrium is completely specified by $ n _ N
$.  Here we do not assume kinetic equilibrium. Instead, we make use of the
fact that the $ N $'s are non-relativistic,
and approximate  the factor $ 1/E _ N $ in front of
the bracket by $ 1/M _ N $. For the first coefficient in Eq.~(\ref{dnNdt})
we thus obtain
\begin{align}
   \Gamma  _ N = \Gamma  _ 0
   \label{GammaN}
\end{align}
at leading order in the Standard Model couplings.
Corrections to Eq.~(\ref{GammaN}) will be discussed
in Sect.~\ref{sec:radiative}.

Now we determine the coefficients in Eq.~(\ref{dnLdt}).
In many cases \footnote{Eq.~(\ref{boltzmannN}) is an exception.}
the Boltzmann equation for a given particle species contains
collision terms which are much larger than the collision terms
contributing to leptogenesis
itself. The processes which give these large contributions
are called
spectator processes
\cite{buchmuller-spectator}. Their role is to bring the
quantum numbers which are not conserved into thermal
equilibrium.
Important examples are scattering by gauge boson exchange,
the anomalous quark-chirality changing QCD-sphalerons
and the  anomalous $ B + L $ changing electroweak sphaleron processes.
Which spectators are in equilibrium depends on the temperature
(cf.\ Table~\ref{tab:spectators}).
All spectator processes conserve $ B - L $. Thus, after
adding the Boltzmann equations for all quarks
and antiquarks, weighted by their respective baryon number,
subtracting the equations for leptons and antileptons, weighted by
their lepton number, and then integrating over momentum the
spectator collision terms drop out. The only violations of
$ B - L $ are due to the interactions of the right-handed neutrinos.

The decays of the right-handed neutrinos contribute to the
coefficient
$ \Gamma _ { B - L, N } $.  The asymmetry arises because the decay
rate of $ N $ into matter differs from the one into antimatter, which
is quantified by a non-zero value of
\begin{align}
   \epsilon  \equiv
   \frac{
   \Gamma  ( N \to \varphi  \ell )
     - \Gamma  ( N \to \overline{ \varphi  } \overline{ \ell } )
     }
     {
   \Gamma  ( N \to \varphi  \ell )
     + \Gamma  ( N \to \overline{ \varphi  } \overline{ \ell } )
     }
     \label{epsilon}
   .
\end{align}
Therefore
\begin{align}

  \Gamma  _ { B - L, N }
  = { \epsilon  \Gamma  _ 0 }
   \label{GammaB-LN}
\end{align}
at leading order in the Standard Model couplings.

Inverse decays of the heavy neutrinos contribute to the washout term
in Eq.~(\ref{dnLdt}) at leading order. At higher orders in the
$ h _ { ij } $
there are also $ 2 \leftrightarrow  2 $
scattering processes with a right handed neutrino exchange
which change lepton number by two units. Unlike
the inverse decays these are not exponentially suppressed
in the non-relativistic regime (see below) which may compensate
the suppression by
the Yukawa couplings and powers of $ T / M _ N $.
The washout factor (\ref{K}) is of order
$ K \sim  h _ { ij } ^ 2 m _ {\rm Pl } / M _ N $, where $ m _ {\rm Pl } $
denotes the Planck mass. Thus at fixed $ K $,  $ h _ { ij } $ increases
with increasing $ M _ N $.  Therefore at very large $ M _ N $
these processes can become important, but we do not consider them here.

Like in the computation of $ \Gamma  _ N $
one can use Maxwell-Boltzmann statistics and neglect
Pauli blocking and Bose enhancement terms, which gives
\begin{align}
   \Gamma _ { B-L }\, n _ { B - L }
   =
   \int \prod _ { a = N, \ell, \varphi  } \frac{ d ^ 3 p _ a }
   { 2 E _ a ( 2 \pi  ) ^ 3 }
   ( 2 \pi  ) ^ 4 \delta  ( p _ \ell + p _ \varphi  - p _ N )
   \left ( f _ \ell f _ \varphi  - f _ { \overline{\ell } }
      f _ { \overline{ \varphi  } } \right )
    \sum | { \cal M } _ 0| ^ 2
    \label{coll}
    .
\end{align}
Here $ \sum | { \cal M } _ 0 | ^ 2 = 16 \pi  M _ N \Gamma  _ 0 $
(see Eq.~(\ref{Gamma0})) is the
tree level matrix element squared for the process $ \ell \varphi  \to N $ summed
over all spins of $ N $ and all isospin components of $ \ell $.
Since we consider small deviations from equilibrium we can expand
in the chemical potentials $ \mu  _ \ell $ and $ \mu  _ \varphi  $
and keep only the linear order,
\begin{align}  f _ \ell f _ \varphi  - f _ { \overline{\ell } }
      f _ { \overline{ \varphi  } } \simeq 2 e ^{ - E _ N / T }
   ( \mu  _ \ell + \mu  _ \varphi  ) /T
   \label{ff}
   .
\end{align}
The chemical potentials
\footnote{Oftentimes the chemical potential of the Higgs bosons is ignored.}
are proportional to $ n _ { B-L }$.
Their precise relation is temperature dependent, because it depends on
which spectator processes are in effect.
One can introduce coefficients $ c _ \ell
$ and $ c _ \varphi   $ to parametrize the relation of lepton and Higgs asymmetries
to $ B - L $,
\footnote{We thank Sacha Davidson
  for clarifying remarks regarding these relations.}
\begin{align}
   n _ \ell - n _ { \overline{ \ell } } =& - c _ \ell  \, n _ { B - L }
   ,\label{cell}
   \\
   n _ \varphi  - n _ { \overline{ \varphi  } } = & - c _ \varphi  \, n _ { B - L }
   \label{cphi}
   .
\end{align}
Results for $  c _ \ell $ and $ c _ \varphi   $ for various temperature
regimes can be found in Table 1 of Ref.~\cite{davidson-review}.
To evaluate the left-hand sides of Eqs.~(\ref{cell})
and (\ref{cphi}) one can expand the  Fermi- and Bose-
distributions for leptons and Higgses up to linear order in the chemical
potentials, and then integrate over momenta.
Note that the momentum integrals are saturated at $ p \sim
T $. That means that, unlike above, one cannot use the Maxwell-Boltzmann
distribution. Using the Fermi- and Bose-distributions instead one
finds
\begin{align}
      \mu  _ \ell  & = \frac{ 3 c _ \ell   }
   { T ^ 2 } \, n _ { B - L }
   \label{muell}
   , \\
   \mu  _ \varphi  & = \frac{ 3 c _ \varphi  }
   { 2 T ^ 2 }\, n _ { B - L }
   \label{muphi}
   .
\end{align}
Combining Eqs.~(\ref{coll}), (\ref{ff}), (\ref{muell}), and (\ref{muphi})
one finally obtains for the washout rate
\begin{align}
  \Gamma  _ { B - L }
  = \frac{ 3 } { \pi  ^ 2 }
  \left ( c _ \ell  + \frac{ c _ \varphi  } { 2 } \right )
  z ^ 2 K _ 1 ( z ) \Gamma  _ 0
  \label{GammaB-L}
   .
\end{align}
Here $ K _ 1 $ is the modified Bessel function of the second kind and
\begin{align}
   z \equiv \frac{ M _ N } { T }
   \label{z}
   .
\end{align}
Note that  we did not make a
non-relativistic approximation for  $ \Gamma _ {B - L } $. Our result
differs from
Ref.~\cite{buchmuller-pedestrians} (where $ c _ \ell = 1 $, $ c _ \varphi  =0 $
was used) and \cite{davidson-review} by a factor $ 12/\pi  ^ 2 $. This
is because we have used quantum statistics to obtain Eqs.~(\ref{muell}) and
(\ref{muphi}). It is suggestive to use
classical statistics for Eqs.~(\ref{coll}) and (\ref{ff}), because
the lepton number washout is due to leptons and Higgs bosons
with energy of order $ M _ N /2 \gg T $. However, the kinetic
equilibration rate (see footnote \ref{stopping} on page \pageref{stopping})
is much larger than the interaction rate of the heavy neutrinos. Therefore
the relation between
the asymmetry and the chemical potentials is determined by the
bulk of kinetically equilibrated leptons and Higgs bosons, which have momenta of order $ T $.

Let us finally comment on the washout due to $ \Delta L = 2 $ processes
mediated by the exchange of a right-handed Majorana neutrino. Unlike
Eq.~(\ref{GammaB-L}) the corresponding rate is not Boltzmann
suppressed with $ e ^{ -M _ N /T } $ for $ T \ll M_N $. On the other hand,
the $ \Delta L = 2 $ rate is of order $ h ^ 4 $. For a given value
of $ K $ large $ h ^2 $ means large $M _N $. Thus the $ \Delta L = 2 $
processes can be important for very heavy right-handed neutrinos.
They were found to be neglible for
$ M _ N \lesssim 10 ^ {13} $ GeV \cite{davidson-review}.

\begin{figure}[t]
  \centerline{
    \epsfxsize=10cm
    \epsffile{washout-over-Hubble.eps}
  } \vspace{0cm}
  \caption[a]
    \label{fg:ratio}
        Ratio of reaction rates in Eqs.~(\ref{dnNdt}), (\ref{dnLdt})
        and the Hubble rate, normalized to the washout factor $ K $.
        $ \Gamma  _ N/H $
        (full black line) increases with $ z $, implying that abundance of
        right-handed neutrinos gets closer and closer to equilibrium with
        increasing time, or, since the equilibrium density drops exponentially,
        gets closer and closer to zero. On the other hand, $ \Gamma  _ { B-L }/H $
        (red dotted and blue dashed line)
        first increases, and then drops exponentially, implying that
        $ B - L $ freezes out.
          }
\end{figure}

\section{Interaction rates and thermal (non-)equilibrium}
\label{sec:neq}

The right-handed neutrinos are out of equilibrium if their interaction
rate is smaller than the Hubble rate. When $ T \mathop{\lsi}  M _ N $,
the interaction rate can be estimated as $ \Gamma  _ 0 $. Therefore
the ratio of the two rates is
\begin{align}
   \frac{   \Gamma  _ N  } {  H } \simeq
   \frac{ \Gamma  _ 0 }{ (T ^ 2/ M _ N ^ 2 ) H | _ { T=M _ N }   }
   =   \left ( \frac{ M _ N } { T  } \right ) ^ 2 K
   \label{Nratio}
   ,
\end{align}
i.e., the deviation from equilibrium becomes smaller
when the temperature decreases. It also means that the deviation from
kinetic equilibrium becomes smaller.
On the other hand, the washout rate is of the same order as $ \Gamma  _ N $
 for $ T \sim M _ N $,
while  for $ T \ll M _ N $
\begin{align}
   \frac{ \Gamma  _ { B - L } } { H } \sim
   \left ( \frac{ M _ N } { T } \right ) ^{ 7/2}
   e ^{ - M _ N / T }  K
   \label{Lratio}
   .
\end{align}
Thus  $ \Gamma  _ { B - L } / H $ should first increase
when the temperature falls below $ M _ N $, and should then drop sharply due to the exponential.
Fig.~\ref{fg:ratio} shows the ratios $ \Gamma  _ N /H $ and $ \Gamma  _ { B-L }/H $,
normalized
to the washout factor $ K $. The ratio $ \Gamma  _ { B-L } /H $ indeed
has a maximum between $ z = 3 $ and $ z = 4 $. Thus one should expect that
this is the most important $ z $-range for leptogenesis. Futhermore, for $ K \mathop{\gsi}  1 $
the deviations from chemical and kinetic equilibrium should be small in
this regime. Note that for
$ z \gg 1 $ there is a hierarchy between the scale for $ B - L $ dissipation
and of the heavy neutrino equilibration, $   \Gamma  _ { B-L } \ll \Gamma  _ N  $.

\section{Relativistic corrections}
\label{sec:relativistic}

There are corrections to Eqs.~(\ref{dnNdt}), (\ref{dnLdt}) due to the
motion of the right-handed neutrinos. In general their momentum
distribution will also deviate from thermal equilibrium.
We expect the first correction to
be related to the kinetic energy density. There can also be relativistic
corrections related to the spin of the right-handed neutrinos, which
we do not consider here. \footnote{We thank M.~Laine for pointing this out.}
At leading order in the gauge and Yukawa couplings we expect that
the correct coefficients in the kinetic equations can be obtained
from Boltzmann equations which contain only the decay and inverse
decay processes.

We have
$ E _ N = M _ N [  1 + O ( v ^ 2 ) ]  $ where $ v $ is a typical
velocity of the $ N $. In thermal equilibrium
$ \vec p ^ 2/M _ N \sim T $ and therefore $ v ^ 2 \sim T/M  _ N
\sim 1/z $.
We expand  the factor $ 1/E _ N $ in front of
the bracket in Eq.~(\ref{boltzmannN}) up to order $ v ^ 2 $, which gives
\begin{align}
  \left ( \frac{ d } { d t } + 3 H \right )
  n _ N = \Gamma _ N \left ( n _ N ^  {\rm eq } - n _ N \right )
  + \Gamma  _ { N,  u  }
     \left (  u - u ^  {\rm eq } \right )
   \label{dnNdtNLO}
   .
\end{align}
Here
\begin{align}
   u \equiv \frac{ 1 } { M _ N} (2 s _ N + 1 ) \int
   \frac{ d ^ 3 p } { ( 2 \pi  ) ^ 3 } \frac{ \vec p ^ 2 } { 2 M _ N}
   f _ N
\end{align}
is the kinetic energy density of the heavy neutrinos devided by their mass.
At lowest order
\begin{align}
   \Gamma  _ { N,  u } = \Gamma  _ 0
   .
\end{align}
The equation
for $ u $ is obtained by multiplying Eq.~(\ref{boltzmannN}) with $ \vec p ^ 2 $
and integrating over $ \vec p $. At leading order in $ v $ this gives
\begin{align}
  \left ( \frac{ d } { d t }  + 5 H \right ) u
 =  \Gamma  _ u \left ( u ^  {\rm eq } - u \right )
 \label{dudt}
 ,
\end{align}
with $ \Gamma_u = \Gamma  _ 0 $ at leading order in $ T/M _ N $ and
in the Standard Model couplings.
The equation for the asymmetry is modified accordingly,
\begin{align}
   \left ( \frac{ d } { d t } + 3 H \right ) n _ { B - L }
   =

   \Gamma  _ { B - L, N } \left ( n _ N  - n _ N ^  {\rm eq } \right )
   +

   \Gamma  _ { B - L, u } \left (      u - u ^  {\rm eq } \right )
   -

   \Gamma  _ { B - L } n _ { B - L }
   \label{dnLdtNLO}
   .
\end{align}
Using the same expansion as for Eq.~(\ref{dnNdtNLO})
one finds at leading order (cf.\ Eq.~(\ref{GammaB-LN}))
\begin{align}
    \Gamma  _ {B-L,  u } = \epsilon  \Gamma  _ 0
   .
\end{align}

\section{Radiative corrections}
\label{sec:radiative}

At leading order only the decays and inverse decays of right-handed neutrinos contribute
to the coefficients in Eqs.~(\ref{dnNdt}) and  (\ref{dnLdt}). In many papers
$ 2 \leftrightarrow 2 $ scattering processes which involve Standard Model
interactions have been included as well, and
their effects were found to be substantial. Recently \cite{salvio} it
was pointed out, that such calculations are not complete because they only
contain $ 2 \leftrightarrow 2 $ particle scattering, while ignoring $
1 \leftrightarrow 3 $ processes and virtual corrections to the leading order $ 1
\leftrightarrow 2 $ scattering.  Infrared contributions
which were found earlier were shown to cancel in the complete answer.

Here we show how  radiative corrections can be incorporated into our approach.
The equilibrium densities $ n _ N ^ { \rm eq } $ and $ u ^ { \rm eq }
$ do not receive radiative corrections from Standard Model interactions.
Therefore they only affect the coefficients $ \Gamma
_ i $ in Eqs.~(\ref{dnNdtNLO}), (\ref{dudt}), and (\ref{dnLdtNLO}).

Radiative corrections to the production rate $ d n _ N / d t $ at
vanishing $ n _ N $ have been computed
in Refs.~\cite{salvio,laine-nlo}.  In \cite{laine-nlo} also the
radiative corrections to the differential rate  $ \partial f _ N /
\partial t $ at $ n _ N = 0 $ have been computed. It can be written
as
\begin{align}
   \left .
     \frac{ \partial  f _ N ( t, \vec p ) } { \partial t }
     \right | _ { f _ N = 0 }
   =
   f _ {\rm F } ( E _ N ) \Gamma  _ 0 \frac{ M _ N } { E _ N }
   \left \{ a + \frac{ \vec p^ 2 } { M _ N ^ 2 } \,b

     + O \left ( \frac{ \vec  p ^ 4 } { M _ N ^ 4 } \right )
    \right \}
   \label{radcorr}
   .
\end{align}
The coefficients $  a $, $ b $ are temperature-dependent and can be expanded
in $ T / M _ N $.

Assuming that Eq.~(\ref{dnNdt}) is still valid for $ n _ N = 0 $,
one can read off the  radiative corrections to
the coefficients in the kinetic equations for $ n _ N $ (Eq.~(\ref{dnNdtNLO})),
and for $ u $ (Eq.~(\ref{dudt})) from Eq.~(\ref{radcorr}).
Expanding the factor $ 1/E _ N $ as in Eq.~(\ref{boltzmannN}), and
integrating over $ \vec p $ we obtain
\begin{align}
  \Gamma  _ N = \Gamma _ u = a \Gamma  _ 0   , \qquad
  \Gamma  _ { N , u } =    ( a - 2 b ) \Gamma  _ 0
   .
   \label{Gamma-w-radcorr}
\end{align}
Radiative corrections that would enter the coefficients in Eq.~(\ref{dnLdtNLO})
for the asymmetry have not been computed so far.

The  coefficients in Eq.~(\ref{radcorr}) are~\footnote{ The $ O ( g ^ 2 ) $ and
the  $ O ( \lambda  T ^ 2 /M _ N ^ 2 ) $
corrections in $ a $ were first computed in Ref.~\cite{salvio}. The $ O ( g ^ 2 ) $
contributions in Refs.~\cite{salvio} and \cite{laine-nlo}  agree, but the
 $ O ( \lambda  T ^ 2 /M _ N ^ 2 ) $ in Ref.~\cite{salvio}
is 4 times larger than the one
in Ref.~\cite{laine-nlo} (see footnote \ref{fuss} on page \pageref{lambda}).
For our numerical results in Sect.~\ref{sec:numerical} we have used the expression in
Ref.~\cite{laine-nlo}. The other terms in
$ a $ and $ b $ were computed in Ref.~\cite{laine-nlo}.}

\begin{align}
 a = {} &
   1 - \frac{\lambda T^2}{M_N^2}
   \, - \,
   |h_t|^2
   \biggl[
     \frac{21}{2(4\pi)^2} +
     \frac{7\pi^2 }{60}
      \frac{T^4}{M_N^4}
   \biggr]
     + \,
   (g_1^2 + 3 g_2^2)
   \biggl[
     \frac{29}{8(4\pi)^2} -
     \frac{\pi^2 }{80}
     \frac{T^4}{M_N^4}
     \biggr]
        \nonumber \\ & {}
     + O \left (  g ^ 2 \frac{ T ^ 6 } { M _ N ^ 6 } ,
      g ^ 3 \frac{ T ^ 2 } { M _ N ^ 2 }\right )
    \label{a}
    \,
    ,
 \end{align}
 \begin{align}
   b = &
   \, - \, \left [
   |h_t|^2
     \frac{7\pi^2 }{45}

     + \,
   (g_1^2 + 3 g_2^2)
     \frac{\pi^2 }{60}
     \right ]  \frac{T^4}{M_N^4}
     + O \left (  g ^ 2 \frac{ T ^ 6 } { M _ N ^ 6 } ,
      g ^ 3 \frac{ T ^ 2 } { M _ N ^ 2 }\right )
     \label{b}
     ,
\end{align}
where $ h _ t $ is the top Yukawa coupling, $ g _ 2 $, $ g _ 1 $ are the
weak SU(2) and U(1) gauge couplings,
$ \lambda = m _ {\rm Higgs }  ^ 2 G _ {\rm F } /\sqrt{ 2 } $
is the Higgs self-coupling~\footnote{\label{fuss}This definition
of $  \lambda $   is the same as the one used
in Ref.~\cite{laine-nlo}.  Ref.~\cite{salvio} writes the rate in terms of $ \lambda _ h \equiv
 m ^ 2 _ {\rm Higgs}/ v _ h ^ 2 $ with $ v _ h = 174 $GeV ($ = v $ in the notation of
Ref.~\cite{salvio}) which is 4 times larger
than $ \lambda  $. In Ref.~\cite{salvio} the rate depends on $ \lambda  _ h $ the same
way as it does depend on $ \lambda  $ in Ref.~\cite{laine-nlo}.}\label{lambda}
and $ G _ {\rm F } $ is the Fermi constant.
The neutrino Yukawa couplings are defined in the $ \overline{\mbox{MS}} $
scheme with renormalization  scale $  M _ N $.

The leading temperature-dependent term in $ a $
is of order
$ \lambda  T ^ 2/M _ N ^ 2 \sim \lambda  v ^ 4 $, where $ v $
is a typical thermal velocity of the heavy neutrinos (cf.\ Sect.~\ref{sec:relativistic}).
If we think of our approach as an expansion in $ v $, and if we
assume $ \lambda   $ to be small, then this term is
of higher order than the one we included in Sect.~\ref{sec:relativistic}.
The other temperature dependent radiative corrections are of even
higher
order ($  \sim g ^ 2 v ^ 8 $), where  $ g $ denotes a generic
Standard Model gauge or Yukawa coupling. It would be straightforward to extend the
equations in Sect.~\ref{sec:relativistic} to higher orders in $ v $
as well. We did not do so, because, as we show in  Sect.~\ref{sec:numerical},
the order $  v ^ 2 $ corrections are quite small in the
parameter region of interest.

\section{Numerical results}
\label{sec:numerical}

In this section we solve the equations for leptogenesis in the
non-relativistic limit
(Sect.~\ref{sec:non-relativistic}), and we add relativistic (Sect.~\ref{sec:relativistic})
and radiative (Sect.~\ref{sec:radiative}) corrections. We parametrize our results
by the effective light neutrino mass~\cite{plumacher}
\begin{align}
   \widetilde{ m } _ 1 \equiv
   \frac{ \left ( m _ {\rm D } ^\dagger m _ {\rm D } \right )  _ { 11 }  }
   { M _ N }
   ,
\end{align}
where $ m _ {\rm D } $ is the Dirac mass matrix of the neutrinos.
It is related to the  washout pa\-ra\-meter $ K $ by \cite{buchmuller-pedestrians}
\begin{align}
   \widetilde{m } _ 1  = K m _ \ast
   \label{m1tilde}
   .
\end{align}
We have used the Standard Model value
\begin{align}
    m _ \ast \simeq 1.08  \times  10 ^{ -3 } {\rm eV }
    .
\end{align}
It
can be shown \cite{fujii} that $  \widetilde{ m } _ 1 $ is
larger than the smallest light neutrino mass.
The range $( \Delta m ^ 2_ {\rm solar } ) ^{ 1/2 } < \widetilde{ m } _
1 < ( \Delta m ^ 2 _ {\rm atmospheric } ) ^{ 1/2 } $ corresponds to $
7.4 < K < 46 $.

\begin{figure}[t]
  \centerline{
    \epsfxsize=10cm
    \epsffile{NLO-LO_ic-thermal.eps}
  } \vspace{0cm}
  \caption[a]
    Relative size of the relativistic corrections to the
    $ B - L $ asymmetry.
Here we used thermal initial conditions. The temperature ranges
correspond to the different values of $ c _ \ell $ and $ c _ \varphi  $. For
$ T \mathop{\lsi}  10 ^{ 12 } $GeV these values correspond to the right-handed neutrinos
 decaying into
$ \tau  $-leptons.
    \label{fg:K-dependence}
          }
\end{figure}

We have started the evolution of $ n _ N $, $ n _ { B - L } $,
and $ u $  at $ z = 1 $ with vanishing $ B - L $ asymmetry. Since
$ v \sim 1 $ for $ z \sim 1 $, the non-relativistic expansion cannot
be expected to work for smaller $ z $.
Following common practice
we express the final value of $ B - L $ asymmetry in terms of the final
efficiency factor $ \kappa $ defined by
\begin{align}
    \lim _ { z \to \infty }
   \frac{ n _ {B-L}
   } { n _ \gamma ^ {\rm eq} }
   \equiv
   \frac{3}{4} \epsilon \kappa
   \label{kappa}
   .
\end{align}
Here $ n _ \gamma ^ {\rm eq} = 2 \zeta  ( 3 ) T ^ 3/\pi  ^ 2 $
is the photon equilibrium density, and
$ \epsilon $ is defined in Eq.~(\ref{epsilon}).

In Fig.~\ref{fg:K-dependence} we show the size of the $ O ( v ^ 2 ) $
relativistic
corrections normalized to the $ O ( v ^ 2 ) $ corrected result
for several temperature ranges by varying $ c _ \ell $
and $ c _ \varphi $ according to Table 1 of Ref.~\cite{davidson-review}.
We find that for $ K \gtrsim  5 $ the size of the relativistic corrections is
already smaller than $ 3 \% $. A detailed overview of
the results for $ K =  8 $ can be found in Table~\ref{tab:spectators}.

Next we study the size
of the relativistic corrections for different initial conditions for the
temperature range $ 10 ^ 8 \mathop{\lsi}  T/\mbox{GeV} \mathop{\lsi}  10 ^{ 11 } $. We compare
thermal initial conditions with the extreme case of
zero initial values for $ n _ N $ and $ u $
at $ z=1 $. The latter initial condition is somewhat unphysical, because
for $ K \mathop{\gsi}  1 $ a substantial number of right handed neutrinos
will have been thermally produced by the time when $ z=1 $. In Fig.~\ref{fg:zero}
we compare the results for these two cases. For $ K \mathop{\lsi} 4 $ the non-relativistic
approximation clearly breaks down for zero initial densities. For
$ K \mathop{\gsi}  5 $ the relativistic corrections
remain small ($ \mathop{\lsi} 1.6\% $ for the parameters in Fig.~\ref{fg:zero})
for zero initial values.

It is remarkable that the corrections are so small. This clearly indicates
that the non-relativistic approximation works very well, and that it is not necessary
to use a Boltzmann equation to take into account the momentum distribution of
the heavy neutrinos.

\begin{table}
\centering
\renewcommand{\arraystretch}{2}
\begin{tabular}{ccccccc}
 $ T $ (GeV) & spectators & $c_{\ell}$ & $c_{\varphi}$ & $ \kappa_{\rm{NR}} $ & $ \kappa $ & ($ \kappa - \kappa_{\textnormal{NR}} $) / $ \kappa $ \\
 \hline
  $\gg 10^{13}$ & $h_t$, gauge & 1 & $\nicefrac{2}{3}$ & 0.0253 & 0.0255 & +0.7\% \\
  $\sim 10^{13}$ & + QCD sphalerons & 1 & $\nicefrac{14}{23}$ & 0.0260 & 0.0262 & +0.7\% \\
  $10^{12}$ - $10^{13}$ & + $h_b, h_{\tau}$ & $\nicefrac{3}{4}$ & $\nicefrac{1}{2}$ & 0.0363 & 0.0366 & +0.9\% \\
  $10^{11}$ - $10^{12}$ & + $h_c, h_s, h_{\mu}$ & $\nicefrac{78}{115}$ & $\nicefrac{56}{115}$ & 0.0403 & 0.0406 & +1.0\% \\
  $10^{8}$ - $10^{11}$ & + EW sphalerons & $\nicefrac{344}{537}$ & $\nicefrac{52}{179}$ & 0.0495 & 0.0500 & +1.1\% \\
   - & - & 1 & 0 & 0.0363 & 0.0366 & +0.9\%
\end{tabular}
  \caption{Final efficiency factors in the non-relativistic
approximation ($ \kappa  _ {\rm NR } $) and including $ O ( v ^ 2 ) $  corrections
($ \kappa  $). Here we have used $ K = 8 $ and thermal initial conditions.
The values of $ c_{\ell} $
and $ c_{\varphi} $ are taken from Ref.~\cite{davidson-review}. We also include
the case $ c_{\ell} = 1 $, $ c _ {\varphi} = 0 $ which has been used in many papers.}
  \label{tab:spectators}
\end{table}

In our approach a deviation from kinetic equilibrium
enters the difference $ u - u ^{\rm eq} $. Thus the fact that the relativistic
corrections are so small also indicates that kinetic equilibrium is a good
approximation. We have checked this by comparing the non-relativistic approximation
with the kinetic equilibrium approximation. The difference between the two is
indeed quite small, and similar
in size as the $ O ( v ^ 2 )  $  corrections. What turns out to be
even smaller is the difference
between our non-relativistic approximation plus $ O ( v ^ 2 ) $ corrections
and the kinetic equilibrium approximation. For example, for $ K = 8 $ and $ T= 10 ^
{ 10 } $
GeV the kinetic equilibrium approximation deviates from the non-relativistic one
by 0.87\%, and it deviates from the $ O ( v ^ 2 ) $ corrected one by only 0.3\%.

\begin{figure}[t]
  \centerline{
    \epsfxsize=10cm
    \epsffile{dependence-on-initial-conditions.eps}
  } \vspace{0cm}
  \caption[a]
 Final efficiency factors for two different initial conditions imposed at $ z = 1 $.
 For thermal initial conditions the non-relativistic approximation remains fairly
 reliable down to very small $ K $. For vanishing initial densities it breaks down
 for $ K \mathop{\lsi} 4 $. For larger $ K $ the dependence on the initial conditions is
 rather weak. The
 temperature is in the range $ 10 ^ 8 \mathop{\lsi}  T/\mbox{GeV} \mathop{\lsi}  10 ^{ 11 } $.
    \label{fg:zero}
          }
\end{figure}

It is interesting to compare our results with those
of Refs.~\cite{basboll,garayoa,hahn}. There Boltzmann equations were
used to compute the momentum spectrum of the heavy neutrinos without assuming
kinetic equilibrium.
Ref.~\cite{basboll} compares results obtained with  quantum
statistics and the momentum dependent Boltzmann equation  with results obtained
with classical statistics and kinetic equilibrium.
For $ K > 5 $ a discrepancy of about $ 15\% $ was found. Ref.~\cite{garayoa}
considers the same setup and confirms a discrepancy of 20\% for  $ K > 1 $.
Our results, on the other hand, indicate that the deviation from kinetic
equilibrium should have a much smaller effect,
which would mean that the discrepancy found in Ref.~\cite{basboll,garayoa}
is due to the statistics. We have checked this switching from quantum to
Boltzmann-statistics for Eqs.~(\ref{muell}), (\ref{muphi}) in the non-relativistic
approximation. Results are shown in Fig.~\ref{fg:boltzmann}, where we see that
using Boltzmann distributions
would underestimate  the asymmetry by at least 20\% in the phenomenologically
interesting region, confirming that the discrepancy is due to the statistics.
Let us stress again that for Eqs.~(\ref{muell}) and
(\ref{muphi}) the correct quantum statistics has to be used even in the non-relativistic
regime,  because  the corresponding momentum integrals are for relativistic particles
and they are saturated at momenta of order $ T $.

\begin{figure}[t]
  \centerline{
    \epsfxsize=10cm
    \epsffile{quantum-over-classical.eps}
  } \vspace{0cm}
  \caption[a]
    Final efficiency factor $ \kappa  $
    computed with Bose- and Fermi-distributions in the washout term, divided
    by the final efficiency factor with the classical
    Maxwell-Boltzmann statistics. In both cases we
    have used the non-relativistic
    approximation. We have chosen
    $ c _ \ell = 1 $, $ c _ \varphi  =0 $ to facilitate comparison with
    Ref.~\cite{basboll}. This figure shows that it is important to
    use the correct quantum statistics. Even at very large $ K $ the error
    caused by using classical statistics is of order 20\%.
    \label{fg:boltzmann}
          }
\end{figure}

The radiative corrections
discussed in Sect.~\ref{sec:radiative}
depend not only on the washout factor, but, through the running
of the Standard Model couplings, also on the neutrino mass $ M _ N $.
In Fig.~\ref{fg:radcorr} we show results for the corrected efficiency factors,
normalized  to the non-relativistic approximation without radiative corrections
$ \kappa  _ {\rm LO} $,
for   $ M _ N = 10 ^ { 10 } $
and $ 10 ^{ 8 } $ GeV.
All curves contain the relativistic $ O ( v ^ 2 ) $ corrections.
They were obtained with  $ \Gamma _ N $, $ \Gamma _ {
N, u }  $, and $ \Gamma  _ u $ computed at different orders in the Standard Model
couplings and in $ v ^ 2 \sim T / M _ N $. In the favored
regime $ 7.4 \lesssim K \lesssim 46 $ the effects of radiative corrections
are
smaller than the $ O ( v ^ 2 ) $ corrections. For smaller
values of $ K $ the effect of the $ O ( g ^ 2) $ and  the $ O ( \lambda  v ^ 4 ) $
corrections to the $ \Gamma  $'s
remain small.
The  fact that the $ O ( \lambda  v ^ 4 ) $
has a  small effect is partly due to the smallness of the Higgs self-coupling
at these scales. The effect of the
$ O ( g ^ 2 v ^ 8 ) $  contributions grows significantly with decreasing $ K $ (see also
Table \ref{tab:radcorr}). This indicates
that in the regime where $ v $-dependent corrections
are important,
the expansion of radiative corrections in powers of $ v ^ 2 $ does not converge
(cf.\ Ref.~\cite{laine-nlo-relativistic}).

\begin{figure}[t]
  \centerline{
    \epsfxsize=14cm
    \epsffile{all-in-one.eps}
  } \vspace{0cm}
  \caption[a]
    Final efficiency factors containing the
    relativistic $ O ( v ^ 2 ) $ corrections,
    normalized to the non-relativistic approximation without radiative corrections
    $ \kappa  _ {\rm LO } $. The different curves correspond to
    different orders included in the coefficients $ \Gamma  _ N $, $ \Gamma  _ { N, u }
    $, and $ \Gamma  _ u  $.
    The renormalization scale of the Standard Model coupling is chosen as the mass
    of the heavy neutrinos. Thermal initial
    conditions for $ n _ N $ and $ u $ were used.
    \label{fg:radcorr}
          }
\end{figure}

\section{Summary and  discussion}
\label{sec:discussion}

We have obtained the rate equations for single flavor leptogenesis in the non-relativistic
limit which are valid to all orders in the Standard Model couplings.
We proposed a systematic expansion around the non-relativistic limit.
It can be used to improve the non-relativistic approximation, and to
assess its range of validity. The coefficients in the rate equations
and the first relativistic corrections were determined
at leading order in the Standard Model couplings. We computed the
$ B - L $ asymmetry from these equations, and we
found that the relativistic  corrections
are quite small in the parameter region favored by
atmospheric and solar neutrino oscillations, which lies in
the so-called strong washout regime.
They are much smaller than differences
between the non-relativistic approximation and the standard result which
uses the assumption of kinetic equilibrium for the
heavy neutrinos as well as classical statistics
for all particle species.
We found that this difference is mainly due to the statistics used
for computing the washout term, and not so much due
to the assumption of kinetic equilibrium.

The smallness of relativistic corrections should be very useful, because
one can obtain precise results without having to compute the momentum
spectrum of the right-handed neutrinos, and also
because it significantly  simplifies the task to take into account Standard Model radiative
corrections. We have taken a step in this
direction by including the production rate of right handed neutrinos
at next-to-leading order in the Standard Model couplings. We find that the radiative
corrections are smaller than the tree-level
relativistic corrections in the favored region
of the washout factor $ K $. For smaller values of $ K $ they become larger
than the tree-level relativistic corrections. This is due to temperature dependent
gauge and top Yukawa interactions which we consistently included in
a leptogenesis calculation for the first time.

\begin{table}
\centering
\begin{tabular}{c|cc}
  & \multicolumn{2}{c}{$ \left( \kappa_{\textnormal{NLO}} -
      \kappa _{\textnormal{LO}} \right) /  \kappa_{\textnormal{LO}} $} \\
 \hline
 $ K $ & \small{$ M_N=10^{8} $ GeV} & $ M_N=10^{10} $ GeV \\
 \hline
  5 & $+3.7\%$ & $+2.5\%$ \\
  8 & $+0.5\%$ & $0.0\%$ \\
 10 & $+0.1\%$ & $-0.4\%$ \\
 20 & $-0.1\%$ & $-0.5\%$ \\
 50 & $-0.1\%$ & $-0.5\%$
\end{tabular}
 \caption{Influence of the radiative corrections on the final efficiency factor
 for two different values of $ M _ N $. $ \kappa  _ {\rm NLO } $ contains
 all relativistic corrections discussed in Sects.~\ref{sec:relativistic} and
 \ref{sec:radiative} and corresponds to the dotted lines in Fig.~\ref{fg:radcorr}.
 Thermal initial
   conditions for $ n _ N $ and $ u $ were used. }
 \label{tab:radcorr}
\end{table}

To conclude, the non-relativistic expansion proposed in this paper is a convenient
tool for computing the lepton asymmetry, and to assess the validity of the non-relativistic
approximation. In case the relativistic corrections are important, also radiative
corrections become important, and then the non-relativistic expansion breaks down.
In this case one has to use equations which are valid in the fully relativistic regime.
This conclusion is based on the radiative corrections to the equilibration rate
of right-handed neutrinos. It would be interesting to see whether they are also
valid if radiative corrections to the other terms in the kinetic equations are
included.

{\bf  Note added:} After this paper was submitted for publication, we were informed by
the authors of Ref.~\cite{biondini}  that they have confirmed the results of
Ref.~\cite{laine-nlo} for the radiative corrections to the production rate of
right-handed neutrinos.

\section*{Acknowledgements}
 We are grateful to Wilfried Buchm\"uller, Sacha Davidson, and Mikko Laine for
 helpful discussions, suggestions and comments, and to Nora Brambilla and
 Antonio Vairo for correspondence.
 This work was supported in part through the DFG funded Graduate School GRK 881.


text:
Hotel workers to rally for higher pay, better working conditions
/wp-content/uploads/newlogo_512.jpg 0 0 UNITE HERE Local 11 /wp-content/uploads/newlogo_512.jpg UNITE HERE Local 112023-01-24 16:15:342023-01-27 14:55:59Hotel workers to rally for higher pay, better working conditions
Rent's Too High?
Wage needed to afford a 2-bedroom apartment in Los Angeles from the National Low-Income Housing Coalition. See also the Los Angeles Hotel Worker Minimum Wage chart.
Hotels rooms under construction per "Hospitality Market Report – Los Angeles" prepared by Elyse Kirby for STR, accessed January 4, 2023 [login required]. See also affordable housing units approved by the City of Los Angeles.
/wp-content/uploads/newlogo_512.jpg 0 0 UNITE HERE Local 11 /wp-content/uploads/newlogo_512.jpg UNITE HERE Local 112023-01-22 15:57:242023-01-27 14:54:38Rent's Too High?
UNITE HERE Local 11 Housekeepers Submit 100k Signatures on Groundbreaking Initiative to City of Los Angeles
Initiative would follow lead of neighboring cities to mandate panic buttons and raise minimum wage for hotel workers
Los Angeles – Over one hundred housekeepers and other hospitality workers today turned in the petitions they have collected since late January to qualify their initiative for the November 2022 ballot. The initiative mirrors protections they have secured in Long Beach, Santa Monica and most recently West Hollywood.
"I am one of thousands of housekeepers in Los Angeles who will finally have panic buttons and other protections on the job", said Martha Moran, a laid off housekeeper from the storied Chateau Marmont. "My coworkers and I deserve to be treated with dignity and respect, and to receive fair compensation for the work we do. This initiative provides those things."
Over one hundred thousand Angelenos have signed on to the measure that provides:
Panic buttons and other security measures to protect hotel housekeepers from sexual assault and threatening conduct
Fair compensation for heavy workloads
Automatic daily room cleaning throughout the industry
Expansion of minimum wage law for hotel workers
"My heart is always with the workers, like my mother, who worked her fingers to the bone," said Councilman Kevin De Leon. "The hard-working immigrant women and men who make up the hospitality industry in our city are the backbone of our economy and I'm proud to stand with them today as they submit their historic initiative petition. I'm ready to work with my colleagues on the L.A. City Council to transform this initiative into law. The people have spoken, and it's our job to listen."
"Even though I don't work at a hotel, I understand that raising the standards for some of the lowest paid workers in the hospitality industry will bring up standards for all of us," said Isha Kallay, food server from the Hollywood Park and Casino. "I wanted to collect signatures for this initiative because we need to stick together in order for all of us to get ahead."
The housekeeping measure comes in response to the hotel industry's attempt to cut labor costs and increase workloads by eliminating daily room cleaning during the pandemic. It also provides vital protections against sexual assault for housekeepers when cleaning guest rooms alone. The workers call on the Los Angeles City Council to outright adopt the law.
"The hotel industry has wanted to get rid of daily room cleaning for years, and the pandemic gave them the perfect excuse," notes Kurt Petersen, co-president of UNITE HERE Local 11. "Angelenos have just shown the industry, led by the Chateau Marmont, that they see through the greedy pandemic profiteering tactics and stand with the housekeepers. I am hopeful that the Los Angeles City Council will do the same and outright adopt the ordinance. Los Angeles is a leading tourist destination; that should mean good jobs for Angelenos."
Watch the entire press conference here
View more photos here
/wp-content/uploads/newlogo_512.jpg 0 0 UNITE HERE Local 11 /wp-content/uploads/newlogo_512.jpg UNITE HERE Local 112022-05-02 16:18:332022-05-11 11:45:00UNITE HERE Local 11 Housekeepers Submit 100k Signatures on Groundbreaking Initiative to City of Los Angeles
LA Times Podcast: Hotel housekeeping is dirtier than ever
Cristina Velasquez, a housekeeper at Hilton Garden Inn in Los Angeles, says that since the pandemic started, she has had to clean three to four days' worth of trash, dirty linens and towels in the same amount of time as before. Click here to listen to the LA TImes Podcast.
/wp-content/uploads/newlogo_512.jpg 0 0 UNITE HERE Local 11 /wp-content/uploads/newlogo_512.jpg UNITE HERE Local 112022-03-05 10:23:342022-03-18 10:29:59LA Times Podcast: Hotel housekeeping is dirtier than ever
JW Marriot Santa Monica to Pay Housekeepers $25 an Hour Following Agreement With Labor Union
"Hats off to Stockdale Capital for being a responsible employer and for embracing Santa Monica's values of community and justice. Thank you to the heroes of this struggle: the courageous room attendants who never lost faith, fought every day against a nefarious Columbia Sussex, and won an extraordinary union agreement," said Kurt Petersen, Co-President of the union.
/wp-content/uploads/newlogo_512.jpg 0 0 UNITE HERE Local 11 /wp-content/uploads/newlogo_512.jpg UNITE HERE Local 112022-01-10 11:35:322022-01-10 11:37:44JW Marriot Santa Monica to Pay Housekeepers $25 an Hour Following Agreement With Labor Union
How the pandemic made hotel housekeeping more difficult — and disgusting
/wp-content/uploads/newlogo_512.jpg 0 0 UNITE HERE Local 11 /wp-content/uploads/newlogo_512.jpg UNITE HERE Local 112022-01-06 22:31:362022-06-01 13:30:42How the pandemic made hotel housekeeping more difficult — and disgusting
Women have been hit hardest by COVID's economic toll
/wp-content/uploads/newlogo_512.jpg 0 0 UNITE HERE Local 11 /wp-content/uploads/newlogo_512.jpg UNITE HERE Local 112021-05-07 16:20:052022-06-01 13:55:34Women have been hit hardest by COVID's economic toll
Industry-Favored Hotel Mr. C Vetoes Panic Buttons for Employees
/wp-content/uploads/newlogo_512.jpg 0 0 UNITE HERE Local 11 /wp-content/uploads/newlogo_512.jpg UNITE HERE Local 112021-05-03 16:51:462021-05-07 16:53:44Industry-Favored Hotel Mr. C Vetoes Panic Buttons for Employees

redpajama_set_name: RedPajamaCommonCrawl


Detached three bedroom cottage with a secure garden, in a friendly little community. Near to shops and beaches, lots of historic places to visit. On a bus route, spots centre with ten pin bowling, for a day with a different type of activity.


Enable tracking Bangladesh's progress towards attainment of SDGs and other national development goals through a web-based information repository. This unique, searchable database provides a snapshot of what those global and national priorities are. Users can get the latest updates of the status of implementation of those goals along with the facilities of data visualization in multiple ways.
The sustainable development goals (SDGs) are a new, universal set of goals, targets and indicators that UN member states will use to frame their development agendas and political policies over the next 15 years. On September 25th 2015, countries adopted SDGs to end poverty, protect the planet, and ensure prosperity for all as part of a new sustainable development agenda. The commitment to "leave no-one behind" is a key feature of the SDGs. The goals are not considered to have been met if those who are most vulnerable and hard to reach are not included.
To ensure Sustainable Development Goals in Bangladesh by leaving no one behind in most possible short time, a set of 39 indicators has been selected under the instructions of SDG Working Committee of The Prime Minister's Office. Under this indicators, some of the indicators are selected from the global Sustainable Development Goals and some of the indicators are selected after modification on Bangladesh perspective. All relevant ministries are connected with this process.
Since 1971 after liberation, the economy of Bangladesh has been premised on the concept of planning. This has been carried through the Five-Year Plans, developed, executed, and monitored by the Planning Commission. In 1973, the First Five Year Plan was formulated under the guidance of the Father of the Nation Bangabandhu Sheikh Mujibur Rahman. Accordingly, between FY1973 and FY2002 Bangladesh implemented 5 successive Five Year Plans. The sixth Five Year Plan was implemented during 2011-16 and the Seventh Five Year Plan will be implemented during the period 2016-20.


Sing me softly to deep sleep.
With your song, my heart to keep.
That to my soul a sorrow brings.
That all my love I must regret.
And so do I with half-blinded sight.
Guided by your voice, into eternity.
1 Comment on "Poem For Marigold"


Can I say I am more Canadian since landing on the East Coast of Canada for the first time?
Last night I tried molasses on my biscuit, apparently a Nova Scotian thing. Strange, but surprisingly good.
I woke up at 10 o'clock this morning... who does that when they're travelling? So little time, so much to do. But I guess the fact that it was only 7am at home might explain things. When I looked out the window this morning, I was rather confused. Had we left Vancouver? Fog filled the sky, much like your average morning in Vancouver.
So we layered up in our rain jackets and took off for the day. First stop: Maritime Museum of the Atlantic. To be honest, it was boring. I'm not really a museum person, but overall, just not an interesting museum.
The one topic that I was interested in was an exhibit called "Hello Sailor!" Gay Life on the Ocean Wave. My initial thought was that I'd be learning about the tragedies that homosexuals on board the ships had to endure, but I was pleasantly surprised. The exhibit was in fact about the "gay heavens" that were these boats. Many men looked to the sea for an escape from the homophobic attitudes they were experiencing on land. There were stories about men who were openly gay at sea, but upon returning to their wives they fell into a state of depression. It was explained that for entertainment, gay and straight men alike put on drag shows for their personal fun at first, and later as shows for their passengers. On some boats, over 50% of the crew was gay, and they were accepted for it. Lesbian women, however, not so much.
After not being too impressed with that museum, what do we do but head to a second museum. This one was much more interesting however. It was the Canadian Museum of Immigration at Pier 21 where thousands upon thousands of immigrants arrived from overseas. Blake and I got our picture under the door where our great grandparents made their first steps onto Canada (known as the land of milk and honey apparently?). Am I ever grateful that they decided to make that move. Canada Day is tomorrow and my Canadian pride is apparent.
Probably my all time favorite exhibit was in that museum; Revolutionizing Cultural Identity. Kip Fulbeck, a professor at the University of California is the creator of this fascinating piece called "The Hapa Project". Hapa: once a derogatory term stemming from the Hawaiian word "half", but is now embraced by many whose mixed racial heritage includes Asian descent.
Kip took pictures of the bare head and shoulders of over 1200 people from all walks of life. With nothing covering their shoulders and little to no makeup hiding their face, we could see these people for who they are. Under their picture was their ethnic backgrounds and a handwritten blurb from each of them answering the question "What are you?". The answers were diverse. Many straightforward and many clever. One of the cuter ones was a young Chinese, Danish boy, "I am part Chinese and part Danish. I don't like to tell people I am Danish though, because they think I'm a pastry."
A Chinese, Japanese, German, Hungarian, English wrote "I am a person of colour. I am not half-"white". I am not half-"Asian". I am whole "other"."
One Japanese, French, Irish lady replied "I am millions of particles fused together making up a far less than perfect masterpiece. I am the big bang."
It might have been the simplicity of this exhibit, or the fact that it is relevant today; not history from 70 years before I was born, but there was something about this exhibit that made it fascinating.
If I had to answer the question, "What are you?" I would probably answer with a Doctor Suess quote "Today you are You, that is truer than true. There is no one alive who is Youer than You." Gotta love Doctor Suess. He says things like they are.
From there we went on a nice long trek (after Dad had promised it was a short little walk) to the Dalhousie campus. A good friend of mine, Taylor Quinn, is heading there in September to study international relations. I was thinking of him as we walked through the campus of the school founded in 1818, the largest school in the Maritimes and one of the oldest post-secondary institutions in Canada. But I was also thinking of him all day actually, because tonight, as the world tunes in to watch William and Kate arrive for their Canadian tour, Taylor will be in Ottawa to have dinner with not only them, but the Prime Minister, the Governor General and Craig Kielburger, founder of Free the Children. Taylor was invited to represent the youth of Canada. I couldn't think of a better person to give this opportunity to; he is more than deserving of it.
And to finish off the day- because the Halifax Citadel we were hoping to visit was closed- I sat on the deep green, and freshly watered grass of the hill that overlooks the city and it's harbour listening to Counting Crows and repeating the motion of threading a red and white bracelet. Serenity.
Tomorrow is Canada and we are off to Peggy's Cove.
Sounds like a great first day!!
Snorkelling Cats, Flying Elephants, Snake Charming, Sail Boating.


text:
Refritrans 2001 Named Carrier Transicold Spain's 2015 Service Partner of the Year
COSLADA, Spain - Jun. 21, 2016
Refritrans 2001, S.L.L. has been named Service Partner of the Year 2015 as part of Carrier Transicold Spain's annual Performance Assessment and Competitive Excellence (PACE) awards. Carrier Transicold, which operates in Spain as Carrier Transicold Espana, S.A., is a part of UTC Climate, Controls & Security, a unit of United Technologies Corp. (NYSE: UTX).
Refritrans, which provides transport refrigeration maintenance service in the Murcia area, achieved the highest platinum score in the 2015 PACE program, a direct result of the company's consistently high standard of service quality and delivery.
Initially introduced in collaboration with Carrier Transicold Europe's service partners and launched in Spain in 2014, the PACE program measures the performance of Carrier Transicold Spain's network service partners against 26 rigorous key performance indicators (KPIs) spanning 40 individual service standards. KPIs include planned maintenance completion rates, breakdown response times, first-time fixes and spare parts availability.
"Every year, we adjust the PACE performance criteria to raise the bar even higher to ensure the program continues to develop in line with segment requirements," said François Riveiro, managing director, Carrier Transicold Spain. "It gives our customers the confidence that our service network will deliver high standards of quality and service at all times."
Carrier Transicold Spain's 24/7 service network covers 27 locations throughout the country. Each service partner is an independent company, providing service for Carrier Transicold equipment. Together, these service partners employ more than 130 mobile service engineers who are responsible for the installation, service and repair of temperature-controlled equipment fitted to vans, trucks and trailers. Their roles include annual temperature control testing and certification as well as F-gas regulation compliance.
For more information on Carrier Transicold, please visit www.carriertransicold.es. Follow Carrier Transicold on Twitter: @SmartColdChain.
About Carrier Transicold
Carrier Transicold helps improve transport and shipping of temperature controlled cargoes with a complete line of equipment and services for refrigerated transport and cold chain visibility. For more than 45 years, Carrier Transicold has been an industry leader, providing customers around the world with advanced, energy-efficient and environmentally sustainable container refrigeration systems and generator sets, direct-drive and diesel truck units, and trailer refrigeration systems. Carrier Transicold is a part of UTC Climate, Controls & Security, a unit of United Technologies Corp., a leading provider to the aerospace and building systems industries worldwide. For more information, visit www.transicold.carrier.com. Follow Carrier on Twitter: @SmartColdChain.
Aline Lucas
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text:
Q: selectbox click and send ajax i have selectbox and data send by ajax. this html

<script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script>
<select id="kode_hanca" onchange="showModel(this)" name="id_vendor_detail[]" class="form-control vendor" required="">
                                    <option value="">- PILIH -</option>
                                    <script>
                                        $("select#kode_hanca").on("click", function () {
                                            $.ajax({
                                                type: "POST",
                                                url: "vendor_inout/vendor_inout_crud.php",
                                                dataType: 'json',
                                                data: {type: "get_kode_hanca"}, //get_kode_hanca
                                                success: function (data) {
                                                    console.log(data);
                                                    var teks = "";
                                                    $.each(data.kode_hanca, function (index, val) { //looping table detail bahan
                                                        var id_vendor_detail = val.id_vendor_detail;
                                                        var kode_hanca = val.kode_hanca;

                                                        teks += "<option class='target_append' value='" + id_vendor_detail + "'>" + kode_hanca + "</option>";
                                                    });
                                                    $("#kode_hanca:last").append(teks);
                                                }
                                            });
                                        });
                                    </script>
                                </select>

data is convert to json and always dinamic. So every i select the selectbox data must load again. when i use click or change its not working to send ajax. how to solve it ?

A: "select" tags cannot contain any other tags, it can only contain "option" tag.
try below code :-
<script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script>
<select id="kode_hanca" onchange="showModel(this)" name="id_vendor_detail[]" class="form-control vendor" required="">
<option value="">- PILIH -</option>
</select>

<script type="text/javascript">
    $("#kode_hanca").on("click", function () {
    $.ajax({
           type: "POST",
           url: "vendor_inout/vendor_inout_crud.php",
           dataType: 'json',
           data: {type: "get_kode_hanca"}, //get_kode_hanca
           success: function (data) {
               console.log(data);
               var teks = "";
               $.each(data.kode_hanca, function (index, val) { //looping table detail bahan
               var id_vendor_detail = val.id_vendor_detail;
               var kode_hanca = val.kode_hanca;

               teks += "<option class='target_append' value='" + id_vendor_detail + "'>" + kode_hanca + "</option>";
               });
                  $("#kode_hanca:last").append(teks);
             }
           });
        });
   </script>

You can also use "change", instead of "click."
Edited:-
Instead you can use below code:-
    <script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script>
        <select id="kode_hanca" onchange="myFunction();" name="id_vendor_detail[]" class="form-control vendor" required="">
        <option value="">- PILIH -</option>
        </select>

        <script type="text/javascript">

function myFunction()
{

showModel(this);

 $.ajax({
                   type: "POST",
                   url: "vendor_inout/vendor_inout_crud.php",
                   dataType: 'json',
                   data: {type: "get_kode_hanca"}, //get_kode_hanca
                   success: function (data) {
                       console.log(data);
                       var teks = "";
                       $.each(data.kode_hanca, function (index, val) { //looping table detail bahan
                       var id_vendor_detail = val.id_vendor_detail;
                       var kode_hanca = val.kode_hanca;

                       teks += "<option class='target_append' value='" + id_vendor_detail + "'>" + kode_hanca + "</option>";
                       });
                          $("#kode_hanca:last").append(teks);
                     }
                   });
}

           </script>

A: select dropdown you should try onchange functionality.
<script type="text/javascript">
    $("#kode_hanca").on("change", function () {
    $.ajax({
           type: "POST",
           url: "vendor_inout/vendor_inout_crud.php",
           dataType: 'json',
           data: {type: "get_kode_hanca"}, //get_kode_hanca
           success: function (data) {
               console.log(data);
               var teks = "";
               $.each(data.kode_hanca, function (index, val) { //looping table detail bahan
               var id_vendor_detail = val.id_vendor_detail;
               var kode_hanca = val.kode_hanca;

               teks += "<option class='target_append' value='" + id_vendor_detail + "'>" + kode_hanca + "</option>";
               });
                  $("#kode_hanca:last").append(teks);
             }
           });
        });
   </script>

A: Try this:
<script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script>
<script type="text/javascript">
function changeSelect() {
     $.ajax({
               type: "POST",
               url: "vendor_inout/vendor_inout_crud.php",
               dataType: 'json',
               data: {type: "get_kode_hanca"}, //get_kode_hanca
               success: function (data) {
                   console.log(data);
                   var teks = "";
                   $.each(data.kode_hanca, function (index, val) { //looping table detail bahan
                   var id_vendor_detail = val.id_vendor_detail;
                   var kode_hanca = val.kode_hanca;

                   teks += "<option class='target_append' value='" + id_vendor_detail + "'>" + kode_hanca + "</option>";
                   });
                      $("#kode_hanca:last").append(teks);
                 }
               });
}
</script>

<select id="kode_hanca" onClick="changeSelect(this.selectedIndex);" name="id_vendor_detail[]" class="target form-control vendor" required="">
<option value="">- PILIH -</option>
</select>

redpajama_set_name: RedPajamaStackExchange


Hanukkah (Hanu-Koh) is a stop on the path.
It is my wish for you to stop for this break and think: What train would I like to get on, where, to what goal, will it take me as it toots…?
The direct, natural reaction of the desire for the Light is programmed and known in advance, so there is no creation here. After all, "creation" (Beria) means "outside of state" (bar), that is, something new. The creation begins with the opportunity given to the desire to respond in an unnatural way, not instinctively, but according to different, extraneous criterion. The desire has to love the Light not for all kinds of pleasant fillings, but for the fact that the Light is elevated, special, giving, and has exceptional, noble qualities, peculiar only to the Light.
It turns out that the created being appreciates the Light precisely for the fact that it behaves in the opposite way to the nature of the desire: as giving and able to act outside its interest. The desire appreciates this quality. While enjoying the filling, the desire simultaneously realizes that the Light brings pleasure to it, fills it, cares for, and tends to it. It begins to appreciate this attitude like a child who grew up and appreciates his parents' love and care. In this way, the Kli begins to appreciate the Creator.
In order to elevate the creation to its degree of bestowal, the Creator takes direct pleasures away from it, and enables it to fill itself and enjoy bestowal and not receiving.
In our life, we implement the same scenario. In infancy, we enjoy receiving from our parents, using them and demanding from them. Yet as we grow older, we gradually become separated from our parents, to this extent we begin to understand what they have done for us, and we treat our children in the same way.
The Creator leads us to independence, which appears when we move from the property of receiving to the property of bestowal, that is, when we use our desire to enjoy for the sake of bestowal in order to bring enjoyment to the Creator and not to ourselves.
This desire should come from us and should not be imposed by the Light. However, this is possible only under the condition that we are in the ten and bestow to each other, internally connecting with each other. Then through this connection, we receive the Light. We demand through the ten this special power of bestowal and use it. Although this energy of bestowal does not belong to us, we can form the need for it in ourselves.
Instead of the instinctive need for filling and enjoying filling ourselves, we want the power of bestowal and enjoyment from it. This is the purpose of creation: that all created beings enjoy bestowal, following the example of the Creator.
The prayer of the common man is to feel pleasant at every moment, and all inanimate nature, plants, and animals ask the same. It is an ordinary, instinctive prayer felt by a person as natural self-care or as the influence of a religion that teaches a person to turn to the Creator with a request for filling.
However, there is a method that teaches us to demand not an egoistic filling in this and in the next world, but the opportunity to become similar to the Creator, to His actions, and enjoy bestowal. Also, you can enjoy bestowal only when you love the one to whom you bestow, as parents enjoy bestowing to their child. By virtue of their love for the children, a parent receives from bestowing to a child even more than if he would take for himself.
This is a proof that love can be stronger than nature. Indeed, we see in inanimate nature, plants, and animals many examples when parents die for their offspring. Their instinct pushes them to sacrifice themselves. It turns out that bestowal illuminated by the power of love is stronger than the desire to enjoy.
Question: How should we look at the friends: Are they higher than me, or are we all equal?
Answer: I do not care if they are higher or lower; I am prepared to work in all the stages! I have to relate to them differently.
After all, in the states of reception and bestowal in which we must achieve our unity, there is no higher and lower. There, each one is both higher and lower than the others since each of us has his own special qualities. I must give them to the whole large Kli (vessel), to the entire soul, and receive from it what I need for correct functioning.
Therefore, only when we care about the general state of the soul that is called either Adam or the Creator (depending on what we are talking about—either about the structure of the system itself or about its filling), can we talk about correct work. Only the connection with the Creator, or aspiration for this connection, gives us the basis for the correct work.
Question: Do I need to pay attention to the fact that friends' aspiration to the Creator is stronger than mine?
Answer: If a person feels this way, it drives him forward.
It is written: "The end of an act is in the preliminary thought"; that is, it determines the beginning of the action. Therefore, the main thing for me is to determine the Creator: what it is, what I am looking for, where I can go, what I aspire to, what state I see as my goal, and based on this, I will already be looking at what I need for achieving this goal.
Question: When I hold a goal before me, in what am I lower than my friends in the spiritual work in the group?
Answer: If any great Kabbalist finds himself in the smallest group, he will still be in something lower than them because he needs to have a connection with them so that they would influence him with their desires, hopes, and thoughts. He must receive from them their desires and their aspirations.
Without this, he will not be able to actively participate in the group, even if he is the greatest. He needs to be lower than them, to serve them in order to receive their aspiration to the Creator. Then, he will be able to work with these aspirations.
Right now is a special period of time of the holiday of Hanukkah. But, in essence, Hanukkah is a process that exists until the very end of correction and is experienced at the beginning of every new degree.
Each time new informational records (Reshimot) from the shattering are revealed in us, we need to make a restriction, a screen, and reflected light, all of which is the essence of the Hanukkah correction.
We want to experience what is happening in the desire to bestow, which is called faith above reason, deciding that everything comes from one upper force, the good that does good. And so each moment.
There are moments we are able to do this and other times we are not successful, but this war goes on continuously. Essentially, this is the war of the Maccabees against the Greeks. This war is being waged within reason, within our experience of this world, our rational minds and our philosophies, within our egoistic mind with which we built so many systems of connections between us.
We are fighting to establish a different form of existence, acknowledging that all our sensory experiences and perceptions are wrong. This is a false picture, a false perception of reality, because it appears within the egoistic mind.
Through the perception of reality within our rational minds, we need to try to feel that everything comes from only one, absolutely benevolent upper force. And every form perceived within the mind is only given to us in order to raise us to different perception, to faith above reason.
In other words, we agree that everything comes from the Creator and this is the specific form with which He can bring us to the true perception of reality, to the world of truth. We must try to establish the truth at every moment 24 hours every day, despite all the impressions in one's earthly feelings and rational mind, which are not subject to doubt, that is, within reason.
Also, we do not run from this rational perception, because "…one against the other He created them." We do not cross anything out, we only try to process it correctly: one against the other until we are able to develop understanding above reason for each understanding within reason.
That is how we attain the perception of spiritual reality and begin to function on two levels: on the material level, observing all its laws and rules accepted among the people of this world, and also in the true spiritual reality.
Then the upper Light is revealed because we are now able to light the jug of oil according to all the conditions, and in this way, we attain the revelation of the Creator and the upper world.
During the day, one day after the next, and every moment, we must try to see ourselves in this battle, imagining the picture within reason and juxtaposed against it, the picture above reason. This is the war of the Maccabees against the Greeks: Greeks are within reason and the Maccabees are above it.


Explore our wonderland in the Historic Clifton Arcade. We offer a "Click & Collect" service. This beats the frustration of parking !
My English Home opened its doors in July 2015 as an official stockist of the Annie Sloan Chalk paint in Bristol. My English Home encapsulates the culture and style of rustic France with the classic styles of an English home.
Come and explore our wonderland of furniture, soft furnishings and creative wonderfulness in our shop in the Historic Clifton Arcade, we are very much looking forward to meeting you soon.
Our online shop – coming soon, will hopefully inspire you to create a warm, stylish and comfortable home.
Don't throw away your furniture or buy new - re-love every piece you currently own to create a modern stylish interior to be proud of at a fraction of the price!
We are situated inside the Historic Clifton Arcade. The main entrance is on Boyces Avenue by the delicious Primrose Cafe and the Albion pub. The side entrance is on the Kings Road by the newly taken over Nettle & Rye Pub.
Discounted Starter kit also available.
Welcome Coffee and pastries on arrival with refreshment breaks throughout the session and cake and a glass of prosecco to finish!
Ideally you will have completed the Essential Technique - Introduction to Annie Sloan Chalk Paint™.
Welcome Coffee and pastries on arrival with refreshment breaks throughout the session, a two course lunch and cake and a glass of prosecco to finish!


text:
Q: How can I format a MySQL TIMEDIFF without seconds? I'm running this SELECT statement:
TIMEDIFF(NOW(), posts.date_modified) as time_ago

And getting results in the format 02:58:32. The last set of digits, :32, are the number of seconds.
How can I omit the seconds?

A: SELECT TIME_FORMAT(TIMEDIFF(NOW(), posts.date_modified), '%H:%i') AS time_ago

Consult the manual for more on TIME_FORMAT() and a table of allowed specifiers.


Robbin was such an amazing makeup artist. Her skills are unparalleled. I have never felt more beautiful in my life! Not only are her skills excellent but she is also such a nice and fun person! Everyone in my wedding party ranted and raved about her. So pleased that I chose her to do my makeup on my big day!!
Robbin was so sweet and great to work with! I felt at ease immediately after meeting her! She did an amazing job on my makeup and made me feel beautiful and still look and feel like myself. Sarah also did an amazing job on my hair and styled it to look exactly how I envisioned. They were both so easy to work with throughout the process and Robbin is very prompt and quick to respond to any questions/emails.
I planned my Michigan wedding from out of town. Not being familiar with the area or vendors, I was given Robbin's name through another make-up artist I had found online. I am so happy that the first person I found was unavailable because Robbin did an absolutely AMAZING job! She is so talented and made me feel like the most beautiful girl in the world on my wedding day! She is extremely organized and professional and was very prompt when responding to my thousands of e-mails! I was a little nervous because I was unable to do a hair and make-up trial due to my own schedule and the long distance. My hair and make-up turned out literally exactly how I wanted it to. She truly is an artist! I highly recommend her to any bride-to-be!
Robbin was so wonderful for my wedding! It started back during my trial when she listened to everything I wanted and executed flawlessly. The morning of the wedding she arrived perfectly on time and everything went so smoothly. Sara, her hair assistant, did absolutely wonderful work with each of my bridesmaids and my Mom. I seriously can't say enough about them. My day went so perfectly, and I have them both to thank for starting the day off so beautifully.
Robbin and her hair associate were amazing!!! Very friendly, easy to work with and did a phenomenal job. They worked fast without sacrificing quality. I highly recommend these two wonderful ladies!
Robbin was outstanding on our wedding day and in the weeks leading up to the event! On the day of, both Robbin and Sarah (hair stylist) were flexible and happy to make adjustments to the schedule without any issue. Everybody's makeup and hair looked incredible- natural but still glamorous, and it lasted all night. Having them come to the house to do hair and makeup was a significant investment, but very worthwhile as it reduced my stress level as I got ready that day, particularly having done a trial with them and knowing that the makeup was tear and sweat proof! I highly recommend this team!
Robbin did a beautiful job with the makeup for my wedding. At my trial she worked with me to figure out exactly what I wanted and took notes and pictures so that the look could be repeated on my wedding day. She was very accommodating and took the time to make sure that myself and each of my bridesmaids were happy with our overall looks. The makeup was elegant, yet natural ... exactly what I was looking for. She is a true professional and her years of experience and passion for what she does shines through in her work!
Robbin/Configure was one of the best things to happen to our wedding! We originally had another makeup artist booked, but she had to drop out extremely last minute. Robbin was able and willing to fill in on-site at our venue in the thumb with five days' notice! She and Sarah (hair artist) were wonderful to work with, communicate with, and have with us on the wedding morning - starting at 6am to accommodate our early afternoon ceremony! And of course, they did an extraordinary job. My wedding had a specific, vintage theme and their work was spot-on and gorgeous!! I cannot recommend them highly enough.


text:
Sonya Mclean
Robber of EBS branch jailed after garda gave chase
A judge has praised a garda for his bravery in chasing and arresting a robber he believed to be armed, while also praising the manager of the raided p(...)
Burglar sold jewellery worth €190,000 for €500 worth of heroin
A burglar who stole a safe containing €190,000 worth of jewellery which he sold "as scrap" for €500 worth of heroin has been jailed for four years.Ger(...)
Medical school dropout sentenced for sending tutors threatening letters
A Cork medical school dropout who sent threatening letters to 10 of his past supervisors and tutors at the Royal College of Surgeons in Ireland (RCSI)(...)


text:
Q: Ambiguous sentences Consider the following two examples:

*

*I've heard more vicious rumors.

*I've heard less vicious rumors.

Which of these examples, if any, can be considered ambiguous in interpretation?

A: Both of them are ambiguous; in fact, your first sentence has three potential interpretations. Let's look at that first.
Imagine you've just heard a nasty rumour about a friend, who knows that people have been spreading rumours about him. You might go to him and say

I've heard more vicious rumours [about you].

What you mean is that you have heard vicious rumours about him in addition to the ones you've heard already.
Now imagine someone you know has stated in conversation how many vicious rumours they have heard to date, and you have heard a higher number. You might say

I've heard more vicious rumours [than you have].

What you mean is the the quantity of vicious rumours you've heard to date exceeds that of your friend.
Now imagine that  you've just found out someone has been spreading a nasty rumour about you, but it isn't as bad as other ones that have been spread about you in the past. You might say

I've heard more vicious rumours [than that one].

In this case, rather than "more" being used for quantity, you're using "more vicious" as a modifier for "rumours", saying that you have heard rumours before which are more vicious than the one being discussed.
Some ambiguity exists with the second sentence as well; however, this is mainly because people usually mistakenly use "less" when referring to countable nouns (which rumours are) so if you meant you had heard a lower quantity of rumours than someone else, you'd actually say

I've heard fewer vicious rumours [than someone].

This eliminates the ambiguity as when you use fewer, it's clear you mean quantity, and in an ideal world where people knew the difference, saying

I've heard less vicious rumours [than that one].

Would be obviously interpreted as meaning the same as the third example from the first sentence.


text:
MEMORA Processos Inovadores
Industry: Government, Banking, Public sector
Memora's mission: "Understanding our customers, building and delivering management and IT solution, in a way to help them to achieve their mensurable goals." Memora is a specialist in modeling and optimizing processes using ARIS. In the past 15 years, the firm has developed an integrated solution that's been a great success for our customers, including:
Brazilian Army
The Brazilian Army aims to achieve a new doctrine by 2022 using technologically advanced defense products, highly trained and motivated professionals. Memora worked closely with the institution to help achieve this goal by adapting the organizational structure, improving the business processes, optimizing the structure of positions and personnel, and releasing personnel for employment in the final activity.
For this Ministry, Memora used ARIS to implement an enterprise architecture solution. Challenged to create a solution that integrates the business and IT vision, Memora used the platform as an architecture tool to map marketing strategies between a strategy and IT assets with a business process view. Reports were also used as a tool for the analysis of impacts and changes in the IT architecture at a tactical level.
One initiative of the Ministry of Finance is the Innovation Front, guided by the Offices of Processes and Projects. In the implementation and operation of the Process Office, Memora established the Value Added Chain as a governance instrument, covering the fronts of the PMIMF, namely: strategy, processes and projects (innovation), risks, people, technology, infrastructure and costs. In developing solutions, Memora focused on the dissemination of a process management culture coupled with the values of the Ministry. Economic growth, federal tax collection, reduced litigation, fiscal balance and transparency of public accounts were analyzed. The inclusion of the Brazilian State in the economic scenario, the solidity, balance and efficiency of the markets, the strengthening of foreign trade and the protection of society were included in the evaluation. In the end, 350 processes were documented, 140 were redesigned, and 212 were diagnosed. In addition, 50 staff were trained. Processes were rethought in an integrated way and as a governance tool.
The Ministry of Health is an agency of the Federal Executive Branch responsible for the organization and elaboration of plans and public policies aimed at the promotion, prevention and healthcare of Brazilians. In order to fulfill this purpose, the Ministry of Health understood the need for consolidation and to increase capacity to continuously improve and promote innovation. The agency hired Memora to support structuring its Office of Management Projects and Processes called EGPP. Memora also helped the Ministry team analyze and optimize processes for warehousing and the distribution and assistance to lawsuits. Joint process modeling work enabled the agency to identify and realize efficiency gains in the execution of these activities.
Mauricio Cauville
eMail: mauricio.cauville@memora.com.br
Address: SIG Quadra 04, Lote 625, Parte A · Brasilia, DF - Brazil - ZIP 70.610-440 · Brazil


text:
Q: How to capture the output of a process in Scala I'm trying to capture the output of a main class launch by sbt via a ProcessBuilder but that doesn't work. What am I missing here ?
The result is printed in the console but not by the println
val result = Seq("sbt", "runMain example.Hello")
  .#<(new ByteArrayInputStream("a".getBytes()))
  .!!
println(s"result=$result")

A: We had a similar situation when trying to use beeline.
This worked for us:
val stdout = new StringBuilder
val stderr = new StringBuilder

var commandSeq = Seq("beeline"
  , "-u"
  , ConfigProvider.config.BeelineConfig.Url
  , "-n"
  , ConfigProvider.config.BeelineConfig.UserName
  , "-p"
  , ConfigProvider.config.BeelineConfig.Password
  , "--outputformat=xmlattr"
  , "-f"
  , filePath)

hiveConfArgs.foreach(arg => {
  commandSeq = commandSeq :+ "--hivevar"
  commandSeq = commandSeq :+ arg
})

commandSeq ! ProcessLogger(stdout append _, stderr append _ + "\n")

(stdout.toString, stderr.toString)


Digital transformation radically changes communication and re-defines how the world connects.
Key digital innovations include immersive technologies (i.e. 360° films, holography, 3D sound experience and Virtual Reality/VR and their various applications), technologies fusing the physical and the digital (such as Augmented Reality/AR, Mixed Reality/MR or the Internet of Things), Artificial Intelligence-based story-telling systems as well as sharing platforms and interactive media channels.
The "Future Narratives and Immersive Experiences" symposium addresses the challenges posed by these developments by bringing together interdisciplinary co-creators in the audiovisual sector, companies with a background in media, VR/AR/MR, games, 3D sound, in the staging of productions, in communication and advertising as well as cultural and heritage institutions and players from other fields with the urge of delivering high-quality storytelling in the digital era.
How to communicate content with the help of digital immersive technologies?
The growing presence of immersive participatory technology and mixed interfaces challenges the development of coherent narratives, as narration is increasingly dispersed and diffused across digital media, with enhanced possibilites for interaction. From webseries, narrative games, interactive storyworlds to AI-powered storytelling, supply is fast-growing.
How to co-create experiences at the time of the Digital?
Digital production and distribution technologies have changed the possibilities of storytelling, with viewers becoming co-creators. As a result, boundaries defining genre and format are increasingly dissolving, for example, blurring the line between films and games. This highlights the need not only for high-quality original content but also for the radical rethinking of formats.
What are the opportunities and challenges of individually tailored immersive communication?
Such dynamism and interactivity of communication can promote audience's engagement with brands, companies and cultural institutions, making them relevant for the recipient's context, for an individually tailored involvement.
Pervasive and immersive technologies such as AR, VR, MR and 3D sound systems open up new possibilities and realms of perception. They support multisensory, 'smart' staging and exhibition, making stories and facts more accessible and crafting new (cultural) experiences.
Connecting digital content to physical objects paves the way to an innovative (cultural) programming. As such, new ways for people to navigate the stories behind objects become possible. This is enhanced by ubiquitous technologies embedding digital into the physical, for example, into objects, enabling them to interact and exchange information.
This is relevant for communication agencies and marketing and advertising experts designing customer experiences as well as for curators working in the context of heritage and historical sites. Therefore, the symposium explores how interactive, immersive and pervasive technologies are reshaping communication and tackles topical questions on what is demanded of the broad range of involved professionals.
It also showcases practical possibilities for uses of digital innovation, opening up new perspectives on their application, making visionary ideas tangible.
The program also addresses trans-disciplinary perspectives on future engagement of audiences and public. It investigates the tension between strategies based on 'story-telling' and 'story-finding', between makers who try to communicate a particular story and users who are free to discover and find their own, often different, stories.
Furthermore, the symposium discusses challenges connected with catching the public's attention in experiential or immersive advertising as well as for museums using immersive digital experiences. Lastly, an important underlying theme discussed at the symposium is the European copyright law reform that addresses the challenges for intellectual ownership posed by the virtually unlimited access and involvement in media content production warranted by technology.
The program takes place on May 22nd 2019 from 4pm to 7pm at the Film University Babelsberg KONRAD WOLF in Potsdam (Marlene-Dietrich-Allee 11, 14482 Potsdam) and is followed from 7:30pm to 10:30pm by a networking dinner at Restaurant Remise Schloss Glienicke (Königstraße 36, Berlin-Wannsee).
Transport from the Film University to Restaurant Remise Schloss Glienicke as well as from Restaurant Remise Schloss Glienicke to Potsdam main railway station is provided.
His Majesty King Willem-Alexander and Her Majesty Queen Máxima pay to Brandenburg and Mecklenburg-Vorpommern.
Prof. Elizabeth Sikiaridi and Prof. Frans Vogelaar of Hybrid Space Lab curate the "Future Narratives and Immersive Experiences" symposium, together with the Embassy of the Kingdom of the Netherlands in Berlin.


Hollywood – the dream factory where the really big stars from all over the world meet and shoot films that touch us deeply.
Stars and starlets, glitter and glamour, scandals and big emotions – that's what Hollywood is all about. Walk the Walk of Fame, reach for the stars, and experience an evening for all the senses!
The VDP (Association of German "Prädikat" Wine Estates) will serve the best from their cellars and our star chefs will conjure up great cinema on your plates. Enjoy your personal "Hollywood Moments"!


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