\section{Introduction}

Following the recent convincing evidence for neutrino oscillations,
leptogenesis\cite{FY}
has become a
well motivated possible
explanation of the origin of the baryon asymmetry of the universe.
In these proceedings, after a short introduction on the three basic
ingredients of leptogenesis,
we will extend the discussion
of Ref.\cite{MR} along two main directions. First, in the framework of
the usual seesaw model
with three right-handed
neutrinos (``type-I'' seesaw model\cite{type-I}), we will study
the neutrino mass constraints, in particular the neutrino
mass upper bound, which exist in order that leptogenesis
can successfully explain the baryon asymmetry of the universe.
We will show how this neutrino mass upper bound can be relaxed
by no longer assuming that the right-handed neutrinos
have a hierarchical mass spectrum.
Secondly, other models of leptogenesis which might also be attractive
for various reasons will be introduced: ``type-II'' seesaw
model with a scalar Higgs triplet, ``type-III'' seesaw model with
fermion triplets, and combinations of them such as the
``type-I'' plus ``type-II'' model which is motivated by
left-right or SO(10) models. The
corresponding neutrino mass bounds will be discussed for all these models.
Finally, we will also briefly consider
the case of radiative neutrino mass models.

\section{The Three Basic Ingredients of Leptogenesis}

There are essentially three ingredients to take
into account for leptogenesis
to work. In this section we will discuss only the case of
the ``type-I`` seesaw model
with three heavy right-handed neutrinos $N_i$
which is based on the following Lagrangian
\begin{equation}
\label{Lseesaw}
L  = L_{\rm SM} +\bar N_i i\partial\hspace{-1.3ex}/\, N_i +
(\lambda ^{ij} ~ H^\dagger N_i L_j   +
\frac{M_{N_i}}{2}   N_i N_i  +\hbox{h.c.})\,,
\end{equation}
with $L_{j}= (\nu_{jL}$, $l_{jL})^T$, $H=(H^0,H^-)^T$.
In Eq.~(\ref{Lseesaw})
we made the choice to work in the basis where the $N_i$ mass
matrix is real and diagonal. This model has 18 parameters; 9 combinations
of them enter in the neutrino mass
matrix $M_{\nu}^I=-\lambda^T M_N^{-1} \lambda v^2$, and nine of them
decouple from it. A very
useful parametrization (in
term of a orthogonal complex $R$ matrix) which allows to span easily
all the parameter space of this model
in agreement with the data, and which is very useful
to determine the neutrino mass bounds below,
can be found in Ref.\cite{CI,HLNPS}. For the following we order
the neutrino masses as: $m_{\nu_3}>m_{\nu_2}>m_{\nu_1}\ge0$ and
$M_{N_3} \ge M_{N_2} \ge M_{N_1}\ge0$.

\subsection{The CP Asymmetry}

At a temperature well above their masses one can expect
the right-handed neutrinos to be in thermal equilibrium
in the universe thermal bath,
due to Yukawa interactions or other possible interactions (such
as gauge interactions in the right-handed sector). However, once
the temperature drops below their masses the right-handed
neutrinos disappear from the thermal bath by decaying to
leptons and Higgs bosons.
The crucial quantity for leptogenesis is the CP-asymmetry, that is to say the
averaged $\Delta L$
which is produced each time one $N_i$ decays. At lowest order, that is to say
at one-loop order, for $N_1$ (and similarly for $N_{2,3}$),
it is given by
\begin{equation}\label{eps}
\varepsilon_{N_1} \equiv \frac{\Gamma({N_1} \rightarrow l H^*) -
\Gamma( N_1 \rightarrow \bar{l} H)}
{\Gamma({N_1} \rightarrow l H^*) + \Gamma( N_1 \rightarrow \bar{l} H)}=
-\sum_{j=2,3}\frac{3}{2}
  \frac{M_{N_1} }{M_{N_j} }\frac{\Gamma_{N_j} }{M_{N_j} }
  I_j\frac{2 S_j + V_j}{3}
\end{equation}
where
\begin{equation}\label{IGamma}
I_j = \frac{ \hbox{Im}\,[ (\lambda  \lambda ^\dagger)_{1j}^2 ]}
{|\lambda \lambda ^\dagger |_{11} |\lambda \lambda ^\dagger |_{jj}}
 \, ,\qquad
\frac{\Gamma_{N_j}}{M_{N_j}} = \frac{|\lambda \lambda ^\dagger |_{jj}}{8\pi}
\equiv \frac{\tilde{m}_j M_{N_j}}{8\pi v^2}
\,,
\end{equation}
and where
\begin{equation}
S_j = \frac{M^2_{N_j}  \Delta M^2_{1j}}{(\Delta M^2_{1j})^2+M_{N_1} ^2
   \Gamma_{N_j} ^2} \,, \,\,\,\,\,
V_j = 2 \frac{M^2_{N_j} }{M^2_{N_1}}
\bigg[ \big(1+\frac{M^2_{N_j} }{M^2_{N_1}}\big)\log\big(1+
\frac{M^2_{N_1}}{M^2_{N_j} }\big)
- 1 \bigg],
\end{equation}
with $\Delta M^2_{ij}=M^2_{N_j}-M^2_{N_i}$.
The factors $S_j$ ($V_j$) comes from the one-loop
self-energy (vertex) contribution to the
decay widths, Fig.~1.
The $I_j$ factors are the CP-violating coupling combinations entering
in the asymmetry.
\begin{figure}[b]
\begin{center}
\begin{picture}(200,60)(0,0)
\Line(0,30)(30,30)
\DashArrowLine(60,60)(90,60){5}
\ArrowLine(60,0)(90,0)
\Line(60,60)(60,0)
\ArrowLine(60,60)(30,30)
\DashArrowLine(30,30)(60,00){5}
\Text(5,22)[]{$N_i$}
\Text(35,12)[]{$H$}
\Text(37,51)[]{$ l_l$}
\Text(69,30)[]{$N_j$}
\Text(85,52)[]{$H^\ast$}
\Text(85,8)[]{$ l_{k}$}
\Line(100,30)(120,30)
\DashArrowArc(135,30)(15,180,360){5}
\ArrowArc(135,30)(15,0,180)
\Line(150,30)(170,30)
\DashArrowLine(170,30)(200,60){5}
\ArrowLine(170,30)(200,0)
\Text(105,22)[]{$N_i$}
\Text(135,6)[]{$H$}
\Text(136,55)[]{$ l_l$}
\Text(160,22)[]{$N_j$}
\Text(187,55)[]{$H^\ast$}
\Text(187,4)[]{$ l_{k}$}
\end{picture}
\end{center}
\caption{One-loop diagrams contributing to the asymmetry
from the $N_i$ decay.}
\label{fig1}
\end{figure}

\subsection{The Efficiency Factor}

Once the averaged $\Delta L$ produced per decay has been calculated, the
second ingredient to consider
is the efficiency factor $\eta$. This factor allows to
calculate the lepton asymmetry produced from the CP-asymmetry,
\begin{equation}
\frac{n_L}{s}=\varepsilon_{N_i} Y_{N_i}|_{T >> M_{N_i}} \eta \,,
\end{equation}
where $Y_{N_i}=n_{N_i}/s$ is the number density
of $N_i$ over the entropy density,
with $Y_{N_i}|_{T >> M_{N_i}}=135 \zeta(3)/(4 \pi^4 g_*)$
where $g_*=112$ is the number of degrees of freedom
in thermal equilibrium in the ``type-I'' model before the $N_i$ decayed. If
all right-handed neutrinos
decay out-of-equilibrium, the lepton asymmetry produced is
just given by the CP asymmetry times the number of $N_i$ over the entropy
density before the $N_i$ decayed, i.e.~$\eta=1$.
However, the efficiency
factor can be much smaller than one, if
they are not fully out-of-equilibrium while decaying,
and/or if there are at this epoch
L-violating processes partly in thermal equilibrium.
The processes which can put the $N_i$ in thermal equilibrium and/or violate
L are the inverse decay process and $\Delta L=1,2$ scatterings.
To avoid a large damping effect, it is necessary
that these processes are not too fast with respect to the Hubble
constant. For the
inverse decay process (which is the
most dangerous process, see e.g. the discussion of Ref.\cite{DiBari}),
this gives the
condition: $\Gamma_{N_i}/H(T \simeq M_{N_i}) \le 1$
with $H(T)=\sqrt{4 \pi^3 g_* / 45} \, T^2/M_{\hbox{\footnotesize Planck}}$.
In practice to calculate $\eta$ we need to put all these processes
in the Boltzmann equations\cite{plum,GNRRS} which allow a precise
calculation of the produced lepton asymmetry
as a function of the temperature $T$. The corresponding efficiency factor
including finite temperature effects can be found in Ref.\cite{GNRRS}
in the limit where
the right-handed neutrinos have a
hierarchical spectrum $M_{N_1} << M_{N_{2,3}}$. In this limit
only the
asymmetry produced by the decay of the lightest
right-handed neutrino $N_1$ survives
and is important, which simplifies greatly the calculations.

\subsection{The $B$ to $L$ Asymmetry Conversion}

Once the $L$ asymmetry $n_L/s$ has been produced and determined,
the last step is to calculate the corresponding baryon asymmetry
produced due to the partial conversion of the $L$
asymmetry to a $B$ asymmetry by
the Standard Model non-perturbative sphaleron processes.
The conversion factor is $n_B/s=-(28/79) n_L/s$.\cite{conversfactor}
It comes from taking into account the fact
that these processes conserve $B-L$, violate
$B+L$ and are
fast (i.e.~in thermal equilibrium) above the electroweak scale.
From this conversion factor we finally get the total baryon
asymmetry produced $n_B/s=-1.38 \cdot 10^{-3} \varepsilon_{N_1} \eta$
which has to be compared
with the experimental BBN and WMAP value for this
ratio: $n_B/s=(8.7 \pm 0.4 ) \cdot 10^{-11}$.

\section{The Neutrino Mass Constraints}

In addition to the three ingredients above there is a fourth crucial
ingredient which makes the all interest of the leptogenesis mechanism:
the neutrino mass constraints. These constraints come from the fact
these are the same interactions
which produce both the neutrino masses and leptogenesis.
There are two types of neutrino mass constraints on leptogenesis.

\subsection{The Neutrino Mass Constraint on the Efficiency}

The first constraint
is the neutrino mass constraint on the
size of the washout, that is to say on $\eta$.
It comes from the fact that, in full generality,
the ratio $\Gamma_{N_i}/H$, which has to be smaller than unity to have
no washout suppression, is always larger
than the ratio of the lightest neutrino mass $m_{\nu_1}$ over
the $m_*= 16 \pi^2 v^2
\sqrt{g_* \pi/45}/M_{\hbox{\footnotesize Planck}}\,$$\sim 10^{-3}$~eV
scale:\cite{BP2}
\begin{equation}
\frac{\Gamma_{N_i}}{H} \ge \frac{m_{\nu_1}}{10^{-3}\,\hbox{eV}}\,.
\label{outofeq}
\end{equation}
This inequality comes simply from Eq.~(\ref{IGamma})
and the $v^2 | \lambda \lambda^\dagger |^2_{jj} / M_{N_j} \ge m_{\nu_1}$
inequality. It
means that, if $m_{\nu_1}> 10^{-3}$~eV, there will
be some washout and the larger is $m_{\nu_1}$ above this
scale the larger is the washout, i.e.~the smaller is $\eta$.
For example for $m_{\nu_1}=\sqrt{\delta
m^2_{\hbox{\footnotesize atm}}} \simeq 0.05$~eV
the value of $\eta$ is $\sim 0.01$ for
$M_{N_1} \le 10^{14}$~GeV. For
$m_{\nu_1}=0.5$~eV this value becomes few $10^{-4}$ for
$M_{N_1} \le 10^{12}$~GeV and decreases for larger values of $M_{N_1}$.
The fact that $m_*$, which is a function of the electroweak and Planck
scales, is of order the neutrino
masses is a quite remarkable fact since it means that for leptogenesis
the $N_i$ are naturally only slightly in thermal equilibrium or
out-of-equilibrium.
Note that Eq.~(\ref{outofeq}) is often rewritten in term of the so-called
effective masses of Eq.~(\ref{IGamma})
as $\tilde{m}_i \ge 10^{-3}$~eV.

\subsection{The Neutrino Mass Constraint on the Size of $\varepsilon_{N_i}$}

The second neutrino mass constraint is on the size of
the CP-asymmetries $\varepsilon_{N_i}$.
It generally applies but, contrary to the first constraint above, it
do not always applies.
This depends on the type of mass spectrum the right-handed
neutrino have: very hierarchical, ``normally''
hierarchical or quasi-degenerate.

\subsubsection{Very hierarchical right-handed
neutrinos: $M_{N_1} << M_{N_{2,3}}$}

If right-handed neutrino masses differ by several
orders of magnitude, the size of the
$\varepsilon_{N_1}$ asymmetry is quite
constrained by the size of the neutrino masses. There exists an upper
bound\cite{epsbound,TH,exactepsbound}
which in its exact form was given by Ref.\cite{exactepsbound}:
\begin{equation}
\label{epsDI}
|\varepsilon_{N_1}| \le
  \frac{3}{16\pi}\frac{M_{N_1}}{v^2}(m_{\nu_3} - m_{\nu_1})=
\frac{3}{16\pi}\frac{M_{N_1}}{v^2}
\frac{\Delta m^2_{\hbox{\footnotesize atm}}}{m_{\nu_3} + m_{\nu_1}}\,.
\end{equation}
Since $\Delta m^2_{\hbox{\footnotesize atm}}\simeq 2 \cdot 10^{-3}$~$eV^2$ is
fixed experimentally, this bound
decreases as the neutrino masses increase.
Therefore as the neutrino masses increase there are two suppression
effects arising: the washout effect increases and the upper bound on the
asymmetry decreases. Successful leptogenesis leads therefore to the upper
bound:\cite{epsP,GNRRS,HLNPS}
\begin{equation}
\label{mnu3bound}
m_{\nu_3} < 0.12-0.15~\hbox{eV} \,.
\end{equation}
For more details see Ref.\cite{MR} in these proceedings.
Note that in order to derive this bound Eq.~(\ref{epsDI}) is not sufficient.
As the final produced lepton asymmetry depends not only on $m_{\nu_3}$
and $M_{N_1}$ but also, through
the efficiency factor,
on $\Gamma_{N_1}$ (or $\tilde{m}_1$),
it is necessary\cite{epsP,HLNPS} to have an upper
bound
for fixed values of these 3 parameters and not only as a
function of $m_{\nu_3}$ and $M_{N_1}$ as in Eq.~(\ref{epsDI}).
This bound can be found in Ref.\cite{HLNPS}.
Note also that, due to the fact that
the upper bound on the CP-asymmetry
is proportional to $M_{N_1}$, successful leptogenesis with
hierarchical right-handed neutrinos
implies also a lower bound on this
mass:\cite{exactepsbound,epsbound,TH,GNRRS}
\begin{equation}
\label{MN1bound}
M_{N_1} > 5 \times 10^{8}\,\hbox{GeV} \,.
\end{equation}
This result holds for the case where  the $N_1$ are in thermal equilibrium
before decaying. Starting instead (due to inflation dynamics)
from a universe with no (with only) right-handed neutrinos at a
temperature above their mass, this bound becomes\cite{GNRRS}: $M_{N_1} > 2
\times 10^9$~GeV, ($2 \times 10^7$~GeV).

\subsubsection{``Normally'' hierarchical right-handed neutrinos}

If right-handed neutrinos have a hierarchy similar to the ones of the
charged leptons or quarks, that is to say if $M_{N_1} \simeq (10-100) M_{N_2}$
with $M_{N_3} > M_{N_2}$, the L-asymmetry production is still dominated by
the decays of the lightest right-handed neutrino $N_1$. In this case the
upper bound on
the CP-asymmetry is the same as for very-hierarchical neutrinos, except
that there are extra corrections\cite{HLNPS}
in $M_{N_1}^2/M_{N_{2,3}}^2$ to be added
in Eq.~(\ref{epsDI}):
\begin{equation}
\delta\varepsilon_1 \simeq
  \frac{3}{16\pi}\frac{M_{N_1}}{v^2} \tilde{m}_{2,3}
\frac{M_{N_1}^2}{M_{N_{2,3}}^2} \,.
\end{equation}
These corrections generically are small so that all the bounds obtained
in the previous section are still valid, but not always.
There exist configurations of the Yukawa couplings which lead to
large corrections.
The point is that, contrary to the leading term in
Eq.~(\ref{epsDI}), the corrections do not decrease
when the neutrino masses decrease and they do not necessarily vanish
for degenerate light-neutrino masses.
As a result\cite{HLNPS}, for special
configurations giving large $\tilde{m}_{2,3}$
but small neutrino masses, one can have successful leptogenesis
with $M_{N_1}$ well below the lower bound of Eq.~(\ref{MN1bound})
and with neutrino masses well above the bound of Eq.~(\ref{mnu3bound}).
An explicit example of such  Yukawa coupling configuration leading to
successful leptogenesis with $M_{N_1} \simeq 10^6$~GeV has
been recently considered in Ref.\cite{RST}.

\subsubsection{Quasi-degenerate right-handed neutrinos}

If at least two right-handed neutrinos have masses very close to each other,
$M_{N_1} \sim M_{N_2}$, the situation is changing drastically with respect
to the two previous cases.
This is due to the fact that the one loop self-energy diagram of Fig.1
displays
in this case
a resonance behaviour\cite{flanz,HLNPS,HMW} coming
from the propagator of the virtual
right-handed neutrino in this diagram. This effect can be seen from the $S_j$
factors in Eq.~(\ref{eps}).
Since in the seesaw model the decay widths of the $N_i$ are generically
much
smaller than their masses, this resonance effect can lead to a
several order of magnitude enhancement of the asymmetry.
At the resonance, that is to say for $M_{N_2}-M_{N_1}=\Gamma_{N_2}/2$,
the $S_{2}$ factor, which is unity in the
$M_{N_1} << M_{N_{2,3}}$ limit, is as large as
$\frac{1}{2}M_{N_2}/\Gamma_{N_2}$.
In this case, up to a $1/2$ factor, the $S_2$ factor
compensates the $\Gamma_{N_2}/M_{N_2}$ prefactor in Eq.~(\ref{eps}) which
gives $\varepsilon_{N_1}=\frac{1}{2} I_2$.
Moreover in this case one can show that the asymmetry is not
bounded anymore by an expression depending on the neutrino masses. In
particular the upper bound is not proportional to
$\Delta m^2_{\hbox{\footnotesize atm}}/(m_{\nu_3}+ m_{\nu_1})$ as
in the hierarchical
case,\footnote{One can check\cite{exactepsbound,HLNPS} that this neutrino
mass factor comes from the fact that
for hierarchical right-handed neutrinos $S_2=S_3$. This equality
has no reason to be true anymore with
quasi-degenerate right-handed neutrinos.} Eq.~(\ref{epsDI}).
The Yukawa coupling factor $I_2$ turns out to be bounded just by unity
so that the upper bound on $\varepsilon_{N_1}$ is just $\frac{1}{2}$
independently of light and heavy neutrino masses\cite{HLNPS}.
Together with $\varepsilon_{N_2}$, which is equal to $\varepsilon_{N_1}$
in this case and has also to be taken into account,
CP-violation is just bounded by unity.
As a result, since neither the maximal asymmetry nor the washout
effect (coming from inverse decay\footnote{Note that there is
a $M_{N_i}$ mass dependence in the washout coming from $\Delta L =2$
scatterings. This effect however can be large only for large
values of the $M_{N_i}$ close to the GUT scale.}) depend on $M_{N_1}$,
successful leptogenesis can be obtained at any scale except
that the $L$ to $B$ conversion from sphalerons still needs to be
effective. This requires $M_{N_1}$ to be above the electroweak
scale (i.e.~typically above $\sim 1$~ TeV).
Moreover since the upper bound on the CP asymmetry
is independent of neutrino masses, there is no more suppression of
the asymmetry for large neutrino masses. The only remaining suppression
effect arising for large neutrino masses is the one of section 3.1 above
coming
from the washout.
Therefore the upper bound on the neutrino masses in this case gets
considerably relaxed. This is shown in Fig.2.a where is plotted, as a
function of $m_{\nu_3}$, the level of degeneracy which is needed to
have successful leptogenesis. Values far above the eV are possible
which means that {\it in full generality there is no more relevant upper
bound on neutrino masses coming from leptogenesis}.
The value $m_{\nu_3} \simeq 1$~eV can lead to successful leptogenesis
with a level of degeneracy of
order $(M_{N_2}-M_{N_1})/M_{N_2}\simeq 4 \cdot 10^{-2}$ which
is quite moderate.

\begin{figure}[t]
\centerline{\epsfxsize=2.0in{\epsfbox{mdeg.eps}} \hspace{0.1cm}
\epsfxsize=2.0in{\epsfbox{degdeg.eps}}}
\caption{Fig.2.a shows$^5$
how much the
maximal value of neutrino mass compatible with thermal
leptogenesis increases
when right-handed neutrinos are allowed to be
quasi-degenerate. The dashed line is what we obtain if we take into
account only the resonance enhancement effect, not the neutrino
mass effect,
see text and Ref.$^5$.
Fig.1.b holds assuming Eqs.~(\ref{splitsol})-(\ref{splitatm})
as a generic example
that everything is as degenerate as neutrinos,
considering
the natural possibility that the $\tilde{m}_i$ should be
quasi-degenerate with the neutrino masses, taking
$\tilde{m}_{1,2,3} < m_{\nu_1}+n(m_{\nu_3}-m_{\nu_1})$.
}
\vspace*{-2mm}
\end{figure}

One could argue that such a degeneracy of right-handed neutrinos
is unnatural and that we would
anyway generally expect a hierarchical spectrum for the right-handed neutrino
by analogy with charged leptons or quarks.
However, since the bound on neutrino masses from leptogenesis is relevant
  only for a quasi-degenerate spectrum of light neutrinos, when
  calculating this bound one can
  wonder what would be the most natural right-handed neutrino mass
  spectrum to explain such correlations between the light-neutrino
  masses.
Presumably, three
quasi-degenerate right-handed neutrinos are
the most natural spectrum in this case (with Yukawa couplings
correlated among each other). Such a quasi degenerate spectrum,
unlike a hierarchical spectrum, wouldn't require in addition precise
correlations between the Yukawa couplings and the right-handed
neutrino masses (in order that the ratio
between the $M_{N_i}$ is compensated in the seesaw mass
formula by the same ratio
between the Yukawa couplings). Therefore
the bound given in Fig.~2.a is the fully relevant one.

To complete this section note
that there exists nevertheless one specific pattern,
which is probably the most realistic one, which leads
to more stringent bounds.\cite{HLNPS}
In fact, if neutrinos were quasi-degenerate,
the degeneracy would presumably not be accidental but due to
some reason:
a broken SO(3) flavour symmetry is probably the simplest possibility.
One expects that in such framework all quantities, and not only
neutrino masses, are close to the ideal limit
where three degenerate right-handed neutrinos
give equal masses to three orthogonal combinations of left-handed
neutrinos.
Therefore one expects something
like $\tilde{m}_i - \tilde{m}_j\approx m_{\nu_i} - m_{\nu_j}$ and
\begin{eqnarray}
&&\frac{M_{N_2}-M_{N_1}}{M_{N_1}}\sim \frac{m_{\nu_2}-m_{\nu_1}}{m_1}\approx
\frac{\tilde{m}_2-\tilde{m}_1}{\tilde{m_{\nu_1}}}
 \approx \frac{\Delta m^2_{\rm sun}}{2 m_{\nu_1}^2}\approx
0.5~10^{-4} \bigg(\frac{\hbox{eV}}{m_{\nu_2}}\bigg)^2 \label{splitsol}
 \,\,\,\,\,\,\,\,{}    \\
&&\frac{M_{N_3}-M_{N_2}}{M_{N_2}}\sim \frac{m_{\nu_3}-m_{\nu_2}}{m_{\nu_2}}
\approx
\frac{\tilde{m}_3-\tilde{m}_2}{\tilde{m_2}}
\approx \frac{\Delta m^2_{\hbox{\footnotesize atm}}}{2 m_{\nu_3}^2}\approx
10^{-3} \bigg(\frac{\hbox{eV}}{m_{\nu_2}}\bigg)^2.\,\,\,\,\,{}
\label{splitatm}
\end{eqnarray}
Right-handed neutrinos can be more degenerate than in the above
estimates
if only the neutrino Yukawa couplings
deviate from the symmetric limit, and can be less degenerate only
if there are accidental cancellations between non-universal
Yukawa couplings and non-degenerate $M_{N_{1,2,3}}$
in the see-saw prediction for neutrino masses.

Assuming the relations of Eqs.~(\ref{splitsol})-(\ref{splitatm}),
the bound on the neutrino masses can be read off from Fig.~2.b
where we give the maximal
baryon asymmetry we obtain as a function of $m_{\nu_3}$ for values of
$\tilde{m}_{1,2,3} < m_{\nu_1}+n(m_{\nu_3}-m_{\nu_1})$
with $n=\{1/2,1,2,4\}$. Taking
$\tilde{m}_{1,2,3} < m_{\nu_3}$ ($n=1$),
as the generic example for the case that the $\tilde{m}$ would be
precisely of order the neutrino masses, gives the constraint
\begin{equation}
m_{\nu_3} < 0.6 \,\, \hbox{eV} \,,  \label{m3reali}
\end{equation}
which is stronger than in the fully general case of Fig.~2.a.
The suppression effect comes essentially from the fact that for values
of $\tilde{m}_{1,2,3}$ close to the neutrino masses the Yukawa
coupling factors $I_j$ are suppressed\cite{HLNPS,epsP}.
The result is quite sensitive to how these quantities are close to each
other. For example taking n=4 (which could be considered as a quite moderately
fine-tuned case) leads already to an upper bound as large as 1 eV.

In summary without a predictive flavour model which would show how
the correlations between the seesaw parameters
at the origin of the degenerate spectrum occur,
in order to have a safe bound we must consider the fully general case
of Fig.~2.a
 (where $n$ was left as a free parameter in order to maximize the asymmetry).
Even in a very constrained situation the
neutrino masses
can be as large as 0.6~eV, Eq.~(\ref{m3reali}), or 1~eV.\footnote{Stronger
constraints will arise if supersymmetry exists
and if right-handed neutrinos lighter than $10^{10}$~GeV
will be needed to avoid gravitino overproduction.}

\section{Leptogenesis in the Framework of Other Seesaw Models}

The type-I seesaw mechanism is probably the most
direct extension of the Standard Model
we can consider in order to explain the neutrino
masses and is in this sense the most attractive. However it is not the only
attractive seesaw model.
Beside the type I seesaw, one can think about
two other basic seesaw mechanisms. The first one is the type-II seesaw
model\cite{type-II}
where
neutrino masses are due to the exchange of an heavy scalar Higgs triplet.
The second one is from the exchange of three heavy self-conjugated
$SU(2)_L$ triplets of fermions\cite{type-III}, a
model which according to us should be
called type-III seesaw model as it induces
the neutrino mass from a third type of heavy particles.
In addition to these three seesaw basic mechanisms one can also think about
combinations of them. In the following we
consider these various alternatives and see whether they can lead
to successful leptogenesis and what are the corresponding mass bounds.

\subsection{The Type-II Seesaw Model}

The type-II seesaw model with just one heavy scalar triplet $\Delta_L$ which
couples to 2 leptons doublets and to two Higgs doublets is
a particularly minimal model. It is based on the Lagrangian
\begin{equation}
\label{Lseesaw2}
 L \owns
-M^2_\Delta Tr\Delta_L^\dagger \Delta_L
 - (Y_\Delta)_{ij} L_{i}^T C i \tau_2
\Delta_L L_{j} + \mu H^T i \tau_2 \Delta_L H + h.c. \,,
\end{equation}
with
\begin{equation}
\Delta_L=
\begin{pmatrix}
\frac{1}{\sqrt{2}}\delta^+ & \delta^{++}  \\
\delta^0 & - \frac{1}{\sqrt{2}} \delta^+
\end{pmatrix} \,.
\end{equation}
It leads to the neutrino mass matrix: $M_\nu^{II} = 2 Y_\Delta
v_\Delta \simeq 2 Y_\Delta \mu^* v^2/M^2_\Delta$.
This model in full generality has only 11 parameters, the
triplet mass, its $\mu$ coupling and 6 real parameters plus
3 phases
in the $Y_\Delta$ Yukawa coupling matrix.
This model has the attractive property that the knowledge of
the full low energy neutrino mass matrix $M_\nu$
would allow to determine the full
flavour high energy structure in $Y_\Delta$, both matrices being just
proportional to each other.
However for leptogenesis it turns out that this model is too minimal.
As explained in Ref.\cite{MS1,HMS1,HLNPS,DHHRR},
since the triplet is not
a self-conjugated
particle, there is no vertex diagram and leptogenesis could
come only from a self-energy diagram involving two leptons in the final state
and two Higgs doublets in the self-energy, third diagram of Fig.3.
This diagram with just one triplet
is real and therefore doesn't bring any CP-violation. At two loops the
asymmetries are too suppressed.
Therefore the  standard model
extended by just one scalar triplet leads in a ``minimal''
way to neutrino masses but do not lead to
successful leptogenesis.

However, based on the type-II model with
just one triplet coupling to leptons, there is one framework which
can work. It is the supersymmetric version of this model, i.e.~the MSSM
model extended by
a pair of scalar triplets with opposite hypercharges.
In a way similar to the soft leptogenesis mechanism with right-handed
neutrinos\cite{SOFTN,MR}, the
supersymmetry breaking terms involving the triplets
can remove the mass degeneracy between the two triplets and lead to
resonant
leptogenesis from the self-energy diagram involving the two
triplets\cite{DHHRR}.
This requires triplet mass between $\sim 10^{3}$~GeV and
$\sim 10^{9}$~GeV.\footnote{Note that triplets, unlike right-handed
neutrinos have gauge scatterings, which tend to put them in closer
thermal equilibrium, reducing the efficiency. The exact
efficiency for a scalar triplet is yet to be calculated but the one for
a fermion triplet is known\cite{HLNPS}. All lower bounds
on the scalar triplet masses we give here are
obtained using the
fermion triplet efficiency, assuming that both efficiencies are same, as
gauge scatterings are expected to be similar up to factors of order one.
This should be checked explicitly.}

\subsection{The Heavy Triplet of Fermion Model}

The third seesaw basic way to induce neutrino masses is by adding to
the Standard Model 3 self-conjugated $SU(2)_L$ triplets of
fermions.\cite{type-III}
The Lagrangian keeps the same structure as the one of the type-I seesaw
model but with different $SU(2)_L$ contractions:
\begin{equation}
L  = L_{\rm SM} +\bar N^a_i iD\hspace{-1.3ex}/\, N^a_i +
(\lambda ^{ij} \tau^a_{\alpha \beta} N^a_i L_j^\alpha H^{\dagger \beta}  +
\frac{M_{N_i}}{2}   N^a_i N^a_i  +\hbox{h.c.})\,,
\end{equation}
with $a=1,2,3$; $\alpha, \beta=1,2$.
As a result it leads to the same seesaw formula than with
singlets: $M_\nu=-\lambda^T M_N^{-1} \lambda v^2$.
For leptogenesis, the one-loop diagrams are exactly the same as
with right-handed neutrinos, Fig.~1, which lead to the same asymmetries up
to $SU(2)_L$ factors of order unity,\cite{HLNPS}
\begin{equation}\label{epsT}
\varepsilon_{N_1}=\sum_{j=2,3}\frac{3}{2}
  \frac{M_{N_1} }{M_{N_j} }\frac{\Gamma_{N_j} }{M_{N_j} }
  I_j\frac{V_j-2 S_j }{3} \,,
\end{equation}
and is therefore 3 times smaller in the
hierarchical limit where $V_j=S_j=1$, see Eq.~(\ref{eps}).
The final amount of baryon asymmetry is given by
the CP-asymmetry times the efficiency factor $\eta$ times a
numerical coefficient which is 3 times bigger than in the singlet case
because now $N_1$ has three
components: $\frac{n_{B}}{s} =-4.1 \cdot
10^{-3} \varepsilon_{N_1} \eta$.
The decay width of each of the 3 components of $N_1$
is given by the same expression as in the singlet
case, Eq.~(\ref{IGamma}).
Scatterings are same as in the type-I model except for $SU(2)_L$
factors\cite{HLNPS}.
The only important difference is that the triplets have
$SU(2)_L \times U(1)$ gauge scatterings the singlets do not have.
This reduces the efficiency factor\cite{HLNPS}.
As a result,
in the hierarchical limit $M_{N_1} << M_{N_{2,3}}$,
the bounds for successful leptogenesis
are slightly more stringent than for singlets:
\begin{equation}
M_{N_1} >1.5 \cdot 10^{10}~\hbox{GeV}\,, \,\,\,\,\,\,\,m_{\nu_3} < 0.12
\hbox{eV} \,.
\label{boundfermions}
\end{equation}
In the quasi-degenerate case the discussion is similar to the one of
the type-I model, see section 3.2.3 above. A value of $M_{N_1}$ as low as
$\sim 1$~TeV is possible, very close to the resonance.

\subsection{The Type-I plus Type-II Model}

The case where we add to the standard model three right-handed neutrinos
and one scalar triplet is quite interesting because it is
the situation of the ordinary left-right models and of the
renormalizable $SO(10)$ models such as defined in Ref.\cite{Senj}.
In this case the relevant Lagrangian is just the sum of the Lagrangians
of Eqs.~(\ref{Lseesaw})
and (\ref{Lseesaw2}).
To discuss this possibility it is necessary to consider two cases,
depending on which particle is the lightest one, the scalar triplet
$\Delta_L$
or the lightest right-handed neutrino $N_1$.

\subsubsection{The $M_\Delta < M_{N_1}$ case}

If the triplet is lighter than $N_1$, the production of the
asymmetry will be naturally dominated
by the decay of the triplet to two leptons.
The CP-asymmetry comes in this case
from the difference between the decay width
of $\Delta_L^*$ to two leptons and of $\Delta_L$ to two anti-leptons.
The leptogenesis one loop diagram
is a vertex diagram involving both the decaying triplet
and a virtual
right-handed neutrino,
first graph of Fig.~3. This diagram was first displayed
in Ref.\cite{OS}.
Calculating explicitly its contribution we get\cite{HS}:
\begin{eqnarray}
\varepsilon_\Delta &=& 2\cdot
\frac{\Gamma (\Delta_L^* \rightarrow l + l) - \Gamma (\Delta_L \rightarrow
\bar{l} + \bar{l })}{\Gamma_{\Delta_L^*}
    +\Gamma_{\Delta_L}}\\
&=&  -\frac{1}{8 \pi}
\frac{M^2_\Delta}{ (\sum_{ij} |(Y_\Delta)_{ij}|^2 M^2_\Delta +|\mu|^2)}
\frac{1}{v^2}
Im[(M^{I\ast}_\nu)_{il} (Y_\Delta)_{il} \mu^*]
\,,
\label{epsD}
\end{eqnarray}
where $M_\nu^{I}$ is the type-I contribution to the neutrino mass
matrix (given in section 2). For each of
the 3 triplet components the total decay width is:
\begin{equation}
\Gamma_\Delta=\frac{1}{8 \pi} M_\Delta \Big( \sum_{ij}|(Y_\Delta)_{ij}|^2 +
\frac{|\mu|^2}{M^2_\Delta} \Big) \,.
\label{gammaD}
\end{equation}
\begin{figure}[t]
\begin{center}
\begin{picture}(310,60)(0,0)
\DashArrowLine(0,30)(30,30){5}
\ArrowLine(60,60)(90,60)
\ArrowLine(60,0)(90,0)
\Line(60,60)(60,0)
\DashArrowLine(30,30)(60,60){5}
\DashArrowLine(30,30)(60,0){5}
\Text(3,22)[]{$\Delta_L^*$}
\Text(35,12)[]{$H$}
\Text(37,51)[]{$H$}
\Text(69,30)[]{$N_k$}
\Text(85,52)[]{$l_i$}
\Text(85,8)[]{$l_{l}$}
\Line(110,30)(140,30)
\DashArrowLine(170,60)(200,60){5}
\ArrowLine(170,0)(200,0)
\DashArrowLine(170,0)(170,60){5}
\DashArrowLine(140,30)(170,60){5}
\ArrowLine(170,0)(140,30)
\Text(115,22)[]{$N_k$}
\Text(145,12)[]{$l_l$}
\Text(147,51)[]{$H$}
\Text(182,30)[]{$\Delta_L$}
\Text(195,52)[]{$H^\ast$}
\Text(195,8)[]{$l_{i}$}
\DashArrowLine(210,30)(230,30){5}
\DashArrowArc(245,30)(15,180,360){5}
\DashArrowArcn(245,30)(15,180,0){5}
\DashArrowLine(260,30)(280,30){5}
\ArrowLine(280,30)(310,60)
\ArrowLine(280,30)(310,0)
\Text(215,22)[]{$\Delta^*_{L_i}$}
\Text(245,6)[]{$H$}
\Text(246,55)[]{$H$}
\Text(272,22)[]{$\Delta^*_{L_j}$}
\Text(297,55)[]{$l_{k}$}
\Text(297,4)[]{$ l_{l}$}
\end{picture}
\end{center}
\caption{One-loop diagrams contributing to the asymmetry
from the $\Delta_L$ and $N$ decays.}
\label{fig3}
\end{figure}
In this case leptogenesis works in a way similar to the type-I model
apart from 3 important differences. The first is that,
like for triplet of fermions above, there are
gauge scatterings which tend to put the $\Delta_L$ closer to
thermal equilibrium. The effect is similar to the one encountered for
triplet of fermions above. The second difference is that here there is no
one-loop self-energy diagram, so there is no possible
resonance effect. The third one
is that the interplay between the neutrino
masses and the size of the washout and the size of the asymmetry
is completely different from the type-I case.
The decay width is
proportional to the triplet couplings but the asymmetry is essentially
proportional to the right-handed neutrino couplings (taking into
account the fact that there are two triplet couplings in both
numerator and denominator of the asymmetry). As a result one may
consider the possibility that the type-II contribution to neutrino masses is
small enough to avoid large washout and that the type-I
contribution to neutrino masses is large. By
increasing in this way the type-I neutrino mass contribution,
the washout remains unchanged but the asymmetry increases
proportionally to the neutrino masses\cite{HS}. Therefore, there is
no upper bound on neutrino masses in this model to have successful
leptogenesis\cite{HS}. Note that
there is nevertheless a lower bound
on the triplet mass because
the asymmetry is proportional to this mass.
For a hierarchical spectrum of light
neutrinos (i.e.~$m_{\nu_3} \sim
\sqrt{\Delta m^2_{\hbox{\footnotesize atm}}}$) the
 bound is essentially the same$^d$ as for the triplet of
fermions, Eq.~(\ref{boundfermions}).
As there is no possible resonance effect
this bound is an absolute bound. For larger values of $m_{\nu_3}$ this
bound decreases because in this case the asymmetry increases.
However, assuming $m_{\nu_3}$ below 1 eV, this bound cannot
decrease by more than one order of magnitude.

\subsubsection{The $M_\Delta > M_{N_1}$ case}

If the triplet is sizeably
heavier than at least one right-handed neutrino, then it is the decay
of the right-handed neutrino to a lepton
and a Higgs boson which dominates the production of the L asymmetry.
In this case leptogenesis can be produced from the pure type-I model
just as in sections 2 and 3.
However here, in addition to this pure type-I contribution, there is
a new contribution to leptogenesis coming from a
diagram\cite{OS,laz1,HS,AK} with a real
right-handed neutrino and a virtual triplet, second diagram of Fig.~3.
Disregarding the pure type-I contribution (assuming that it has a small
contribution because the type-I Yukawa couplings and/or
their phases are small)
this new diagram can perfectly lead to successful leptogenesis alone.
For the asymmetry defined in Eq.~(\ref{eps}) one obtains from this
diagram\cite{HS,AK}
\begin{equation}
\varepsilon_{N_1}^\Delta=\frac{3}{16 \pi} \frac{M_{N_1}}{v^2}
\frac{\sum_{il} Im[\lambda_{1i} \lambda_{1l}(M^{II*}_\nu)_{il}]}
{\sum_i |\lambda_{1i}|^2} \,,
\label{epsNDbis}
\end{equation}
where $M_\nu^{II}$ is the neutrino mass matrix induced by the type-II
contribution (given in section 4.1 above).
The discussion is similar to the one of
the  $M_\Delta < M_{N_1}$ case above inverting
the role of type-I and type-II two contributions.
It is now the decay of the right-handed neutrino to lepton and Higgs,
induced by the type-I couplings, Eq.~(\ref{IGamma}), which essentially
determines
the washout. To increase the asymmetry without increasing
the washout, one can then consider the possibility of keeping
the type-I contribution small, increasing
the type-II contribution which increases the asymmetry but not the
washout.
As a result, here too, there is no more upper bound on neutrino
masses for leptogenesis\cite{HS}.
For hierarchical light neutrinos the lower
bound on the lightest right-handed neutrino mass is to a good approximation
the same as
in the pure type-I model\cite{HS}, Eq.~(\ref{MN1bound}).
As the asymmetry is linear in both $M_{N_1}$ and the neutrino masses,
for larger values
of $m_{\nu_3}$ the asymmetry linearly increases,
so the bound on $M_{N_1}$ linearly decreases. But here too
assuming $m_{\nu_3}$ below 1 eV,
this bound cannot decrease by more than one order of magnitude.
The precise bound is given in Ref.\cite{AK} as a function of the
efficiency factor $\eta$.

Note that, if instead of considering a dominant type-II
contribution, we consider the case where both type-I and type-II contributions
are important this lower bound on $M_{N_1}$ doesn't get relaxed.
In this case both contributions
appear to be just proportional to their respective
contributions to the neutrino masses so that, baring
a possible cancellation of CP-violating phases, leptogenesis
is expected to be dominated by the contribution which dominates the
neutrino masses\cite{HS}.
From Eq.~(\ref{eps}) the pure type-I contribution to leptogenesis
for hierarchical right-handed neutrinos turns out to be
given by Eq.~(\ref{epsNDbis}) replacing $M_{\nu}^{II}$ by
$M_{\nu}^I$, a fact which can be nicely understood by
using effective dimension five neutrino mass operators.\cite{AK}

For a quasi-degenerate spectrum of right-handed neutrino the pure type-I
contribution to leptogenesis is expected dominant and the triplet
contribution which doesn't display any resonant behaviour is in this case
negligible.

\subsection{The Multiple Type-II Models}

If there is more than one heavy scalar triplet, leptogenesis can be
easily induced by the decay of the triplets to two leptons with
a one-loop self-energy diagram involving two different
triplets\cite{MS1,HMS1},
third diagram of Fig.3.
The asymmetry in this case is given by\cite{MS1}
\begin{equation}
\varepsilon_{\Delta_i} =  -\frac{1}{ \pi} M_{\Delta_i}
\frac{ Im[(\mu^*_i \mu_j (Y_{\Delta_i})_{kl}
 (Y^*_{\Delta_j})_{kl} ] }{  |(Y_{\Delta_i})_{kl}|^2 M^2_{\Delta_i}
+|\mu_i|^2}  \frac{M^2_{\Delta_j}
\Delta M^2_{1j}}{(\Delta M^2_{1j})^2+M_{\Delta_1} ^2
   \Gamma_{\Delta_j} ^2}
\,,
\label{epsDi}
\end{equation}
where there is now a triplet scalar index on the couplings
of Eqs.~(\ref{Lseesaw2}) and (\ref{gammaD}).
For hierarchical triplets, under the assumption of footnote d above,
successful leptogenesis leads to a triplet mass bound similar to
the one of Eq.~(\ref{boundfermions}) for hierarchical fermion triplets,
in accordance with the estimate of Ref.\cite{HMS1}.
This bound is higher than
for right-handed neutrinos due to gauge scatterings. Similarly for
quasi-degenerate triplets, one can go down to 1-10 TeV (very close to
the resonance). And,
here too, even with hierarchical triplets,
there is no more upper bound on neutrino masses for successful
leptogenesis because one can always increase the neutrino mass contribution
of the virtual triplet in the self-energy diagram leading to a
larger asymmetry without increasing the washout.

\subsection{The Multiple Type-I Seesaw Case}

To add extra fermion singlets $S_i$ on top of the 3 right-handed
neutrinos $N_i$ is
also a possibility one might consider,\cite{AB,Ut,AAL} especially
if these extra
$SU(2)_L \times U(1)$ singlets are
also singlets of SO(10) in case one can build a
renormalizable SO(10) model\cite{AB,Ut} without a 126
scalar multiplet to give mass to the $N_i$'s.
In this case, as there is no 126, there is no triplet.
This may lead to successful leptogenesis just from the same diagrams as
in the type-I model, Fig.~1,
except that there are now more than 3 singlets to be
put in these diagrams. There are diagrams where both
real and virtual singlets are $N_i$'s or are $S_i$'s.
Clearly, for example if the lightest heavy particle is a $S$, one can
increase the Yukawa couplings of the $N$ leaving the $S$ Yukawa
couplings unchanged. In this way the neutrino masses increase but not
the washout which is due the $S$ couplings, so that,
here too, there is no relevant upper bound on neutrino masses for successful
leptogenesis.

Note that extra diagrams with a real $N$ and a
virtual $S$ in Fig.1 (or viceversa)
are also possible if they couple to the same scalar multiplets.
In the SO(10) models of  Ref.\cite{AB,Ut} the $S$ and $N$ do not
couple to the same multiplets because the $S$ which is a singlet of SO(10)
couples to the 16 of matter $\psi_{16}$ and a scalar
$H_{\overline{16}}$
but the $N$ which is in $\psi_{16}$ couples to an other $\psi_{16}$
and a scalar $H_{10}$.
However through mixing of the $H_{\overline{16}}$
and $H_{10}$ from a coupling
involving the vev of another $H_{\overline{16}}$ such a diagram may exist, a
possibility still to consider.
It is straightforward to check that such a diagram can lead to
successful leptogenesis and here
too without leading to a relevant upper bound on the neutrino masses.

Note also that there is an other possibility with extra singlets\cite{AAL},
simply by assuming
more than three generations of fermions and by assuming a huge hierarchy of
Yukawa couplings (to have $m_{\nu_4}> 45$~GeV). This may also lead to
successful leptogenesis (e.g.~at a low scale).

\section{The Radiative Models}

An other class of models one might consider to explain the neutrino
masses and mixings is the one where neutrino masses are induced by
loop diagrams. This is quite interesting as in most of these models
the scale where they are generated is not far beyond the reach of
present particle accelerators.
However to generate leptogenesis at the $\sim$~TeV scale in this framework
turns out to be a quite hard task for various reasons.
A first reason is that in these models, such as the ones based on
violation of R-parity\cite{Rparity} or
based on the presence of a charged scalar
$SU(2)_L$ singlet (Zee model\cite{Zee}), the
heavy states whose decays could generate the
lepton asymmetry are not gauge singlets. As a result there is a large
washout suppression coming from gauge scatterings involving these
heavy states which are very fast at low scale. A second reason is that
at the TeV scale the Hubble constant is much smaller than for example
at $10^{10}$~GeV
since it is proportional to the square of the temperature.
Since the decay width is only linear in the mass, the condition $\Gamma < H$
requires therefore couplings much smaller at the TeV scale than
at $10^{10}$~GeV.
Since the asymmetry is proportional to these tiny
couplings, this leads in general to too small
lepton asymmetry, see Ref.\cite{TH}.

One solution to these problems is to consider
three body decays instead of two-body decays.\cite{TH}
An other solution is to consider
the seesaw extended MSSM, that is the say the MSSM extended by
three right-handed (s)neutrinos. In this framework
it has been shown\cite{BHS} that a large enough asymmetry can be obtained
from the L-violating soft
supersymmetry breaking terms involving the right-handed sneutrinos.
With sneutrino masses of order a few TeV, successful
leptogenesis together with light neutrino masses (induced
radiatively from the same soft terms) can be induced.

\section{Summary}

In summary there are quite a few models which from the same interactions
can lead to successful generation of neutrino masses and leptogenesis:
type-I model with right-handed neutrinos, type-II model (although only if
involving the soft terms), type-III model with triplet of
fermions, type-I plus type-II model, multiple type-I or type-II models,
or even a radiative model (in the seesaw extended MSSM from soft terms).

The upper bound on neutrino masses is quite sensitive to the model
considered as well as on the assumptions made on the heavy mass spectrum
in each model.
In the type-I model (and similarly in the type-III model)
a stringent bound, Eq.~(\ref{mnu3bound}), can
be found only
assuming a hierarchical spectrum of right-handed neutrinos. However
this assumption
doesn't appear
to be the most natural assumption one can make
when considering this bound, for which
the light neutrinos have a quasi-degenerate
spectrum. A probably more natural
quasi-degenerate spectrum of right-handed neutrinos
leads instead in full generality to an upper bound on neutrino masses
far beyond the eV scale. Even in a highly constrained model, such as the
one based on
a SO(3)
symmetry in section 3.2.3,
a value around 1 eV appears to be possible for successful
leptogenesis.
Only with extra assumptions (such as low reheating temperature) one might
get a more stringent bound.
In the other models (multiple type-I and/or type-II) the upper bound
is far above the eV scale even for a hierarchical spectrum of heavy states.

Similarly the lower bound on the scale of leptogenesis is sensitive to
the model and/or the heavy mass spectrum considered. However for this
scale there is at least one firm conclusion one can draw:
in all the seesaw models we considered here, if
the masses of the heavy states differ by orders of magnitude,
this scale has to be orders of magnitude above the
TeV scale, that is to say
above $10^7-10^{10}$ GeV depending on the model.

\section*{Acknowledgments}
It is a pleasure to thank L. Boubekeur, G. D'Ambrosio, A. Hektor,
Y. Lin, E. Ma, A. Notari,
M. Papucci, M. Raidal, A. Rossi,
U. Sarkar, G. Senjanovic and A. Strumia with whom part of the work presented
here was done. We also thank A. Faraggi and U. Sarkar for discussions on
multiple type-I models. This work was supported by the Marie Curie
HPMF-CT-01765 and EU MRTN-CT-2004-503369 (The quest for unification)
contracts.


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

The Kondo effect\cite{Kondo} in quantum dots has been
an active research field over a decade.
The early prediction about a characteristic
gate-voltage dependence of
the linear-response conductance\cite{NgLee,GlatzmanRaikh,Kawabata}
has been confirmed experimentally in semiconductor devices,
\cite{Goldharber,Cronenwett,Simmel,VanDerWiel}
and other new features of the Kondo physics
are also being studied extensively in various situations,
such as an AB ring, Josephson junction, ferromagnetic leads, etc.

The equilibrium and linear-response properties of a single quantum dot
connected to normal leads can be explained basically
based on the knowledge of the Kondo problem
in dilute magnetic alloys,\cite{Hewson}
although there exists some differences
in experimental geometry (configuration)
between the impurity in magnetic alloys and
quantum dots in semiconductor devices.
Therefore, the low-energy properties can be described
by the local Fermi liquid theory,
\cite{Nozieres,YamadaYosida,Shiba}
and the nonperturbative approaches developed
for the alloys, such as the quantum Monte Carlo\cite{ao6,Sakai}
 and numerical renormalization group (NRG) methods,\cite{Izumida}
can be applicable to the quantum dots.
Particularly, the NRG has been used successfully to calculate
the linear-response conductance of the quantum dots.\cite{Izumida}

The nonlinear transport under a finite bias voltage $V$, however,
is still not fully understood, despite of a number of theoretical efforts.
\cite{WM,HDW2,TheoriesII,Schiller,KNG,KSL,ao10,ao13,FujiiUeda}
Among a variety of aspects of the nonequilibrium properties,
in this report we focus our attention mainly on the low-energy properties.
Specifically, we describe the Fermi-liquid behavior
of the first nonlinear term of the differential conductance $dI/dV$
using the Ward identities, which is derived by applying
the perturbation theory in the Coulomb interaction $U$
of Yamada and Yosida to the Keldysh Green's function.
We show that the low-energy asymptotic form of
the order $U^2$ self-energy\cite{HDW2} is
essentially retained in all orders in $U$,
and the contributions of the higher-order terms
are absorbed into the coefficients which  can be written
in terms of the {\em local-Fermi-liquid\/} parameters such as
the width of the Kondo resonance $\widetilde{\Delta}$
and Wilson ratio $R$.\cite{ao10}
The proof was provided previously in ref.\ \citen{ao10}.
In the present report, however, we give another derivation,
using the property of the impurity
Green's function $\mbox{\boldmath $G$}(\omega)$
as a functional of a nonequilibrium distribution
function $f_{\text{eff}}(\omega)$,
through which the dependence of $\mbox{\boldmath $G$}(\omega)$
on $eV$ and $T$ arises.
This property also allows us to deduce
some exact results in the limit of large $eV$.\cite{ao13}
In the present report, we re-examine the low-energy properties
with an emphasis on this  aspect of the Green's function as a functional.

We also present an alternative description of
the low-voltage Fermi-liquid behavior
using the renormalized perturbation theory (RPT).\cite{HewsonRPT}
The unperturbed Green's function of the RPT
in the Keldysh formalism consists of
the propagators of the free quasiparticles,
which are determined by
the renormalized resonance of the width $\widetilde{\Delta}$
and the nonequilibrium distribution $f_{\text{eff}}(\omega)$.
To second order in the residual interaction $\widetilde{U}$,
which is  defined in eq.\ (\ref{eq:renorm_parm}),
it gives the exact low-energy  $(eV)^2$ coefficient of $dI/dV$.
The higher order terms in $\widetilde{U}$
determine the high-energy properties.
It has recently been confirmed that in equilibrium a combination
of the RPT and NRG gives an efficient way of calculating the temperature
dependence of the susceptibility,\cite{HewsonNRG}
so that the RPT seems to be one possible approach
to the nonequilibrium properties beyond the Fermi-liquid regime.

 In \S \ref{sec:model},
 we describe the Keldysh formalism for the Anderson impurity
in order to describe clearly the properties of
  $\mbox{\boldmath $G$}(\omega)$ as
  a functional of $f_{\text{eff}}(\omega)$.
In \S \ref{sec:low-voltage},
we consider the low-energy behavior of
the self-energy at small $eV$ using the Ward identities,
and give an exact low-energy
expression of $dI/dV$ in the electron-hole symmetric case.
In \S \ref{sec:RPT}, the RPT is applied to the
low-voltage Fermi-liquid regime,
and the procedure of the perturbation expansion
in  $\widetilde{U}$ in the Keldysh formalism is provided.
In the appendix, details of the Ward identities
and properties of $\mbox{\boldmath $G$}(\omega)$ as
  a functional of $f_{\text{eff}}(\omega)$ are given.

\section{Keldysh Formalism for the Anderson Model}
\label{sec:model}

We start with the single Anderson impurity connected
to two leads at the left ($L$) and right ($R$):
\begin{align}
H \  &= \  H_{c} \, + \, H_{d} \, + \,  H_{\rm mix} \, + \, H_{U} \;,
\label{eq:H}
\\
 H_{c}  &=
\sum_{\lambda=L,R}
 \sum_{k\sigma}
  \epsilon_{k \lambda}^{\phantom{0}}\,
         c^{\dagger}_{k \lambda \sigma}
         c^{\phantom{\dagger}}_{k \lambda \sigma}
\,,
\label{eq:Hc}
\\
 H_{d}  & =  \sum_{\sigma}   E_d \,  n_{d\sigma} \;, \quad
 H_{U}  =  \frac{U}{2} \biggl(\sum_{\sigma}n_{d\sigma} -1\biggr)^2 \;,
\label{eq:Hd}
\\
 H_{\rm mix} \,  &=
 \sum_{\lambda=L,R}
 \sum_{\sigma} v_{\lambda}^{\phantom 0}  \left(\,
             d^{\dagger}_{\sigma}  \psi^{\phantom{\dagger}}_{\lambda \sigma}
      +\,   \psi^{\dagger}_{\lambda \sigma}  d^{\phantom{\dagger}}_{\sigma}
             \,\right)   ,
\label{eq:Hmix}
\end{align}
where $d_{\sigma}$ annihilates an electron with spin
$\sigma$ at the dot,
$n_{d\sigma} = d^{\dagger}_{\sigma} d^{\phantom{\dagger}}_{\sigma}$,
and $E_d = \epsilon_d + U/2$.
We assume that the onsite potential $\epsilon_d$ is a constant
independent of the bias voltage, and take
the Fermi level at equilibrium $\mu$ to be
the origin of the energy, i.e.,  $\mu = 0$.
In the lead at $\lambda$ ($= L,\, R$),
the energy spectrum is given by
$\epsilon_{k \lambda} = \epsilon_{k}  + eV_{\lambda}$.
To specify how the bias voltage $V$ is applied to each of the leads,
we introduce a parameter $\alpha_{\lambda}$ such that
$V_L = \alpha_L V$ and $V_R = - \alpha_R V$ with $\alpha_L + \alpha_R =1$.
In eq.\ (\ref{eq:Hmix}),
 $v_{\lambda}$ is the tunneling matrix element
between the dot and lead at $\lambda$, and
 $\psi_{\lambda \sigma}^{\phantom{\dagger}}
= \sum_k c_{k \lambda \sigma}^{\phantom{\dagger}}
/\sqrt{N}$.
 We will use units $\hbar=1$.

In the thermal equilibrium, we know that the density matrix is given by
 $\rho_{\rm eq}^{\phantom{0}} \propto \text{e}^{-\beta H}$,
 and thus the Hamiltonian determines both the time evolution
 and statistical weight.
 However, in  a nonequilibrium steady state
 the density matrix cannot be determined simply by $H$,
 and it depends on how the system has been driven to the steady state.
 The Keldysh formalism has been used widely for this purpose
 to determine the density matrix $\widehat{\rho}(t)$
 for nonequilibrium states.\cite{Keldysh,Caroli,Landau}

The method uses the procedure of an adiabatic switching on,
which is described by the operator
 ${\cal U}(t,t_0)  =  \text{T} \exp
        [\,- \text{i} \int_{t_0}^t \text{d}t' \,
        \widetilde{H}_2 (t') \,] $.
Here, $\widetilde{O}(t)  \equiv
      \text{e}^{\text{i} H_1 t}\, O\, \text{e}^{-\text{i} H_1 t}$
is an operator in the interaction representation with respect
to $H_1$, which is a time-independent part of the total Hamiltonian
$H(t)=H_1 + H_2\, \text{e}^{-\delta |t|}$.
In the interaction representation the density matrix
defined by
$\widetilde{\rho}(t) \equiv
 \text{e}^{\text{i} H_1 t} \, \widehat{\rho}(t)\, \text{e}^{-\text{i} H_1 t}$
can be rewritten in the form
\begin{align}
\widetilde{\rho}(t) \, = \, {\cal U}(t,-\infty) \,
                         \widetilde{\rho}(-\infty) \,
                         {\cal U}(-\infty,t)
\label{eq:rho} \;,
\end{align}
where $\widetilde{\rho}(-\infty)$ represents
the initial statistical weight.
The average value of
a Heisenberg operator ${\cal O}_H(t) = {\cal U}(0,t) \,
\widetilde{\cal O}(t) \, {\cal U}(t,0)$ is given by
\begin{align}
& \langle {\cal O}_H(t) \rangle \ \equiv \
            \mbox{Tr} \left[\,\widehat{\rho}(0) \,{\cal O}_H(t) \, \right]
\nonumber
\\
  &=
  \text{Tr}
       \left[\, \widetilde{\rho}(-\infty) \,
      {\cal U}(-\infty,+\infty) \,
      {\cal U}(+\infty,t) \, \widetilde{\cal O}(t)
      \, {\cal U}(t,-\infty) \,\right].
\label{eq:average_H}
\end{align}
The stream of time seen in this expression
is usually illustrated as the Keldysh contour shown
in Fig.\ \ref{fig:contour}:
the $+$ branch corresponds
to the time evolution by the operator
$ {\cal U}(-\infty,+\infty) =
 \widetilde{\text{T}} \exp
[\, \text{i} \int_{-\infty}^{\infty} \text{d}t'
\, \widetilde{H}_2 (t') \,] $,
where $\widetilde{\text{T}}$ denotes
the anti-time-ordering operator.
If one chooses $H_1$ to be bilinear,
the Feynman-diagrammatic approach is applicable
for the Green's functions defined by
\begin{align}
 G^{--}_{\sigma}(t) & \ = \  -\text{i} \,
\langle \text{T}\,
d^{\phantom{\dagger}}_{\sigma}(t)\, d^{\dagger}_{\sigma}(0)
\rangle  \;,
\label{eq:G^--}
\\
G^{-+}_{\sigma}(t) & \ = \     \text{i}\,
\langle
d^{\dagger}_{\sigma}(0) \, d^{\phantom{\dagger}}_{\sigma}(t)
\rangle \;,
\label{eq:G^-+}
\\
G^{+-}_{\sigma} (t) & \ = \  - \text{i} \,
\langle
d^{\phantom{\dagger}}_{\sigma}(t) \,
d^{\dagger}_{\sigma}(0)
\rangle \;,
\label{eq:G^+-}
\\
G^{++}_{\sigma}(t) & \ = \ -\text{i} \,
\langle \widetilde{\text{T}}\,
d^{\phantom{\dagger}}_{\sigma}(t)\, d^{\dagger}_{\sigma}(0)
\rangle\;.
\label{eq:G^++}
\end{align}
These functions are linearly dependent
 $G^{-+}+G^{+-}=G^{--}+G^{++}$.
Furthermore, the retarded and advanced Green's functions
can be written as $G^r = G^{--}-G^{-+}$ and $G^a = G^{--}-G^{+-}$,
 respectively.

\begin{figure}[tb]
\setlength{\unitlength}{0.75mm}
\hspace{-0.8cm}
\begin{picture}(120,20)(-5,0)
\thicklines

\put(100,10){\oval(12,9)[r]}

\put(60,5.5){\vector(1,0){1}}
\put(58.5,14.5){\vector(-1,0){1}}

\put(20,14.5){\line(1,0){80}}
\put(20,5.5){\line(1,0){80}}

\put(7,14){\makebox(0,0)[bl]{$-\infty$}}
\put(7,4){\makebox(0,0)[bl]{$-\infty$}}
\put(110,9){\makebox(0,0)[bl]{$+\infty$}}

\put(66,-1){\makebox(0,0)[bl]{$-$ branch}}
\put(66,17){\makebox(0,0)[bl]{$+$ branch}}

\end{picture}

\caption{The Keldysh contour for the time evolution.}
\label{fig:contour}
\end{figure}

\subsection{Traditional Formulation}
\label{subsec:traditional}

 To describe a nonequilibrium steady state
 under a finite bias voltage,
 Caroli {\em et al.\ }\cite{Caroli} has
 introduced the initial statistical weight of the form
\begin{equation}
\widetilde{\rho}(-\infty)
\, \propto  \,
   \text{e}^{-\beta (\,H_d \,+\, H_c \,-\, \mu_L N_L \, -\, \mu_R N_R \,)}
    \;,
\label{eq:rho_ini}
\end{equation}
where
$N_{\lambda} = \sum_{k\sigma}
c^{\dagger}_{k \lambda \sigma} c^{\phantom{\dagger}}_{k \lambda \sigma}$.
The two chemical potentials,
 $\mu_{L} \equiv eV_{L}$ and  $\mu_{R} \equiv eV_{R}$,
 are defined with respect to the isolated systems
 described by $H_1 = H_{d} + H_{c}$,
  and  the remaining part $H_2 = H_{\rm mix} + H_U$ is
  switched on adiabatically.
Specifically, in the noninteracting case $U=0$,  the Green's functions
can be written in the form
\begin{align}
 G_{0}^{--}(\omega) &=
 \left[ 1-f_{\text{eff}}(\omega) \right]
     G_{0}^r(\omega)
     \, + \,  f_{\text{eff}}(\omega)\,G_{0}^a(\omega)
\;,
\label{eq:G_0^--}
\\
G_{0}^{-+}(\omega) & \,= \,
-\,f_{\text{eff}}(\omega)
        \left[\,G_{0}^r(\omega)- G_{0}^a(\omega)\,\right] ,
\label{eq:G_0^-+}
\\
G_{0}^{+-}(\omega) & \,=  \,
\left[ 1-f_{\text{eff}}(\omega) \right]
        \left[\,G_{0}^r(\omega)- G_{0}^a(\omega)\,\right] ,
\label{eq:G_0^+-}
\\
 G_{0}^{++}(\omega) & \, = \,
 -\left[ 1-f_{\text{eff}}(\omega) \right]
     G_{0}^a(\omega)
     \, - \,  f_{\text{eff}}(\omega)\,G_{0}^r(\omega)
,
\label{eq:G_0^++}
\end{align}
where $G_{0}^{r}(\omega)  =
  \bigl[\, \omega - E_d + \text{i} \Delta \,\bigr]^{-1}$,
$G^a(\omega) = \{G^r(\omega)\}^*$,
and $\Delta = \Gamma_L + \Gamma_R$ with
 $\Gamma_{\lambda} = \pi \rho_{\lambda}^{\phantom{0}} v_{\lambda}^2$.
 We assume that the density of
 states $\rho_{\lambda}^{\phantom{0}}(\omega)
 = \sum_k
\delta(\omega-\epsilon_{k \lambda}^{\phantom{0}})/N$ is a constant,
and the band width is very large.
One important feature we see in eqs.\
(\ref{eq:G_0^--})--(\ref{eq:G_0^++}) is that
all the information about the nonequilibrium distribution
is contained in the distribution function,\cite{HDW2}
\begin{equation}
f_{\text{eff}}(\omega) \, = \,
{  f_L(\omega) \,\Gamma_L
  + f_R(\omega) \,\Gamma_R
  \over
 \Gamma_L +\Gamma_R } \;.
\label{eq:f_0}
\end{equation}
Here  $f_{\lambda}(\omega)
= f(\omega - \mu_{\lambda}^{\phantom{0}})$,
and $f(\omega)=[\,\text{e}^{\omega/T}+1\,]^{-1}$.
At $T=0$ the distribution function  $f_{\text{eff}}(\omega)$  has two steps,
at $\omega=\mu_L$ and $\mu_R$,
as shown in Fig.\ \ref{fig:distribution}.
At $eV=0$,
it coincides with the usual Fermi function $f(\omega)$.

The interacting Green's function $\mbox{\boldmath $G$}(\omega)$
satisfies the Dyson equation,
\begin{align}
& \{\mbox{\boldmath $G$}(\omega)\}^{-1}
    =
 \{\mbox{\boldmath $G$}_{0}(\omega)\}^{-1}
-  \mbox{\boldmath $\Sigma$}(\omega) \;,
\label{eq:Dyson}
\\
& \mbox{\boldmath $G$}_{0}
\, = \,
\left[\,
 \begin{matrix}
  G^{--}_{0} & G^{-+}_{0}   \cr
  G^{+-}_{0} & G^{++}_{0}  \cr
 \end{matrix}
 \, \right] ,
\quad
\mbox{\boldmath $\Sigma$} \, = \,
 \left[ \,
  \begin{matrix}
   \Sigma^{--} & \Sigma^{-+}  \cr
   \Sigma^{+-} & \Sigma^{++}  \cr
  \end{matrix}
 \,\right] .
\label{eq:Keldysh_Matrix}
\end{align}
Here $\mbox{\boldmath $\Sigma$}(\omega)$ is the self-energy due to $H_U$:
the four elements are linearly
dependent $\Sigma^{-+}+ \Sigma^{+-}= -\Sigma^{--}-\Sigma^{++}$,
and we have also two extra relations in the $\omega$-space,
$\Sigma^{a}(\omega)= \left\{\Sigma^{r}(\omega)\right\}^*$
and $\Sigma^{--}(\omega)= - \left\{\Sigma^{++}(\omega)\right\}^*$.
Using these relations,
the retarded Green's function is written in the form
\begin{align}
G^{r}(\omega) \, = \,
  {1 \over \omega -E_d + \text{i} \Delta - \Sigma^r(\omega) }
\end{align}
with $\Sigma^r = \Sigma^{--} + \Sigma^{-+}$.
The four elements of $\mbox{\boldmath $G$}(\omega)$ are also
written in the forms similar to eqs.\ (\ref{eq:G_0^--})--(\ref{eq:G_0^++}),
for which $G^{r}_0$ and $G^{a}_0$ are replaced by the interacting ones
and $f_{\text{eff}}(\omega)$ is replaced by a correlated distribution
defined by
\begin{align}
f_{\text{eff}}^U(\omega) & =
{ f_L(\omega) \,\Gamma_L
  + f_R(\omega) \,\Gamma_R -
  {\displaystyle \mathstrut 1 \over
  \displaystyle \mathstrut 2\text{i}}\,\Sigma^{-+}(\omega)
  \over
 \Gamma_L +\Gamma_R - \text{Im} \Sigma^r(\omega)} \;.
\label{eq:f_int}
\end{align}
This function was introduced
by Hershfield {\em et al.}, and was studied
using the order $U^2$ self-energy.\cite{HDW2}
Note that $\Sigma^{-+}(\omega)$ is pure imaginary,
and at $eV=0$ it takes the form
 $\left.\Sigma^{-+}(\omega)\right|_{eV=0} =
 2\text{i}f(\omega)
 \text{Im}\left.\Sigma^{r}(\omega)\right|_{eV=0}$.
Thus, in the nonequilibrium state, the distribution function
$f_{\text{eff}}^U(\omega)$ generally depends on the interaction $U$,
while  in equilibrium  $eV=0$ it coincides with the Fermi function.

\begin{figure}[tb]
\setlength{\unitlength}{0.7mm}
\begin{picture}(105,49)(-56,-10)
\thicklines

\put(-45,0){\vector(1,0){110}}
\put(0,0){\vector(0,1){40}}

\put(-45,25){\line(1,0){30}}
\put(-15,15){\line(0,1){10}}
\put(-15,15){\line(1,0){40}}
\put(25,0){\line(0,1){15}}

\put(0,-5){\makebox(0,0){\large $0$}}
\put(-15,-5){\makebox(0,0){\large $\mu_R^{\phantom{0}}$}}
\put(27,-5){\makebox(0,0){\large $\mu_L^{\phantom{0}}$}}
\put(54,-7){\makebox(0,0){\large $\omega$ }}
\put(11,37){\makebox(0,0){\large $f_{\text{eff}}(\omega)$}}
\put(-4,26){\makebox(0,0){\large $1$}}
\put(47,9){\makebox(0,0)
   {\Large $\frac{\mathstrut \Gamma_L}{\mathstrut \Gamma_L+\Gamma_R}$}}

\thinlines

\put(32,10){\vector(0,-1){10}}
\put(32,10){\vector(0,1){5}}

\end{picture}
\caption{
The nonequilibrium distribution $f_{\text{eff}}(\omega)$
at $T=0$.}
\label{fig:distribution}
\end{figure}

\subsection{Alternative Formulation}
\label{subsec:alternative}

As described in the above,
the noninteracting Green's function for $H_0 \equiv H_c + H_d +H_{\rm mix}$
can be calculated analytically taking all contributions
of the tunneling matrix element $H_{\rm mix}$ into account.
A question arises:
do we always have to start with the isolated systems to
obtain eqs.\ (\ref{eq:G_0^--})--(\ref{eq:G_0^++})?
The answer is no.
An alternative description was given by Hershfield.\cite{Hershfield}
The basic idea is to assign the two different chemical potentials
directly to the left- and right-moving scattering states
which are written formally, using the Lippmann-Schwinger equation, as
\begin{align}
\gamma_{k \lambda \sigma}^{\dagger}
\ = \
c_{k \lambda \sigma}^{\dagger}
\,+\,
\frac{1}{\varepsilon_k - H_0 +\text{i} \delta}
\,H_{\rm mix} \,
c_{k \lambda \sigma}^{\dagger} \;,
\label{eq:L-S}
\end{align}
where $\lambda = L,\, R$. The incident wave comes in
the left for $\gamma_{kL\sigma}^{\dagger}$,
and in the right for $\gamma_{kR\sigma}^{\dagger}$.
These scattering states are the eigenstates,
by which $H_0$ can be diagonalized as
\begin{align}
H_0 \, = \, \sum_{\lambda=L,R}\sum_{k\sigma}
 \varepsilon_{k}^{\phantom{0}}
 \gamma_{k\lambda\sigma}^{\dagger}
 \gamma_{k\lambda\sigma}^{\phantom{0}} \;.
\label{eq:H0_Jost}
\end{align}
Note that generally the bound states and continuum states
without the degeneracy (for the left and right movers) are present.
Such states are not distinguished from the degenerate scattering states
in eq.\ (\ref{eq:H0_Jost}) for simplicity.
With these scattering states,
the density matrix for $U=0$ can be expressed explicitly as\cite{Hershfield}
\begin{equation}
\widehat{\rho}_0^{\phantom{0}}(0)
\, \propto  \,
   \text{e}^{ -\beta (\,H_d \,+\, H_c \,+\, H_{\rm mix}
  \, -\, \mu_L \mathcal{N}_L
  \,-\, \mu_R  \mathcal{N}_R
\, )}
    \;,
\label{eq:rho_0}
\end{equation}
where
$  \mathcal{N}_{\lambda\sigma} = \sum_{k}
\gamma_{k\lambda\sigma}^{\dagger}
 \gamma_{k\lambda\sigma}^{\phantom{0}}
$.
One can confirm that the noninteracting Green's functions
eqs.\ (\ref{eq:G_0^--})--(\ref{eq:G_0^++})
can be calculated directly from eq.\ (\ref{eq:rho_0}).
Therefore, the Coulomb interaction  $H_U$ can be switched on
starting from the connected system
taking $\widehat{\rho}_0(0)$, given in eq.\ (\ref{eq:rho_0}),
to be the initial statistical weight.
It is carried out by using eq.\ (\ref{eq:average_H})
and redefining the initial condition as
 $H_1 \Rightarrow H_0$, $H_2 \Rightarrow H_U$,
 and  $\widetilde{\rho}(-\infty) \Rightarrow
  \text{e}^{ -\beta (\,H_0
  \, -\, \mu_L \mathcal{N}_L
  \,-\, \mu_R  \mathcal{N}_R
\, ) }
$.

The perturbation series in $U$ of
the nonequilibrium Green's
function $\mbox{\boldmath $G$}$
can be generated automatically using the path
integral representation,\cite{PathIntegral}
\begin{align}
\mathcal{Z}  &\, =
\int \!
{\cal D}\eta^{\dagger} {\cal D}\eta \  \text{e}^{\text{i} S}
\;,
\label{eq:ZJ}
\\
G^{\nu\nu'}_{\sigma}(t,t') &\, = \,
{ -\text{i}\over \mathcal{Z} }
\int \!
{\cal D}\eta^{\dagger} {\cal D}\eta \  \text{e}^{\text{i} S}
\,\eta_{\sigma \nu}^{\phantom{\dagger}}(t)
\,
\eta_{\sigma \nu'}^{\dagger}(t')
\;,
\label{eq:Green_path}
\end{align}
where $\eta_{\sigma \nu}^{\phantom{\dagger}}(t)$
is a Grassmann number for the branch $\nu$ ($=-,\,+$) in
the Keldysh contour.
The action $S$ is defined by
\begin{align}
   S & \ = \
   S_0
      \, + \,
   S_U \;,
  \label{eq:action_tot}
   \\
   S_0 & \,=\,
   \sum_{\sigma} \int_{-\infty}^{\infty} \! \text{d}t\,\text{d}t'\
 \mbox{\boldmath $\eta$}_{\sigma}^{\dagger}(t)\,
       \mbox{\boldmath $K$}_0(t,t')\,
 \mbox{\boldmath $\eta$}_{\sigma}(t')
 \;,
 \label{eq:action_0}
   \\
   S_U & \, = \,
   -\, \frac{U}{2}\! \int_{-\infty}^{\infty} \! \text{d}t \,
 \Biggl[\,
 \biggl(\, \sum_{\sigma}
 \eta_{\sigma -}^{\dagger}(t)\,
 \eta_{\sigma -}^{\phantom{\dagger}}(t)
 \,-\,1  \,\biggr)^2
\nonumber \\
 & \qquad \qquad \qquad  -   \,
\biggl(\,\sum_{\sigma}
 \eta_{\sigma +}^{\dagger}(t)\,
 \eta_{\sigma +}^{\phantom{\dagger}}(t)
 \,-\,1  \,\biggr)^2
 \,\Biggr]  ,
  \label{eq:action_U}
\end{align}
where
 \begin{align}
    \mbox{\boldmath $K$}_0(t,t') & \,  =
    \int_{-\infty}^{\infty} \! {\text{d}\omega \over 2 \pi}\,
    \left\{\mbox{\boldmath $G$}_0(\omega)\right\}^{-1}
     \text{e}^{-\text{i}\omega (t-t')} \;,
\label{eq:K_0}
 \end{align}
and
$
 \mbox{\boldmath $\eta$}_{\sigma}^{\dagger}(t)
 = \bigl(\,
 \eta_{\sigma -}^{\dagger}(t) \,, \,
 \eta_{\sigma +}^{\dagger}(t) \,\bigr)
 $.
 In the Keldysh formalism
the perturbation expansion works
with the real frequencies (or real times).
Therefore, eq.\ (\ref{eq:Green_path})  shows that
the dependence of $\mbox{\boldmath $G$}(\omega)$
on the bias voltage and temperature
arises through $f_{\text{eff}}(\omega)$
in the noninteracting
Green's function $\mbox{\boldmath $G$}_0(\omega)$
which determines $S_0$ via eqs.\ (\ref{eq:action_0}) and (\ref{eq:K_0}).
Thus, the full Green's function $\mbox{\boldmath $G$}(\omega)$,
can be regarded as a functional of $f_{\text{eff}}(\omega)$.
The precise form of the functional
is obtained by expanding $\text{e}^{\text{i} S}$ in eq.\ (\ref{eq:Green_path})
in the power series of $U$,
and substituting eqs.\ (\ref{eq:G_0^--})--(\ref{eq:G_0^++})
into every single $\mbox{\boldmath $G$}_0$'s in the series.
Therefore, the change in the self-energy,
$\delta \mbox{\boldmath $\Sigma$}$,
caused by a small variation
in the distribution function,  $\delta \! f_{\text{eff}}$,
can be expressed in the form
\begin{align}
 \delta \mbox{\boldmath $\Sigma$}_{\sigma}(\omega)
& \, = \,
 \sum_{\nu\nu'\sigma'}
\int_{-\infty}^{\infty} \! {\text{d}\omega'}
\biggl.
{
\delta \mbox{\boldmath $\Sigma$}_{\sigma}(\omega)
\over
\delta G_{0\sigma'}^{\nu\nu'}(\omega')
}
\biggr|_{\delta \! f_{\text{eff}}=0}
 \delta G_{0\sigma'}^{\nu\nu'}(\omega')
\nonumber \\
& \quad  + \
\Bigl[\, \mbox{higer order terms in $\delta \! f_{\text{eff}}$} \, \Bigr]
\; ,
\label{eq:functional_f1}
\end{align}
where
\begin{align}
 \delta G_{0\sigma'}^{\nu\nu'}(\omega')
&\,=\,
- \left[\,G_{0\sigma'}^r(\omega')- G_{0\sigma'}^a(\omega')\,\right]
\,\delta \! f_{\text{eff}}(\omega') \;.
\label{eq:delta_G0}
\end{align}
The functional derivative
 $\delta \mbox{\boldmath $\Sigma$}
 _{\sigma}/
\delta G_{0\sigma'}^{\nu\nu'}$ can be related
to the vertex corrections in the Keldysh formalism.
The functional aspect discussed here
is analogous to the functional approach
of Luttinger and Ward.\cite{LuttingerWard}
However, in eq.\ (\ref{eq:functional_f1}),
the functional derivative is taken with respect to
noninteracting Green's function.
 At finite temperatures $T \neq 0$
the distribution function $f_{\text{eff}}(\omega)$ dose not have
the discontinuities,
and thus it can be treated as a regular function in general discussions.
Nevertheless, the singularities appearing in the limit of $T \to 0$
play an important role, for instance,
as we see in eq.\ (\ref{eq:Ward_2_Im_diagram}).

\section{Fermi-Liquid Behavior at Small Voltages}
\label{sec:low-voltage}

In equilibrium and linear-response regime,
the low-energy properties at $\omega$, $T \ll T_K$
can be described by the local Fermi liquid theory,\cite{Nozieres}
where $T_K$ is the Kondo temperature.
The Fermi liquid theory can also describe
the nonlinear response at small bias-voltages $eV \ll T_K$.\cite{ao10}
Our proof uses the Ward identities\cite{YamadaYosida,Shiba,Yoshimori}
in the Keldysh formalism.
In this section, we describe the outline of the derivation
of the identities, and then determine the low-voltage behavior
of the differential conductance $dI/dV$ up to terms of order $(eV)^2$
in the electron-hole symmetric case.

\subsection{Ward identities}

We first of all consider the behavior of
 $\mbox{\boldmath $G$}_0(\omega)$ at small $eV$.
 The first derivative at $eV=0$ is written in the form
\begin{align}
  &
 \!\!\!\!
 \left.
  {\partial
 G_0^{\nu\nu'}(\omega)
 \over
 \partial ({\sl e}V)}
 \right|_{eV=0}
 = \,
  -\, \alpha
 \left(
 {\partial \over \partial \omega}
 + {\partial \over \partial E_d}
 \right)
 G_{0:\text{eq}}^{\nu\nu'}(\omega)\,,
  \label{eq:derivative_1b}
\end{align}
where
$\alpha \equiv (\alpha_L \Gamma_L - \alpha_R \Gamma_R)/
   (\Gamma_L+ \Gamma_R)$,
   and the label \lq\lq eq" in the subscript
   stands for the \lq\lq equilibrium", so that
 $
 G_{0:\text{eq}}^{\nu\nu'}
 \equiv
 \bigl.G_{0}^{\nu\nu'}\bigr|_{eV=0}$.
Owing to the properties of $f_{\text{eff}}(\omega)$,
the differential coefficient with respect to $eV$
can be related to the equilibrium quantities
in the right-hand side eq.\  (\ref{eq:derivative_1b}).
From the discussions in \S \ref{sec:model},
the self-energy can also be regarded a functional
of $\mbox{\boldmath $G$}_{0}(\omega)$.
Thus, the differential coefficients of
$\mbox{\boldmath $\Sigma$}(\omega)$ with respect to $eV$ can be
calculated taking the derivative
of $\mbox{\boldmath $G$}_{0}$'s appearing
in the perturbation series in $U$, as described in the appendix.
Then, using eq.\ (\ref{eq:derivative_1b}), we obtain
\begin{align}
\left.
{\partial
\mbox{\boldmath $\Sigma$}(\omega)
\over
\partial ({\sl e}V)}\right|_{eV=0}
 &= \
 -\,
 \alpha
\left(
{\partial \over \partial \omega}
+ {\partial \over \partial E_d}
\right)
\mbox{\boldmath $\Sigma$}_{\text{eq}}(\omega)
\;,
\label{eq:derivative_self_1}
\\
\left.
{\partial^2
\mbox{\boldmath $\Sigma$}(\omega)
\over
\partial ({\sl e}V)^2}\right|_{eV=0}
& \, =
\  \alpha^2
\left(
{\partial \over \partial \omega}
 + {\partial \over \partial E_d}
\right)^2
\mbox{\boldmath $\Sigma$}_{\text{eq}}(\omega)
 \nonumber \\
 & \quad \ \  + \,
{ \Gamma_L\,\Gamma_R
  \over \left( \Gamma_L+ \Gamma_R \right)^2}
   \  \widehat{D}^2
\mbox{\boldmath $\Sigma$}_{\text{eq}}(\omega)
\;,
\label{eq:derivative_self_2_with_CII}
\end{align}
where
$\mbox{\boldmath $\Sigma$}_{\text{eq}}(\omega)
\equiv \left. \mbox{\boldmath $\Sigma$}(\omega) \right|_{eV=0}$.
The operator $\widehat{D}^2$ acts on
the noninteracting Green's functions in the perturbation series
for $\mbox{\boldmath $\Sigma$}_{\text{eq}}(\omega)$,
and it takes the second derivative
$(\partial /\partial \omega' + \partial/ \partial E_d )^2$, as
\begin{align}
&   \widehat{D}^2
\mbox{\boldmath $\Sigma$}_{\text{eq},\sigma}(\omega)
 \   =
 \nonumber \\
& \quad  \sum_{\nu\nu'\sigma'}
\int \! {\text{d}\omega'}
{
\delta \mbox{\boldmath $\Sigma$}_{\text{eq},\sigma}(\omega)
\over
\delta G_{0:\text{eq},\sigma'}^{\nu\nu'}(\omega')
}
\left(
{ \partial
  \over
  \partial \omega' }
+{ \partial
  \over
  \partial E_d }
  \right)^2 G_{0:\text{eq},\sigma'}^{\nu\nu'}(\omega') .
\end{align}
Using these relations,
the low-bias behavior of the self-energy
can be deduced from the equilibrium quantities.

\begin{figure}[t]
\leavevmode
\hspace{0.5cm}
\begin{minipage}{0.72\linewidth}
\includegraphics[width=1.2\linewidth,
, clip, trim = 0cm 0cm 0cm 0cm]{vertex04x.eps}
\end{minipage}
\caption{The diagrams contribute to the singularities.}
\label{fig:vertex}
\end{figure}

Specifically, at $T =0$ and $eV = 0$,
the usual zero-temperature formalism
is applicable for the causal Green's function
defined with respect to the equilibrium ground state,
\begin{align}
G_{\rm eq}^{--}(\omega) \ =  \
 G_{\rm eq}^{r}(\omega) \, \theta(\omega)
\,+\, G_{\rm eq}^{a}(\omega)\, \theta(-\omega)
\;,
\end{align}
where $\theta (\omega)$ is the Heaviside step function.
Thus at $T=0$ the causal matrix-element
in the right-hand side of eq.\ (\ref{eq:derivative_self_1})
can be related to the vertex corrections,\cite{Yoshimori}
\begin{align}
& \left( {\partial \over \partial \omega}
+ {\partial \over \partial E_d} \right)
 \Sigma_{{\rm eq},\sigma}^{--}(\omega)
\ = \
\nonumber
\\
& \qquad  \qquad  \quad -
\sum_{\sigma'} \,
\Gamma_{\sigma\sigma';\sigma'\sigma}(\omega,0;0,\omega)
\,
A_{{\rm eq},\sigma'}(0)
\;,
\label{eq:Ward_1}
\end{align}
where $\Gamma_{\sigma\sigma';\sigma'\sigma}(\omega,\omega';\omega',\omega)$
is the vertex function for the causal Green's function
in the $T=0$ formalism, and
 $A_{{\rm eq},\sigma}(\omega)
= -\text{Im}\, G^{r}_{{\rm eq},\sigma}(\omega)/\pi$.
Similarly,
the causal element of $\widehat{D}^2
\mbox{\boldmath $\Sigma$}_{\text{eq},\sigma}$
can be written as\cite{ao10}
\begin{align}
& \widehat{D}^2  \Sigma_{{\rm eq},\sigma}^{--}(\omega)
\ =
\nonumber
\\
& \quad
\sum_{\sigma'}
{\partial \over \partial \omega'}\,
\Bigl.
\Gamma_{\sigma\sigma';\sigma'\sigma}(\omega,\omega';\omega',\omega)
\,
A_{{\rm eq},\sigma'}(\omega')
 \,\Bigr|_{\omega'=0} \;.
\label{eq:D2_T0}
\end{align}

Eliashberg\cite{Eliashberg} has shown quite generally
by using the Lehmann representation that
the imaginary part of the vertex function has
some singularities.\cite{Shiba,Yoshimori,ao11}
 For small frequencies, the singularities
relevant to eqs.\ (\ref{eq:Ward_1}) and (\ref{eq:D2_T0})
 arise from the diagrams shown in Fig.\ \ref{fig:vertex}.
The asymptotic form of the imaginary part of eq.\ (\ref{eq:D2_T0})
at small $\omega$ and $\omega'$ can be calculated as\cite{ao10}
\begin{align}
&   \ \
\sum_{\sigma'}
{\partial \over \partial \omega'}
\, \mbox{Im}\,
\Gamma_{\sigma\sigma';\sigma'\sigma}(\omega,\omega';\omega',\omega)
\nonumber \\
& =
-\,
\left|\Gamma_{\uparrow\downarrow;\downarrow\uparrow}(0,0;0,0)
\right|^2
\nonumber \\
& \quad
\times \, \mbox{Im} \Biggl[ \,
2 \int {d \omega'' \over 2\pi \text{i}}\,
G_{\rm eq}^{--}(\omega'') \,
{\partial  \over \partial \omega'} G_{\rm eq}^{--}(\omega -\omega'+ \omega'')
\nonumber \\
& \qquad \qquad
+ \int {d \omega'' \over 2\pi \text{i}}\,
G_{\rm eq}^{--}(\omega'') \,
{\partial  \over \partial \omega'} G_{\rm eq}^{--}(\omega +\omega'-\omega'')
\,\Biggr]
\nonumber
\\
& = \,
-\,
\pi \left\{A_{\rm eq}(0)\right\}^2\, \left|
  \Gamma_{\uparrow\downarrow;\downarrow\uparrow}(0,0;0,0)\right|^2 \,
\nonumber\\
& \qquad \times
\Bigl[\,
- 2\,\mbox{sgn}(\omega'-\omega) \,
+ \,\mbox{sgn}(\omega' +\omega) \,
\Bigr] \;.
\label{eq:Ward_2_Im_diagram}
\end{align}
Here the first term in the last line corresponds to the
contributions of the diagram (a) and (b),
and the second term corresponds to that of the diagram (c).
Due to these singularities,
the value of eq.\ (\ref{eq:Ward_2_Im_diagram})
in the limit of $\omega,\, \omega' \to 0$ depends
on which frequency is taken first to be zero.
Taking the limit for small frequencies carefully,
the low-energy behavior of $\mbox{Im}\, \Sigma^r(\omega)$
are determined up to terms of order $\omega^2$, $(eV)^2$, and $T^2$,
\begin{align}
&\mbox{Im}\, \Sigma^r(\omega)
  =
   \, -\,
  { \pi \over 2 } \left\{A_{\rm eq}(0)\right\}^3\, \left|
  \Gamma_{\uparrow\downarrow;\downarrow\uparrow}(0,0;0,0)\right|^2
\,
\nonumber \\
 &  \times
        \left[\,
            \left(\,\omega -
              \alpha\, {\sl e}V\,
              \right)^2
              +  { 3\,\Gamma_L \Gamma_R
                 \over \left( \Gamma_L + \Gamma_R \right)^2}
                \,(eV)^2
              +(\pi T)^2
           \,\right] .
\label{eq:self_imaginary}
\end{align}
The result at equilibrium $eV=0$ has been
provided by Yamada and Yosida,\cite{YamadaYosida}
and it is extended to the nonequilibrium steady state here
up to terms of order $(eV)^2$.
Note that we have not assumed the electron-hole symmetry so far.

\subsection{Results in the electron-hole symmetric case}

In this subsection we consider the low-energy behavior
of $G^r(\omega)$  and $dI/dV$  using
the result of $\mbox{Im}\, \Sigma^r(\omega)$
obtained in eq.\ (\ref{eq:self_imaginary}).
Specifically, we concentrate on
the electron-hole symmetric case, where
$\epsilon_d = -U/2$, $\Gamma_L = \Gamma_R$, and $\alpha_L=\alpha_R=1/2$.
 In this case $A_{\rm eq}(0) = 1/(\pi\Delta)$,
and the real part of the self-energy takes the form
\begin{align}
\mbox{Re}\, \Sigma^r(\omega) &\,= \,\left( 1 - z^{-1} \right)
 \omega \, + \, O(\omega^3) \;,
\label{eq:re_S}
\\
z &\,\equiv \,
\left(\,
1- \left.{\partial \Sigma_{\rm eq}^r(\omega) \over \partial \omega }
\right|_{\omega=0} \,\right)^{-1} \;.
\label{eq:wave_remorm}
\end{align}
Thus,
 $G^r(\omega)$ can be calculated
exactly up to terms of order $\omega^2$, $T^2$ and $(eV)^2$
 using eq.\ (\ref{eq:self_imaginary}),
\begin{align}
G^r(\omega) \simeq
{z \over
\omega + \text{i}\, \widetilde{\Delta}
      +\,\text{i}\,
 {\displaystyle \mathstrut \widetilde{U}^2
 \over \displaystyle
 2 \widetilde{\Delta}
 \rule{0cm}{0.4cm} \mathstrut
       (\pi \widetilde{\Delta})^2}
\left[ \, \omega^2 +
{\displaystyle \mathstrut 3  \over \displaystyle \mathstrut 4}
\,({\sl e}V)^2 +(\pi T)^2
       \right]
       }         ,
 \label{eq:Gr_symmetric_case}
\end{align}
where the renormalized parameters are defined by
\begin{align}
\widetilde{\Delta}\, &\equiv \,z \Delta \;, \qquad
\widetilde{U} \, \equiv\,  z^2\,
\Gamma_{\uparrow\downarrow;\downarrow\uparrow}(0,0;0,0) \;.
\label{eq:renorm_parm}
\end{align}
The  order $U^2$ result of Hershfield {\em et al.}\cite{HDW2}
can be reproduced from eq.\ (\ref{eq:Gr_symmetric_case})
replacing $\widetilde{U}$ by the bare Coulomb interaction $U$,
and using the order $U^2$ result for
the renormalization factor\cite{YamadaYosida,ZlaticHorvatic}
$z = 1 - (3-\pi^2/4) \,u^2 + \cdots$, where $u= U/(\pi \Delta)$.

Thus, in the symmetric case
the low-voltage behavior
is characterized by the two parameters $\widetilde{\Delta}$
and $\widetilde{U}$.
These parameters
are defined with respect to the equilibrium ground state,
for which the exact Bethe ansatz results exist
\cite{KawakamiOkiji,WiegmanTsvelick,ZlaticHorvatic}
as shown in Fig.\ \ref{fig:Bethe}.
The width of the Kondo resonance $\widetilde{\Delta}$ decreases
with increasing $U$, and
for $u \gtrsim 2.0$ it is approximated well
by the asymptotic form $\widetilde{\Delta} \simeq (4/\pi) T_K$,
where the Kondo temperature is defined by
\begin{align}
T_K \, = \, \pi \Delta \sqrt{u / (2 \pi)}\, \exp[-\pi^2 u/8 + 1/(2u)] \;.
\label{eq:T_K}
\end{align}
The Wilson ratio is usually defined by
$R \equiv \widetilde{\chi}_s/\widetilde{\gamma}$,
where $\widetilde{\gamma}$ and $\widetilde{\chi}_s$ are
the enhancement factors for the $T$-linear specific heat
and spin susceptibility,  respectively.\cite{YamadaYosida}
Alternatively, it can be written in terms
of $\widetilde{\Delta}$ and $\widetilde{U}$, as\cite{HewsonRPT}
\begin{align}
R -1 \, = \,  \widetilde{U}/(\pi \widetilde{\Delta}) \;.
\label{eq:wilson_vs_renorm}
\end{align}
The Wilson ratio increases with $u$ from the noninteracting value
$R=1$ to the strong-coupling limit value $R = 2$.
The charge excitations at the impurity site
are still surviving for $u \lesssim 2.0$,
and it makes the value of $R$ smaller than $2$.

The nonequilibrium current $I$ can be calculated from
the retarded Green's function,\cite{MW}
\begin{align}
 I\,  &  =\,  {2 {\sl e} \over h}  \int_{-\infty}^{\infty}
 \! \text{d}\omega
   \left[\, f_L - f_R \, \right]
\frac{4\, \Gamma_L \Gamma_R}{\Gamma_R + \Gamma_L}
 \left[\, - {\rm Im}\, G^r(\omega) \,\right] .
\label{eq:caroli}
\end{align}
Substituting
eq.\
(\ref{eq:Gr_symmetric_case})
into eq.\ (\ref{eq:caroli}),
the differential conductance $dI/dV$ can be determined exactly
up to terms of order $T^2$ and $(eV)^2$,
\begin{align}
& {dI \over dV}
\ = \ {2 e^2 \over h}
  \Biggl[\, 1 \,
  -
                \, { 1
                  + 2\, (R-1)^2
                  \over 3}
                 \left( {\pi T \over \widetilde{\Delta}}\right)^2
   \nonumber \\
 &  \qquad \qquad \qquad \;
 \,
      -
                \, { 1
                  + 5\, (R-1)^2
                  \over 4}
                 \left( {eV \over \widetilde{\Delta} } \right)^2
 + \cdots
 \,\Biggr] .
\label{eq:dI_dV}
\end{align}
The result shows that the nonlinear $(eV)^2$ term is also scaled
by the resonance width $\widetilde{\Delta}$,
and the coefficient generally depends on
the parameter $(R-1)^2$, or $\widetilde{U}^2/(\pi \widetilde{\Delta})^2$.
As mentioned, in the strong-coupling limit $u \to \infty$,
the two characteristic parameters
become $\widetilde{\Delta} \to (4/\pi)\, T_K$ and
$R \to 2$.

\begin{figure}[t]
\leavevmode
\begin{center}
\begin{minipage}{0.9\linewidth}
\includegraphics[width=1\linewidth]{wilson5.eps}
\end{minipage}
\caption{The $U$ dependence of
$\widetilde{\Delta}/\Delta$
and
$R-1$,
which can also be interpreted as
 $z$ and $\widetilde{U}/(\pi\widetilde{\Delta})$, respectively.
These parameters were calculated using
the Bethe ansatz solution summarized in ref.\ \citen{ZlaticHorvatic}. }
\label{fig:Bethe}
\end{center}
\end{figure}

\subsection{Comparison with other approaches }

To our knowledge, similar attempts to calculate
the coefficient $c_V$ of the $(eV)^2$ term of $dI/dV$ have been made
by two groups\cite{KNG,KSL} in the strong-coupling limit $u \to \infty$:
\begin{align}
\frac{dI}{dV} \,= \,
\frac{2e^2}{h} \Biggl[\, 1 - c_T \left(\frac{\pi T}{T_K}\right)^2
-  c_V \left(\frac{eV}{T_K}\right)^2 + \cdots \Biggr] .
\end{align}
To compare the results, the difference in a numerical factor
of order $1$ in the definition $T_K$ must be
taken into account. To avoid this uncertainty,
we use $T_K$ defined in eq.\ (\ref{eq:T_K}), and rescale
the results presented in refs.\ \citen{KNG} and \citen{KSL}
such that the coefficient for the linear-response $T^2$ term
agrees with the result of Yamada and Yosida $c_T = (\pi/4)^2$.
Kaminski, Nazarov, and Glazman\cite{KNG} have carried out
a perturbation expansion
around the strong-coupling fixed point to
obtain $c_V^{\rm KNG} = (3/8)\,c_T$.
 Konik, Saleur, and Ludwig\cite{KSL} have used
 the equilibrium Bethe ansatz solution,
and then made some extra assumptions for calculating
the nonlinear coefficient to obtain $c_V^{\rm KSL} = 4\, c_T$
(the parameters corresponding to $\alpha_L$ and  $\alpha_R$
used by KSL seem to be different from ours).
Our result eq.\ (\ref{eq:dI_dV}),
which is obtained using the Ward identities,
shows $c_V^{\rm Ward} = (3/2) \,c_T$ in the strong-coupling limit.

Although the Hamiltonian is somewhat different,
we also note for comparison that
Schiller and Hershfield\cite{Schiller} obtained the result
corresponding to $c_V^{\rm SH} = 3\,c_T$ for a special
parameter set which can be related to the Emery-Kivelson solution of
the two-channel Kondo model.

\section{Renormalized Perturbation Theory at Finite Bias Voltages}
\label{sec:RPT}

Although the description of the low-energy properties
discussed in \S \ref{sec:low-voltage} is exact,
the underlying physics of the quasiparticles
might not be seen directly in the microscopic derivation.
In the case of the three-dimensional Fermi liquid,
the vertex function played a central role to clarify
a link between the intuitive picture of the quasiparticles
and Green's functions.
Specifically,
the residual interaction between two quasiparticles,
which had been introduced phenomenologically,
was shown to be connected to the forward scattering amplitude.\cite{AGD}

For the Anderson impurity the vertex function at Fermi energy,
 $z^2 \Gamma_{\uparrow\downarrow;\downarrow\uparrow}(0,0;0,0)$,
corresponds to the scatter amplitude,
and it is equal to $\widetilde{U}$ by the
 definition in eq.\ (\ref{eq:renorm_parm}).
The perturbation expansion in $\widetilde{U}$,
which has been formulated precisely
in the equilibrium case by Hewson,\cite{HewsonRPT}
provides the link between the quasiparticles
and microscopic theory of the local Fermi liquid.
All the basic Fermi-liquid behavior have been shown to be
reproduced in the expansion up to terms of
order $\widetilde{U}^2$. Furthermore,
the approach is not limited to low energies.
To carry out the expansion systematically, however,
one has to take account of the renormalization conditions
that are necessary to avoid overcounting of the many-body effects,
because in the renormalized perturbation theory (RPT)
the expansion parameter already contains some
contributions of the Coulomb interaction.

In this section we apply  the RPT to the nonequilibrium steady state.
It reproduces the result of $dI/dV$ in the Fermi-liquid regime,
and gives us one possible way to calculate the corrections needed
at high voltages.
For simplicity, we concentrate on the electron-hole symmetric case;
$\epsilon_d = -U/2$, $\Gamma_L = \Gamma_R$, and $\alpha_L=\alpha_R=1/2$.
The unperturbed Green's function is defined
such that it describes the Kondo resonance with
the renormalized level width
$ \widetilde{G}_0^{r}(\omega) = \bigl[\,\omega
  + \text{i} \widetilde{\Delta}\,\bigr]^{-1}$,
  as that in the equilibrium case.\cite{HewsonRPT}
However,
in the nonequilibrium case,
it is not obvious how the distribution function
for the free quasiparticles should be given by.
We simply assume here that it is given by
the noninteracting one, which in the electron-hole symmetric case
takes the form
$f_{\rm eff}(\omega)=\left[\, f(\omega-eV/2)
+ f(\omega+eV/2) \,\right]/2$.
Hence, the four elements of the unperturbed Green's functions,
$\widetilde{\mbox{\boldmath $G$}}_{0}$,
take the forms
\begin{align}
 &\!
 \widetilde{G}_0^{--}(\omega) =
 \bigl[ 1-f_{\text{eff}}(\omega) \bigr]\,
     \widetilde{G}_0^r(\omega)
      +   f_{\text{eff}}(\omega)\,
      \widetilde{G}_0^a(\omega) ,
\label{eq:G_^--}
\\
&\!
\widetilde{G}_0^{-+}(\omega) =
-\,f_{\text{eff}}(\omega)
        \left[\,\widetilde{G}_0^r(\omega)-
        \widetilde{G}_0^a(\omega)\,\right] ,
\label{eq:G_^-+}
\\
&\!
\widetilde{G}_0^{+-}(\omega) =
\bigl[ 1-f_{\text{eff}}(\omega) \bigr]
        \left[\,\widetilde{G}_0^r(\omega)
        - \widetilde{G}_0^a(\omega)\,\right] ,
\label{eq:G_^+-}
\\
 & \!
 \widetilde{G}_0^{++}(\omega) =
 -\bigl[ 1-f_{\text{eff}}(\omega) \bigr]\,
     \widetilde{G}_0^a(\omega)
      -   f_{\text{eff}}(\omega) \,\widetilde{G}_0^r(\omega) \;.
\label{eq:G_^++}
\end{align}
Correspondingly,
the full propagator of the quasiparticles,
which includes all effects of $\widetilde{U}$,
is defined by $\widetilde{\mbox{\boldmath $G$}}(\omega) \equiv z^{-1}
\mbox{\boldmath $G$}(\omega)$.
Therefore, in terms of the renormalized quantities,
the nonequilibrium current, eq.\ (\ref{eq:caroli}),
is written as
\begin{align}
 I\,  &  =\,  {2 {\sl e} \over h}  \int_{-\infty}^{\infty}
 \! \text{d}\omega
    \left[\, f_L - f_R \, \right]
 \left[\, -\,\widetilde{\Delta}\,
 {\rm Im}\, \widetilde{G}^r(\omega) \,\right] \;.
\label{eq:caroli_ren}
\end{align}
The self-energy correction due
to $\widetilde{U}$ satisfies the Dyson equation of the form
$ \widetilde{\mbox{\boldmath $\Sigma$}}(\omega)\,
    \equiv
 \{\widetilde{\mbox{\boldmath $G$}}_{0}(\omega)\}^{-1}
 -
\{\widetilde{\mbox{\boldmath $G$}}(\omega)\}^{-1}$.
Using eq.\ (\ref{eq:Dyson}), we have
\begin{align}
\widetilde{\mbox{\boldmath $\Sigma$}}(\omega)\,
&\, =
 \{\widetilde{\mbox{\boldmath $G$}}_{0}(\omega)\}^{-1}
\, - \, z \left[\,\{\mbox{\boldmath $G$}_{0}(\omega)\}^{-1}
-\mbox{\boldmath $\Sigma$}(\omega)\,\right]
\nonumber \\
&\, = \,
   z\,\mbox{\boldmath $\Sigma$}(\omega)
 \, + \, (1 - z )  \mbox{\boldmath $\tau$}_3 \,\omega
\;,
\label{eq:self_ren}
\end{align}
where
$\mbox{\boldmath $\tau$}_i$ for $i=1,2,3\,$ is the Pauli matrix.

\begin{figure}[tb]
\leavevmode
\begin{center}
\begin{minipage}{0.3\linewidth}
\includegraphics[width=1.0\linewidth,
clip, trim = 0cm 0cm 0cm 0cm]{2ndS2004z.eps}
\end{minipage}
\caption{
The second-order diagram of $\widetilde{\mbox{\boldmath $\Sigma$}}$.
The dashed represents $\widetilde{U}$,
and solid lined represents
the free-quasiparticle propagator
$\widetilde{\mbox{\boldmath $G$}}_{0}$.
}
\label{fig:2nd_graph}
\end{center}
\end{figure}

\subsection{Low-energy behavior up to terms of order $\omega^2$, $T^2$
and $(eV)^2$}

At low-energies,  the $\omega$-linear contributions
in the right-hand side of eq.\ (\ref{eq:self_ren})
cancel out owing to eq.\ (\ref{eq:re_S}).
The contributions of order $\omega^2$, $T^2$ and $(eV)^2$
arise from the second-order diagram
for $\widetilde{\mbox{\boldmath $\Sigma$}}(\omega)$
shown in Fig.\ \ref{fig:2nd_graph},
where the solid and dotted lines represent
 $\widetilde{\mbox{\boldmath $G$}}_{0}(\omega)$ and $\widetilde{U}$,
 respectively. Calculating the contributions from the diagram,
 and then taking the cancellation of the $\omega$-linear
 term into account,
 we have
\begin{align}
\widetilde{\Sigma}^r(\omega)
\,= \, -\, \text{i}\,
  {\displaystyle \mathstrut \widetilde{U}^2
 \over \displaystyle
 2 \widetilde{\Delta}
 \rule{0cm}{0.4cm} \mathstrut
       (\pi \widetilde{\Delta})^2}
\left[ \, \omega^2 +
{\displaystyle \mathstrut 3  \over \displaystyle \mathstrut 4}
\,({\sl e}V)^2 +(\pi T)^2
       \right] \,+\, \cdots .
\label{eq:self_U2ren}
\end{align}
It simply reproduces the renormalized Green's function
 $\widetilde {G}^r(\omega)$ corresponding
 to eq.\ (\ref{eq:Gr_symmetric_case}).
Furthermore, using eq.\ (\ref{eq:caroli_ren}),
the $(eV)^2$ term of $dI/dV$ in eq.\ (\ref{eq:dI_dV})
is also reproduced in the expansion up to terms
of order $\widetilde{U}^2$.
Since eqs.\ (\ref{eq:Gr_symmetric_case}) is asymptotically exact,
the higher-order terms in $\widetilde{U}$ do not
change the low-energy behavior in eq.\ (\ref{eq:self_U2ren}).

Note that eq.\ (\ref{eq:self_U2ren}) follows from
the fact that we have used $f_{\text{eff}}(\omega)$ for
the distribution function of the free quasiparticles.
This assumption seems to be justified also from
the fact that the many-body effects on
 the correlated distribution $f_{\text{eff}}^U(\omega)$
defined in eq.\ (\ref{eq:f_int}) arise first in
the order $U^2$ contribution.

\subsection{Beyond the Fermi-liquid regime}

To study the higher-energy behavior
at large $\omega$, $T$, and $eV$ with the RPT, one needs to calculate
the higher order terms in $\widetilde{U}$.
In the following, we describe the outline of the procedure of the expansion.

At high energies the renormalization factor $z$ cannot be defined
 with respect to $T=0$ and $eV=0$ no longer.
 This is because the coefficient of the $\omega$-linear term
of the self-energy  depends on $T$ and $eV$.
 For instance, in the next order, the terms of
the form $T^2\omega$ and $(eV)^2\omega$ exist.
Therefore,  $z$ is redefined such that
the $\omega$-linear contributions in eq.\ (\ref{eq:self_ren})
cancel out
\begin{align}
\biggl.
{
\partial
\widetilde{\Sigma}^r(\omega) \over
\partial
\omega }
\biggr|_{\omega=0} \,=\,0 \;.
\label{eq:norm_condition1}
\end{align}
Hence $z$ generally depends on  $T$ and $eV$.
The perturbation expansion in $\widetilde{U}$ can be carried out
following that in the equilibrium case.\cite{HewsonRPT}
We first of all rewrite the action $S$  in the form
\begin{align}
S\, &= \, z^{-1} \widetilde{S}_0
\,+ \, z^{-2} S_U  \widetilde{U}/\,U
\,-\, S_{\rm cou}
\label{eq:action_rewritten}
\\
S_{\rm cou} &= \,
\lambda\, z^{-2} S_U \widetilde{U} /\,U
\, + \,
 z^{-1} \widetilde{S}_0 - S_0
\;,
\label{eq:action_counter}
\end{align}
where $\widetilde{S}_0$ is the action for
the free quasiparticle
corresponding to
the propagator $\widetilde{\mbox{\boldmath $G$}}_{0}(\omega)$, and
$\lambda \equiv 1 - z^2\,U/\widetilde{U}$.
In eqs.\ (\ref{eq:action_rewritten})
and (\ref{eq:action_counter})
the factor $1/U$ is introduced just to cancel
the bare Coulomb interaction $U$ included in $S_U$ by
the definition in eq.\ (\ref{eq:action_U}).
The perturbation series in $\widetilde{U}$ is generated
by taking  $z^{-1}\widetilde{S}_0$ in eq.\ (\ref{eq:action_rewritten})
to be the unperturbed part
and taking the remaining terms
$ z^{-2} S_U  \widetilde{U}/U - S_{\rm cou}$ to be the perturbed part.
Here, $S_{\rm cou}$ is
the counter-term which avoids overcounting of the many-body effects.
Specifically, the last two terms in the right-hand side of
eq.\ (\ref{eq:action_counter}), which can be rewritten in the form
 \begin{align}
 z^{-1}\widetilde{S}_0  - S_0
 &=   \left(z^{-1} - 1\right)
    \sum_{\sigma} \int_{-\infty}^{\infty} \!\! \text{d}\omega\,
 \mbox{\boldmath $\eta$}_{\sigma}^{\dagger}(\omega)
 \mbox{\boldmath $\tau$}_3 \,\omega\,
 \mbox{\boldmath $\eta$}_{\sigma}(\omega) \,.
 \end{align}
It corresponds to the counter-term for the renormalization factor $z$.
In the RPT, the two parameters $z$ and $\lambda$ are regarded as
functions of the renormalized parameters
 $\widetilde{\Delta}$ and $\widetilde{U}$,
and are expanded as series in the powers of $\widetilde{U}$.
Then the expansion coefficients for $z$ and $\lambda$
can be determined such that the two renormalization conditions,
eqs.\ (\ref{eq:norm_condition1}) and (\ref{eq:norm_condition2}),
are satisfied by each order in $\widetilde{U}$;
\begin{align}
\widetilde{\Gamma}_{\uparrow\downarrow;\downarrow\uparrow}(0,0;0,0)
\,= \,\widetilde{U} \;.
\label{eq:norm_condition2}
\end{align}
Here  $\widetilde{\Gamma}_{\uparrow\downarrow;\downarrow\uparrow}
\equiv z^2 \,\Gamma_{\uparrow\downarrow;\downarrow\uparrow}
$ is the vertex part for the four external causal Green's
functions $\widetilde{G}^{--}$.
In the RPT,
$\widetilde{\Gamma}_{\uparrow\downarrow;\downarrow\uparrow}(0,0;0,0)$
is calculated in the power series in $\widetilde{U}$,
and at high-energies it generally depends on $T$ and $eV$.
Note that the contribution of the parameter $\lambda$ first arises
in the order $\widetilde{U}^3$ terms.\cite{HewsonRPT}
For this reason, the condition
of $\lambda$ is not necessary to be taken into account in the expansion
up to order $\widetilde{U}^2$.

As already mentioned, higher-order terms in $\widetilde{U}$
are needed to study the high-energy behavior of
 $\widetilde{\Sigma}^r(\omega)$ and $dI/dV$
beyond the $\omega^2$- and $(eV)^2$-terms.
One possibility is to include the contributions
up to terms of order $\widetilde{U}^4$.
The corresponding calculations in the bare-$U$ expansion
have been carried out by Fujii and Ueda.\cite{FujiiUeda}
Alternatively, in the equilibrium case at $eV=0$,
a combination of the RPT and NRG has been examined recently,
and the results reproduce the $T$-dependence of
the spin susceptibility accurately
in a wide temperature range.\cite{HewsonNRG}
Such a combination would be another possibility to go beyond
the Fermi-liquid regime at large bias voltages.

\section{Summary}
\label{sec:SUMMARY}

 We have studied the low-energy properties of
 the Anderson model under a finite bias voltage $V$
 using the properties of
 the Keldysh Green's function at the impurity site
  $\mbox{\boldmath $G$}(\omega)$
 as a functional of the nonequilibrium distribution
 function $f_{\text{eff}}(\omega)$.
 Through the distribution function $f_{\text{eff}}(\omega)$,
 the $T$- and $eV$-dependence of $\mbox{\boldmath $G$}(\omega)$ arise.
  The Ward identities for the derivative of the self-energy
  with respect to $eV$ follow from these properties
  that can be summarized in the form of eq.\ (\ref{eq:functional_f1}).
 Using the Ward identities,
 the differential conductance $dI/dV$  has been
 determined up to terms of order $(eV)^2$ in eq.\ (\ref{eq:dI_dV})
 in the electron-hole symmetric case.
 The coefficients are determined by
 two characteristic parameters $\widetilde{\Delta}$ and $R$.

 We have also described the low-energy properties using
 the renormalized perturbation theory in the Keldysh formalism.
  To second order in $\widetilde{U}$,
 it reproduces the exact $(eV)^2$ coefficients for $dI/dV$.
 The Fermi-liquid behavior
 of $dI/dV$ follows from the assumption that the distribution function for
 the free quasiparticles are the same as that of the noninteracting
 electrons $f_{\text{eff}}(\omega)$.
 In order to study the corrections to the Fermi liquid theory
 at large bias voltages with the RPT,
 one needs to calculate the higher order terms in $\widetilde{U}$.

\section*{Acknowledgements}

I wish to thank A. C. Hewson for helpful discussions on
the renormalized perturbation theory.
This work was supported
by the Grant-in-Aid for Scientific Research from JSPS.