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\preprint{CFNUL/02-01}

\title{Leptogenesis in a prompt decay scenario}% Force line breaks with \\

\author{Lu{\'{\i}}s Bento}
%\author{Lus Bento}
\email{lbento@fc.ul.pt}
%\thanks{cccc}
%\altaffiliation[Also at ]{Physics Department, XYZ University.}%Lines break automatically or can be forced with \\
%%\author{Second Author}%
%%\email{Second.Author@institution.edu}
\affiliation{%
Centro de F\'{\i}sica Nuclear da Universidade de Lisboa, \\
Avenida Prof.\ Gama Pinto 2, 
1649-003 Lisboa, Portugal
%%This line break forced with \textbackslash\textbackslash
}%

%%\author{Charlie Author}
%% \homepage{http://www.Second.institution.edu/~Charlie.Author}
%%\affiliation{
%%Second institution and/or address\\
%T%his line break forced% with \\
%%}%

\date{October, 2002}% It is always \today, today,
             %  but any date may be explicitly specified

\begin{abstract}
Leptogenesis is studied within the seesaw neutrino mass model in a regime where all sterile neutrinos have prompt rather than delayed decays.
Leptogenesis occurs during thermal production of ultrarelativistic neutrinos in Higgs boson decays.
The large $B-L$ asymmetry is slowly pumped into the chemically decoupled right-handed quarks and leptons, most notably, $b_R$, which later protect $B-L$ 
from fast $L$ violating processes.
The scale of the final baryon asymmetry is controlled by the sum of light neutrino square masses, $\munu$.
For $\munu$ between the atmospheric neutrino mass gap and $0.1 \evs$, it naturally agrees with the observed baryon number.
\end{abstract}

%\pacs{98.80.Cq}% PACS, the Physics and Astronomy
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\maketitle


Leptogenesis is an atractive way of generating the baryon number of the 
Universe~\cite{Buchmuller00}. 
The main idea put forward by Fukugita and Yanagida~\cite{Fukugita86}
is that a lepton number asymmetry can be produced in the decays of heavy sterile neutrinos into leptons and Higgs bosons and such an asymmetry 
%Then, part of the $B-L$ number so generated 
is partially transferred to the baryon sector through electroweak sphaleron processes~\cite{Kuzmin85}
that violate $B$ and $L$ but not $B-L$.
The mechanism requires lepton number and $CP$ violation, ensured by 
%the Majorana masses of the heavy sterile neutrinos and their 
neutrino Majorana masses and
complex Yukawa couplings.
% with the standard model lepton doublets. 
Both masses and couplings form the well known 
seesaw model~\cite{Gellmann79}
of light neutrino masses and thus establish a close relation between baryogenesis and 
low energy phenomenology.
The third Sakharov condition~\cite{Sakharov67}, 
the departure from thermal equilibrium, is satisfied in the delayed decay scenario.
It means here that the Majorana neutrinos live longer than the age of the Universe when they approach non-relativistic temperatures and their production is Boltzmann suppressed.
It turns out that such condition puts an upper bound on the natural order of magnitude of the light neutrino masses~\cite{Fischler91,Buchmuller00}
well below the mass scale suggested by the atmospheric neutrino 
anomaly~\cite{skatm00}.

The seesaw mechanism supplements the standard model with singlet neutrinos,
$N_a$,
with heavy Majorana masses and  Yukawa couplings to the standard lepton and Higgs doublets,
$l_i$, $\phi$, of the form
\be{Yuk} 
h_{ia}l_i N_a \phi + \frac{1}{2} M_{a} N_a N_a  
+  {\rm H.C.}  \; .
\eeq
%%\end{equation} 
Spontaneous breaking of SU(2) $\times$ U(1) yields the light neutrino mass matrix
($v= \langle \phi^0 \rangle$)
\be{mij}
m_{ij} =- (h M^{-1} h^{T} )_{ij} \, v^{2}
			\; .
\eeq
%where $v= \langle \phi^0 \rangle$.
The proper decay rate of $N_a$ into leptons and Higgs is
%$N_a \rightarrow l\phi,\bar{l},\bar{\phi}$ 
% $N_a \rightarrow l\phi,\bar{l},\bar{\phi}$:
$\Gamma_a = (h ^{\dagger} h)_{aa} M_a /8\pi$.
Delayed decay occurs when the ratio to the Hubble expansion rate, 
$K_a = \Gamma_a /H$,
is small at the temperature $T=M_a$.
In the radiation era, 
$H = 1.66\, g_\ast^{1/2}T^2/M_{P}$, 
where the number of relativistic degrees 
of freedom, $g_\ast$, equals 107.5 in the standard model.
It is enough to compare the sum
%%$K =\sum K_a$,
\be{k}
K = \sum K_a =
 (10^3 \; \mathrm{ eV}^{-1} )
  \tr [  h M^{-1} h ^{\dagger} ] v^2 
\end{equation}
with the light neutrino scale 
$\tr [ m ]$
to conclude that the delayed decay condition, $K_a <1$, is in general in conflict with the atmospheric neutrino mass gap~\cite{skatm00},
$\Delta m^2 \approx 3 \cdot 10^{-3} \evs$,
which implies $K > 50$. 
Strictly speaking, the delayed decay scenario only requires that the lightest of the heavy neutrinos satisfies $K_a <1$, and larger $K_a$ values are tolerated for a certain choice of parameters that maximize the $CP$ 
asymmetry~\cite{davidson02,Buchmuller02}. 
But that is not the most general picture and this sort of tuned hierarchy is certainly not possible if the light neutrinos are 
quasi-degenerate~\cite{davidson02,Buchmuller02}.

As the seesaw provides a simple and elegant neutrino mass mechanism, it would be nice to have a successfull leptogenesis in the case where all heavy neutrinos have prompt rather than delayed decays. 
That is the subject of our paper. 
As turns out, there is a simple and mild modification of the original leptogenesis mechanism that gets the job done.

Given that neutrino Majorana masses and Yukawa couplings provide the necessary $B-L$ and $CP$ violation the first concern is the out-of-equilibrium condition.
According to the inflation paradigm~\cite{Kolb90}, 
the radiation era starts when a scalar field, so-called inflaton, decays and transforms its energy into the form of particles so constituting the initial thermal bath. 
If sterile neutrinos do not couple to the inflaton,
 they can only be thermally produced from standard lepton and Higgs particles.
These processes are naturally out-of-equilibrium at very high  temperatures and it takes some time before sterile neutrinos reach thermal equilibrium densities.
During that period there is a clear departure from equilibrium and a $B-L$ asymmetry can be generated.
This has been observed, actually, in numerical calculations~\cite{Buchmuller00}.

The second key issue is how to protect $B-L$.
In the prompt decay scenario $B-L$ is rapidly violated when the temperature falls to the Majorana mass scale and is completely washed out, unless there is some chemically decoupled sector of particles capable of storing a fraction of the asymmetry.
Such decoupled sector exists in the standard model,
the  right-handed leptons, $e_i = e_R, \mu_R, \tau_R$.
Their capacity of protecting $B-L$ has been recognized~\cite{Cline93}
and results from the fact that the Yukawa mediated transitions become effective considerably late in time,
%at flavor dependent temperatures, 
depending on the flavor
well after the possible lepto-baryogenesis and/or subsequent washout periods.
Another question is how to generate $e_i$ asymmetries.
Right-handed quarks, with exception of the top, also have weak Yukawa couplings. Yet, they can be produced out of left-handed quarks through QCD instantons.
These, however, are universal in flavor and conserve the differences between the baryon asymmetries of any two right-handed quarks.
Such differences are violated by Yukawa processes only and are thus protected from washout effects.

In our scenario a lepton asymmetry is generated in the lepton doublet and sterile neutrino sectors.
A fraction of it is transferred to the bottom $b_R$, tau $\tau_R$, charm $c_R$ and other right-handed fermions through slow Yukawa interactions.
It is crucial for the survival of the $b_R$, $\tau_R$, or $c_R$
% any other flavor asymmetry
asymmetries 
that the heavy Majorana neutrinos decay and vanish from the Universe before $b_R$, $\tau_R$ ($c_R$) enter in chemical equilibrium at a temperature around $10^{12} \gev$ ($10^{11}$ GeV).
By that time $\Delta L=2$ reactions mediated by off-shell Majorana neutrinos should be also out-of-equilibrium.
Their rate relates to the sum of light neutrino square masses
$\munu =\tr [ m m^\dagger ] $ as
$\Gamma_2 \approx  \munu T^3 /(\pi^3 v^{4})$~\cite{Fukugita90}.
They are out-of-equilibrium, i.e., 
$\Gamma_2 /H < 1$ at $T\leq 10^{12}- 10^{11} \gev$
% ($T\leq 10^{11} \gev$), 
for masses $\munu < 0.04 - 0.4 \evs$,
% ($\munu < 0.4  \evs$),
well above the atmospheric neutrino mass gap.
%$\Delta m^2 \approx 3 \cdot 10 ^{-3} \evs$.

Let us now examine the leptogenesis processes in detail.
We assume that the Universe is initially empty of sterile neutrinos,
not produced in the inflaton decay
but only thermally from standard leptons and Higgs.
The dominant production reaction is the Higgs boson decay into leptons and neutrinos, $\barphi \rightarrow l_i N_a$,
allowed by a Higgs thermal mass, 
$m_\phi = x_\phi T \sim \frac{1}{2} T$~\cite{Cline93},
much higher than the lepton thermal masses.
The proper decay rate into $N_a$ per $\barphi$ isospin state is 
$\Gamma_\phi^a = (h ^{\dagger} h)_{aa} m_\phi /16\pi$.
Taking for definiteness a reheating temperature much higher than the neutrino masses, the neutrino densities $n_a$ converge to the equilibrium density
%$\nequi $  
$\nequi = 0.90\, T^3/\pi^2$  
in the radiation era as
\be{na}
n_a = \nequi \left( 1 - e^{-T_a /T} \right)		\; ,
%T^{-1}} \right)		\; .
\eeq
with relaxation temperatures
$T_a \approx \frac{1}{6} K_a M_a$.
As a consequence of the prompt decay assumption, $K_a \gtrsim 60$, sterile neutrinos reach thermal equilibrium while they are still ultrarelativistic.

Leptogenesis is dominated by the following $CP$ asymmetric reactions:
Higgs decays into leptons and sterile neutrinos, inverse decays and scatterings of leptons off neutrinos.
The asymmetries result from the interference between the diagrams on the left side of Fig.~\ref{fig1} with the absorptive part of the respective diagram on the right-hand side.
Unitarity and $CPT$ invariance impose constraints on the reaction rates such that when all particles are in thermal equilibrium the asymmetry source terms cancel each other~\cite{Weinberg79,Kolb80,bento02}.
Generically denoting
the difference between the total rates of a reaction 
$X \rightarrow Y$ and its $CP$-conjugate $\bar{X} \rightarrow \bar{Y}$
as $\Delta \gamma (X \rightarrow Y)$
(in a fixed comoving volume),
the relevant constraints are expressed in terms of thermal equilibrium rates 
as:
\bsub{CPT}
\dgama (\bar{\phi} \ra N_a l_i) _\equi + \dgama  ( N_a l_i \ra \barphi ) _\equi =0
 	\;,	\\
\dgama ( N_a l_i \ra  N_b l_j )_\equi + \dgama ( N_b l_j  \ra N_a l_i ) _\equi =0
 	\;,	\\
\dgama  ( N_a l_i \ra \bar{\phi} )_\equi + \sum_{bj} \dgama  ( N_a l_i \ra  N_b l_j  ) _\equi =0
 	\;.	
\esub
%fffffffffffffffffffffffffffffffffffffffffffffffffffff
%fffffffffffffffffffffffffffffffffffffffffffffffffffff
\begin{figure}[t]
\includegraphics[width=70mm]{promptFig}% Here is how to import EPS art
\caption{\label{fig1} Diagrams contributing to the CP-asymmetries in decays, inverse decays and scatterings.}
\end{figure}
%fffffffffffffffffffffffffffffffffffffffffffffffffffff
%fffffffffffffffffffffffffffffffffffffffffffffffffffff
The evolution of the set of observables $Q_\alpha$ is governed by the  Boltzmann equations, whose general structure is,
integrating over a constant comoving volume,
$\dot{Q}_\alpha = - \Gamma_{\alpha \beta} Q_\beta + (\dot{Q}_\alpha)_S $.
The first are transport terms and $(\dot{Q}_\alpha)_S$ stand for source terms.
The particle asymmetry sources vanish in thermal equilibrium 
but, as long as the $N_a$ neutrinos stay rarefied, scatterings and inverse decays do not match Higgs decays and particle asymmetries develop in the various lepton flavors, sterile neutrinos, and Higgs boson as well, as enforced by hypercharge conservation.
The calculation of the source terms is simplified by employing Boltzmann rather than quantum statistics as is common practice.
Relating the out-of-equilibrium and thermal equilibrium reaction rates as $\gamma = \gamma_\equi \, n_a/\nequi $
for $N_a X \ra Y$, and  $\gamma = \gamma_\equi $
for $X \ra N_a Y$,
one derives from the above $CPT$ and unitarity constraints the $l_i$ lepton number sources,
\bea{lis}
% \dot{L}^S_i 
(\dot{L}_i )_S
= \sum_{abj} \frac{n_a - \nequi}{\nequi}
%(n_a /\nequi -1)  
\, \dgama (N_a l_j\ra  N_b l_i )_\equi 
	\;,	\\
 \dgama (N_a l_j\ra  N_b l_i )_\equi = 
 - \frac{c_l}{32 \pi^2} \frac{Y_\phi}{T} J_{ijab }
	\;,	%\nonumber
\eea
where $c_l \approx 0.1$,
$J_{ijab} = \mathrm{Im} \{ h_{ia} h_{ja} h_{ib}^\ast h_{jb}^\ast  \} M_a M_b$
and $Y_\phi$ is the Higgs abundance per isospin state.

Sterile neutrinos also have lepton number, $L=-1$ for left-handed $N_a$ fields and $L=+1$ for the conjugate $\bar{N}_a$.
But what matters for particle propagation and transport of lepton number are the spin eigenstates rather than chiral states.
When neutrinos are ultrarelativistic, helicity and chiral states are almost identical, but
when they are non-relativistic Majorana masses take over and lepton number vanishes.
In fact, a positive (negative) helicity eigenstate has a well defined lepton number expectation value equal to the neutrino speed $u$ ($-u$).
Helicity is not an invariant quantity, however, there is a privileged reference frame, the comoving thermal bath rest frame.
In that frame, isotropy ensures that the spin density matrix is diagonal in the helicity basis.
This is not true in any other frame, and means that neutrinos can be divided in two populations of opposite helicities.

Under the non-equilibrium conditions of the leptogenesis era the two neutrino helicity populations develop unequal abundances and net lepton numbers in much the same way as the standard leptons.
From the same decay, inverse decay and scattering processes one derives the $N_b$ lepton number source terms,
\be{lbs}
(\dot{L}_b)_S = - r_N \sum_{aij} \frac{n_a - \nequi}{\nequi}
%(n_a /\nequi -1)  
\, \dgama (N_a l_j\ra  N_b l_i )_\equi 
	\;,	
\eeq
where $r_N$ is a coefficient estimated to be of the order but larger than unit.
Summing both lepton and neutrino contributions, one obtains the
total $B-L$ asymmetry as a function of the temperature:
\be{BmL}
B-L = c\, \yequi \sum_{ab} 
\frac{\mathrm{Im} [(h^\dagger h)_{ba}]^2}{(h^\dagger h)_{aa}}
% H_{ab} 
\frac{M_a M_b}{ T_a^{2}}
%%f(\frac{T_a}{T}) 
%%f({T_a}/{T})
f\left(\frac{T_a}{T} \right) ,
\eeq
where $c \approx 2 c_l (r_N -1)/\pi \sim 0.1$, 
%%$H_{ab} = \mathrm{Im} [(h^\dagger h)_{ba}]^2 /(h^\dagger h)_{aa}$,
%$\yequi =\nequi/s$ 
$\yequi $
is the massless fermion thermal abundance
and $f(x)= 2-(x^2 +2x +2 ) e^{-x}$.

The other question is how fast Majorana masses erase lepton number.
In a decay $\bar{\phi} \rightarrow l_i N_a$, lepton number suffers a variation $\Delta L = 2$ if $N_a$ has positive helicity,
which occurs with a branching fraction about $10\, M_a^2/T^2 $.
It makes an average variation $\Delta L \approx 20\,M_a^2/T^2$ per decay.
On the other hand, $N_a$ enter in equilibrium at temperatures 
$T_a \approx \frac{1}{6} K_a M_a$.
Then, one estimates that lepton number violation enters in equilibrium at a temperature 
$T_\ast \approx \sum 20\, T_a M_a^2 /T_\ast^2$,
i.e., $T_\ast^3 \approx \sum 20\, T_a M_a^2$.
For constants $K_a$ as large as 60, $T_\ast$ is still close to the largest of the Majorana masses, $M_3$: $T_\ast \approx 6\, M_3 < T_a$.
The larger $K_a$ are, the smaller is $T_\ast$ in comparison with the equilibrium temperatures $T_a$.

At temperatures above $T_\ast$, $B-L$ is not significantly damped and is slowly transferred via Yukawa interactions to the right-handed leptons and quarks, $e_i$, $u_i$ and $d_i$
(with the exception of the top, $t_R$, very early in chemical equilibrium).
The dominant processes are Higgs boson decays into 
leptons~\cite{Cline93}, 
$\bar{\phi} \leftrightarrow e_i \bar{l}_i$,
kinematically forbidden in the case of quarks due to comparable quark thermal masses~\cite{Davidson94}, 
and top quark scatterings like
$\bar{\phi} t_R \leftrightarrow b_R \phi$,
$ b_L t_L \leftrightarrow b_R t_R$.
These processes generate the otherwise zero asymmetries 
$Q_{e_i} \equiv L_{e_i}$
and $Q_{u_i} = B_{u_i} - B_{u_R}$, $Q_{d_i} = B_{d_i} - B_{u_R}$.
For right-handed quarks,
the contrast asymmetries with respect to the up quark $u_R$ are the appropriate charges because, contrary to the individual flavor asymmetries, they are conserved by QCD instantons and are only violated by Yukawa couplings.
Their growth rates are determined by the total Yukawa production rates, $\Gamma_\chi \yequi$, of
$\chi =e_i$, $u_i = c_R$, $d_i$, and degeneracy parameters $\eta = \mu/T$ as,
\bsub{qeiqdi}
\dot{Q}_{e_i} = 2 (\eta_{l_i} - \eta_{\phi} - \eta_{e_i})\,
\Gamma_{e_i} \yequi
		\; ,			\label{qei}	\\
\dot{Q}_{u_i} = 2 (\eta_{q_i} + \eta_{\phi} - \eta_{u_i})\,
\Gamma_{u_i} \yequi
		\; ,			\label{qui}	\\
%\Delta \dot{B}_{d_i} = 2 (\eta_{q_i} - \eta_{\phi} - \eta_{d_i})\,
\dot{Q}_{d_i} = 2 (\eta_{q_i} - \eta_{\phi} - \eta_{d_i})\,
\Gamma_{d_i} \yequi
		\;,			\label{qdi}
\esub
neglecting flavor violation and  $u_R$ Yukawa coupling.
%(for up quarks, with opposite sign in front of $\eta_{\phi}$).
$\Gamma_\chi$ scale as the temperature and the ratios $\Gamma_\chi /H = T_\chi /T$ 
define equilibrium temperatures, $T_\chi$, under which the respective detailed balance equations are enforced.

Integration of Eqs.~(\ref{qeiqdi}) requires knowing the relations between the chemical potentials and the leptogenesis sources given in Eqs.~(\ref{lis}) and (\ref{lbs}). 
In general there is no direct relation not even for the leptons $l_i$ due to fast flavor changing interactions.
As soon as the first neutrino, say $N_c$, is in equilibrium ($T < T_c$), the lepton flavors $l_i$ are rapidly violated 
enforcing their chemical potentials $\eta_{l_i}$ to be identical: 
$\eta_{l_i}+\eta_{N_c}+\eta_{\phi}=0$.
The Higgs communicates also its asymmetry to $t_R$ and top quark doublet:
$\eta_{q_t}+\eta_{\phi}-\eta_{t_R}=0$.
Fast electroweak sphaleron processes cause an additional transport between leptons and quarks.
Because the $Q$ asymmetries increase rapidly with time, initially as $T^{-4}$,
we concentrate on the latest period before $T_\ast$, where sterile neutrinos are already in equilibrium but Majorana masses do not violate lepton number fast enough to yield null neutrino chemical potentials.
Eqs.~(\ref{qeiqdi}) can be written as 
$\dot{Q}_\chi = \alpha_\chi \Gamma_\chi (B-L)$ 
with coefficients $\alpha_\chi$ determined by the all set of detailed balance equations~\cite{Khlebnikov88}.
One obtains
\be{qchi}
%Q_\chi = c_\chi 
Q_\chi = c\, \alpha_\chi \lambda_\chi^2
\yequi \sum_{ab} 	%J_{ab} 
\frac{\mathrm{Im} [(h^\dagger h)_{ba}]^2}{[(h^\dagger h)_{aa}]^2}
\frac{M_a M_b}{T_a T} g\left( \frac{T_a}{T} \right)
		%,	
\eeq
where 
%$J_{ab} = \mathrm{Im} [(h^\dagger h)_{ba}]^2 /[(h^\dagger h)_{aa}]^2$
%$c_\chi = c\, \alpha_\chi \lambda_\chi^2$ and 
$g(x) = 2-6/x + (x+4+6/x)e^{-x}$.
$\lambda_\chi$ are the $e_i$ Yukawa couplings or, in
the case of quarks, quantities scaling with the $u_i$ and $d_i$ reaction rates as 
$\lambda_\chi^2 = \lambda_{\tau}^2 T_\chi/ T_{\tau}$.
For example, one estimates 
$T_{\tau} \approx \frac{1}{2} \cdot 10^{12} \gev$ for $\lambda_{\tau} = 10^{-2}$, and
$T_{b} \approx 2 \cdot 10^{12} \gev$. 
Then,
$\lambda^2_{b} \approx 4 \cdot 10^{-4}$. 

The $Q$ asymmetries grow with 
$T^{-1}$ ($g \rightarrow 2 $) 
down to the temperature $T_\ast < T_a$
where lepton number starts to be rapidly violated.
Then, a new constraint is enforced,
$\eta_{l_i} + \eta_\phi =0$, 
that causes $B-L$ to fall down to the level of the right-handed fermion asymmetries,
which in turn remain marginally damped as long as the respective Yukawa processes are slow, as implied by Eqs.~(\ref{qeiqdi}).
Assuming that the QCD and electroweak sphalerons were or enter in equilibrium at some point below $T_\ast$, one obtains, neglecting other flavor asymmetries,
\be{bmlstar}
B-L = \frac{60}{29}\, Q^{\ast}_b + \frac{6}{29}\, Q^{\ast}_\tau
%	 -\frac{67.5}{29}\, Q^{\ast}_c
		\;.
\eeq
The stars indicates that the asymmetries are calculated from Eq.~(\ref{qchi}) at the temperature $T_\ast$ and remain practically constant after that.

Let us assume that the heavy neutrinos decay and the $\Delta L =2$ reactions go out of equilibrium before $b_R$ and $\tau_R$ enter in chemical equilibrium.
From then on $B-L$ is exactly conserved.
As the temperature goes down and all right-handed leptons and quarks gradually enter in equilibrium $B-L$ is redistributed among particles to eventually yield
the present baryon number asymmetry
$B_0 =({12}/{37}) (B-L)$~\cite{Khlebnikov88}.
%\be{B}
%B_0 = \frac{32 + 4 n_H }{98 + 13 n_H} (B-L)	\,,
%\eeq
%where $n_H$ are the number of Higgs doublets.
The coefficients $\alpha_X$ in Eq.~(\ref{qchi}) depend on which reactions are in equilibrium above $T_\ast$.
If QCD and electroweak sphalerons are not yet in equilibrium, 
$\alpha_b = 0.24 $ and $\alpha_\tau = 0 $.
Otherwise, 
$\alpha_b = 0.14$ and $\alpha_\tau = 0.032$.
In both cases one obtains $B-L$ and baryon asymmetries at
exactly the observed order of magnitude, 
$B-L \gtrsim 10^{-10}$ per unit of entropy,
taking precisely the prompt decay ratio values indicated by the atmospheric neutrino evidence, $K_a =\Gamma_a/H \sim 10^2$,
without assuming any particular hierarchy between Yukawa couplings.

In the solution described above $B-L$ is mainly originated from the $b_R$ asymmetry $Q_b = B_{b_R} - B_{u_R}$. 
It assumes that $\Delta L = 2$ reactions are not in equilibrium when $b_R$ is.
The implications of this are better assessed by studying the evolution of $B-L$.
During and in the period following rapid lepton number violation,
and before $\tau_R$ enters in equilibrium,
 all $l_i$ have identical chemical potentials.
In that case,
$\dot{B}-\dot{L} = 4 (\eta_l + \eta_\phi) \Gamma_2 \yequi$
where $\Gamma_2 \yequi$ is the total rate of $\Delta L = -2$ reactions and
$ \eta_l \equiv \eta_{l_i}$.
When $b_R$ is in chemical equilibrium this gives
$\dot{B}-\dot{L} = - 0.44\, \Gamma_2 (B-L)$.
It means that $B-L$ is damped at a rate $\Gamma = 0.44\, \Gamma_2$,
directly related to the light neutrino mass scale $\munu$ by
%light neutrino spectrum:
$\Gamma/H \approx  \munu \, T /(10^{11}\, \gev \evs )$.
It is clear that $B-L$ is essentially preserved at the $b_R$ equilibrium epoch, $T < T_b \approx 2 \cdot 10^{12} \gev$,
for $\munu$ up to $0.1 \evs$, one order of magnitude higher than the atmospheric neutrino mass gap.

If $\munu$ goes beyond $0.1 \evs$, the charm $c_R$ replaces $b_R$ in the role of protecting $B-L$ because then, 
$B-L = - \frac{3}{2} Q^\ast_c$, whether or not $\tau_R$ is in equilibrium.
However, $Q_c^\ast \approx 0.03 \, Q_b^\ast$ is too small if not compensated by a strong hierarchy in the Yukawa coupling structure.

To conclude, leptogenesis can operate in a regime where all sterile neutrinos,
including the lightest,
decay promptly when they vanish from the Universe.
The light neutrino mass scale $\munu = \sum m_\nu^2$
determines the natural order of magnitude of the final baryon asymmetry.
First, $\munu$ values between the atmospheric neutrino mass gap and $0.1 \evs$
indicate that the heavy neutrinos decay promptly with ratios 
$K_a = \Gamma_a /H \sim 10^{2}$,
and reach thermal equilibrium at temperatures 10 times or more their masses.
The dominant neutrino production processes,
ignored in the literature so far,
are the decays of the Higgs boson, with finite thermal mass.
%are the Higgs boson decays (with a finite thermal mass).
During thermal production, $B-L$ is generated in both lepton and sterile neutrino sectors reaching the $10^{-7}$ level.
Later, when the Majorana masses rapidly violate lepton number, $B-L$ is effectively protected by the off-equilibrium $b_R$ quark if, again, 
$\munu \lesssim 0.1 \evs$, and the Majorana masses lie above the $b_R$
equilibrium temperature $T_b \approx 2 \cdot 10^{12} \gev$.
Then, $B-L$ falls down 3 orders of magnitude, which meets exactly the observed baryon number asymmetry without any further conditions.


%\newpage
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\end{thebibliography}
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