hep-ph0411380.tex
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{\hbox to\hsize{
\hfill \bf hep-ph/0411380 }}
{\hbox to\hsize{\hfill November 2004 }}

\bigskip
\vspace{3\baselineskip}

{\Large \bf

Anomaly-induced baryogenesis \\}

\bigskip

\bigskip

{\bf Archil Kobakhidze \\}
\smallskip

{ \small \it
Department of Physics and Astronomy, UNC-Chapel Hill, NC 27599-3255, USA \\}


\bigskip



\vspace*{.5cm}

{\bf Abstract}\\
\end{center}
\noindent We propose a new mechanism for dynamical generation of the observed baryon asymmetry within the minimal Standard model extended by massive Majorana neutrinos and non-vanishing electroweak Chern-Simons term. We show that electroweak Chern-Simons number is produced in the expanding universe due to the conformal anomaly and subsequently converted into baryon number through the triangle anomaly. 
\bigskip

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\end{titlepage}
%\baselineskip=16pt

In the Standard Model the conservation of $B+L$ (baryonic + leptonic) current is violated due 
to the quantum triangle anomaly,  
\begin{equation}
\partial_{\mu}j^{\mu}_{B+L}=
\frac{n_g}{16\pi^2}W^a_{\mu\nu}\tilde{W}^{a\mu\nu} , 
\label{1}
\end{equation}
while classically the above current is conserved. In (\ref{1}) $n_g$ is the number of quark-lepton generations and 
$\tilde{W}^{a\mu\nu}=1/2\epsilon^{\mu\nu\rho\sigma}W^a_{\rho\sigma}$ is a dual of the $SU(2)_L$  field 
strength tensor $W^a_{\mu\nu}$. We have ignored in (\ref{1}) the weak hypercharge and gravitational 
contributions  which are irrelevant for the subsequent discussion. The equation (\ref{1}) tells us that the generation of non-zero $B+L$ charge, $N_{B+L}=\int d^3xj^{0}_{B+L}$, is related to the generation of the electroweak Chern-Simons number,
\begin{equation}
N_{CS}=-\frac{n_g}{8\pi^2}\int d^3x\epsilon_{ijk}{\rm Tr}\left(W_i\partial_jW_k+\frac{2i}{3}W_iW_jW_k\right).  
\label{2}
\end{equation}  
Here $W_{\mu}=W_{\mu}^aT^a$ is $SU(2)_L$ vector potential with $T^a$ being the generators of $SU(2)_L$ group, ${\rm Tr}(T^aT^b)=\frac{1}{2}\delta^{ab}$.

This interrelation between $B+L$ charge and the electroweak Chern-Simons number is an essential ingredient of the electroweak baryogenesis \cite{Kuzmin:1985mm}. Two remaining Sakharov's conditions \cite{Sakharov:1967dj} for the dynamical baryogenesis are also satisfied, at least qualitatively, in the Standard Model: CP is violated through the complex Cabibbo-Kobayashi-Maskawa (CKM) matrix, and departure from thermal equilibrium is provided during the electroweak phase transition. Quantitatively, however, the electroweak baryogenesis within the minimal Standard Model cannot account for the observed baryon-to-entropy density ratio,
\begin{equation}
\eta=\frac{n_B}{s}\approx 10^{-10} .  
\label{3}
\end{equation}
There are two basic reasons for this. First, the electroweak phase transition is not sufficiently strongly first-order in order to prevent wash out of the generated baryon asymmetry, unless the electroweak Higgs mass is below $\sim 45$ GeV, which is ruled out by collider experiments. Second, the CKM CP violation is too small for successful baryogenesis because its contribution to $\eta$ is suppressed by higher powers of Yukawa couplings. Thus, in order to generate the desired baryon asymmetry (\ref{3}) one must look for models beyond the standard particle physics and/or cosmology (see e.g. \cite{Riotto:1999yt} for a recent review and references therein).       

So far we have only one experimentally established evidence for the physics beyond the Standard Model. Namely, the anomalies in neutrino experiments most naturally can be explained by their oscillations due to the tiny but non-zero masses. From the theoretical perspectives it is also more natural to assume that massive neutrinos are Majorana particles. Irrespective of the actual mass generation mechanism, Majorana masses for neutrinos can be assumed to be originated from the effective interactions
\begin{equation}
\frac{1}{2M}(LH)^2 ,  
\label{4}
\end{equation}
where $L$ and $H$ denote left-handed lepton and the electroweak Higgs doublet fields (the gauge and family structure is suppressed in (\ref{4})), respectively, while $M$ is the effective mass scale. This parameter must be such that upon the electroweak Higgs condensation, $<H>\equiv M_{EW}\simeq 174$ GeV Eqn. (\ref{4}) reproduces the range of neutrino masses dictated by the oscillation experiments. That is to say, the lowest possible value is $M\approx 10^{15}$ GeV for the heaviest neutrino, e.g. $m_{\nu_\tau}\approx 0.1$ eV. The effective interactions (\ref{4}) violate lepton number, $\Delta L=2$, while conserving baryon number and hence violate $B+L$. 

Beside the interactions (ref{4}), we assume the existence of non-vanishing electroweak Chern-Simons theta-term,
\begin{equation}
\frac{\theta_{EW}}{16\pi^2}W^a_{\mu\nu}\tilde{W}^{a\mu\nu},
\label{5}
\end{equation}
in the low-energy effective Lagrangian. This term provides for a new source of $CP$ violation in the Standard Model. Within the bosonic sector of the Standard Model, i.e. ignoring the fermions, (\ref{5}) appears in the effective Lagrangian due to the inter-vacua tunnelings provided by $SU(2)_L$ instantons. If we add to the model only leptons with massive Majorana neutrinos the $\theta_{EW}$-term will be physical similar to the more familiar $\theta_QCD$-term in the QCD with massive quarks. Contrary to the case of $\theta_QCD$, however, inclusion of massive quarks allows to rotate away $\theta_{EW}$. This is related to the existence of zero-modes of massive quark states in the background of $SU(2)_L$ instantons \cite{Krasnikov:1979kz}, \cite{Anselm:1993uj}. Speaking on the language of symmetries, clasically conserved baryon number is responsible for the fact that $\theta_{EW}$ is unphysical. Hence assuming physical effects from the $\theta_{EW}$-term we keep in mind the existence of certain new physics beyond the Standard Model at high energies that supports $\theta_{EW}$. Actually, it is more natural to expect the presence of $\theta_{EW}$-term in the low-energy effective theory than its absence. Indeed, almost all popular extensions of the Standard Model, such as Grand Unified Theories, left-right symmetric models, etc., generally admit rather than forbid $\theta_{EW}$-term. Also, quantum gravity effects are expected to violate global numbers, including $B$ and $L$, hence removing zero-modes and making $\theta_{EW}$ physical. Finally, one can speculate that fermionic zero-modes are removed because of non-trivial spatial topology of the universe \cite{me}. In such a universe $\theta_{EW}$ would be non-vanishing as well. All these possibilities might indeed be realized in nature. Here we are not interested in the actual physics which stands behind of non-zero $\theta_{EW}$-term. In what follows we simply assume that this new physics itself is essentially irrelevant for the generation of the observed baryon asymmetry. Its main effect is to support non-vanishing $\theta_{EW}$-term (\ref{5}).  

Adopting the above modification of the Standard Model which seems to be absolutely minimal and perhaps even necessary in the view of current neutrino experiments and theoretical arguments given above, we are ready now to discuss a new scenario for the dynamical generation of the observed baryon asymmetry. Namely, we will show that topological fluctuations of $SU(2)_L$ weak gauge bosons in the expanding universe generate non-zero Chern-Simons number (\ref{2}) due to the quantum conformal anomaly. The Chern-Simons number subsequently produces non-zero baryonic number through the triangle anomaly (\ref{1}). 

To see this more explicitly let us start by considering the Lagrangian for the $W$-bosons with electroweak $\theta_{EW}$ term included, 
\begin{equation}
L_W=\sqrt{-g}\left[-\frac{1}{2g^2}{\rm Tr}W_{\mu\nu}W^{\mu\nu}-\frac{\theta_{EW}}{8\pi^2}{\rm Tr}W_{\mu\nu}\tilde{W}^{\mu\nu}\right],  
\label{6}   
\end{equation}
in homogeneous, isotropic and spatially flat spacetime with line element given by, 
\begin{equation}
ds^2=a(\tau)\left[d\tau^2-d\bf{x}^2\right].
\label{7}   
\end{equation} 
In (\ref{7}) $\tau$ is the conformal time which is related to the cosmic time by $dt=a(\tau)d\tau$. Tree level Lagrangian (\ref{6}) have the same form in flat Minkowski space and in conformally flat spacetime (\ref{7}) due to the conformal invariance. However, the conformal invariance is anomalous, so that the (one-loop) quantum corrections modify (\ref{6}) as \cite{Calzetta:1985ji}-\cite{Mazzitelli:1995mp}:
\begin{eqnarray}
L_W=-\frac{\sqrt{-g}}{2}\left(\frac{1}{g^2_{EW}}-
\frac{b}{8\pi^2}\ln(\frac{a}{a_0})\right){\rm Tr}W_{\mu\nu}W^{\mu\nu}- \nonumber \\
\frac{\sqrt{-g}\theta_{EW}}{8\pi^2}{\rm Tr}W_{\mu\nu}\tilde{W}^{\mu\nu} .  
\label{8}   
\end{eqnarray}  
The effect of conformal anomaly is seen in (\ref{8}) as a scale-factor dependent running of effective gauge coupling constant, $g(a)$,  
\begin{equation}
\frac{dg(a)}{d\ln a}=\frac{b}{16\pi^2}g(a)^3 ,
\label{9}   
\end{equation}  
where $b$ is nothing but the group-theoretical factor one-loop beta-function, which accounts for the contributions of all $SU(2)_L$-doublet fields in the loop with masses less than the expansion rate of the universe, i.e. $m<<H=\frac{a^{\prime}}{a^2}$ ($a^{\prime}=da/d\tau$) ($b=\frac{4n_g}{3}+\frac{n_h}{6}-\frac{22}{3}$, where $n_g$ is the number of quark-lepton generations and $n_h$ is the number of electroweak Higgs bosons). Once the expansion rate drops below the mass of a given particle this particle decouples, and, eventually, when $b=0$, the effective gauge coupling $g(a)$ freezes to a constant value which we observe at colliders, i.e. $\alpha(M_{EW})_{EW}^{-1}=\frac{4\pi}{g_{EW}^2}\approx 29$. This happens at a scales corresponding to the electroweak phase transition, $M_{EW}\simeq 174$ GeV. Since within the standard hot big bang theory $H\approx 17T^2/M_p$, in the epoch where time variation of the gauge coupling stops the universe was still rather hot, $T_0\approx 10^{10}$ GeV.
The time variation of the electroweak coupling constant described by Eq. (\ref{9}) turns out to be absolutely crucial for our mechanism of baryogenesis. It provides out-of-equilibrium generation of non-zero Chern-Simons number. To see this explicitly let us write down the equation of motion which follows from the Lagrangian (\ref{8}) in the linearized approximation: 
\begin{equation}
\left[\partial^2_{\tau}-\partial_l^2\right]W_i
+aH
\frac{b\theta_{EW}}{\pi}\alpha^2(a)\epsilon_{ijk}\partial^{j}W^{k}=0
\label{10}   
\end{equation}
where we have used the rescaled field strength $W_{\mu\nu}\to \left(\frac{1}{g^2}-\frac{b}{8\pi^2}\ln(\frac{a}{a_0})\right)W_{\mu\nu}$ and have chosen the gauge $W_0=\partial_iW_i=0$. In this form the above equation formally is similar to the equation of motion of the Abelian gauge field coupled to axionic field. The latter model has been studied in the context of generation of cosmic magnetic field (see e.g., \cite{Turner:1987bw}-\cite{Prokopec:2001nc}). The equation for the circular polarized mode functions, $F_{\beta}(\tau)$, 
\begin{eqnarray}
W_i(\tau, {\bf x})=\Sigma_{\beta}\int \frac{d^3{\bf k}}{(2\pi)^{3/2}}\times \nonumber \\
\left[F_{\beta}(\tau)\epsilon_{i\beta}a_{\beta}({\bf k})e^{i{\bf kx}}+
F^{*}_{\beta}(\tau)\epsilon^{*}_{i\beta}a^{+}_{\beta}({\bf k})e^{-i{\bf kx}}\right], \nonumber 
\\
\label{11}
\end{eqnarray}
($\beta =+,-$ labels two circular polarizations) can be written as:
\begin{equation}
F_{\pm}^{\prime \prime}+\left(k^2\mp aH\frac{b\theta_{EW}}{\pi}\alpha^2(a)k\right)F_{\pm}=0.
\label{12}
\end{equation}
The creation and annihilation operators, $a^{+}_{\beta}({\bf k})$ and $a_{\beta}({\bf k})$ in (\ref{11}), satisfy bosonic commutation relations, $[a_{\beta}({\bf k}), a^{+}_{\beta^{\prime}}({\bf k}^{\prime})]=\delta_{\beta \beta^{\prime}}\delta({\bf k}-{\bf k}^{\prime})$ and $[a_{\beta}({\bf k}), a_{\beta^{\prime}}({\bf k}^{\prime})]=[a^{+}_{\beta}({\bf k}), a^{+}_{\beta^{\prime}}({\bf k}^{\prime})]=0$.

In radiation dominated era we have $a(\tau)\propto \tau$, $aH=1/\tau$. Ignoring the logarithmic scale-variation of gauge coupling constant in (\ref{12}) we set $\alpha(a)\approx\alpha_{EW}$. The solutions then can be expressed in terms of Whittaker functions:
\begin{equation}
F_{\pm}(\tau)=\frac{e^{\mp \frac{\pi}{2}\Theta}}{\sqrt{2k}}W_{\pm\Theta, 1/2}(2ik\tau), 
\label{13}
\end{equation}
where $\Theta=\frac{b}{2\pi}\theta_{EW}\alpha^2_{EW}$. When the universe cools down the temperatures $T<10^{10}$ GeV, $b=0$ in (\ref{12}) and the solutions are plane waves, $F_{\pm}(\tau)=\frac{1}{\sqrt{2k}}exp(-ik\tau)$. In order to preserve canonical commutation relations among the creation and annihilation operators $b_{\beta}^{+}$ and $b_{\beta}$, we smoothly match the solutions at $\tau=\tau_0$. Actually, we are interested in large wave-length fluctuations ($k|{\tau}|\to 0$) only,
\begin{equation}
F_{\pm}(\tau)\approx 
=\frac{e^{\mp \frac{1}{4}\theta_{EW}\alpha^2_{EW}}}{\sqrt{2k}}~  ({\rm for}~ \tau<\tau_0),
\label{14}
\end{equation}  
since the small wave-length fluctuations $k|{\tau}|\to \infty$ essentially are plane waves and hence their contribution to the Chern-Simons number density is subdominant. For large wave-length fluctuations the Bogoliubov transformations are particularly simple:
\begin{equation}
b_{\beta}({\bf k})=\frac{e^{-\beta \pi \Theta /2}}{2}\left[a_{\beta}({\bf k})+a^{+}_{\beta}({\bf k})\right]. 
\label{15}
\end{equation}
Now it is straightforward to calculate integrated Chern-Simons number density:
\begin{eqnarray}
n_{CS}(\tau_0)\approx -\frac{n_g\alpha_{EW}}{2\pi}\int_0^{\tau_0}d\tau 
<\epsilon_{ijk}{\rm Tr}W_i\partial_jW_k> \nonumber \\
\approx -\frac{n_g\alpha_{EW}\Theta}{2\pi^2}\int_0^{\tau_0}d\tau\int^{\Lambda}_0 k^2dk\delta(k\tau)= 
-\frac{n_g\alpha_{EW}\Theta \Lambda^3}{6\pi^2}, 
\label{16}
\end{eqnarray}
where $\Lambda$ is a UV (conformal) momentum cut-off. As we have discussed, the Chern-Simons number is fully converted into a $B+L$ charge, i.e. $n_{B+L}=n_{CS}$.

Now let us discuss the faith of $B+L$ generated through the electroweak Chern-Simons number. When the universe cools down below $T_0\approx 10^{10}$ GeV, the effective gauge coupling freezes out and hence $B+L$ production stops. However, equilibrium $\Delta L=2$ processes described by the interactions (\ref{4}), i.e. $L+H\to\bar{L}+H^{+}$, are also in action at certain stage \cite{Fukugita:2002hu}. This happens when the rate of $\Delta L=2$ reactions, $\Gamma_{\Delta L=2}\approx \frac{0.12T^3}{4\pi M^2}$, exceeds the expansion rate of the universe, $\Gamma_{\Delta L=2}>H$, i.e. when the temperature of the universe is $T> 10^{14}$ GeV. An important point is that at such higher temperatures the rate of the electroweak shpaleron-induced $B+L$ violating processes are relatively suppressed. This means that only L-number will be washed out, leaving baryonic charge, $n_B\approx n_{B+L}/2$, untouched in the anomaly-generated $n_{B+L}$. Finally, at temperatures $T< 10^{12}$ GeV the rate of $B+L$ reactions due to the shpalerons, $\Gamma_{shp}\approx 2\cdot 10^2\alpha_{EW}^5T$, overcome the expansion rate of the universe and become dominating, while $\Delta L=2$ reactions decouple. The effects of shpalerons  in equilibrium is to convert part of the baryonic charge into leptonic charge. The amount of the survived baryonic charge is given by \cite{Khlebnikov:1988sr}:  
\begin{equation}
n_B=\frac{4n_g+2n_h}{22n_g+13n_h}n_{B+L}
\label{17}
\end{equation}
In radiation dominated universe the entropy density is $s=\frac{2\pi^2}{45}N_{eff}T^3$, where $N_{eff}$ is an effective number of massless ($m\ll T$) degrees of freedom. Hence, in the Standard Model with $n_g=3$ generations of quarks and leptons, $n_h=1$ electroweak Higgs doublet, the effective number of degrees of freedom is $N_{eff}\approx 100$. With these numbers at hand, and assuming no significant entropy production during the matter dominated era, we estimate for the baryon number (\ref{17}) to entropy density ratio to be:
\begin{eqnarray}
\frac{n_B}{s}=\frac{5}{16\pi^5}\frac{n_g(4n_g+2n_h)(8n_g+h_h/2-44)}{22n_g+13n_h}\times \nonumber \\
N_{eff}\alpha^3_{EW}\theta_{EW}\left(\frac{\Lambda}{T}\right)^3\approx
4.1\cdot 10^{-5}\theta_{EW}\left(\frac{\Lambda}{T}\right)^3 .
\label{18}
\end{eqnarray}
Thus, depending on the ratio $\Lambda/T$ and on electroweak $\theta_{EW}$ parameter, it is possible to create baryon asymmetry of the desired amount (\ref{2}) in our scenario. 

In conclusion, we have proposed new mechanism for the dynamical baryogenesis within the Standard Model with massive Majorana neutrinos. The key effects come from the quantum anomalies. The conformal anomaly is responsible for the out-of-equilibrium generation of the electroweak Chern-Simons number which is subsequently converted into baryonic charge due to the triangle anomaly. Resulting baryonic charge survives during the evolution of the universe and can be easily accomodated with the empirical value of baryon asymmetry.


\subparagraph{Acknowledgments.}

I am grateful to Xavier Calmet, Gia Dvali, Paul Frampton, Laura Mersini-Houghton and Martin Sloth for valuable discussions. 

\begin{thebibliography}{99}

%\cite{Kuzmin:1985mm}
\bibitem{Kuzmin:1985mm}
V.~A.~Kuzmin, V.~A.~Rubakov and M.~E.~Shaposhnikov,
%``On The Anomalous Electroweak Baryon Number Nonconservation In The Early
%Universe,''
Phys.\ Lett.\ B {\bf 155}, 36 (1985).
%%CITATION = PHLTA,B155,36;%%

%\cite{Sakharov:1967dj}
\bibitem{Sakharov:1967dj}
A.~D.~Sakharov,
%``Violation Of CP Invariance, C Asymmetry, And Baryon Asymmetry Of The
%Universe,''
Pisma Zh.\ Eksp.\ Teor.\ Fiz.\  {\bf 5}, 32 (1967)
[JETP Lett.\  {\bf 5}, 24 (1967\ SOPUA,34,392-393.1991\ UFNAA,161,61-64.1991)].
%%CITATION = ZFPRA,5,32;%%

%\cite{Riotto:1999yt}
\bibitem{Riotto:1999yt}  A.~Riotto and M.~Trodden, ``Recent progress in
baryogenesis,'' Ann.\ Rev.\ Nucl.\ Part.\ Sci.\ {\bf 49} (1999) 35
[arXiv:hep-ph/9901362]. 
%%CITATION = HEP-PH 9901362;%%

%\cite{Krasnikov:1979kz}
\bibitem{Krasnikov:1979kz}
N.~V.~Krasnikov, V.~A.~Rubakov and V.~F.~Tokarev,
``Zero Fermion Modes In Models With Spontaneous Symmetry Breaking,''
J.\ Phys.\ A {\bf 12}, L343 (1979).
%%CITATION = JPAGB,A12,L343;%%

%\cite{Anselm:1993uj}
\bibitem{Anselm:1993uj}
A.~A.~Anselm and A.~A.~Johansen,
``Can electroweak theta term be observable?,''
Nucl.\ Phys.\ B {\bf 412}, 553 (1994)
[arXiv:hep-ph/9305271].
%%CITATION = HEP-PH 9305271;%%

%\cite{me}
\bibitem{me}
A.~Kobakhidze, in preparation. 

%\cite{Calzetta:1985ji}
\bibitem{Calzetta:1985ji}
E.~Calzetta, I.~Jack and L.~Parker,
``Curvature Induced Asymptotic Freedom,''
Phys.\ Rev.\ Lett.\  {\bf 55}, 1241 (1985).

%\cite{Dolgov:1993vg}
\bibitem{Dolgov:1993vg}
A.~Dolgov,
``Breaking of conformal invariance and electromagnetic field generation in the
universe,''
Phys.\ Rev.\ D {\bf 48}, 2499 (1993)
[arXiv:hep-ph/9301280].
%%CITATION = HEP-PH 9301280;%%

%\cite{Mazzitelli:1995mp}
\bibitem{Mazzitelli:1995mp}
F.~D.~Mazzitelli and F.~M.~Spedalieri,
``Scalar electrodynamics and primordial magnetic fields,''
Phys.\ Rev.\ D {\bf 52}, 6694 (1995)
[arXiv:astro-ph/9505140].
%%CITATION = ASTRO-PH 9505140;%%

%\cite{Turner:1987bw}
\bibitem{Turner:1987bw}
M.~S.~Turner and L.~M.~Widrow,
``Inflation Produced, Large Scale Magnetic Fields,''
Phys.\ Rev.\ D {\bf 37}, 2743 (1988).
%%CITATION = PHRVA,D37,2743;%%

%\cite{Garretson:1992vt}
\bibitem{Garretson:1992vt}
W.~D.~Garretson, G.~B.~Field and S.~M.~Carroll,
``Primordial magnetic fields from pseudoGoldstone bosons,''
Phys.\ Rev.\ D {\bf 46}, 5346 (1992)
[arXiv:hep-ph/9209238].
%%CITATION = HEP-PH 9209238;%%

%\cite{Giovannini:1999by}
\bibitem{Giovannini:1999by}
M.~Giovannini,
``Hypermagnetic knots, Chern-Simons waves and the baryon asymmetry,''
Phys.\ Rev.\ D {\bf 61}, 063502 (2000)
[arXiv:hep-ph/9906241].
%%CITATION = HEP-PH 9906241;%%

%\cite{Prokopec:2001nc}
\bibitem{Prokopec:2001nc}
T.~Prokopec,
``Cosmological magnetic fields from photon coupling to fermions and  bosons in
inflation,''
arXiv:astro-ph/0106247.
%%CITATION = ASTRO-PH 0106247;%%

%\cite{Fukugita:2002hu}
\bibitem{Fukugita:2002hu}
M.~Fukugita and T.~Yanagida,
``Resurrection of grand unified theory baryogenesis,''
Phys.\ Rev.\ Lett.\  {\bf 89}, 131602 (2002)
[arXiv:hep-ph/0203194].
%%CITATION = HEP-PH 0203194;%%

%\cite{Khlebnikov:1988sr}
\bibitem{Khlebnikov:1988sr}
S.~Y.~Khlebnikov and M.~E.~Shaposhnikov,
``The Statistical Theory Of Anomalous Fermion Number Nonconservation,''
Nucl.\ Phys.\ B {\bf 308}, 885 (1988).
%%CITATION = NUPHA,B308,885;%%
\end{thebibliography}

\end{document}


hep-ph0411380.extracted_bib
\begin{thebibliography}{99}

%\cite{Kuzmin:1985mm}
\bibitem{Kuzmin:1985mm}
V.~A.~Kuzmin, V.~A.~Rubakov and M.~E.~Shaposhnikov,
%``On The Anomalous Electroweak Baryon Number Nonconservation In The Early
%Universe,''
Phys.\ Lett.\ B {\bf 155}, 36 (1985).
%%CITATION = PHLTA,B155,36;%%

%\cite{Sakharov:1967dj}
\bibitem{Sakharov:1967dj}
A.~D.~Sakharov,
%``Violation Of CP Invariance, C Asymmetry, And Baryon Asymmetry Of The
%Universe,''
Pisma Zh.\ Eksp.\ Teor.\ Fiz.\  {\bf 5}, 32 (1967)
[JETP Lett.\  {\bf 5}, 24 (1967\ SOPUA,34,392-393.1991\ UFNAA,161,61-64.1991)].
%%CITATION = ZFPRA,5,32;%%

%\cite{Riotto:1999yt}
\bibitem{Riotto:1999yt}  A.~Riotto and M.~Trodden, ``Recent progress in
baryogenesis,'' Ann.\ Rev.\ Nucl.\ Part.\ Sci.\ {\bf 49} (1999) 35
[arXiv:hep-ph/9901362]. 
%%CITATION = HEP-PH 9901362;%%

%\cite{Krasnikov:1979kz}
\bibitem{Krasnikov:1979kz}
N.~V.~Krasnikov, V.~A.~Rubakov and V.~F.~Tokarev,
``Zero Fermion Modes In Models With Spontaneous Symmetry Breaking,''
J.\ Phys.\ A {\bf 12}, L343 (1979).
%%CITATION = JPAGB,A12,L343;%%

%\cite{Anselm:1993uj}
\bibitem{Anselm:1993uj}
A.~A.~Anselm and A.~A.~Johansen,
``Can electroweak theta term be observable?,''
Nucl.\ Phys.\ B {\bf 412}, 553 (1994)
[arXiv:hep-ph/9305271].
%%CITATION = HEP-PH 9305271;%%

%\cite{me}
\bibitem{me}
A.~Kobakhidze, in preparation. 

%\cite{Calzetta:1985ji}
\bibitem{Calzetta:1985ji}
E.~Calzetta, I.~Jack and L.~Parker,
``Curvature Induced Asymptotic Freedom,''
Phys.\ Rev.\ Lett.\  {\bf 55}, 1241 (1985).

%\cite{Dolgov:1993vg}
\bibitem{Dolgov:1993vg}
A.~Dolgov,
``Breaking of conformal invariance and electromagnetic field generation in the
universe,''
Phys.\ Rev.\ D {\bf 48}, 2499 (1993)
[arXiv:hep-ph/9301280].
%%CITATION = HEP-PH 9301280;%%

%\cite{Mazzitelli:1995mp}
\bibitem{Mazzitelli:1995mp}
F.~D.~Mazzitelli and F.~M.~Spedalieri,
``Scalar electrodynamics and primordial magnetic fields,''
Phys.\ Rev.\ D {\bf 52}, 6694 (1995)
[arXiv:astro-ph/9505140].
%%CITATION = ASTRO-PH 9505140;%%

%\cite{Turner:1987bw}
\bibitem{Turner:1987bw}
M.~S.~Turner and L.~M.~Widrow,
``Inflation Produced, Large Scale Magnetic Fields,''
Phys.\ Rev.\ D {\bf 37}, 2743 (1988).
%%CITATION = PHRVA,D37,2743;%%

%\cite{Garretson:1992vt}
\bibitem{Garretson:1992vt}
W.~D.~Garretson, G.~B.~Field and S.~M.~Carroll,
``Primordial magnetic fields from pseudoGoldstone bosons,''
Phys.\ Rev.\ D {\bf 46}, 5346 (1992)
[arXiv:hep-ph/9209238].
%%CITATION = HEP-PH 9209238;%%

%\cite{Giovannini:1999by}
\bibitem{Giovannini:1999by}
M.~Giovannini,
``Hypermagnetic knots, Chern-Simons waves and the baryon asymmetry,''
Phys.\ Rev.\ D {\bf 61}, 063502 (2000)
[arXiv:hep-ph/9906241].
%%CITATION = HEP-PH 9906241;%%

%\cite{Prokopec:2001nc}
\bibitem{Prokopec:2001nc}
T.~Prokopec,
``Cosmological magnetic fields from photon coupling to fermions and  bosons in
inflation,''
arXiv:astro-ph/0106247.
%%CITATION = ASTRO-PH 0106247;%%

%\cite{Fukugita:2002hu}
\bibitem{Fukugita:2002hu}
M.~Fukugita and T.~Yanagida,
``Resurrection of grand unified theory baryogenesis,''
Phys.\ Rev.\ Lett.\  {\bf 89}, 131602 (2002)
[arXiv:hep-ph/0203194].
%%CITATION = HEP-PH 0203194;%%

%\cite{Khlebnikov:1988sr}
\bibitem{Khlebnikov:1988sr}
S.~Y.~Khlebnikov and M.~E.~Shaposhnikov,
``The Statistical Theory Of Anomalous Fermion Number Nonconservation,''
Nucl.\ Phys.\ B {\bf 308}, 885 (1988).
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