\section{Introduction}

An origin of cosmological baryon asymmetry is one of
 the prime open questions in particle physics as well as in cosmology.
Among various mechanisms of baryogenesis,
 leptogenesis~\cite{FukugitaYanagida}
 is one of the most attractive idea because of its simplicity and
 the connection to neutrino physics.
Particularly, thermal leptogenesis requires
 only the thermal excitation of heavy right-handed Majorana neutrinos
 which generate
 tiny neutrino masses via the seesaw mechanism~\cite{Type1seesaw} and
 provides several implications for the light neutrino mass
 spectrum~\cite{Buchmulleretal}.
The size of CP asymmetry in a right-handed neutrino decay is, roughly speaking,
 proportional to the mass of right-handed neutrino.
Thus,
 we obtain only insufficiently small CP violation for a lighter right-handed neutrino mass.
That is the reason why it has been regarded that
 leptogenesis in low energy scale is in general difficult in the
 conventional Type I seesaw mechanism~\cite{LowerBound,Davidson:2002qv}.

On the other hand,
 in supersymmetric models with conserved R-parity to avoid rapid proton decay,
 thermal leptogenesis faces with ``gravitino problem''
 that the overproduction of gravitinos spoils the success of
 Big Bang Nucleosynthesis (BBN)~\cite{GravitinoProblem},
 whereas the stable lightest supersymmetric particle (LSP) becomes dark matter candidate.
In order not to overproduce gravitinos,
 the reheating temperature after inflation should not be so high
 that right-handed neutrinos can be thermally produced~\cite{GravitinoProblem2}.
In the framework of gravity mediated supersymmetry (SUSY) breaking,
 a few solutions,
 e.g., gravitino LSP with R-parity violation~\cite{Buchmuller:2007ui},
 very light axino LSP~\cite{Asaka:2000ew}
 and strongly degenerated right-handed neutrino masses~\cite{ResonantLeptogenesis},
 have been proposed.

Recently,
 a new class of two Higgs doublet models (THDM)~\cite{2hdm} has been considered
 in Refs.~\cite{Ma,Nandi,Ma:2006km,Davidson:2009ha,Logan:2010ag,
 HabaHirotsu}.
The motivation is as follows.
As mentioned above, seesaw mechanism naturally realizes tiny
 masses of active neutrinos
 through heavy particles coupled with
 left-handed neutrinos.
However,
 those heavy particles are almost decoupled in the low-energy effective
 theory, few observations are expected in collider experiments.
Then, some people consider a possibility of
 reduction of seesaw scale to TeV~\cite{TeVseesaw,Haba:2009sd},
 where effects of TeV scale right-handed neutrinos might be
 observed in collider
 experiments such as Large Hadron Collider (LHC) and
 International Linear Collider (ILC).
However, they must introduce a fine-tuning in order to obtain
 both tiny neutrino mass
 and detectable left-right neutrino mixing
 through which
 experimental evidences can be discovered.
Other right-handed neutrino production processes in extended models
 by e.g.,
 $Z'$ exchange~\cite{Zprime} or Higgs/Higgsino decay~\cite{CerdenoSeto}
 also have been pointed out.
Here, let us remind that
 Dirac masses of fermions are proportional to their Yukawa couplings
 as well as a vacuum expectation value (VEV) of the relevant Higgs field.
Hence, the smallness of a mass might be due to not a small Yukawa coupling
 but a small VEV of the Higgs field.
Such a situation is indeed realized in some THDM.
For example,
 in Type-II THDM with a large $\tan\beta$,
 the mass hierarchy between up-type quark and down-type quark
 can be explained by the ratio of Higgs VEVs, and
 when $\tan\beta \sim 40$,
 Yukawa couplings of top and bottom quark are
 same scale of order of unity~\cite{Hashimoto:2004xp}.
Similarly, there is a possibility that
 smallness of the neutrino masses comparing to those of quarks and
 charged leptons is originating from an extra Higgs
 doublet with the tiny VEV.
This idea is that
 neutrino masses are much smaller than other fermions
 because the origin of them comes from different VEV of different
 Higgs doublet, and then
 we do not need
 extremely tiny neutrino Yukawa couplings.
Let us call this kind of
 model~\cite{Ma,Nandi,Ma:2006km,Davidson:2009ha,Logan:2010ag,HabaHirotsu}
 neutrinophilic Higgs doublet model.
Especially, in models in Refs.\cite{Ma, HabaHirotsu},
 tiny Majorana neutrino masses are obtained
 through a TeV scale Type-I seesaw mechanism
 without requiring tiny Yukawa couplings.

Notice that neutrino Yukawa couplings in neutrinophilic Higgs doublet models
 do not need to be so small.
This fact has significant implication to leptogenesis.
The CP violation of right-handed neutrino decay
 is proportional to neutrino Yukawa coupling squared.
We can obtain a large CP violation for such large neutrino Yukawa couplings.
This opens new possibility of low scale thermal leptogenesis.
In this paper, we will show that
 CP asymmetry is enhanced and thermal leptogenesis suitably works
 in multi-Higgs models with a neutrinophilic Higgs doublet field,
 where
 the tiny VEV of the neutrinophilic Higgs field has
 equivalently larger neutrino Yukawa couplings, and
 then TeV-scale seesaw works well.
We will show that
 the thermal leptogenesis suitably works at low energy scale as
 avoiding enhancement of lepton number violating wash out effects.
We will also point out that
 thermal leptogenesis in gravity mediated SUSY breaking works well
 without confronting gravitino problem in a supersymmetric model.

\section{Neutrinophilic Higgs doublet models}

\subsection{Minimal neutrinophilic THDM }
\label{subsec:Minimal}

Let us show a two Higgs doublet model,
 which we call neutrinophilic THDM model,
 originally
 suggested in Ref.~\cite{Ma}.
In the model, one additional Higgs doublet $\Phi_{\nu}$, which gives only neutrino Dirac masses,
 besides the SM Higgs doublet $\Phi$ and a discrete $Z_2$-parity are introduced.
The $Z_2$-symmetry charges (and also lepton number) are assigned
 as the following table.
\begin{table}[h]
\centering
\begin{center}
\begin{tabular}{|l|c|c|} \hline
fields  &  $Z_{2}$-parity & lepton number \\ \hline\hline
SM Higgs doublet, $\Phi$  &  $+$ &  0 \\ \hline
new Higgs doublet, $\Phi_{\nu}$
 &  $-$ & 0 \\ \hline
right-handed neutrinos, $N$  &  $-$ & $1$ \\ \hline
others  &  $+$ & $\pm 1$: leptons, $0$: quarks \\ \hline
\end{tabular}
\end{center}
\end{table}
Under the discrete symmetry,
 Yukawa interactions are given by
\begin{eqnarray}
{\mathcal L}_{yukawa}=y^{u}\bar{Q}_L \Phi U_{R}
 +y^d \bar{Q}_{L}\tilde{\Phi}D_{R}+y^{l}\bar{L}\Phi E_{R}
 +y^{\nu}\bar{L}\Phi_{\nu}N+ \frac{1}{2}M \bar{N^{c}}N
 +{\rm h.c.}\;
\label{Yukawa:nuTHDM}
\end{eqnarray}
where
 $\tilde{\Phi}=i\sigma_{2}\Phi^{\ast}$, and
 we omit a generation index.
$\Phi_\nu$ only couples with $N$ by the $Z_2$-parity so that
 flavor changing neutral currents (FCNCs)
 are suppressed.
Quark and charged lepton sectors are the same as Type-I THDM,
 but notice that
 this neutrinophilic THDM is quite different from
 conventional Type-I, II, X, Y THDMs~\cite{2hdm}.

The Higgs potential of the neutrinophilic THDM is given by
\begin{align}
V^\text{THDM}
&
= m_\Phi^2 \Phi^\dag \Phi + m_{\Phi_\nu}^2 \Phi_\nu^\dag \Phi_\nu
-m_3^2\left(\Phi^\dag \Phi_\nu+\Phi_\nu^\dag \Phi\right)
+\frac{\lambda_1}2(\Phi^\dag \Phi)^2
+\frac{\lambda_2}2(\Phi_\nu^\dag \Phi_\nu)^2\nonumber \\
&\qquad+\lambda_3(\Phi^\dag \Phi)(\Phi_\nu^\dag \Phi_\nu)
+\lambda_4(\Phi^\dag \Phi_\nu)(\Phi_\nu^\dag \Phi)
+\frac{\lambda_5}2\left[(\Phi^\dag \Phi_\nu)^2
+(\Phi_\nu^\dag \Phi)^2\right].
\label{Eq:HiggsPot}
\end{align}
The $Z_2$-symmetry is softly broken by $m_3^2$.
Taking a parameter set,
\begin{equation}
m_\Phi^2 < 0, ~~~ m_{\Phi_\nu}^2 > 0, ~~~ |m_{3}^2| \ll m_{\Phi_\nu}^2,
\end{equation}
we can obtain the VEV hierarchy of Higgs doublets,
\begin{equation}
v^2 \simeq \frac{-m_\Phi^2}{\lambda_1},
 ~~~ v_{\nu} \simeq \frac{-m_{3}^2 v}{ m_{\Phi_\nu}^2 + (\lambda_3 + \lambda_4 + \lambda_5 ) v^2} ,
\end{equation}
 where we have decomposed the SM Higgs doublet $\Phi$ and
 the extra Higgs doublet $\Phi_{\nu}$  as
\begin{eqnarray}
\Phi =
\left(
\begin{array}{c}
v+ \frac{1}{\sqrt{2}}\phi^{0}\\
\phi^{-}
\end{array}
\right)
,\;\;
\Phi_{\nu}=
\left(
\begin{array}{c}
v_{\nu}+\frac{1}{\sqrt{2}}\phi^{0}_{\nu} \\
\phi^{-}_{\nu}
\end{array}
\right).
\end{eqnarray}
When we take values of parameters as
 $m_\Phi \sim 100$ GeV,
 $m_{\Phi_\nu} \sim 1$ TeV,
 and $|m_{3}^2| \sim 10$ GeV$^2$,
 we can obtain $v_\nu \sim 1$ MeV.
The smallness of $|m_{3}^2|$ is guaranteed by the
 ``softly-broken'' $Z_2$-symmetry.

For a very large $\tan \beta=v/v_{\nu} (\gg 1)$ limit we are interested in,
the five physical Higgs boson states and those masses are respectively given by
\begin{eqnarray}
 H^\pm \simeq \ [\phi_\nu^\pm] ,       && ~~~ m^2_{H^\pm} \simeq m_\nu^2 + \lambda_3 v^2 , \\
 A \simeq  {\rm Im} [\phi_{\nu}^0] ,   && ~~~ m^2_A \simeq m_\nu^2 +
  (\lambda_3 + \lambda_4+ \lambda_5) v^2 ,       \\
 h \simeq {\rm Re} [\phi^0] ,          && ~~~ m^2_{h} \simeq 2 \lambda_1 v^2  , \\
 H \simeq {\rm Re} [\phi_\nu^0] ,      && ~~~ m^2_{H} \simeq m_\nu^2 + (\lambda_3 + \lambda_4+\lambda_5) v^2 ,
\end{eqnarray}
 where negligible ${\cal O}(v_{\nu}^2)$ and ${\cal O}(m_3^2)$ corrections are omitted.
Notice that
 $\tan\beta$ is extremely large so that
 the SM-like Higgs $h$ is almost originated from $\Phi$,
 while other physical Higgs particles, $H^\pm, H, A$, are almost
 originated from $\Phi_\nu$.
Since $\Phi_\nu$ has Yukawa couplings only with neutrinos and
 lepton doublets,
 remarkable phenomenology can be expected which
 is not observed in other THDMs.
For example, lepton flavor violation (LFV) processes
 and oblique corrections are
 estimated in Ref.~\cite{Ma}, and
 charged Higgs processes in collider experiments
 are discussed
 in Refs.~\cite{Davidson:2009ha, Logan:2010ag}~\footnote{
The model deals with Dirac neutrino version
 in neutrinophilc THDM, but
 phenomenology of charged lepton has a similar region in part.}.

The neutrino masses including one-loop radiative corrections~\cite{Ma:2006km} are estimated as
\begin{equation}
(m_\nu)_{ij} = \sum_k\frac{y^{\nu}_{ik} v_\nu y^{\nu T}{}_{kj} v_\nu}{M_k} + \sum_k {y^\nu_{ik} y^{\nu T}{}_{kj} M_{k} \over 16
 \pi^{2}}
\left[
{m_R^{2} \over m_R^{2}-M_{k}^{2}} \ln {m_R^{2} \over M_{k}^{2}} -
{m_I^{2} \over m_I^{2}-M_{k}^{2}} \ln {m_I^{2} \over M_{k}^{2}} \right],
\end{equation}
where $m_R$ and $m_I$ are the masses of $ {\rm Re} [\phi^{0}]$ and
 ${\rm Im} [\phi_\nu^{0}]$ respectively.
It is easy to see the tree level contribution gives ${\cal O} (0.1)$ eV neutrino masses
 for $M_k \sim 1$ TeV, $v_\nu \sim 1$ MeV and $y^{\nu} = {\cal O}(1)$.
The one-loop contribution is induced for a nonvanishing $\lambda_5$.
When
 $m_R^{2} - m_I^{2} = 2 \lambda_5 v^{2} \ll m_0^{2} = (m_R^{2} + m_I^{2})/2$,
\begin{equation}
({m}_\nu)_{ij} = {\lambda_5 v^{2} \over 8 \pi^{2}}
\sum_k {y^\nu_{ik} y^\nu_{jk} M_{k} \over m_0^{2} - M_{k}^{2}} \left[
1 - {M_{k}^{2} \over m_0^{2}-M_{k}^{2}} \ln {m_0^{2} \over M_{k}^{2}}
 \right],
\end{equation}
and it shows
\begin{eqnarray}
&&({m}_\nu)_{ij} = {\lambda_5 v^{2} \over 8 \pi^{2}}
\sum_k {y\nu_{ik} y^nu_{jk} \over M_{k}} \left[
\ln {M_{k}^{2} \over m_0^{2}} - 1 \right], \;\; (M_{k}^{2} \gg m_0^{2}), \\
&&
({m}_\nu)_{ij} = {\lambda_5 v^{2} \over 8 \pi^{2} m_0^{2}}
\sum_k y^\nu_{ik} y^\nu_{jk} M_{k}, \;\; (m_0^{2} \gg M_{k}^{2}), \\
&&
({m}_\nu)_{ij} \simeq {\lambda_5 v^{2} \over 16 \pi^{2}}
\sum_k {y^\nu_{ik} y^\nu_{jk} \over M_{k}}, \;\; (m_0^{2} \simeq M_{k}^{2}).
\end{eqnarray}
Thus, when the masses of Higgs bosons (except for $h$) and right-handed neutrinos are
 ${\mathcal O}(1)$ TeV,
 light neutrino mass scale of order ${\mathcal O}(0.1)$ eV is
 induced with
 $\lambda_5 \sim 10^{-4}$.
Thus, whether tree-level effect is larger than
 loop-level effect or not
 is determined by
 the magnitude of $\lambda_5$ (and $m_A, m_H$),
 which contribute one-loop diagram.

\subsection{A UV theory of neutrinophilic THDM }
\label{subsec:HabaHirotsu}

Here let us show a model in Ref.~\cite{HabaHirotsu} as
 an example of UV theory of the
 neutrinophilic THDM.
This model is constructed by introducing
 one gauge singlet scalar field $S$,
 which has a lepton number,
 and $Z_3$-symmetry shown
 as the following table.
\begin{table}[h]
\centering
\begin{center}
\begin{tabular}{|l|c|c|} \hline
fields  &  $Z_{3}$-charge & lepton number \\ \hline\hline
SM Higgs doublet, $\Phi$  &  1 & 0 \\ \hline
new Higgs doublet, $\Phi_{\nu}$
 &  $\omega^{2}$ & 0 \\ \hline
new Higgs singlet, $S$
 &  $\omega$  &
 $-2$ \\ \hline
right-handed neutrinos, $N$  &  $\omega$  & 1 \\ \hline
others  &  1  &  $\pm 1$: leptons, $0$: quarks \\ \hline
\end{tabular}
\end{center}
\end{table}
Under the discrete symmetry,
 Yukawa interactions are given by
\begin{eqnarray}
{\mathcal L}_{yukawa}=y^{u}\bar{Q}^{L}\Phi U_{R}
 +y^d \bar{Q}_{L}\tilde{\Phi}D_{R}+y^{l}\bar{L}\Phi E_{R}
 +y^{\nu}\bar{L}\Phi_{\nu}N+\frac{1}{2}y^{N}S\bar{N^{c}}N +{\rm h.c.} .
\label{22}
\end{eqnarray}
The Higgs potential can be written as
\begin{eqnarray}
V=&m_\Phi^{2}|\Phi|^{2}+m_{\Phi_{\nu}}^{2}|\Phi_{\nu}|^{2}-m_S^{2}|S|^{2}

 -\lambda S^{3}-\kappa S\Phi^{\dagger}\Phi_{\nu}\nonumber\\
     &+\frac{\lambda_{1}}{2}|\Phi|^{4}+\frac{\lambda_{2}}{2}|\Phi_{\nu}|^{4}+\lambda_{3}|\Phi|^{2}|\Phi_{\nu}|^{2}+\lambda_{4}|\Phi^{\dagger}\Phi_{\nu}|^{2}\nonumber\\
     &+\lambda_{S}|S|^4+
  \lambda_{\Phi}|S|^{2}|\Phi|^{2}+\lambda_{\Phi_{\nu}}|S|^{2}|\Phi_{\nu}|^{2} + {\it h.c.} .
\label{Potential:HabaHirotsu}
\end{eqnarray}
$Z_3$-symmetry forbids
 dimension four operators,
 $(\Phi^\dagger \Phi_{\nu})^{2}$,
 $\Phi^\dagger \Phi_{\nu}|\Phi|^2$,
 $\Phi^\dagger \Phi_{\nu}|\Phi_{\nu}|^2$,
 $S^4$, $S^2|S|^2$, $S^2|\Phi|^2$,
 $S^2|\Phi_{\nu}|^2$,
 and dimension two or three operators,
 $\Phi^\dagger \Phi_{\nu}$,
 $S|\Phi|^{2}$, $S|\Phi_\nu|^{2}$.
Although there might be introduced small soft breaking
 terms such as $m_3'^2\Phi^\dagger \Phi_{\nu}$ to avoid
 domain wall problem,
 we omit them here, for simplicity.
It has been shown that, with $\kappa \sim 1$ MeV, the desirable hierarchy of
 VEVs
\begin{eqnarray}
&& v_s \equiv \langle S \rangle \sim  1 \;\hbox{TeV},\;\;\;
  v \sim  100 \;\hbox{GeV}, \;\;\;
  v_{\nu} \sim  1 \;\hbox{MeV},
\end{eqnarray}
 and neutrino mass
\begin{equation}
 m_\nu \simeq \frac{y^{\nu2}  v_{\nu}^{2}} {M_N } .
\end{equation}
 with Majorana mass of right-handed neutrino $M_N = y^N v_s$
 can be realized~\cite{HabaHirotsu}.
This is so-called
 Type-I seesaw mechanism in a TeV scale, when
 coefficients
 $y^\nu$ and $y^N$ are assumed to be
 of order one.
The masses of scalar and pseudo-scalar mostly from $S$ are
 given by
\begin{eqnarray}
 m^{2}_{H_{S}} &=& m_3^{2}+2\lambda_{S} v_s^2 , \;\;\;\;\;\;
 m^{2}_{A_{S}} = 9 \lambda v_s ,
\end{eqnarray}
 in the potential Eq.~(\ref{Potential:HabaHirotsu}) without CP violation.
For parameter region with $v_s \gg 1$ TeV,
 both scalar and pseudo-scalar are heavier than other particles.
After integrating out $S$, thanks to the $Z_3$-symmetry,
 the model ends up with an effectively neutrinophilic THDM
 with approximated $Z_2$-symmetry, $\Phi \to \Phi, \Phi_\nu \to -\Phi_\nu$.
Comparing to the neutrinophilic THDM,
 the value of $m_3^2$, which is a soft $Z_2$-symmetry breaking
 term, is expected to be
 $\kappa v_s$.
$\lambda_{5}$ is induced by integrating out $S$, which is
 estimated as ${\mathcal O}(\kappa^2/m_S^2)\sim 10^{-12}$.
Thus, the neutrinophilic THDM has an approximate $Z_2$-symmetry.

As for the neutrino mass induced from one-loop diagram \footnote{
We would like to thank J. Kubo and H. Sugiyama
 for letting us notice this topic.
},
 UV theory induces small $\lambda_5\sim 10^{-12}$
 due to $Z_3$-symmetry,
 so that radiative induced neutrino mass from
 one-loop diagram
 is estimated as $\lambda_5 v^2/(4\pi)^2M \sim 10^{-4}$
 eV.
This can be negligible comparing to light neutrino mass
 which is induced from
 tree level Type-I seesaw mechanism.
The tree level neutrino mass is
\begin{equation}
m_\nu^{tree}
 \sim {y_\nu^2 v_\nu^2 \over M}\sim
{y_\nu^2 \kappa^2v^2 \over v_s^2M},
\end{equation}
where we input
 $v_\nu \sim {\kappa v \over v_s}$.
On the other hand, one-loop induced neutrino mass is
 estimated as
\begin{equation}
m_\nu^{loop} \sim {\lambda_5 y_\nu^2 \over 16\pi^2}{v^2 \over M}
\sim {y_\nu^2 \over 16\pi^2}{\kappa^2 v^2 \over M^2 M}.
\end{equation}
Putting $M\sim v_s$,
\begin{equation}
{m_\nu^{loop} \over m_\nu^{tree}}\sim
{1 \over 16\pi^2},
\end{equation}
which shows loop induced neutrino mass
 is always smaller than tree level mass
 if UV theory is the model of Ref.~\cite{HabaHirotsu}.

\subsection{Supersymmetic extension of neutrinophilic Higgs doublet model}
\label{subsec:Super}

Now let us show the supersymmetric extension of the neutrinophilic
 Higgs doublet model.
The supersymmetric extension is straightforward by
 extending its Higgs sector to be a four Higgs doublet model.
The superpotential is given by
\begin{eqnarray}
{\mathcal W}&=&y^{u}\bar{Q}^{L}H_u U_{R}
 +y^d \bar{Q}_{L}{H_d}D_{R}+y^{l}\bar{L}H_d E_{R}
 +y^{\nu}\bar{L}H_{\nu}N+M {N^{}}^2 \nonumber \\
&& +\mu H_uH_d + \mu' H_\nu H_{\nu'}
+\rho H_u H_{\nu'} + \rho' H_\nu H_d,
\end{eqnarray}
where $H_u$ ($H_d$) is Higgs doublet which gives mass of
 up- (down-) sector.
$H_\nu$ gives neutrino Dirac mass and $H_{\nu'}$ does not
 contribute to fermion masses.
For the $Z_2$-parity,
 $H_u, H_d$ are even, while $H_\nu, H_{\nu'}$ are odd.
The $Z_2$-partity is softly broken by the $\rho$ and $\rho'$.
We assume that

 $|\mu|, |\mu'| \gg |\rho|, |\rho'|$, and
 SUSY breaking soft squared masses can trigger
 suitable electro-weak symmetry breaking.
The Higgs potential is given by
\begin{eqnarray}
 V &=& (|\mu|^2 +|\rho|^2) H_u^\dag H_u + (|\mu|^2+|\rho'|^2) H_d^\dag H_d
      + (|\mu'|^2 +|\rho'|^2) H_{\nu}^\dag H_{\nu} + (|\mu'|^2+|\rho|^2) H_{\nu'}^\dag H_{\nu'}  \nonumber \\
  && + \frac{g_1^2}{2} \left( H_u^\dag \frac{1}{2} H_u - H_d^\dag\frac{1}{2} H_d
     + H_{\nu}^\dag \frac{1}{2} H_{\nu} - H_{\nu'}^\dag \frac{1}{2}H_{\nu'} \right)^2  \nonumber \\
  && + \sum_a \frac{g_2^2}{2} \left( H_u^\dag \frac{\tau^a}{2} H_u + H_d^\dag\frac{\tau^a}{2} H_d
     + H_{\nu}^\dag \frac{\tau^a}{2} H_{\nu} + H_{\nu'}^\dag \frac{\tau^a}{2}H_{\nu'} \right)^2  \nonumber \\
  && + m_{H_u}^2 H_u^\dag H_u  + m_{H_d}^2 H_d^\dag H_d
     + m_{H_\nu}^2 H_{\nu}^\dag H_{\nu}+ m_{H_{\nu'}}^2 H_{\nu'}^\dag H_{\nu'} \nonumber \\
  && + B \mu H_u \cdot H_d + B' \mu' H_{\nu}\cdot H_{\nu'}
 + \hat{B} \rho H_u \cdot H_{\nu'} +
 \hat{B}' \rho' H_{\nu}\cdot H_{d}\nonumber\\
&& + \mu^* \rho H_d^\dag H_{\nu'}+\mu^* \rho' H_u^\dag H_{\nu}+
 \mu'^* \rho' H_{\nu'}^\dag H_{d}+\mu'^* \rho H_\nu^\dag H_{u}
 + {\it h.c.} ,
\end{eqnarray}
 where $\tau^a$ and dot represent a generator of $SU(2)$ and
 its anti-symmetric product respectively.
We assume Max.[$|\hat{B}\rho|, |\hat{B}'\rho'|, |\mu\rho|,
 |\mu'\rho|,|\mu\rho'|,|\mu'\rho'|$] $\sim {\mathcal O}(10)$
 GeV$^2$, which triggers VEV hierarchy between
 the SM Higgs doublet and neutrinophilic Higgs doublets.
Notice that
 quarks and charged lepton have small non-holomorphic Yukawa
 couplings with $H_\nu$, through one-loop diagrams
 associated with small mass parameters of $\hat{B}\rho,
 \hat{B}'\rho', \mu\rho, \mu'\rho, \mu\rho', \mu'\rho'$.
This situation is quite different from non-SUSY model,
 where these couplings are extremely suppressed by
 factor of $v_\nu/v$.
As for the gauge coupling unification,
 we must introduce extra particles, but
 anyhow,
 the supersymmetric extension of neutrinophilic Higgs doublet model
 can be easily
 constructed as shown above.

\section{Leptogenesis}

\subsection{A brief overview of thermal leptogenesis}

In the seesaw model, the smallness of the neutrino masses can be
 naturally explained by the small mixing
 between left-handed neutrinos and
 heavy right-handed Majorana neutrinos $N_i$.
The basic part of the Lagrangian in the SM
 with right-handed neutrinos is described as
\begin{eqnarray}
{\cal L}_{N}^{\rm SM}=-y^{\nu}_{ij}
\overline{l_{L,i}} \Phi N_j
-\frac{1}{2} \sum_{i} M_i \overline{ N^c_i} N_i + h.c. ,
\label{SMnuYukawa}
\end{eqnarray}
where $i,j=1,2,3$ denote the generation indices,
 $h$ is the Yukawa coupling,
 $l_L$ and $\Phi$ are the lepton and the Higgs doublets,
 respectively, and
 $M_i$ is the lepton-number-violating mass term
 of the right-handed neutrino $N_i$
 (we are working on the basis of
 the right-handed neutrino mass eigenstates).
With this Yukawa couplings, the mass of left-handed neutrino is expressed by the well-known formula
\begin{equation}
m_{ij} = \sum_k \frac{y^{\nu}_{ik}v y^{\nu}{}^T_{kj}v}{M_k}.
\end{equation}

The decay rate of the lightest right-handed neutrino is given by
\begin{eqnarray}
\Gamma_{N_1} = \sum_j\frac{y^{\nu}_{1j}{}^\dagger y^{\nu}_{j1}}{8\pi}M_1 = \frac{(y^{\nu}{}^\dagger y^{\nu})_{11}}{8\pi}M_1.
\end{eqnarray}
Comparing to the Friedmann equation for a spatially flat spacetime
\begin{equation}
H^2 = \frac{1}{3 M_P^2}\rho ,
\label{FriedmannEq}
\end{equation}
 with the energy density of the radiation
\begin{equation}
\rho = \frac{\pi^2}{30}g_*T^4 ,
\end{equation}
 where $g_*$ denotes
 the effective degrees of freedom of relativistic particles and $M_P \simeq 2.4 \times 10^{18}$ GeV is the reduced Planck mass,
the condition of the out of equilibrium decay $\Gamma_{N_1} < \left.H\right|_{T=M_1}$ is rewritten as
\begin{eqnarray}
\tilde{m}_1 \equiv (y^{\nu}{}^\dagger y^{\nu})_{11} \frac{v^2}{M_1} <
\frac{8\pi v^2}{M_1^2} \left.H\right|_{T=M_1}
\equiv m_*
\end{eqnarray}
 with $ m_* \simeq 1\times 10^{-3}$ eV and $v=174$ GeV.

In the case of the hierarchical mass spectrum for right-handed neutrinos,
 the lepton asymmetry in the Universe is generated
 dominantly by CP-violating out of equilibrium decay of
 the lightest heavy neutrino, $N_1 \rightarrow l_L \Phi^*$
 and $ N_1 \rightarrow \overline{l_L} \Phi $.
Then, its CP asymmetry is given by~\cite{FandG}
\begin{eqnarray}
\varepsilon &\equiv&
\frac{\Gamma(N_1\rightarrow \Phi+\bar{l}_j)-\Gamma(N_1\rightarrow \Phi^*+l_j)}
{\Gamma(N_1\rightarrow \Phi+\bar{l}_j)+\Gamma(N_1\rightarrow \Phi^*+l_j)} \nonumber \\
 &\simeq&  -\frac{3}{8\pi}\frac{1}{(y^{\nu} y^{\nu}{}^{\dagger})_{11}}\sum_{i=2,3}
\textrm{Im}(y^{\nu}y^{\nu}{}^{\dagger})^2_{1i} \frac{M_1}{M_i}, \qquad \textrm{for} \quad M_i \gg M_1 .
\end{eqnarray}
Through the relations of the seesaw mechanism,
 this can be roughly estimated as
\begin{eqnarray}
\varepsilon
 \simeq \frac{3}{8\pi}\frac{M_1 m_3}{v^2} \sin\delta
 \simeq 10^{-6}\left(\frac{M_1}{10^{10}\textrm{GeV}}\right)
 \left(\frac{m_3}{0.05 \textrm{eV}}\right) \sin\delta,
 \label{epsilon}
\end{eqnarray}
where $m_3$ is the heaviest light neutrino mass normalized by
 $0.05$ eV which is a preferred to account for atmospheric neutrino
 oscillation data~\cite{atm}.

Using the above $\varepsilon$,
 the resultant baryon asymmetry generated via thermal leptogenesis
 is expressed as
\begin{equation}
\frac{n_b}{s} \simeq  C \kappa \frac{\varepsilon}{g_*}  ,
 \label{b-sRatio}
\end{equation}
where $\left. g_*\right|_{T=M_1} \sim 100$ ,
 the so-called dilution (or efficiency) factor $ \kappa \leq {\cal O}(0.1) $ denotes
 the dilution by wash out processes,
 the coefficient
\begin{equation}
C = \frac{8 N_f + 4 N_H}{22 N_f + 13 N_H} ,
\label{C}
\end{equation}
 with $N_f$ and $N_H$ being the number of fermion generation and Higgs doublet~\cite{C}
 is the factor of the conversion from lepton to baryon asymmetry
 by the sphaleron~\cite{KRS}.
In order to obtain the observed baryon asymmetry
 in our Universe $n_b/s \simeq 10^{-10}$~\cite{WMAP},
 the inequality $\varepsilon \gtrsim 10^{-7}$ is required.
This can be rewritten as $M_1 \gtrsim 10^9$ GeV,
 which is the so-called Davidson-Ibarra bound for models with
 hierarchical right-handed neutrino mass spectrum~\cite{LowerBound,Davidson:2002qv}.

\subsection{leptogenesis in neutrinophilic THDM }

Now
 we consider leptogenesis in the neutrinophilic THDM with the extra Higgs doublet $\Phi_{\nu}$
 described in Sec.~\ref{subsec:Minimal}.
The relevant interaction part of Lagrangian Eq.~(\ref{Yukawa:nuTHDM}) is expressed as
\begin{eqnarray}
{\cal L}_{N}=-y^{\nu}_{ij}
\overline{l_{L,i}} \Phi_{\nu} N_j
-\frac{1}{2} \sum_{i} M_i \overline{ N^c_i} N_i + h.c. .
\label{Yukawa:nuTHDM(2)}
\end{eqnarray}
The usual Higgs doublet $\Phi$ in Eq.~(\ref{SMnuYukawa}) is replaced by
 new Higgs doublet
 $\Phi_{\nu}$.
Again, we are working on the basis of
 the right-handed neutrino mass eigenstates.
Then, with these Yukawa couplings,
 the mass of left-handed neutrino is given by
\begin{equation}
m_{ij} = \sum_k \frac{y^{\nu}_{ik}v_{\nu} y^{\nu}{}^T_{kj}v_{\nu}}{M_k}.
\end{equation}
Thus, for a smaller VEV of $v_{\nu}$, a larger $y^{\nu}$ is required.

The Boltzmann equation for the lightest right-handed neutrino $N_1$,
 which is denoted by $N$ here, is given by
\begin{eqnarray}
\dot{n}_N+3Hn_N
&=& -\gamma (N\rightarrow L\Phi_{\nu}) - \gamma (N\rightarrow \bar{L}\Phi_{\nu}^*)
\qquad\textrm{:decay}\nonumber\\
&& +\gamma (L\Phi_{\nu}\rightarrow N) + \gamma (\bar{L}\Phi_{\nu}^* \rightarrow N)
\qquad \textrm{:inverse decay}\nonumber\\
&& -\gamma (N L \rightarrow A \Phi_{\nu})-\gamma ( N \Phi_{\nu} \rightarrow L A)
 -\gamma (N \bar{L} \rightarrow A \Phi_{\nu}^* )-\gamma ( N \Phi_{\nu}^* \rightarrow \bar{L} A)  \nonumber\\
&& +\textrm{inverse  processes} \qquad \qquad \qquad \qquad  : \textrm{s-channel scattering} \nonumber\\
&& -\gamma (N L \rightarrow A \Phi_{\nu})-\gamma ( N \Phi_{\nu} \rightarrow L A) -\gamma ( N A \rightarrow L \Phi_{\nu})  \nonumber\\
&& -\gamma (N \bar{L} \rightarrow A \Phi_{\nu}^* )-\gamma ( N \Phi_{\nu}^* \rightarrow \bar{L} A) -\gamma ( N A \rightarrow \bar{L} \Phi_{\nu}^*)  \nonumber\\
&& +\textrm{inverse  processes} \qquad \qquad \qquad \qquad
: \textrm{t-channel scattering} \nonumber\\
&& -\gamma (N N \rightarrow {\rm Final}) + \gamma ({\rm Final} \rightarrow N N)
: {\rm annihilation} \nonumber\\
&=& -\Gamma_D (n_N-n_N^{eq})-\Gamma_{scat} (n_N-n_N^{eq})
 -\langle\sigma v(\rightarrow \Phi, \Phi_{\nu})\rangle  (n_N^2-n_N^{eq}{}^2)
\label{Boltzman:N}
\end{eqnarray}
 where $\Phi, \Phi_{\nu}$ and $A$ denote the Higgs bosons, the neutrinophilic
 Higgs bosons and gauge bosons, respectively.
Notice that
 usual $\Delta L =1$ lepton number violating scattering processes
 involving top quark is absent in this model,
 because $\Phi_{\nu}$ has
 neutrino Yukawa couplings.
Although the annihilation processes $(N N \rightarrow {\rm Final})$ is noted in Eq.~(\ref{Boltzman:N}),
 in practice, this is not relevant because
 the coupling $y^{\nu}_{i1}$ is so small, as will be shown later,
 to satisfy
 the out of equilibrium decay condition.

The Boltzmann equation for the lepton asymmetry $L \equiv l-\bar{l}$ is given by
\begin{eqnarray}
 && \dot{n}_L+3H n_L  \nonumber \\
&=& \gamma(N\rightarrow l\Phi_{\nu}) - \gamma( \bar{N} \rightarrow \bar{l}\Phi_{\nu}^*) \nonumber \\
 && -\{ \gamma(l\Phi_{\nu}\rightarrow N) - \gamma( \bar{l}\Phi_{\nu}^* \rightarrow \bar{N}) \} \qquad\textrm{:decay and inverse decay} \nonumber \\
&& -\gamma ( l A \rightarrow N \Phi_{\nu} )+\gamma ( \bar{l} A \rightarrow \bar{N} \Phi_{\nu}^* )
 -\gamma (N l \rightarrow A \Phi_{\nu}) \nonumber \\
&& +\gamma ( \bar{N} \bar{l} \rightarrow A \Phi_{\nu}^*)  \quad\textrm{:s-channel $\Delta L=1$ scattering} \nonumber\\
&& -\gamma (N l \rightarrow A \Phi_{\nu})+\gamma (\bar{N} \bar{l} \rightarrow A \Phi_{\nu}^*)-\gamma (l A \rightarrow N \Phi_{\nu})+\gamma (\bar{l} A \rightarrow \bar{N} \Phi_{\nu}^*) \nonumber\\
&&  -\gamma ( l \Phi_{\nu} \rightarrow N A)+\gamma ( \bar{l} \Phi_{\nu}^* \rightarrow \bar{N} A)  \quad\textrm{:t-channel $\Delta L=1$ scattering} \nonumber \\
&& +\gamma( \bar{l}\bar{l} \rightarrow \Phi_{\nu}^*\Phi_{\nu}^*)-\gamma(ll\rightarrow \Phi_{\nu}\Phi_{\nu}) \nonumber \\ &&
  +2 \{ \gamma(\bar{l}\Phi_{\nu}^*\rightarrow l\Phi_{\nu})- \gamma(l\Phi_{\nu}\rightarrow \bar{l}\Phi_{\nu}^*) \} \quad\textrm{:t and s-channel $\Delta L=2$ scattering}  \nonumber  \\
&=& \varepsilon\Gamma_D(n_N-n_N^{eq}) - \Gamma_W n_L
\end{eqnarray}
where
\begin{equation}
\Gamma_W =
 \frac{1}{2}\frac{n_N^{eq}}{n_{\gamma}^{eq}}\Gamma_N + \frac{n_N}{n_N^{eq}}\Gamma_{\Delta L=1,t}
 + 2\Gamma_{\Delta L=1,s}+ 2\Gamma_{\Delta L=2}
\end{equation}
 is the wash-out rate.

The condition of the out of equilibrium decay is given as
\begin{eqnarray}
\tilde{m}_1 \equiv (y^{\nu}{}^\dagger y^{\nu})_{11}\frac{v_{\nu}^2}{M_1} <
\frac{8\pi v_{\nu}^2}{M_1^2} \left.H\right|_{T=M_1}
\equiv m_* \left(\frac{v_{\nu}}{v}\right)^2
\end{eqnarray}
Notice that for $v_{\nu} \ll v$ the upper bound on $\tilde{m}_1$ becomes more stringent,
 which implies that the lightest left-handed neutrino mass is almost vanishing $m_1 \simeq 0$.
Alternatively the condition can be expressed as
\begin{equation}
 ( y^{\nu}{}^{\dagger} y^{\nu})_{11} < 8 \pi \sqrt{ \frac{\pi^2 g_*}{90} }\frac{M_1}{M_P} .
 \label{OoEqDecay}
\end{equation}
Hence, for the TeV scale $M_1$,
 the value of $ (y^{\nu}{}^{\dagger} y^{\nu} )_{11}$ must be very small,
 which can be realized by taking all $y^{\nu}_{i1}$ to be small.
Under such neutrino Yukawa couplings $y^{\nu}_{i1} \ll y^{\nu}_{i2}, y^{\nu}_{i3}$ and
 hierarchical right-handed neutrino mass spectrum,
 the CP asymmetry,
\begin{eqnarray}
\varepsilon
&\simeq & -\frac{3}{8\pi}\frac{1}{(y^{\nu}{}^{\dagger}y^{\nu})_{11}}
\left(\textrm{Im}(y^{\nu}{}^{\dagger}y^{\nu})^2_{12} \frac{M_1}{M_2}
 + \textrm{Im}(y^{\nu}{}^{\dagger}y^{\nu})^2_{13} \frac{M_1}{M_3} \right)  \nonumber \\
& \simeq & -\frac{3}{8\pi}\frac{m_{\nu} M_1}{v_{\nu}^2} \sin\theta \nonumber \\
& \simeq & -\frac{3}{16\pi} 10^{-6} \left(\frac{0.1 {\rm GeV}}{v_{\nu}}\right)^2
  \left(\frac{M_1}{100 {\rm GeV}}\right)
  \left(\frac{m_{\nu}}{0.05 {\rm eV}}\right) \sin\theta ,
\label{CPasym}
\end{eqnarray}
 is significantly enhanced due to
 large Yukawa couplings $y^{\nu}_{2i}$ and $y^{\nu}_{3i}$ as well as
 the tiny Higgs VEV $v_{\nu}$ .
The thermal averaged interaction rate of $\Delta L =2$ scatterings is expressed as
\begin{eqnarray}
 \Gamma^{(\Delta L =2)}
 = \frac{1}{n_{\gamma}}
  \frac{T}{32 \pi (2 \pi )^4}
 \int ds \sqrt{s} K_1\left(\frac{\sqrt{s}}{T} \right) \int \frac{d \cos\theta}{2}\sum \overline{ |{\cal M}|^2}
\end{eqnarray}
 with
\begin{eqnarray}
\sum \overline{|{\cal M}|^2}
 &=& 2 \overline{|{\cal M}_{\rm t}|^2} + 2 \overline{|{\cal M}_{\rm
 s}|^2}
 \simeq \sum_{j,(\alpha, \beta)} 2 |y^{\nu}_{\alpha j} y^{\nu}_{\beta
  j}{}^{\dagger}|\frac{s}{M_{N_j}^2}, \quad  \textrm{ for} \quad s \ll
  M_j^2 \ .
\end{eqnarray}
The decoupling condition
\begin{eqnarray}
 \Gamma^{(\Delta L =2)}
   < \sqrt{\frac{\pi^2 g_*}{90}} \frac{T^2}{M_P},
\end{eqnarray}
 for $T < M_1$ is rewritten as
\begin{eqnarray}
 \sum_i \left(\sum_j \frac{  y^{\nu}_{ij} y^{\nu}_{ji}{}^{\dagger} v_{\nu}^2}{M_j}\right)^2
   < 32 \pi^3 \zeta(3) \sqrt{\frac{\pi^2 g_*}{90}} \frac{v_{\nu}^4}{T M_P} .
 \label{L2DecouplingCondition}
\end{eqnarray}
For lower $v_{\nu}$, $\Delta L =2$ wash out processes are more significant.
Inequality~(\ref{L2DecouplingCondition}) gives
 the lower bound on $v_{\nu}$
 in order to avoid too strong wash out.

We here summarize all conditions for successful thermal leptogenesis,
 and the result is presented in Fig.~\ref{fig:AvailableRegion}.
The horizontal axis is the VEV of neutrino Higgs $v_{\nu}$
 and the vertical axis is the mass of the lightest right-handed
 neutrino, $M_1$.
In the red brown region, the lightest right-handed neutrino decay into Higgs boson $H$ with
 assuming $M_H= 100$ GeV, and lepton
 is kinematically not allowed.
In turquoise region corresponds to inequality~(\ref{L2DecouplingCondition}), $\Delta L=2$ wash out effect is too strong.
The red and green line is contour of the CP asymmetry of $\varepsilon=10^{-6}$ and $10^{-7}$,
 respectively, with
 the lightest right-handed neutrino decay in
 hierarchical right-handed neutrino mass spectrum.
Thus, in the parameter region above the line of $\varepsilon = 10^{-7}$,
 thermal leptogenesis easily works
 even with hierarchical masses of right-handed neutrinos.
For the region below the line of $\varepsilon = 10^{-7}$,
 the resonant leptogenesis mechanism~\cite{ResonantLeptogenesis}, where CP asymmetry is enhanced resonantly
 by degenerate right-handed neutrino masses, may work.
Here we stress that, for $v_{\nu} \ll 100$ GeV,
 the required degree of mass degeneracy is considerably milder than
 that for the original resonant leptogenesis.

\begin{figure}
    \centerline{\includegraphics{Lepto.eps}}
\caption{
 Available region for leptogenesis.
 The horizontal axis is the VEV of neutrino Higgs $v_{\nu}$
 and the vertical axis is the mass of the lightest right-handed neutrino mass $M_1$.
 In the red brown region, the lightest right-handed neutrino decay into Higgs boson $\Phi_{\nu}$ and lepton
 is kinematically forbidden.
 In turquoise region, $\Delta L=2$ wash out effect is too strong.
 The red and green line is contour of the CP asymmetry of $\varepsilon=10^{-6}$ and $10^{-7}$,
 respectively, with the lightest right-handed neutrino decay in
 hierarchical right-handed neutrino mass spectrum.
}
\label{fig:AvailableRegion}
\end{figure}

\subsection{Constraints on an UV theory}

Let us suppose that neutrinophilic THDM is derived from a model reviewed
 in Sec.~\ref{subsec:HabaHirotsu} by integrated out a singlet field $S$.
If $S$ is relatively light, thermal leptogenesis discussed above could be affected.
That is the annihilation processes of $N_1$ which has been justifiably ignored in Eq.~(\ref{Boltzman:N}).
However, the annihilation could take place more efficiently
 via S-channel $S$ scalar exchange processes in the UV theory~\cite{HabaHirotsu}.

For example,
 the annihilation $N_1 N_1 \rightarrow \Phi_{\nu}\Phi_{\nu}^*$
 with the amplitude
\begin{eqnarray}
 \overline{ |{\cal M}|}^2
 = \left| \frac{y^{\nu}_1 \lambda_{\Phi_{\nu}} v_s}{ s - M_{H_S}^2 -i M_{H_S} \Gamma_{H_S}}\right|^2 \frac{s-4 M_1^2}{4} ,
\end{eqnarray}
 would not be in equilibrium, if
\begin{eqnarray}
\lambda_{\Phi_{\nu}}
 \lesssim
 40 \frac{M_{H_S}^2}{M_1^{3/2} M_P^{1/2}}
 \simeq 0.1 \left(\frac{10^5 \, {\rm GeV}}{M_1}\right)^{3/2}\left(\frac{M_{H_S}}{10^7 \, {\rm GeV}}\right)^2 ,
\end{eqnarray}
 for $M_S \gg T > M_1$ is satisfied.
Here, $\Gamma_{H_S}$ denotes the decay width of $H_S$.
Constraints on other parameters such as $\lambda_\Phi$ and
 $\kappa$ can be similarly obtained.

\section{Supersymetric case:
 Reconciling to thermal leptogenesis, gravitino problem and neutralino dark matter}

As we have shown in Sec.~\ref{subsec:Super}, it is possible to construct a supersymmetric model with $\Phi_{\nu}$.
A discrete symmetry, called ``R-parity'', is imposed in many supersymmetric models
 in order to prohibit rapid proton decay.
Another advantage of the conserved R-parity is that it guarantees the absolute stability of
 the LSP, which becomes a dark matter candidate.
In large parameter space of supergravity model with gravity mediated SUSY breaking,
 gravitino has the mass of ${\cal O}(100)$ GeV and decays into LSP (presumablly the lightest neutralino)
 at late time after BBN.
Then, decay products may affect the abundances of light elements produced during BBN.
This is so-called ``gravitino problem''~\cite{GravitinoProblem}.
To avoid this problem,
 the upper bound on the reheating temperature after inflation
\begin{equation}
 T_R < 10^6 - 10^7 \, {\rm GeV},
\label{ConstraintsOnTR}
\end{equation}
has been derived
 as depending on gravitino mass~\cite{GravitinoProblem2}.
By comparing Eq.~(\ref{ConstraintsOnTR}) with
 the CP violation in supersymmetric models with hierarchical right-handed neutrino masses,
 which is about four times larger than that in non-supersymmetric
 model~\cite{SUSYFandG} as,
\begin{eqnarray}
\varepsilon
 \simeq  -\frac{3}{2\pi}\frac{1}{(y^{\nu} y^{\nu}{}^{\dagger})_{11}}\sum_{i=2,3}
\textrm{Im}(y^{\nu}y^{\nu}{}^{\dagger})^2_{1i} \frac{M_1}{M_i},
\label{SUSYepsilon}
\end{eqnarray}
 it has been regarded that thermal leptogenesis
 through the decay of heavy right-handed neutrinos hardly work because of
 gravitino problem.

As we have shown in the previous section,
 a sufficient CP violation $\varepsilon = {\cal O}(10^{-6})$ can be realized
 for $v_{\nu} = {\cal O}(1)$ GeV
 in the hierarchical right-handed neutrino masses with $M_1$ of ${\cal O}(10^5 - 10^6)$ GeV.
This implies
 that the reheating temperature after inflation $T_R$ of ${\cal O}(10^6)$ GeV is high enough
 in order to produce right-handed neutrinos by thermal scatterings.
Thus, it is remarkable that
 SUSY neutrinophilic model with $v_{\nu} = {\cal O}(1)$ GeV
 can realize thermal leptogenesis
 in gravity mediated SUSY breaking with unstable gravitino.
In this setup,
 the lightest neutralino could be LSP and
 dark matter with the standard thermal freeze out scenario.

\begin{figure}
    \centerline{\includegraphic
                                {LeptoSUSY.eps}}
\caption{
 The same as Fig.~\ref{fig:AvailableRegion} but with Eq.~(\ref{SUSYepsilon}).
 The additional horizontal black dashed line represents a reference value of
 the upper bound on reheating temperature after inflation $T_R$ of $10^6$ GeV
 from gravitino overproduction.
 }
\label{fig:SUSYAvailableRegion}
\end{figure}

\section{Conclusion}

We have examined the possibility of thermal leptogenesis
 in neutrinophilic Higgs doublet models,
 whose tiny VEV gives neutrino Dirac mass term.
Thanks to the tiny VEV of the neutrinophilic
 Higgs field,
 neutrino Yukawa couplings are not necessarily small,
 instead, they tend to be large, and
 the CP asymmetry in the lightest right-handed neutrino decay
 is significantly enlarged.
Although
 the $\Delta L = 2$ wash out effect also could be enhanced
 simlitaneously,
 we have found
 the available parameter region where
 its wash out effect is avoided as
 keeping the CP asymmetry large enough.
In addition, in a supersymmetric
 neutrinophilic Higgs doublet model,
 we have pointed out that
 thermal leptogenesis in gravity mediated SUSY breaking works well
 without confronting gravitino problem.
Where the lightest neutralino could be LSP and
 dark matter with the standard thermal freeze out scenario.

\section*{Acknowledgements}
We would like to thank M.~Hirotsu for collaboration
 in the early stage of this work.
We are grateful to S.~Matsumoto, S.~Kanemura and K.~Tsumura
 for useful and helpful discussions.
This work is partially supported by Scientific Grant by Ministry of
 Education and Science, Nos. 20540272, 22011005, 20039006, 20025004
 (N.H.), and
 the scientific research grants from Hokkai-Gakuen (O.S.).


\section{Introduction}

High resolution observations acquired by the new generation of ground-based and space-borne telescopes are allowing us to study more and more details of the photospheric fine structure that characterizes sunspots \citep{Sol03, Sch09}. In particular, the investigation of the bright structures observed inside the dark umbrae, like umbral dots or light bridges (LBs), has become essential to understand the physical mechanisms that are responsible for the heat transport from the convection zone into the photosphere and for the diffusion of the magnetic field.

LBs are structures that rapidly intrude from the leading edge of penumbral filaments into the umbra. They are often linked to fragmentation of sunspots and to the final phases of the sunspot evolution \citep[e.g.,][and references therein]{Fal16,Fel16}, although not all spots exhibit this phenomenon. \citet{Sob97} classified LBs into strong or faint, depending on whether they separate the umbra into distinct umbral cores. Moreover, he distinguished between granular or filamentary LBs, depending on their internal structure. In particular, filamentary LBs have a central dark lane, i.e., a main dark axis along the LB, and branches of dark and bright hairs connected to the central lane \citep[e.g.,][]{Lit04}. Another classification was proposed by \citet{Tho04}, who distinguished between segmented and unsegmented LBs. The former, which are by far the most common, are characterized by bright granules separated by dark lanes perpendicular to the LB axis \citep{Ber03}, although the granular pattern visible in these structures differs in size, lifetime, and brightness from the quiet Sun granulation \citep[see, e.g.,][]{Lag14,Fal17}. On the contrary, unsegmented LBs resemble the elongated filaments forming the penumbra of sunspots, without any evidence of granular cells. In recent observations performed at the GREGOR telescope, \citet{Sch16} identified a new class of LBs, called thick LBs, with small transverse \textsc{Y}-shaped dark lanes similar to dark-cored penumbral filaments.

Doppler velocity measurements show the presence of upward plasma motions above LBs at the photopheric level, specially along the dark lane \citep{Rim97, Gio08, Rou10}. It seems that the plasma convection takes the lead in these regions, which are characterized by a magnetic field weaker and more horizontal than in the surrounding umbra \citep{Lit91, Lek97, Jur06}. However, it is still controversial whether LBs have a magneto-convective origin or are due to the field-free convection that penetrates into the strong umbral magnetic field from below the photosphere and forms a cusplike magnetic field near the visible surface \citep{Tho04, BI11}. Remarkable long-lasting plasma ejections or surge activities are observed in the chromosphere along some LBs \citep[e.g.][]{Bha07, Shi09, Lou14, Tor15}. Magnetic reconnection occurring in the low chromosphere and the upper photosphere is thought to originate this small-scale eruptive activity above LBs \citep{Son17}.

Recently, \citet{Kle13} analyzed unusual filamentary structures observed within the umbra of the very flare-productive active region (AR) NOAA 11302 preceding sunspot. These structures, called umbral filaments (UFs), do not resemble typical LBs in morphology or in evolution and are formed by curled filaments that reach from the penumbra well into the umbra. Furthermore, UFs show counter Evershed flow along them at the photospheric level and energy dissipation phenomena in the higher atmospheric layers.

In this Letter, we describe the observation of an elongated bright structure inside the umbra of the big preceding sunspot of AR NOAA 12529. Curiously, its appearance transformed the original spot into a huge, heart-shaped sunspot, so that it became celebrated in popular media. We investigated this feature, which at first glance resembles a filamentary LB. We find that it is characterized by mixed directions of the plasma motions and a strong horizontal magnetic field, with a portion of the structure having opposite polarity to that of the hosting sunspot. These and other signatures allow us to interpret this structure as a manifestation of a flux rope that is located in higher layers of the solar atmosphere above the umbra. We describe the data set and its analysis in Sect.~2. The observations are presented in Sect.~3 and our conclusions are reported in Section~4.

\begin{figure}[t]
	\centering
	\includegraphics[trim=10 125 120 480, clip, scale=0.6]{Fig1.ps}
	\caption{AR NOAA 12529 as seen in the continuum filtergram, with overlain contours of the simultaneous map of the vertical component of the magnetic field (red/blue = -800/+800~G), taken by HMI in the \ion{Fe}{1} 6173~\AA{} line, at the time of the central meridian passage of the AR. The box frames the portion of the field of view shown in Figure~\ref{fig2}. \label{fig1}}
\end{figure}

\section{Data and analysis}

We have analyzed Space-weather Active Region Patches (SHARPs) data \citep{Hoe14} acquired by the Helioseismic and Magnetic Imager \citep[HMI;][]{Sch12} onboard the \textit{Solar Dynamic Observatory} \citep[SDO;][]{Pes12} satellite from 2016 April 8 to April 19 to study the main sunspot of the AR NOAA 12529. We have used continuum filtergrams, Dopplergrams, and vector magnetograms acquired along the \ion{Fe}{1} line at 6173~\AA{}, with a pixel size of 0\farcs51 and a time cadence of 12~min.

The vector field has been computed using the Very Fast Inversion of the Stokes Vector code \citep{Bor11}, which performs a Milne-Eddington inversion of the observed Stokes profiles, optimized for the HMI pipeline; the remaining $180^{\circ}$-azimuth ambiguity is resolved with the minimum energy code \citep{Met94}. More details about the SHARP pipeline are reported in \citet{Bob14}. The SHARP data have been corrected for the rotation angle of $180^{\circ}$ of HMI images. Finally, the vector magnetic field components have been transformed into the local solar frame, according to \citet{Gar90}.

HMI Dopplergrams have been corrected for the effect of solar rotation, by subtracting the mean velocity averaged over 10 days, deduced from the HMI data series relevant to Carrington rotation 2176. Doppler velocity has been calibrated assuming umbra (i.e., pixels with normalized continuum intensity $I_c < 0.4$) on average at rest \citep[e.g.][]{Rim94}.

We have also used images acquired in the extreme ultraviolet (EUV) by the Atmospheric Imaging Assembly \citep[AIA;][]{Lem12} at 171 \AA{} and 304 \AA{}, with a pixel size of about 0\farcs6 and a time cadence of 12~s.

Furthermore, we have analyzed full disc images of the chromosphere acquired in the H$\alpha$ line at 6562.8~\AA{} by INAF - Catania Astrophysical Observatory. These images have been coaligned together with HMI and AIA data.

\section{Results}

During the entire passage across the solar disc, AR NOAA 12529 was composed by a large preceding sunspot of negative polarity and some following pores of positive polarity, as shown in Figure~\ref{fig1}. The main sunspot was already formed when it appeared at the East limb on 2016 April 8, at heliographic latitude $\sim 10$~N. It occupied an area of about $2000 \,\mathrm{Mm}^2$.

On April 11, when the sunspot was located at $\mu = 0.8$, several bright elongated structures similar to small LBs appeared to the north-west of the spot, intruding into the umbra (see the top-left panel of Figure~\ref{fig2}). They have a curved shape, not showing a granular pattern at the spatial resolution of the HMI continuum images. In the map of the vertical component of the magnetic field taken at the same time (first column, second row of Figure~\ref{fig2}), we note that these structures are characterized by a lower field intensity, in comparison with the other parts of the sunspot umbra. The horizontal component of the magnetic field in the hook-shaped structure observed to the north-east of the sunspot, located at $[-500\arcsec,-490\arcsec] \times [250\arcsec, 260\arcsec]$, is weaker than the surrounding penumbral filaments, whereas in the intruding structures located to the west it is comparable to that of the penumbra (first column, third row of Figure~\ref{fig2}).

The longest of these structures, seen at $[-480\arcsec,-470\arcsec] \times [255\arcsec, 265\arcsec]$ on April 11, reached its maximum length of about 30\arcsec{} on April 13 (top-right panel of Figure~\ref{fig2}), when it was located at $[-80\arcsec,-65\arcsec] \times [65\arcsec, 80\arcsec]$, the sunspot being located at $\mu = 0.96$. At that time, a portion along the axis of the structure has a magnetic field of positive polarity, i.e., opposite to that of the magnetic field of the surroundings (second column, second row of Figure~\ref{fig2}). Moreover, the horizontal field in the structure is $\approx 2000 \,\mathrm{G}$, about 50\% stronger than in the surrounding penumbra (second column, third row of Figure~\ref{fig2}). Many of these bright structures, including the longest one, remained visible till the AR reached the West limb, when the sunspot appeared already fragmenting, in a decay phase.

\begin{figure*}
 \centering

	\includegraphics[trim=10 155 155 290, clip, scale=0.45]{Fig2a.ps}%
	\includegraphics[trim=40 155  40 290, clip, scale=0.45]{Fig2e.ps}
	\includegraphics[trim=10 155 155 330, clip, scale=0.45]{Fig2b.ps}%
	\includegraphics[trim=40 155  40 330, clip, scale=0.45]{Fig2f.ps}
    \includegraphics[trim=10 155 155 330, clip, scale=0.45]{Fig2c.ps}%
    \includegraphics[trim=40 155  40 330, clip, scale=0.45]{Fig2g.ps}
    \includegraphics[trim=10  95 155 330, clip, scale=0.45]{Fig2d.ps}%
    \includegraphics[trim=40  95  40 330, clip, scale=0.45]{Fig2h.ps}
  \caption{Evolution of the main sunspot of AR NOAA 12529 between 2016 April 11 (left) and April 13 (right) as seen in the continuum filtergrams (first row), in the simultaneous maps of the vertical (second row) and horizontal (third row) magnetic field components, and in the simultaneous Dopplergrams (fourth row), taken by HMI in the \ion{Fe}{1} 6173~\AA{} line. Contours represent the umbra-penumbra boundary at $I_c = 0.4$. \label{fig2}}
\end{figure*}

Interestingly, we find a noticeable behaviour of the plasma motions along the line of sight in these bright structures. On April 11, although the AR was near the East limb and the uncertainties due to the projection effects are not negligible, the velocity observed along the bright structures in the umbra (bottom-left panel of Figure~\ref{fig2}) is slightly larger than in the surroundings. In particular, the hook-shaped bright structure located to the north-eastern side of the umbra shows a velocity direction inverse to the Evershed flow in its northern leg. The bright structure to the western side of the umbra clearly also exhibits a plasma velocity higher than the surrounding Evershed flow, in this case along the normal Evershed flow direction. However, on April 13, when the longest of these structures reached its maximum length and the AR was approximating to the central meridian, we note plasma motions with both directions along it (see the arrow in the bottom-right panel of Figure~\ref{fig2}): upflows and downflows alternate from the outer to the inner part of the bright structure with velocities of the order of $1\,\mathrm{km \,s}^{-1}$. Each portion of the bright structure with the same direction of the plasma motion is about 5\arcsec{} long. Note that these alternate, coherent plasma flows may suggest a helical motion along the structure.

This behaviour is mostly visible when the sunspot passed near the central meridian, i.e., on April 13 and 14, but we cannot exclude that these plasma motions may persist during the whole life of the bright structure, i.e., for about 5 days, from April 11 to April 16, although they are not visible due to the foreshortening effects.

\begin{figure}
 \centering
	\includegraphics[trim=10 125 120 480, clip, scale=0.6]{Fig3a.ps}\\
	\includegraphics[trim=10 125 120 490, clip, scale=0.6]{Fig3b_neg.ps}\\
    \includegraphics[trim=10  95 120 490, clip, scale=0.6]{Fig3c_neg.ps}
  \caption{AR NOAA 12529 as seen in the H$\alpha$ line by INAF - Catania Astrophysical Observatory (top panel), at 304 \AA{} (middle panel) and at 171 \AA{} (bottom panel) by AIA. Note that AIA images are shown in a reversed color table. The contour in the middle panel corresponds to the umbra-penumbra boundary at $I_c = 0.4$ in the simultaneous continuum image. Red/blue contours in the bottom panel correspond to -400/+400~G in the simultaneous map of the vertical component of the magnetic field. The arrow in the bottom panel indicates the EUV filament channels corresponding to the filaments observed in the H$\alpha$ line and at 304~\AA. \label{fig3}}
\end{figure}

In the image of the chromosphere acquired on April 13 at 10:26 UT in the H$\alpha$ line at INAF -- Catania Astrophysical Observatory, we note the presence of chromospheric filaments located to the north-western side of the sunspot (top panel of Figure~\ref{fig3}). These filaments are also visible  in the AIA image at 304~\AA{} (middle panel of Figure~\ref{fig3}), connecting the sunspot to opposite polarity flux concentrations observed to the west of the preceding spot. It is worth noting that these filaments have the same curvature as the longest photospheric bright structure observed in the umbra of the sunspot. In particular, the portion of the filament corresponding to the bright structure inside the umbra at photospheric level also appears bright at 304~\AA{}. Moreover, at 171~\AA{} the filaments correspond to EUV filament channels (see the bottom panel of Figure~\ref{fig3}), surrounded by some bright loops with the same curvature, indicated with an arrow in the same figure.

\section{Discussion and conclusions}

In April 2016 a big sunspot passed over the visible hemisphere of the Sun showing bright elongated features apparently similar to unsegmented LBs. In this Letter we analyzed these structures, in particular the longest one, whose characteristics suggest a completely different origin with respect to LBs.

We found that no granular pattern was visible in this intruding filamentary feature at the HMI resolution. Moreover, the magnetic field strength in the structure is not weaker than in the surrounding umbral region. At the same time, we showed the presence of a strong horizontal field component of the magnetic field in the structure and a vertical component opposite to that of the surrounding penumbral filaments in a portion of it. Peculiar line-of-sight plasma motions along this structure are also observed at the photospheric level. In addition, images of the chromosphere at 6562.8~\AA{} and at 304~\AA{} reveal the existence of some small filaments in the nearby, which are characterized by the same curvature of the photospheric structure, corresponding to EUV channels observed at 171~\AA{}.

As regards the magnetic configuration, one has to be cautious with the interpretation of the results deduced from the Milne-Eddington solution implemented in the VFISV code. In fact, it may not account for the complexity of the Stokes profiles present in regions where peculiar processes are at work. Actually, this affects the values of $\chi^2$ of the fits, which are $2-3$ times higher in the bright structure than in the spot. Moreover, the smearing due to the point-spread function of the telescope may dilute into the bright structure part of the magnetic signal belonging to the surrounding umbra and penumbral filaments. However, the average estimated errors for the vertical and horizontal component of the field in the structure are about $\pm 35$~G and $\pm 70$~G, respectively. Thus, the strong horizontal field of about 2000~G appears to be a robust result.

The presence of such a significant horizontal component of the magnetic field allows us to exclude the field-free configuration typically observed in LBs. Conversely, it supports the possibility that the magnetic field of the flux rope forming the filaments around the sunspot may have a counterpart in the middle of the sunspot umbra, where it appears to end in the cospatial chromospheric images. This would also provide an explanation for the detection of opposite polarity in the structure with respect to the umbra, as well as an alternative interpretation for the opposite polarity of an apparently filamentary LB with respect its hosting umbra observed by \citet{Bha07}. Note that this phenomenon is different from the magnetic reversals found by \citet{Lag14} and \citet{Fel16} in LBs, due to the bending of the field lines along the atmospheric stratification of the structure.

Moreover, this scenario is confirmed by the portions of the bright feature characterized by both upward and downward plasma motions along it, which are compatible with plasma flows along the helical field lines of a flux rope. Actually, the length of each portion of the filamentary structure with coherent plasma motion is of about 5\arcsec{}, which could correspond to the pitch of a helically twisted flux tube.

Therefore, we guess that this bright filamentary structure is not a filamentary LB, but it is due to the accumulation of plasma coming from higher solar atmospheric levels into the photosphere. The presence of stably inclined fields belonging to the flux rope, which are touching part of the sunspot umbra, would set in radiatively driven penumbral magneto-convective mode \citep{Jur14, Jur17}. Indeed, high-resolution observations have shown that plasma fallen from chromospheric  layers can lead to the formation of penumbra in sunspots \citep{Shi12, Rom13, Rom14}. Furthermore, penumbral-like structures, i.e., so called orphan penumbrae, have been reported as a photospheric counterpart of a chromospheric filament \citep{Kuc12a, Kuc12b, Bue16}.

Observations of similar bright filaments inside the umbra, called UFs, have been recently reported by \citet{Kle13}. They found that UFs were characterized by a horizontal flow opposite to the Evershed flow and they also found from coronal images the presence of bright coronal loops which seemed to end in UFs. To interpret such phenomena, \citet{Kle13} conjectured two different scenarios. In the first, the bright filament ending in the umbra is formed by a sheet spanning many atmospheric layers and producing a siphon inflow by the pressure difference between the umbra and the network. In this case, the emission observed in the filament could be due to the energy dissipation at the boundary layers between the sheet and the sunspot magnetic field. In the second scenario, they interpreted the emission as an effect of a thick magnetic flux tube with high enough density such that the observed region is formed higher in the atmosphere.

However, in both cases \citet{Kle13} did not take into account the presence of a flux rope whose signatures are clearly visible in our observation, albeit the second scenario they proposed shares some similarities with ours. In fact, we do not detect counter Evershed flow, with the exception of one leg of the hook-shaped bright structure observed on April 11 to the north-eastern side of the umbra. We find, instead, a peculiar pattern of the plasma flows which seems to describe a helical motion along the flux rope of a filament ending in the sunspot umbra. Note that the target of \citet{Kle13} suffered from severe projection effect, due to the position of AR NOAA 11302 on the solar disc when UFs were observed ($\mu \approx 0.6$).

It is also worth mentioning that AR NOAA 12529 was not a flare-rich AR: the strongest event was a unique M-class (M6.7) flare on April 18, while \citet{Kle13} suggested that the UFs could be related to high flare-productivity, as they would influence the structure of the overlying coronal loop leading to magnetic rearrangement.

To support our scenario, further analyses are necessary. In this respect, in a next paper we shall analyze the high-resolution observations performed by the spectropolarimeter aboard the \textit{Hinode} satellite in the photosphere and by the \textit{IRIS} spacecraft during the central meridian passage of AR NOAA 12529. A preliminary analysis of the \textit{Hinode} data shows that a strong linear polarization signal is detected in the structure within the umbra, which also exhibits a portion with circular polarization of sign opposite to that of the surrounding umbra, much larger than that observed with HMI. Note that these polarization signals cannot be due to the contamination induced by the point spread function of the telescope with the signals from the surrounding magnetic elements. This strengthens the interpretation that we are proposing here.

The question about the conditions to observe such a phenomenon remains still open. Why do we not often observe this kind of bright structures similar to LBs but without any granular pattern? Can these structures be caused by the particular proximity of the footpoints of chromospheric filaments to a sunspot? Or can they be due to a peculiar configuration of the flux rope forming these filaments? An answer will likely be obtained by benefitting from the high spatial resolution and continuous temporal coverage provided by the next generation of solar observatories, such as the Solar Orbiter space mission \citep{Muller:13} and the large-aperture ground-based telescopes DKIST \citep{Keil:10} and EST \citep{Collados:10}.

\acknowledgments

The authors wish to thank an anonymous referee for his/her insightful comments. This work was supported by the Istituto Nazionale di Astrofisica (PRIN-INAF 2014), by the Italian MIUR-PRIN grant 2012P2HRCR on The active Sun and its effects on space and Earth climate, by Space Weather Italian COmmunity (SWICO) Research Program, and by the Universit\`a degli Studi di Catania. The research leading to these results has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement no.~739500 (PRE-EST project). The SDO data used in this paper are courtesy of NASA/SDO science team.

\facility{SDO (HMI, AIA)}, \facility{OACT:0.15m}


\section{Introduction}
 \label{section:Intro}

An action formulation for radiation reaction (RR) was presented in \cite{BirnholtzHadarKol2013a}.
While \cite{BirnholtzHadarKol2013a} focused on the post-Newtonian approximation to the two body problem in Einstein's gravity, it stressed the method's generality, and presented detailed calculations also for radiation reaction of scalar and electromagnetic (EM) fields.
Following demand, this paper highlights the method by focusing on the well-known electromagnetic Abraham--Lorentz--Dirac (ALD) force \cite{Abraham,Lorentz1,Lorentz2,Dirac,ALD-Jackson,ALD-Rohrlich}.

The method offers certain advantages over the more standard approach. First, the action formulation allows the application of efficient tools including the elimination of fields through Feynman diagrams, and allows ready formulation and systematic computation of analogous quantities in non-linear theories such as gravity. Secondly, a single effective action would be seen to encode several ALD-like forces and its form would stress a connection between the source and target not seen in the usual force expression.

We start in this section by surveying the background and reviewing the general method and its key ingredients before proceeding to concrete demonstrations in the following sections.
In section \ref{section:mass--rope} the method is illustrated in a simple set-up -- a mass attached to an infinite rope. Section \ref{section:ALD} describes the application of the method to the leading non-relativistic ALD force (also known as the Abraham-Lorentz force), i.e. the force acting on an accelerating electric charge due to its own EM field.
The derivation has a strong analogy with the mass--rope system, especially after a reduction to a radial system through the use of spherical waves; many technicalities of the reduction are deferred to Appendix \ref{sec:gauge}, and may be skipped in a first reading.

\vspace{.5cm} \noindent  {\bf Background}.
Feynman diagrams and effective field theories were first introduced in the context of quantum field theories, but they turn out to be applicable already in the context of classical (non-quantum) field theories.
One such theory is the Effective Field Theory (EFT) approach to general relativity (GR) introduced in \cite{GoldbergerRothstein1}, see also contributions in \cite{CLEFT-caged,NRG}.

Similarly, the development of an action method for dissipative systems was first developed in the early 60's in the context of quantum field theory, and is known as the Closed Time Path formalism \cite{CTP}.
The essential idea is to formally introduce a doubling of the fields which can account for dissipation.
Here too relevant non-quantum problems exist, such as radiation reaction, and hence it was natural to seek a theory for the non-quantum limit.
This was discussed in the context of the EFT approach to GR in \cite{GalleyEFT} and was promoted to general classical non-conservative systems in \cite{GalleyNonConservative}.
We also recommend comparing the mass--rope system presented here with the case of coupling a mass to infinitely many oscillators, presented in \cite{GalleyNonConservative}, and with the action formulations of a damped harmonic oscillator in \cite{Kosyakov:2007qc}.
Essential elements of the theory were reformulated in \cite{BirnholtzHadarKol2013a}.
This formulation was extended from 4d to general dimensions in \cite{BirnholtzHadar2013b}.

\subsection*{Brief review of method}

Here we present the tools generally, before demonstrating their use in the following sections.

{\bf Field doubling}. The theory in \cite{BirnholtzHadarKol2013a} was formulated for classical (non-quantum) field theories with dissipative effects, such as radiation reaction.
The source for the departure from the standard theory was identified to be a \emph{non-symmetric (or directed) propagator}, such as the retarded propagator.
For such theories the fields are doubled together with the sources, so that the propagator always connects a field and its double and thereby assigns a direction to it.
Here the term field should be understood to refer also to dynamical variables in mechanics (viewed as a field theory in 0+1 d).

A generic theory may be described by a field $\phi=\phi(x)$ together with its source $\rho=\rho(x)$ and the equation of motion for $\phi$, $0=EOM_\phi(x)$, where the subscript notation readily generalizes to a multi-field set-up.
The double field action is given by \cite{BirnholtzHadarKol2013a}  \begin{equation}
{\hat S}} \def\hJ{{\hat J}} \def\hQ{{\hat Q}} \def\hx{{\hat x} \[ \phi,{\hat \phi};\, \rho,{\hat \rho} \] :=
	\int d^d x\, \[ EOM_\phi(x)\,{\hat \phi}(x)
	+\int d^d y\, \frac{\delta EOM_\phi (y) }{\delta \rho(x)} {\hat \rho}(x)\, \phi(y) \],
\label{doubled-action}
\end{equation}
where ${\hat \phi}$ is a doubled auxiliary field, ${\hat \rho}$ is the corresponding source, and $d^dx \equiv d^{d-1} {\bf x}\, dt$ is a space-time volume element.
The action is best described in terms of functional variations, but in our examples it will amount to simple concrete expressions shown in the following sections.
The expression is constructed such that $\delta {\hat S}} \def\hJ{{\hat J}} \def\hQ{{\hat Q}} \def\hx{{\hat x}/\delta {\hat \phi} = EOM_\phi$, namely that the equation of motion with respect to ${\hat \phi}$ reproduces the original $EOM_\phi$.
The first part introduces terms of the form $\phi(x)\, {\hat \phi}(x)$, $\rho(x)\, {\hat \phi}(x)$, while the second part introduces terms of the form ${\hat \rho}(x)\, \phi(x)$.

This formulation is applicable to arbitrary equations of motion, not necessarily associated with an action.
When they \emph{can} be derived from an action $S=S[\phi,\rho]$,  such as in the case of radiation reaction, namely if $EOM_\phi=\frac{\delta S}{\delta \phi}$ for some $S$, the double field action is
\begin{equation}
{\hat S}} \def\hJ{{\hat J}} \def\hQ{{\hat Q}} \def\hx{{\hat x} \[ \phi,{\hat \phi};\, \rho,{\hat \rho} \] :=
	\int d^d x\, \[ \frac{\delta S[\phi,\rho]}{\delta \phi(x)}\,{\hat \phi}(x)
	+\frac{\delta S[\phi,\rho]}{\delta \rho(x)}\, {\hat \rho}(x)\,  \] ~.
\label{doubled-action2}
\end{equation}
Here ${\hat \phi},\, {\hat \rho}$ can be assigned the following meaning: ${\hat \phi}$ is the linearized field perturbation, sourced by reverse (advanced) propagation from a generic source ${\hat \rho}$.
They correspond to the Keldysh basis of the Closed Time Path formulation and ${\hat S}} \def\hJ{{\hat J}} \def\hQ{{\hat Q}} \def\hx{{\hat x}$ is a linearization of the CTP action with respect to ${\hat \phi}$.
We remark that this procedure determines the action explicitly, and in fact gives a recipe for the function $K$ mentioned in \cite{GalleyNonConservative}.
It also generalizes the specific field doubling method (using complex conjugation) of \cite{Kosyakov:2007qc}.

{\bf Zone separation}. The EFT approach is characterized by a hierarchy of scales leading to multiple zones (and a matched asymptotic expansion).
In particular when the velocities of the sources are small with respect to the velocity of outgoing waves, such as in the Post-Newtonian approximation to GR, one defines a system zone and a radiation zone.

{\bf Elimination}. Given ${\hat S}} \def\hJ{{\hat J}} \def\hQ{{\hat Q}} \def\hx{{\hat x}$, one may proceed to eliminate fields through Feynman diagrams. \emph{Radiation sources $Q[x]$} are defined diagrammatically by
\begin{eqnarray}
-Q[x] :=
\parbox{10mm} {\includegraphics[scale=0.3]{DiagsGeneralRulesVertex.pdf}}
\label{def:Q}
\end{eqnarray}
where the double heavy line denotes the whole system zone.
Their definition involves the elimination of the system zone field and matching, see \cite{BirnholtzHadarKol2013a}  eq. (2.49), but this shall mostly not be required in the present paper.
The doubled radiation sources are given by
\begin{equation}
 \hQ[x, \hx] = \int dt\, \frac{\delta Q[x]}{\delta x^i(t)}\, \hx^i(t) \, \, . \label{def:Qhat}
 \end{equation}
which is the linearized perturbation of the source and as such does not require the computation of new diagrams beyond (\ref{def:Q}).
Here too this general formula will be seen to reduce to simple expressions in the examples.

Next the \emph{radiation reaction effective action} ${\hat S}} \def\hJ{{\hat J}} \def\hQ{{\hat Q}} \def\hx{{\hat x}_{RR}$ may be computed through elimination of the radiation zone, see \cite{BirnholtzHadarKol2013a} eq. (2.53).
This action describes the effects of dissipation and radiation reaction.

{\bf Gauge invariant spherical waves in the radiation zone}. When the spatial dimension $D>1$ (as in the ALD problem) each zone respects an enhanced symmetry and corresponding field variables and gauge should be chosen:
\begin{itemize}} \def\ei{\end{itemize}
\item In the system zone time-independence (stationarity) is an approximate symmetry, due to the slow velocity assumption, and hence non-Relativistic fields (and a compatible gauge) should be used \cite{NRG}.
\item In the radiation zone the system appears point-like and hence it was recognized in \cite{BirnholtzHadarKol2013a} that this zone is spherically symmetric and accordingly gauge invariant spherical waves (see \cite{AsninKol}) should be chosen as the field variables.
\ei

{\bf Matching within action}. Subsection 2.3 of \cite{BirnholtzHadarKol2013a} demonstrated how the matching equations which are a well-known, yet sometime sticky element of the effective field theory approach, can be promoted to the level of the action.
This is achieved by introducing an action coupling between the two zones, whereby the matching equations become ordinary equations of motion with respect to novel field variables.
These are termed two-way multipoles, and reside in the overlap region.
This is an important ingredient of our method, but is not used directly in this paper.

\section{Mass--rope system}
\label{section:mass--rope}

As a first example we consider a system shown in fig. \ref{fig:mass-rope} composed of a rope stretched along the $x$ axis with linear mass density $\lambda$ and tension $T$ (hence the velocity of waves along it is $c := \sqrt{T/\lambda}$).
Its displacement along the $y$ axis is given by $\phi(x)$.
The rope is attached at $x=0$ to a point mass $m$ which is free to move along the $y$ axis and is connected to the origin through a spring with spring constant $k$.
We assume there are no incoming waves from infinity.
Our goal is to find an effective action describing the evolution of the point mass in time \emph{without direct reference} to the field -- that is, after the field's elimination.
The eliminated field carries waves, and therefore energy, away from the mass; we show directly how this elimination gives a generalized (non-conservative) effective action describing the dynamics of the mass, which is that of a damped harmonic oscillator.

\begin{figure}[t!]
\centering \noindent
\includegraphics[scale=0.5]{mass_spring.pdf}
\caption[]{The mass--rope system.}
 \label{fig:mass-rope}
\end{figure}

The action describing the system (mass+rope) is given by
\begin{equation}
S \, = \,
	\frac{1}{2} \int\!\! dt \left[ m \dot{y}^2 - k y^2 \right]
	+ \frac{T}{2}\!\! \int\!\! dt dx \, \left( \partial \phi \right)^2
	-  \int\!\! dt  \,Q \left[  \phi(0) - y  \right]	 \, \, ,
\label{mass rope action}
\end{equation}
where $\left( \partial \phi \right)^2 = \frac{1}{c^2}\dot{\phi}^2-(\partial_x \phi)^2$ and $\dot{y} := \partial_{t} y$.
The action is composed of the mass's action, a dynamical rope term, and a coupling term.
This coupling enforces the boundary condition $y \equiv \phi(x=0)$ at the level of the action and introduces the Lagrange multiplier $Q(t)$, where the choice of notation will be explained later.

We wish to obtain a long-distance effective action for the rope, and for that purpose we analyze the equations of motion.
Varying the action with respect to $\phi$ and $Q$ we find
\begin{eqnarray}
\square \phi &=& -\frac{Q}{T}\, \delta(x) \label{EOM1} \, \, , \\
y &=& \phi(0) \label{EOM2} \, \, ,
\label{eom full theory}
\end{eqnarray}
where $\Box :=  \frac{1}{c^2} \partial^2_t \, - \partial^2_x$.
Analysis of (\ref{EOM1}) near $x=0$ gives $\left[ \partial_x \phi \right]=Q/T$ where $\left[ \partial_x \phi \right]:=(\partial_x \phi)|_{0^+}-(\partial_x \phi)|_{0^-}$ denotes the jump at the origin.
On the other hand the equations of motion and the outgoing wave condition imply
\begin{equation}
\phi(x,t) =
	\left\{ \begin{array}{c}
		y(t - x/c)  \qquad x>0 \\
		y(t + x/c)  \qquad  x<0
	\end{array} \right.
\label{full theory radiation}
\end{equation}
and hence $\left[ \partial_x \phi \right] = -2 \dot{y}/c$.
Altogether we obtain
\begin{equation}
\frac{Q}{T} = \left[ \partial_x \phi \right] = -\frac{2}{c} \, \dot{y} \, \, .
\label{lambda1}
\end{equation}

At this point we can substitute (\ref{lambda1}) in the eq. of motion for $y$ and obtain a damped harmonic oscillator.
Yet, here we wish to demonstrate how such a dissipative system may be described by a (double field) action.
For that purpose we proceed and find that the following action implies the same equations of motion\footnote
{
In fact there is a slight difference: the effective action (\ref{mass rope far region effective action}) implies $\dot{\phi}(0)=\dot{y}$ which is not exactly the same as (\ref{EOM2}).
}
including (\ref{lambda1})
\begin{equation}
S \, = \,
	\frac{1}{2} \int\!\! dt \left[ m \dot{y}^2 - k y^2 \right]
	+ \frac{T}{2}\!\! \int\!\! dt dx \, \left( \partial \phi \right)^2
	+ 2\, Z \!\! \int\!\! dt \, \dot{y} \,  \phi(0)		 \, \, ,
\label{mass rope far region effective action}
\end{equation}
where  $Z:= \sqrt{T \lambda}$ is the rope's impedance.
We interpret $S$ as a long-distance effective action for $\phi$ (as it incorporates the solution (\ref{full theory radiation}) including the asymptotic boundary conditions).
Comparing (\ref{mass rope far region effective action}) with a standard origin source term $S_{\mathrm{int}} = -\int \! \phi(0,t) \, Q(t) dt \, \,$ justifies the notation $Q$.
In fact, $Q$ is the force exerted on the mass by the rope.

We double the field $\phi$ as well as its source $y$ in (\ref{mass rope far region effective action}) as described in (\ref{doubled-action2}) and find the double field action
\begin{equation}
\hat{S} \, = \, T \int dt\, dx\, \partial \phi\, \partial \hat{\phi} + 2\, Z \!\! \int\!\! dt \[ \dot{y} \,  \hat{\phi}(0) + \dot{\hat{y}}\, \phi(0) \] \, , \,
\label{mass rope action frequency domain}
\end{equation}
Transforming to frequency domain with the convention
$\phi(t) = \!\int\! \frac{d\omega}{2\pi}\, \phi(\omega)\, e^{-i \omega t}$,
complex conjugate fields appear such as $\phi^*(\omega)$ and we obtain the Feynman rules.
The directed propagator from $\hat{\phi}^*$ to $\phi$ is
\begin{align}
\label{Feynman Rule Propagator}
\parbox{20mm} {\includegraphics[scale=0.5]{DiagsGeneralRulesProp_x.pdf}}
=G_{\omega}(x',x) \, = \, - c\,  \frac{e^ {i \frac{\omega}{c} \, |x-x'|}}{i\, \omega \, Z}  ~,
\end{align}
and the sources for the fields $\hat{\phi}^*,\, \phi$ are (compare (\ref{def:Q}),(\ref{def:Qhat}))
\begin{align}
\label{Feynman Rule Vertex}
-Q_\omega \equiv \parbox{20mm} {\includegraphics[scale=0.3]{DiagsGeneralRulesVertex.pdf}}
\!\!\!\!\!\!\!\!\!\!= - i \omega \,  Z \, y_{\omega}~~~~~ ,~~\,
-\hat{Q}^*_\omega \equiv \parbox{20mm} {\includegraphics[scale=0.3]{DiagsGeneralRulesVertexHat.pdf}}
\!\!\!\!\!\!\!\!\!\!=i  \omega \, Z \, \hat{y}^{*}_{\omega}~~ .~~~~~~
\end{align}
Similar Feynman rules hold for $\phi^*,\, \hat{\phi}$.

Now that we have the Feynman rules we can proceed to compute the outgoing radiation and the radiation reaction effective action.
Radiation away from the source, for $x>0$, is given by
\begin{eqnarray}
\phi_\omega(x) = \parbox{20mm}{\includegraphics[scale=0.5]{DiagRadiationScalar_x.pdf}} = c \, y_{\omega} \, e^{ i \omega x /c} ~ ,
\label{rad}
\end{eqnarray}
which in the time domain becomes
\begin{equation}
\phi(t,x) = \int \frac{d \omega}{2 \pi}\,  y_\omega \, e^{ - i \omega (t -  \frac{x}{c})} = y(t -  \frac{x}{c})~ ,
\label{rad solution time}
\end{equation}
reproducing (\ref{full theory radiation}).

The \emph{radiation reaction effective action} is found by eliminating the field $\phi$
\begin{eqnarray}
\hat{S}_{RR} &=& \parbox{20mm}{\includegraphics[scale=0.5]{ActionDiagScalarL.pdf}} ~ + ~ c.c. ~
	= \, \int \frac{d \omega}{2 \pi} \, \, \hat{Q}^{*}_{\omega} \, G_{\omega}(0,0) \, Q_{\omega} \, + c.c. \,
		= \nonumber \\
		&=& \int \frac{d \omega}{2 \pi}  \, i \omega \,  Z \, \hat{y}^{*}_{\omega} \, y_{\omega}
			~+ ~ c.c.
	=  \, -2\, Z \int \hat{y} \, \dot{y} \,  dt ~ .
\label{RR effective action}
\end{eqnarray}
Thus the full generalized effective action for the mass becomes
\begin{equation}
\hat{S}_{tot} \, = \,	\hat{y} \frac{\delta}{\delta y }\left\{ \frac{1}{2} \int\!\! dt \left[ m \dot{y}^2 - k y^2\right]\right\}
	- 2 Z \int \hat{y} \partial_t y \,  dt
\label{final action rope}
\end{equation}
and by taking the Euler--Lagrange equation with respect to $\hat{y}$ we obtain
\begin{equation}
m \ddot{y} = -k y - 2 Z \dot{y} ~ ,
\label{EOM for mass on rope}
\end{equation}
which is the equation of a damped harmonic oscillator, as expected.

We remark that the $1+1$ mass--rope system can be regarded as a specific case of a scalar point charge coupled to a scalar field in $d=D+1$ dimensions, and thus falls under the general considerations of \cite{BirnholtzHadar2013b}, for $d=2$. Excluding relativistic, retardation and higher-multipole effects (irrelevant in our 1 spatial dimension), we recover the form given there for the radiation-reaction force (eq. 2.50), with $G q^2 = Z$, and a factor of 2 because the rope has two sides.

\section{Electromagnetic charge--field system}
\label{section:ALD}

The physical problem of the self-force (radiation-reaction force) on an accelerating electric charge has been treated for over 100 years \cite{Abraham,Lorentz1,Lorentz2,Dirac,ALD-Jackson,ALD-Rohrlich} in different methods.
We wish to show how in its simplest form and to leading order, the Abraham--Lorentz force can be easily found in essentially the same method as in the mass--rope system.
We start likewise with the standard Maxwell electromagnetic action\footnote
{
We use Gaussian units, the speed of light $c=1$, and the metric signature is  $(+,-,-,-)$.
}
\begin{equation}
\label{MaxwellEMAction}
S_{full}=\!-\frac{1}{16\pi}\!\! \int\!\! d^4 x \,F_{\mu\nu} F^{\mu\nu} - \!\!\int\!\! d^4 x\, A_{\mu} J^{\mu},
\end{equation}
where the field of the rope $\phi$ has been replaced by the EM field $A^\mu$ (with $F_{\mu\nu} \!=\! \partial_\nu A_\mu \!-\! \partial_\mu A_\nu$)\footnote
{
As usual $F_{\mu\nu}$ encodes the electric and magnetic fields through $E_i=F_{0i}$, $B_i=-\frac{1}{2}\eps_{ijk}F_{jk}$.
}
and the mass on a spring is replaced by a point-charge $q$ with trajectory ${\bf x}_p(t)$ and current density $J^\mu=q\frac{dx^\mu_p}{d\tau}\delta({\bf x} - {\bf x}_p)$.

As a 3+1\,d problem, this appears more complicated than the mass-rope system.
However, in the radiation zone the system appears point-like and the problem becomes spherically symmetric.
As dissipation and reaction are related to the waves propagating to spatial infinity, we can reduce the problem using the spherical symmetry  to an effective 1+1 dimensional ($r,t$) system, which can be treated analogously to the mass--rope system.
Physically, the reduction amounts to working with spherical wave variables which are described in appendix \ref{sec:gauge}.
For the purpose of the leading non-relativistic ALD force it suffices to consider the electric dipole ($\ell=1$) sector.
We denote the corresponding field variable by ${\bf A}={\bf A}(r,t)$ and the source by $\boldsymbol{\rho}=\boldsymbol{\rho}(r,t)$, both defined in Appendix \ref{sec:gauge}.
In this sector the action, reduced to 1+1 dimensions, is given by (\ref{EM Action leading2}). In the time domain it becomes
\begin{equation}
S= \int \!\! dt \! \int\!\! dr
	\left[
		- \frac{r^4}{12} {\bf A} \!\cdot\! \Box {\bf A}
		- {\bf A} \!\cdot\! \boldsymbol{\rho}
	\right],
\label{EM Action leading}
\end{equation}
where now $\Box :=\partial_t^2 - \partial_r^2 - \frac{4}{r}\partial_r$.
This field content is very similar to that of the rope: the coordinate $r$ replaces $x$, the field ${\bf A}(r)$ replaces the field $\phi(x)$. As in (\ref{mass rope action}) the action has both a kinetic field term and a source-coupling term, though their form here differs.
The double-field action is found using (\ref{doubled-action2}) as in the mass--rope system, and is given by
\begin{equation}
{\hat S}} \def\hJ{{\hat J}} \def\hQ{{\hat Q}} \def\hx{{\hat x} = \int \!\! dt \! \int\!\! dr
	\left[
		- \frac{r^4}{6} \hat{{\bf A}} \!\cdot\! \Box {\bf A}
		- \( \hat{{\bf A}} \!\cdot\! \boldsymbol{\rho} + {\bf A} \!\cdot\! \hat{\boldsymbol{\rho}} \)
	\right].
\label{EM hatted Action leading}
\end{equation}

The method thus proceeds similarly to find the Feynman propagator for the field and the expressions for the source vertices - non-hatted and hatted.
To obtain the propagator for $ {\bf A}$ we transform to the $\omega$ frequency domain and consider the homogenous part of the field equation (\ref{EOM PhiS}), which in dimensionless variables $x:= \omega r$, becomes
\begin{equation}
[ {\partial}} \def\dag{\dagger_x^2 + \frac{4}{x}{\partial}} \def\dag{\dagger_x + 1]\, \tilde{b}_{3/2}=0.
\label{Modified Bessel equation}
\end{equation}
Its solutions are the origin-normalized Bessel functions
\begin{equation}
\tilde{b}_{3/2} := \Gamma(\frac{5}{2}) \, 2^{3/2}\, \frac{B_{3/2}\alpha(x)}{x^{3/2}}~ ,
\end{equation}
where $B \equiv \{J,Y,H^\pm\}$ includes the Bessel functions $J,Y$ and the Hankel functions, $H^\pm=J \pm i\, Y$.
The origin-normalized Bessel $\tilde{j}_{3/2} = 1 + \co\(x^2\)$ is smooth at the origin and $\tilde{h}^+_{3/2} = -\frac{3}{x^2}\, e^{i\,x} \( 1 + \co\(\frac1x\) \)$ is an outgoing wave.
More details on these Bessel functions are given in Appendix \ref{sec:Bessel}.

Thus \emph{the propagator for spherical waves is}
\begin{align}
\label{Feynman Rule Propagator}
\parbox{20mm} {\includegraphics[scale=0.5]{DiagsGeneralRulesProp.pdf}}
=G(r',r) \, = \, -\frac{2i\omega^3}{3} \, \tilde{j}_{3/2}(\omega r_1) \, \tilde{h}^+_{3/2}(\omega r_2) \,;\\
r_1:=\text{min}\{r',r\},\,\,\,r_2:=\text{max}\{r',r\}.	\nonumber
\end{align}

In this sector, the source in the radiation zone is nothing but the electric dipole $ {\bf Q} \equiv {\bf D}$. Its form is identified through matching (\ref{def:Q}) the full theory with the radiation zone \begin{equation}
{\bf A}_{full}(r)
	\!=\! \int\!\! dr' \boldsymbol{\rho}(r') \!\! \( \frac{-2i\omega^{3}}{3}  \tilde{j}_{3/2}(\omega r') \tilde{h}^+_{3/2}(\omega r) \)
	\!\! =  {\bf Q} \frac{-2i\omega^{3}}{3} \tilde{h}^+_{3/2}(\omega r)
	\!=\! {\bf A}_{rad}(r)~. ~~~
\label{EM scalar wavefunction at radiation zone1}
\end{equation}
Using (\ref{def:Q},\ref{def:Qhat},\ref{Modified Bessel equation}, \ref{EM inverse sources}, \ref{EM source scalar Phi}) and integration by parts, we read from (\ref{EM scalar wavefunction at radiation zone1}) the electric dipole source vertex ${\bf Q}$ at leading non-relativistic order
\begin{eqnarray}
\label{Feynman Rule Vertex}
\parbox{10mm} {\includegraphics[scale=0.3]{DiagsGeneralRulesVertex.pdf}}
&=& {\bf Q}\!=\!\!\!\int\!\!dr' \tilde{j}_{3/2}(\omega r') \boldsymbol{\rho} (r')
\!=\!-\frac{q_\omega}{2}\!\!\int\!\! d^3 x'  \tilde{j}_{3/2}(\omega r') {\bf n}' \left[r'^2 \delta({\bf x}' - {\bf x}_p) \right]'
	= q {\bf x}_p + \dots
	\equiv {\bf D}
\nonumber\\
\parbox{10mm} {\includegraphics[scale=0.3]{DiagsGeneralRulesVertexHat.pdf}}
&=&\hat{\bf Q} = \frac{\delta {\bf Q}}{\delta x^{i}}\hat{x}^{i}	= q \hat{\bf x}_p  + \dots
	\equiv \hat{\bf D}
\end{eqnarray}
We remark that this is merely the static electric dipole moment ${\bf D}$ of the source, where the ellipsis denote relativistic corrections.

With the Feynman rules at hand we can proceed to determine the outgoing radiation and the radiation reaction effective action.
Radiation away from the source is given diagrammatically by (compare \ref{rad})
\begin{eqnarray}
{\bf A}(r)=
\parbox{20mm}{\includegraphics[scale=0.5]{DiagRadiationScalar.pdf}}
= 	-{\bf Q}\, G(0,r)
=-\frac{2i\,\omega \, {\bf Q}\, e^{i \omega\, r}}{r^2}~ , ~~~
\label{rad EM}
\label{Radiation EM using feynman scalar}
\end{eqnarray}
where we have used (\ref{Feynman Rule Propagator},\ref{Feynman Rule Vertex},\ref{Bessel H asymptotic2}). In the time domain, we find
\begin{equation}
{\bf A} ({\bf x},t)=\frac{2}{r}\, \partial_t {\bf Q}(t-r) ~.
\label{radiation A}
\end{equation}
The EM radiation reaction effective action is
\begin{eqnarray}
{\hat S}} \def\hJ{{\hat J}} \def\hQ{{\hat Q}} \def\hx{{\hat x}_{EM} =
\parbox{20mm}{\includegraphics[scale=0.5]{ActionDiagScalarL.pdf}} ~ + ~ c.c. ~
	&=& \, \frac{1}{2} \int \! \frac{d \omega}{2 \pi} ~ \hat{{\bf Q}}^{*} \, G(0,0) \, {\bf Q} \, + c.c. \, \nonumber\\
&=&\frac{2}{3}\int\!\!dt \, \hat{\bf Q} \cdot \partial_t^{3}{\bf Q}
\label{RR effective action EM general dipole}
\end{eqnarray}
where the propagator was evaluated at $r=r'=0$, as in (\ref{RR effective action}), and we regulated $ \tilde{h}^+(0)  \to \tilde{j}(0)=1$.
In the process of computing ${\hat S}} \def\hJ{{\hat J}} \def\hQ{{\hat Q}} \def\hx{{\hat x}_{EM}$ the fields were eliminated and only the particle's dipole remains.
Note that the third time derivative originates from the $\omega^3$ term in (\ref{Feynman Rule Propagator}), which in turn originates from the behavior of the Bessel function near its origin.

For a single charge we substitute (\ref{Feynman Rule Vertex}) in (\ref{RR effective action EM general dipole}) to obtain \begin{equation}
{\hat S}} \def\hJ{{\hat J}} \def\hQ{{\hat Q}} \def\hx{{\hat x}_{EM} = \frac{2}{3}q^2\!\!\int\!\!dt \, \hat{\bf x} \cdot \partial_t^{3}{\bf x}   ~.
\label{RR effective action EM}
\end{equation}
This can now be used to find the radiation-reaction (self) force, similarly to (\ref{final action rope},\ref{EOM for mass on rope}), through the Euler-Lagrange equation with respect to $\hat{\bf x}$
\begin{equation}
{\bf F}_{RR} = \frac{2}{3} \, q^2 \dddot{\bf x}.
\label{EOM for charge ALD}
\end{equation}
This of course matches the Abraham--Lorentz result \cite{Abraham,Lorentz1,Lorentz2}, and is the leading order term in the fully relativistic result of Dirac \cite{Dirac} (given in similar form in \cite{BirnholtzHadarKol2013a}, eq. (3.67)).

In addition our approach offers some benefits. Rewriting (\ref{RR effective action EM general dipole}) with the notation ${\bf D} \equiv {\bf Q}$ we have  \begin{equation}
{\hat S}} \def\hJ{{\hat J}} \def\hQ{{\hat Q}} \def\hx{{\hat x}_{EM} = \frac{2}{3}\int\!\!dt \, \hat{\bf D} \cdot \partial_t^{3}{\bf D}
\label{hSD}
\end{equation}
This expression applies not only to single charge, but also to a system of charges if only we set \begin{equation}
{\bf D} \left[ {\bf x}_a \right] := \sum_a q_a\, {\bf x}_a \end{equation}
where the sum is over all the particles in the system. To get the radiation reaction force on any specific charge in the system we need only vary the single object ${\hat S}} \def\hJ{{\hat J}} \def\hQ{{\hat Q}} \def\hx{{\hat x}$  \begin{equation}
F_{RR,a}^i = \frac{\delta {\hat S}} \def\hJ{{\hat J}} \def\hQ{{\hat Q}} \def\hx{{\hat x}}{\delta {\bf x}_a^i}
\end{equation}

Moreover, the form  (\ref{hSD}) and the Feynman diagram (\ref{RR effective action EM general dipole}) reveal that the dipole appears in the self-force twice: once in an obvious way as ${\bf D}$, the source of the radiation and reaction fields, and a second, less obvious, time as $\hat{\bf D}$ which is the ``target'' coupling through which the reaction field acts back on the charges.  In this sense the source and the target are seen to be connected.

Going beyond the non-relativistic limit, an expression for all the relativistic corrections was given in eq. (3.65) of \cite{BirnholtzHadarKol2013a}.
When expanded, the first relativistic correction includes the electric quadrupole term, the magnetic dipole term, and relativistic corrections to the electric dipole.
These were shown to confirm the expansion of Dirac's formula to next-to-leading order (eq. (3.68) there).

\section*{Conclusions}
\label{section:Conclusion}

Inspired by the (quantum) Closed Time Path formalism, we have shown how dissipative systems can be treated with an action principle in classical contexts.
The explicit algorithm to find the generalized action incorporates field doubling, zone separation and  spherical waves.
They were demonstrated by deriving the generalized (dissipative) action for a classical oscillator attached to a rope (\ref{RR effective action})
and for the Abraham--Lorentz--Dirac EM self-force (\ref{RR effective action EM}).
While these two problems may seem remote from each other, and involve different dimensionality, fields, and sources, the treatment of their radiation follows a very similar path.

\subsection*{Acknowledgments}

We thank B. Kosyakov for encouragement and many helpful comments and B. Remez for commenting on a draft.

This research was supported by the Israel Science Foundation grant no. 812/11 and it is part of the Einstein Research Project "Gravitation and High Energy Physics", which is funded by the Einstein Foundation Berlin.
OB was partly supported by an ERC Advanced Grant to T. Piran.