\section{Introduction}
The discovery of  2D  materials has
spurred a surge of experimental and theoretical interests in
understanding their physical and chemical properties.
The piezoelectricity of 2D materials has been a research hotspot in the ever-increasing energy conversion area\cite{q4,q4-1}, which requires a 2D system to be non-centrosymmetrical.  The 2H-$\mathrm{MoS_2}$ is a typical 2D piezoelectric material, which has been confirmed experimentally\cite{q5}.
In theory, piezoelectric properties of many 2D materials have been reported\cite{q11,q7,q10,q9,q13,q17,q16}. Strong in-plane piezoelectric response is common, while large out-of-plane piezoelectric strain coefficient is rare.  Searching for  2D materials  with large out-of-plane piezoelectric response has become an important quest  due to its compatibility with bottom/top gate technologies.  Another interesting feature of 2D systems is nontrivial topology, and TI of them is also called as quantum spin Hall insulator (QSHI) with spin-momentum-locked conducting
edge states. The first QSHI is predicted in graphene\cite{t3}, and then the  HgTe/CdTe and InAs/GaSb quantum wells are  experimentally confirmed as QSHIs \cite{t4,t5}.
Theoretically, many 2D systems have been proposed as QSHIs by first-principle calculations\cite{t6,t7,qt3,t8,t9,t10}.
Searching for QSHI with large gap (larger than 25 meV) is very important for room temperature device applications.
\begin{figure*}

  \includegraphics[width=12.0cm]{Fig1.eps}
  \caption{(Color online)For Janus monolayer VCClBr, the (a) top view and (b) side view of crystal structure, and   the   primitive cell  is marked  by  black  frames;  (c):The phonon band dispersions  by using GGA; (d):The  total energy fluctuations as a function of simulation time at 300 K, and insets show the
 final structures (top view (a) and side view (b)) of VCClBr after 8 ps at 300 K. }\label{t0}
\end{figure*}

Multifunctional 2D materials  are of particular interest due to providing  a potential platform for multi-functional electronic devices.
Compared to individual  piezoelectricity or  nontrivial band topology  in a single 2D material, their combination  can provide a unique opportunity
for intriguing physics and practical device applications. To search PQSHI,  there are two ways :(1) seeking for nontrivial band topology from 2D piezoelectric materials; (2) finding or constructing non-centrosymmetrical QSHI, which may be a executable method, especially for sandwich structure.
It is well known that a  sandwich structure can be used as a parent to construct Janus structure, which lacks central symmetry, giving rise to  piezoelectric effect\cite{gsd1,gsd2,gsd4}.  The typical Janus MoSSe monolayer has been  experimentally synthesized based on $\mathrm{MoS_2}$ with symmetrical upper and lower atomic layers\cite{e1,e2}.
However, the  nontrivial topology may be broken by building Janus structure from a QSHI\cite{gsd4,gsd4-1}.
The  Janus  $\mathrm{SrAlGaSe_4}$ and MoSSe as derivatives of QSHI $\mathrm{SrGa_2Se_4}$ and 1T'-$\mathrm{MoS_2}$ lose nontrivial topology. However, strain   can  make them possess topological properties of their parents\cite{gsd4,gsd4-1}.

Recently, based on  VOCl as a prototype,  20 2D monolayers VXY (X = B, C, N, O and F; Y = F, Cl, Br and I) are predicted by means of first-principles calculations\cite{w-1}.
They  possess  emerging topological, magnetic and dielectric properties, which depend electron occupation, ligand
electronegativity and crystal field strength. Calculated results demonstrate that the VCY
systems exhibit nontrivial topological properties\cite{w-1}.  Their crystal structure possesses  central symmetry,  which leads to disappeared piezoelectricity.  However, this structure can be regarded as a sandwich structure. These provide a possibility to induce  PQSHI by constructing Janus structure  based on VCY monolayer as parent.

In this work, we build a Janus crystal structure VCClBr, which can be attained by  replacing the  top Cl layer in  $\mathrm{VCCl_2}$  monolayer with Br atoms.
Calculated results show that  VCClBr indeed is a PQSHI with  dynamical, mechanical  and thermal  stabilities. It is proved that nontrivial band topology  of VCClBr is
robust against strain and  external electric field. The calculated out-of-plane piezoelectric response is strong compared with ones of many 2D materials\cite{q7,q9}, like Janus transition metal dichalcogenide (TMD) monolayers. Finally, by analysing  out-of-plane piezoelectric coefficients of  VCXY (X$\neq$Y=F, Cl and  Br), $\mathrm{CrX_{1.5}Y_{1.5}}$ (X=F, Cl and  Br; Y=I)\cite{w-2} and  NiXY (X$\neq$Y=Cl, Br and I)\cite{w-3}, the strong  out-of-plane piezoelectric response can be attained  by  constructing Janus monolayer with large electronegativity difference of X and Y atoms.
Our works highlight  multifunctionalities of Janus VCClBr, which is
promising for applications in the next-generation topotronic and  piezoelectric devices.

The rest of the paper is organized as follows. In the next
section, we shall give our computational details. In  the next few sections,  the  crystal structure and stability, electronic structures and  piezoelectric properties of Janus  monolayer  VCClBr are shown.  Finally,  our discussion and conclusions are given.

\begin{table}
\centering \caption{For monolayer VCClBr, VCFBr and VCFCl, the lattice constants $a$ and $b$ ($\mathrm{{\AA}}$), GGA+SOC gaps $E^g_{SOC}$ (meV), band gap category $BGC$ [direct bandgap (D) and indirect bandgap (I)],  topological
invariant $Z_2$ and dynamical stability.}\label{tab}
  \begin{tabular*}{0.48\textwidth}{@{\extracolsep{\fill}}ccccccc}
  \hline\hline
Name&$a$ &$b$&$E^g_{SOC}$&$BGC$&$Z_2$&Dynamics\\\hline
VCClBr & 3.257& 3.959&76&D&1&Y\\\hline
VCFBr & 3.140&3.949&45&I&1&N\\\hline
VCFCl&3.092&3.949 & 20&D&1&N\\\hline\hline
\end{tabular*}
\end{table}

\section{Computational detail}
We perform density functional theory (DFT) calculations\cite{1} using the Vienna Ab initio
Simulation Package (VASP), with projected augmented wave method\cite{pv1,pv2,pv3}.
  The total energy  convergence criterion and cutoff energy
for plane-wave expansion are set
to $10^{-8}$ eV and 500 eV, respectively.  The exchange-correlation potential is adopted within generalized gradient approximation (GGA) of Perdew, Burke and  Ernzerhof\cite{pbe}. Furthermore, Heyd-Scuseria-Ernzerhof screened hybrid functional
(HSE06)\cite{hse} is also employed to confirm nontrivial topology. The SOC is included  to investigate electronic structures and topological properties of VCClBr. To sample the Brillouin zone (BZ),  a Monkhorst-Pack mesh of 16$\times$13$\times$1  is adopted for geometry optimization, and  the residual force on each atom  is required to be  less than 0.0001 $\mathrm{eV.{\AA}^{-1}}$.
 To eliminate
spurious interactions with its periodic images, a vacuum region
of more than 18 $\mathrm{{\AA}}$ is adopted in the direction perpendicular
to the VCClBr monolayer.

The phonon dispersion
spectrums  are attained  by finite displacement method  with a supercell
of 5$\times$5$\times$1, as implemented in Phonopy code\cite{pv5}.  To calculate the second order interatomic force constants (IFCs), a 2$\times$2$\times$1 k-mesh is employed with kinetic energy cutoff of 500 eV. The elastic stiffness tensor $C_{ij}$ and  piezoelectric stress coefficients $e_{ij}$ are attained by using  strain-stress relationship (SSR) and density functional perturbation theory (DFPT) method\cite{pv6}, respectively.
The 2D elastic coefficients $C^{2D}_{ij}$
 and   piezoelectric stress coefficients $e^{2D}_{ij}$
have been renormalized by the the length of unit cell along $z$ direction ($L_z$):  $C^{2D}_{ij}$=$L_z$$C^{3D}_{ij}$ and $e^{2D}_{ij}$=$L_z$$e^{3D}_{ij}$.
A 16$\times$13$\times$1 Monkhorst-Pack  mesh  is adopted to calculate  $C_{ij}$ and $e_{ij}$ by using  GGA.
WannierTools code\cite{w1} is used to calculate  surface state and $Z_2$ topological invariant, based on the tight-binding Hamiltonians constructed from maximally localized Wannier functions  by employing $d$-orbitals of V atoms and  $p$-orbitals of C, Cl and Br atoms, as  implemented in Wannier90 code\cite{w2}

\begin{figure}

  \includegraphics[width=8cm]{Fig2.eps}
\caption{(Color online) Angular dependence of the Young's modulus $C_{2D}(\theta)$ (Left) and Poisson's
ratio $\nu_{2D}(\theta)$ (Right) of  monolayer VCClBr.}\label{cv}
\end{figure}

\begin{figure}

  \includegraphics[width=8cm]{Fig3.eps}
\caption{(Color online)Top:The energy band structures  of  VCClBr  using GGA  and GGA+SOC.}\label{band}
\end{figure}

\begin{figure*}

   \includegraphics[width=12cm]{Fig4.eps}
\caption{(Color online) The nontrivial topological edge states (Left) and evolution of WCC (Right) of VCClBr monolayer. }\label{band-s}
\end{figure*}

\begin{figure*}

  \includegraphics[width=12cm]{Fig5.eps}
\caption{(Color online)The energy band structures (Left)  of  VCClBr  using HSE06 along with the corresponding nontrivial topological edge states (Right).}\label{band-hse}
\end{figure*}

\begin{figure}

  \includegraphics[width=8cm]{Fig6.eps}
\caption{(Color online)The schematic layered crystal structures of  $\mathrm{VCCl_2}$ (a) and VCClBr (b). Applying  an external electric field $E$ on $\mathrm{VCCl_2}$  can be used to simulate Janus structure VCClBr. }\label{e}
\end{figure}

\begin{figure}

   \includegraphics[width=7.0cm]{Fig7.eps}
  \caption{(Color online) Top plane: the energy band gaps of  $\mathrm{VCCl_2}$ monolayer  as a function of applied electric field  by using both GGA and GGA+SOC; Middle plane: the energy band gaps of  VCClBr monolayer  as a function of applied electric field along +$z$ (up) and -$z$ (dn) directions  by using  GGA+SOC; Bottom plane: the energy band gaps of  VCClBr monolayer  as a function of   applied strain, including biaxial (ab) and uniaxial cases (a and b).  }\label{gap}
\end{figure}
\begin{figure}

  \includegraphics[width=8cm]{Fig8.eps}
\caption{(Color online) The  energy bands of   $\mathrm{VCCl_2}$ monolayer  under 0.00 eV/$\mathrm{{\AA}}$ and 0.04
eV/$\mathrm{{\AA}}$ electric field  using GGA (Left)  and GGA+SOC (Right).}\label{cl-e-band}
\end{figure}

\section{Crystal structure and stability}
As shown in \autoref{t0} (a) and (b), the middle
VC layer of VCClBr is sandwiched by Cl and Br layers, which can be considered as a Janus structure.  The VCClBr  can be constructed by  replacing  top Cl atom layer  in  $\mathrm{VCCl_2}$  monolayer with Br atoms. Its each V atom is 6-
fold coordinated with four C , one Cl and one  Br atoms, giving rise to a strongly distorted octahedron unit.
The symmetry of VCClBr (No.25) is lower than that of  $\mathrm{VCCl_2}$  (No.59) due to
the lack of  horizontal mirror symmetry. Namely, the $\mathrm{VCCl_2}$ has centrosymmetry with disappeared piezoelectricity, while the VCClBr lacks  inversion symmetry, possessing piezoelectricity. The optimized lattice
constant of VCClBr is  a=3.257 $\mathrm{{\AA}}$ and  b=3.959 $\mathrm{{\AA}}$. Due to different atomic sizes and electronegativities of Cl and Br atoms,  inequivalent  bond lengths can be observed, for example $d_{Cl-V}$=2.449 $\mathrm{{\AA}}$ and  $d_{Br-V}$=2.575 $\mathrm{{\AA}}$, which  can induce an electrostatic potential gradient,  giving rise to a built-in electric field.

To validate its dynamic stability, the phonon dispersions  of VCClBr  are calculated, which is plotted in \autoref{t0} (c).
No imaginary vibrational frequency with three acoustic
and fifteen optical phonon branches is observed,  suggesting that VCClBr is
dynamically stable. In addition,  by performing ab-initio molecular dynamics (AIMD) simulations,  the thermal stability  of VCClBr with a supercell of  4$\times$4$\times$1 for more than 8000 fs  at 300 K is examined. The total energy fluctuations as a function of simulation time along with the final
configurations are plotted in \autoref{t0} (d). Calculated results show that VCClBr undergoes  no structural
reconstruction with  small energy
 fluctuates,   indicating its thermal stability.

 To further check
the mechanical stability of VCClBr, its elastic constants are calculated.
By using Voigt notation, the elastic tensor with $mm2$ point-group symmetry  can be written as:
\begin{equation}\label{pe1-4}
   C=\left(
    \begin{array}{ccc}
      C_{11} & C_{12} & 0 \\
     C_{12} & C_{22} &0 \\
      0 & 0 & C_{66} \\
    \end{array}
  \right)
\end{equation}
The calculated  $C_{11}$=130.67 $\mathrm{Nm^{-1}}$, $C_{22}$=215.81 $\mathrm{Nm^{-1}}$, $C_{12}$=17.08 $\mathrm{Nm^{-1}}$  and  $C_{66}$=53.45 $\mathrm{Nm^{-1}}$.
The calculated elastic constants satisfy the  Born  criteria of mechanical stability:
 \begin{equation}\label{pe1-4-0}
  C_{11}>0,~~C_{22}>0,~~ C_{66}>0,~~C_{11}-C_{12}>0
\end{equation}
 Therefore, VCClBr is  mechanically stable.

Since the $C_{11}$ and  $C_{22}$ of  VCClBr are very different, the  anisotropy
of its Young's moduli $C_{2D}(\theta)$ and
Poisson's ratios $\nu_{2D}(\theta)$ should be very distinct. The direction-dependent $C_{2D}(\theta)$ and $\nu_{2D}(\theta)$  can be attained by  the following two formulas\cite{ela,ela1}:
 \begin{equation}\label{pe1-4-1}
  C_{2D}(\theta)=\frac{C_{11}C_{22}-C_{12}^2}{C_{11}m^4+C_{22}n^4+(B-2C_{12})m^2n^2}
\end{equation}
 \begin{equation}\label{pe1-4-2}
  \nu_{2D}(\theta)=\frac{(C_{11}+C_{22}-B)m^2n^2-C_{12}(m^4+n^4)}{C_{11}m^4+C_{22}n^4+(B-2C_{12})m^2n^2}
\end{equation}
where the $\theta$ is  the angle of direction with $x$/$y$ direction defined as as $0^{\circ}$/$90^{\circ}$, $m=sin(\theta)$, $n=cos(\theta)$ and $B=(C_{11}C_{22}-C_{12}^2)/C_{66}$.
 As plotted in \autoref{cv}, the  $C_{2D}(\theta)$ and
 $\nu_{2D}(\theta)$ of  VCClBr  show very strong anisotropy.
The $C_{2D}$ is 129.32/213.58  $\mathrm{Nm^{-1}}$ along $x$/$y$ direction, and 0.079/0.131 for  $\nu_{2D}$.

\begin{figure*}

   \includegraphics[width=16cm]{Fig9.eps}
\caption{(Color online) The energy band structures of VCClBr with SOC  at representative applied strain, including biaxial (Top panel), uniaxial-$a$ (Middle panel) and  uniaxial-$b$ (Bottom panel) cases. }\label{band-strain}
\end{figure*}

\begin{figure}

  \includegraphics[width=8cm]{Fig10.eps}
\caption{(Color online) The energy band structures  of VCFCl and VCFBr monolayers by using  GGA+SOC. }\label{band-1}
\end{figure}

\begin{figure*}

   \includegraphics[width=12cm]{Fig11.eps}
\caption{(Color online)  The nontrivial topological edge states of VCFCl (a) and VCFBr (b).}\label{band-s-1}
\end{figure*}

\section{Electronic structures}
The energy band structures of VCClBr by using GGA and GGA+SOC are plotted in \autoref{band}.
 In the absence of SOC, the VCClBr is a direct gap semiconductor with the gap of  57 meV, and the valence band maximum (VBM) and conduction band minimum (CBM) locate  at the one point along $\Gamma$-Y path. It clearly seen that the very strong anisotropy is observed for energy band structures along $\Gamma$-X and $\Gamma$-Y paths.
  When including SOC, the VCClBr is still  a direct gap semiconductor, and the gap is improved to 76 meV.
The previous calculated results show  that $\mathrm{VCCl_2}$ is a QSHI\cite{w-1}.  The Janus monolayer  $\mathrm{SrAlGaSe_4}$ as a derivative of centrosymmetric QSHI $\mathrm{SrGa_2Se_4}$ is a normal insulator, which can be reverted by strain\cite{gsd4}. However, Janus  monolayer  $\mathrm{CSb_{1.5}Bi_{1.5}}$ still maintains nontrivial topological properties from QSHI $\mathrm{CX_3}$ (X=Sb and Bi) with inversion symmetry\cite{gsd5}. To confirm the topological properties of VCClBr,  the $Z_2$ topological
invariant is calculated. For a material with  inversion symmetry,  $Z_2$ topological
invariant can be attained from the product of parities of all
occupied states at four time-reversal-invariant-momentum
 points in the 2D BZ. However, VCClBr is non-centrosymmetrical,  and the $Z_2$  can be verified  by the calculation of
Wannier charge center (WCC). If $Z_2$ equals 1/0, a material is  a topologically nontrivial/trivial state.
As plotted in \autoref{band-s},   an arbitrary reference line crosses  the evolution lines of WCC  an odd number of times, giving rise to $Z_2$=1, which indicates
that VCClBr is a QSHI. Furthermore, a QSHI should  exhibit non-trivial topological
edge states.  We further calculate one-dimensional edge states on (100) edge, which are plotted in \autoref{band-s}.
It is clearly seen that topological helical edge states with the appearance of the Dirac cone  connect the conduction and valence bands.
The HSE06 is also employed to check the reliability of nontrivial topological properties of VCClBr. Within HSE06+SOC, the energy band structure and one-dimensional edge states are plotted in \autoref{band-hse}. Calculated results show that the gap of VCClBr is improved to 87 meV from 76 meV with GGA+SOC, and the nontrivial topological properties are still kept with topological helical edge states.

The $\mathrm{VCCl_2}$ is a  Dirac semimetal without considering  SOC, and no opened gap is observed\cite{w-1}.
Compared to  $\mathrm{VCCl_2}$,  VCClBr lacks  horizontal mirror symmetry,  possessing a built-in electric field, which induces the opened gap.
To confirm this, we apply  an external electric field ($E$) perpendicular to the $\mathrm{VCCl_2}$, which should induce a gap.
 Since the electric field is a  vector (not a pseudovector),  the horizontal mirror symmetry will be broken by electric field.
 This is different from the magnetization as a pseudovector, which preserves the horizontal mirror symmetry.
Thus, applying  an external electric field can be used to simulate Janus structure (see \autoref{e}) in a sense.

For $\mathrm{VCCl_2}$ monolayer, the evolution of  band gap as a function of applied electric field is provided in \autoref{gap} by using GGA and GGA+SOC, and the representative energy bands    under 0.00 eV/$\mathrm{{\AA}}$ and 0.04
eV/$\mathrm{{\AA}}$ electric field are plotted in \autoref{cl-e-band}.  In the absence of SOC,  the band gap increases  monotonously from 0 meV with increasing electric field for $\mathrm{VCCl_2}$. This implies that applied electric field can induce the gap for $\mathrm{VCCl_2}$,   which is  equivalent to a built-in electric field by constructing Janus structure, like VCClBr. For example 0.04
eV/$\mathrm{{\AA}}$ electric field, the gap of $\mathrm{VCCl_2}$  is  73 meV without considering SOC, and the GGA+SOC gap is reduced to 67 meV.
It is noted that the gap with GGA+SOC is larger than one with GGA under small electric field, and then the opposite situation is observed with sequentially increasing electric field.  Finally, the edge states are calculated within considered $E$ range.  It is found that the nontrivial topological properties of $\mathrm{VCCl_2}$ are robust against applied electric field, and the representative topological helical edge states under  0.40
eV/$\mathrm{{\AA}}$ electric field are plotted in FIG.1 of electronic supplementary information (ESI).

The electric field effects on electronic and topological properties of VCClBr are also investigated.  Janus VCClBr has different atomic species on its upper and lower facets (see \autoref{e}), which means that applying $+z$ and $-z$ directional electric field are not equivalent.  This is different from $\mathrm{VCCl_2}$, which  has the same atomic species on its upper and lower facets (see \autoref{e}), giving rise to equivalent effects  caused by applying both $+z$ and $-z$ directional electric field.
 The energy band gaps of  VCClBr monolayer  as a function of applied electric field along +$z$  and -$z$  directions  by using  GGA+SOC are plotted in \autoref{gap}.
The gap increases with increasing  $+z$  directional $E$, while increasing  $-z$  directional $E$ reduces the gap.  However, the gap change is very small with $E$ from 0.00 eV/$\mathrm{{\AA}}$ to 0.40 eV/$\mathrm{{\AA}}$, and only
5.1/-3.6 meV for $+z$ and $-z$ directional $E$, which is very smaller than one (37.3 meV) of  $\mathrm{VCCl_2}$.
It is also proved that the nontrivial topological properties of VCClBr are robust against applied electric field.

Strain is an effective method to tune electronic properties of 2D systems, and further affects their topological properties\cite{h1,h2,h3}.
Here, both biaxial  and uniaxial strains are considered to investigate the robustness of nontrivial topological properties of VCClBr.
 The $L/L_0$  is used to simulate compressive/tensile strain, where $L$ and $L_0$ are the strained and  unstrained lattice constants, respectively.   The $L/L_0$$<$1 ($L/L_0$$>$1) means compressive (tensile) strain. For biaxial strain,  the $a$  and $b$ lattice parameters simultaneously change, and atomic positions are allowed to relax. For uniaxial strain,  the $a$/$b$ lattice parameter changes, and the $b$/$a$ lattice parameter and atomic positions are allowed to relax  in response to
an applied in-plane uniaxial strain.

The electronic structures  versus $L$/$L_0$ are calculated by using GGA+SOC. The variation of band gaps with biaxial, uniaxial-$a$ and  uniaxial-$b$  cases are plotted in \autoref{gap}, and the energy band structures at representative applied strain are shown in \autoref{band-strain}.
For biaxial strain, with increasing strain, the gap firstly increases, and then decreases, which is due to the transition of CBM from one point  along $\Gamma$-S to  one point along $\Gamma$-Y. It is found that both compressive and tensile strains can induce semiconductor-metal transitions, which are due to the variation of CBM and VBM.
Our calculated results show that the nontrivial topological properties are robust within considered strain range.
For  0.96 or 1.04 strain, VCClBr is a metal, but the gap along  $\Gamma$-Y path is still maintained, and topological helical edge states still exist (see FIG.2 of ESI). Partly, the  uniaxial-$a$ compressive/tensile strain is equivalent to  uniaxial-$b$ tensile/compressive one, which can be confirmed by their gaps versus $L/L_0$.
It is found that  uniaxial strain can also induce the change of CBM or VBM, giving rise to nonmonotonic dependence of gap on $L/L_0$. Semiconductor-metal transitions can also be observed at  uniaxial-$a$ 0.96 strain.  However, the gap along  $\Gamma$-Y path always exists. The topological helical edge states can always be observed with considered uniaxial strain range, which are plotted in FIG.3 and FIG.4 of ESI at 0.96 and 1.04 strains.

\begin{table*}
\centering \caption{For monolayer VCClBr, VCFBr and VCFCl, the elastic constants $C_{ij}$ ($\mathrm{Nm^{-1}}$), piezoelectric stress coefficient $e_{ij}$ along with electronic part $e_{ij_e}$ and  ionic part $e_{ij_i}$ ($10^{-10}$ C/m), and piezoelectric strain coefficient $d_{ij}$ (pm/V). }\label{tab0}
\begin{tabular*}{0.96\textwidth}{@{\extracolsep{\fill}}ccccccccccccc}
  \hline\hline
Name& $C_{11}$&	$C_{12}$&	$C_{22}$&	$C_{66}$&	$e_{31_e}$&	$e_{31_i}$&	$e_{31}$&	$e_{32_e}$&	$e_{32_i}$&	$e_{32}$&	$d_{31}$&	$d_{32}$\\\hline
VCClBr& 130.67&	17.08&	215.81&	53.45&	-0.612&	0.019&	-0.593&	-0.513&	-0.032&	-0.545&	-0.425&	-0.219\\\hline
VCFBr& 151.06&	21.42&	229.97&	54.47&	-1.957&	0.115&	-1.842&	-1.669&	-0.243&	-1.912&	-1.116&	-0.727\\\hline
VCFCl& 161.47&	23.10&	229.17&	55.06&	-1.359&	0.165&	-1.194&	-1.135&	-0.176&	-1.311&	-0.667&	-0.505\\\hline\hline
\end{tabular*}
\end{table*}

\section{Piezoelectric properties}
 For  noncentrosymmetric material, an  applied strain or stress can induce  electric
dipole moments and produce electricity, called  piezoelectric effect. The pristine  $\mathrm{VCCl_2}$ is non-piezoelectric, while VCClBr with particular Janus structure  possessess piezoelectric effect.
The third-rank piezoelectric stress tensor  $e_{ijk}$ and strain tensor $d_{ijk}$ can be used to describe piezoelectric response of a material.  The relaxed piezoelectric tensors ($e_{ijk}$ and $d_{ijk}$) are obtained as the sum of ionic and electronic contributions:
 \begin{equation}\label{pe0}
      e_{ijk}=\frac{\partial P_i}{\partial \varepsilon_{jk}}=e_{ijk}^{elc}+e_{ijk}^{ion}
 \end{equation}
and
 \begin{equation}\label{pe0-1}
   d_{ijk}=\frac{\partial P_i}{\partial \sigma_{jk}}=d_{ijk}^{elc}+d_{ijk}^{ion}
 \end{equation}
Where $P_i$, $\varepsilon_{jk}$ and $\sigma_{jk}$ are polarization vector, strain and stress, respectively, and the superscripts $elc$ and $ion$ mean electronic and ionic contributions. The  $e_{ijk}^{elc}$ and $d_{ijk}^{elc}$  ($e_{ijk}$ and $d_{ijk}$) are also called clamped-ion (relax-ion) piezoelectric coefficients.
The $e_{ijk}$ and $d_{ijk}$ can be connected  by elastic tensor $C_{mnjk}$:
 \begin{equation}\label{pe0-1-1}
    e_{ijk}=\frac{\partial P_i}{\partial \varepsilon_{jk}}=\frac{\partial P_i}{\partial \sigma_{mn}}.\frac{\partial \sigma_{mn}}{\partial\varepsilon_{jk}}=d_{imn}C_{mnjk}
 \end{equation}

For 2D materials,  these conditions ( $\varepsilon_{ij}$=$\sigma_{ij}$=0 for i=3 or j=3) are assumed, which implies that only the in-plane strain and stress are taken into account. However, the polarization can be allowed  to remain out-of-plane. We
define the in-plane directions $x_1$ and $x_2$ as the  short edge and long edge direction of  primitive cell,
and $x_3$ as perpendicular to the 2D layer.
Employing Voigt notation,  the  piezoelectric stress   and strain tensors  due to $mm2$ point group symmetry  can be reduced into :
 \begin{equation}\label{pe1-1}
 e=\left(
    \begin{array}{ccc}
     0 & 0 & 0 \\
     0 & 0 & 0 \\
      e_{31} & e_{32} & 0 \\
    \end{array}
  \right)
    \end{equation}

  \begin{equation}\label{pe1-2}
  d= \left(
    \begin{array}{ccc}
      0 & 0 & 0 \\
       0 & 0 & 0 \\
      d_{31} & d_{32} &0 \\
    \end{array}
  \right)
\end{equation}
When   a uniaxial in-plane strain is applied,  only a  vertical piezoelectric polarization  exists ($d_{31}$$\neq$0 or $d_{32}$$\neq$0).
Moreover,  with applied  biaxial in-plane strain,  the superposed
out-of-plane  polarization  will arise ($d_{31}$$\neq$0 and $d_{32}$$\neq$0).
The $e_{31}$ and  $e_{32}$  can be directly  calculated by DFPT, and the
$d_{31}$ and  $d_{32}$   can be  derived by \autoref{pe1-4}, \autoref{pe0-1-1}, \autoref{pe1-1} and \autoref{pe1-2}:
\begin{equation}\label{pe2-0}
     d_{31}=\frac{e_{31}C_{22}-e_{32}C_{12}}{C_{11}C_{22}-C_{12}^2}
\end{equation}
\begin{equation}\label{pe2-1}
     d_{32}=\frac{e_{32}C_{11}-e_{31}C_{12}}{C_{11}C_{22}-C_{12}^2}
\end{equation}

The calculated $e_{31}$ ($e_{32}$) is -0.593$\times$$10^{-10}$ C/m (-0.545$\times$$10^{-10}$ C/m)  with ionic part 0.019$\times$$10^{-10}$ C/m (-0.032$\times$$10^{-10}$ C/m)  and electronic part -0.612$\times$$10^{-10}$ C/m (-0.513$\times$$10^{-10}$ C/m).
The electronic and ionic polarizations  of $e_{31}$ have  opposite signs,  while they have the same signs for $e_{32}$. However, the electronic contribution
 dominates the  piezoelectricity of both $e_{31}$ and $e_{32}$.  Based on \autoref{pe2-0} and \autoref{pe2-1}, the $d_{31}$  and $d_{32}$  can be attained  from previous calculated $C_{ij}$ and $e_{ij}$. The calculated  $d_{31}$ ($d_{32}$)  is  -0.425 pm/V (-0.219 pm/V),  which are  higher than or  compared with ones of  other 2D materials\cite{q7,q9}.   The $d_{31}$ is almost twice as large as $d_{32}$, which is mainly  due to very different $C_{11}$  and $C_{22}$.  Thus,  VCClBr  is a potential PQSHI.

\begin{figure}

   \includegraphics[width=8cm]{Fig12.eps}
\caption{(Color online)  The $d_{31}$  and  $d_{32}$ of (a): VCXY (X$\neq$Y=F, Cl and  Br), (b):$\mathrm{CrX_{1.5}Y_{1.5}}$ (X=F, Cl and  Br; Y=I) and (c): NiXY (X$\neq$Y=Cl, Br and I) . The electronegativity difference of X and Y atoms are given in parentheses.}\label{d}
\end{figure}
\section{Discussions and Conclusion}
 The $\mathrm{VCY_2}$ (Y=F, Cl, Br and I) monolayers have been predicted to be topological Dirac semimetals without SOC\cite{w-1}. However, for $\mathrm{VCI_2}$,  the Fermi level deviates from Dirac point. Thus, based on $\mathrm{VCY_2}$ (Y=F, Cl and Br) monolayers, we construct another two Janus monolayer VCFBr and VCFCl of VCXY (X$\neq$Y=F, Cl and  Br).
 The optimized lattice constants $a$ and $b$ are listed in \autoref{tab}. It is found that  $a$ increases with the size of  X and Y anions
(from VCFCl to VCFBr to VCClBr), while $b$ is less affected, which is because the
X and Y anions are directly bonded to V atoms along  $x$ direction. The energy band structures of VCFBr and VCFCl within SOC are plotted in \autoref{band-1}.
Calculated results show that VCFCl and VCClBr are direct gap semiconductors, while VCFBr is an indirect gap semiconductor. The CBM of VCFBr is at one point along $\Gamma$-S path, while the VBM locates at one point along $\Gamma$-Y path. For  VCFCl and VCClBr, both CBM and VBM locate at one point along $\Gamma$-Y path.
As summarized in \autoref{tab}, the gap increases with the size of X and Y anions
(from VCFCl to VCFBr to VCClBr). It is clearly seen that topological helical edge states connect the valence and conduction bands of VCFBr and VCFCl (see \autoref{band-s-1}), confirming their nontrivial topological properties.

 For monolayer VCClBr, VCFBr and VCFCl, the calculated elastic constants $C_{ij}$, piezoelectric stress coefficient $e_{ij}$ along with electronic   and  ionic parts, and piezoelectric strain coefficient $d_{ij}$ are listed in \autoref{tab0}. For these monolayers, calculated elastic constants  satisfy   mechanical stability criteria, confirming their mechanical stabilities.
For all these monolayers, the electronic and ionic contributions  of $e_{31}$   have  opposite signs,  while they have the same signs for $e_{32}$, and the electronic part dominates  $e_{31}$  and  $e_{32}$. For three monolayers, the $d_{31}$ is larger than $d_{32}$.  The $d_{31}$ and $d_{32}$ of VCFBr are the largest  among these monolayers, which may be due to very different electronegativities of F and Br atoms compared with F and Cl or Cl and Br atoms. The $d_{31}$ and $d_{32}$ (absolute value) of VCXY (X$\neq$Y=F, Cl and  Br) along with the electronegativity difference of X and Y atoms are plotted in \autoref{d}.
  Similar phenomenon  can be found in monolayer $\mathrm{CrX_{1.5}Y_{1.5}}$ (X=F, Cl and Br; Y=I) and NiXY (X$\neq$Y=Cl, Br and I)\cite{w-2,w-3}, and  their $d_{31}$  along with the electronegativity difference of X and Y atoms are also plotted in \autoref{d}. It is clearly seen that the large  electronegativity difference of X and Y atoms is related with large out-of-plane piezoelectric response. It is noted that the $d_{31}$ of VCFBr is larger than 1.0 pm/V, which is rare for out-of-plane piezoelectric response of 2D systems.   Finally, the phonon dispersions of  VCFBr and VCFCl are calculated, which are plotted in FIG.5 of ESI. Unfortunately, the two monolayers are dynamically unstable due to existing imaginary frequencies. However, these comparisons provide some thoughts and scientific basis for  searching large out-of-plane piezoelectric response in Janus 2D materials.

In summary, we have demonstrated coexistence of piezoelectricity and robust nontrivial band topology in Janus VCClBr monolayer with dynamical, mechanical and thermal  stabilities based on the reliable DFT calculations. The  nontrivial band topology in Janus VCClBr monolayer is confirmed by topological helical edge states, which is
robust against strain and external electric field.
 Only out-of-plane piezoelectric response exists due to broken horizontal mirror symmetry. The predicted   out-of-plane $d_{31}$ and $d_{32}$  are  higher than  or comparable with ones of  many 2D known materials, which is highly desirable for ultrathin piezoelectric devices. It is found that the elastic, electronic and piezoelectric properties of  VCClBr monolayer show very strong anisotropy due to the Cl and Br anions  directly bonded to V atoms along the $x$ direction.
Our predicted VCClBr provides a possible platform  to future development
of multifunctional piezoelectronics, and these findings open
new idea to realize  PQSHI with large out-of-plane piezoelectric response.

\begin{acknowledgments}
This work is supported by Natural Science Basis Research Plan in Shaanxi Province of China  (2021JM-456). We are grateful to the Advanced Analysis and Computation Center of China University of Mining and Technology (CUMT) for the award of CPU hours and WIEN2k/VASP software to accomplish this work.
\end{acknowledgments}


\section{Introduction}\label{sec:intro}
Thermodynamic properties of gauge theories are of common interest
in various fields of physics ~\cite{LeBellac,Kapusta},
including the baryogenesis and leptogenesis,
the physics of quark-gluon plasma (QGP),
neutrinos physics in big-bang nucleosynthesis
and in compact stars, and many others.
A novel ingredient in the dynamics at finite temperature ($T$)
is due to the existence of thermally excited particles,
with which a probe particle is scattered in the classical level,
leading to a damping known as Landau damping and a modification
of the spectral properties of the probe particle.
In this paper, partly motivated by the recent development of
the resonant leptogenesis scenario \cite{Pilaftsis,Pilaftsis:2005rv},
we study the spectral properties of neutrinos at finite $T$,
specifically in the electroweak-scale, in a way where
the gauge (in)dependence is manifestly checked.

Although the standard model (SM) seems to fail to explain
the observed baryon asymmetry of the universe (BAU),
an extension of the SM by adding right-handed Majorana neutrinos
($N_{\sub{R}}$) may have a chance to account for the BAU.
Indeed, a thermal leptogenesis scenario
based on such an extended model
\cite{Fukugita:1986hr,LGene_Review}
gives a simple and promising account of the BAU.
In this scenario,
a decay of $N_{\sub{R}}$ out of equilibrium
via Yukawa interactions
generates a net lepton number,
which is partially converted into the baryon numbers
through sphaleron processes
\cite{Kuzmin:1985mm,Klinkhamer:1984di,Arnold:1987mh}
in the electroweak phase transition.
Furthermore, if the mass difference between
two right-handed neutrinos
is on the order of their $CP$-violating decay width,
the $CP$ asymmetry is
dynamically enhanced and so is leptogenesis
\cite{Pilaftsis,Pilaftsis:2005rv}.
This resonant leptogenesis scenario implies that
$N_{\sub{R}}$ with a TeV-scale or electroweak-scale mass
is relevant to the BAU and thus
has opened a possibility to create
a lepton number at electroweak-scale temperature.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=3.0cm]{NR_decay.eps}
\caption{A decay process of
a right-handed Majorana neutrino ($N_{\mathrm{R}}$)}
to a standard model lepton $L=(\nu,l_{\sub{L}})^{T}$ and
a standard model Higgs doublet $\Phi$.
\label{Fig:NR_decay}
\end{center}
\end{figure}

Figure~\ref{Fig:NR_decay} shows
the $CP$-violating decay process of $N_{\sub{R}}$,
which gives one of the most important contributions
to lepton number creation, where
both the Higgs particle and leptons are involved.
In the present work,
we suppose that the relevant processes occur
at finite $T$ of the electroweak-scale.
In this $T$ region,
the following two finite-$T$ effects come into play:
1)~the Landau damping of the Higgs particle and leptons
by thermally excited particles including massive gauge bosons;
2)~the $T$ dependence of the masses of
the involved particles,
generated
by the Englert-Brout-Higgs-Guralnik-Hagen-Kibble mechanism
\cite{EBHGHK},
which will be called simply
the Higgs mechanism from now on in this paper.
If the standard model leptons have nontrivial
spectral properties due to the aforementioned two effects,
lepton number creation
via the $N_{\sub{R}}$ decay in Fig.~\ref{Fig:NR_decay}
will be significantly modified.

The important role of the Landau damping for
modifying the fermion spectral properties
is known and established
in QED and QCD in the context of QGP physics.
A fermion interacting with thermally excited gauge bosons and antifermion
admits a novel collective excitation mode
known as a (an anti) plasmino
as well as the normal quasiparticle excitation with thermal mass
\cite{quasi_particle}.
Such collective excitations in the QGP at extremely high $T$
are shown to exist and their dispersion relations are obtained in a gauge-invariant way,
at least at one-loop order,
which is now known as the hard thermal loop (HTL)
approximation \cite{HTL}.
A full spectral property beyond the dispersion relations
of massive electrons in QED plasma
is investigated in Ref.~\cite{Baym:1992eu},
where a physical account of the interplay
between the electron-mass and temperature-effects
that leads to a Landau damping were presented.

As first discussed in Ref.~\cite{DOlivo:1992vm},
we expect that
the leptons appearing in Fig.~\ref{Fig:NR_decay}
should also admit nontrivial collective excitations at finite $T$.
Recently, the neutrino spectral properties
at electroweak-scale temperature
in the broken phase have been investigated
by using the HTL approximation in the unitary gauge
\cite{Boyanovsky:2005hk}.
A remarkable finding there was the possible existence of
a novel branch of the dispersion relation with
a collective nature in the ultrasoft-energy region.
Here, an important question is whether
such an ultrasoft mode is physically significant with regards to
the gauge invariance, which cannot be checked within the unitary gauge.
In fact, the gauge (in)dependence of the spectral properties of neutrinos
in the ultrasoft-energy region
has not been fully investigated yet,
except for those
 at $T=1$--$10$ MeV~\cite{DOlivo:1992vm}.

Here, we note that the gauge (in)dependence of
the spectral properties of massless fermions (quarks)
in the  ultrasoft-energy region
has been discussed in the physics of QGP.
It was shown \cite{3peakQGP,Kitazawa:2007ep} that the spectral function of
massless quarks coupled to
massive vector soft modes in QGP
can have  a novel peak in the  ultrasoft-energy region
as well as those corresponding to the normal and the antiplasmino modes,
and it was later demonstrated~\cite{Satow:2010ia} that
their properties are gauge independent
by using the Stueckelberg formalism~\cite{Stueckel_O,Stueckel_R}.

Now, one should recognize that
the relevant ingredients for the description
of hot neutrinos at the electroweak-scale temperature
are the same as those in Ref.~\cite{Satow:2010ia},
{\em i.e.}, a massless fermion and a massive vector boson with
a mass comparable with $T$.
Thus, a natural and intriguing question is whether
the neutrino spectral density at $T$ of the electroweak-scale
should also develop a three-peak or three-bump structure
including an ultrasoft mode without gauge dependence,
as quarks do in the QGP
\cite{3peakQGP,Kitazawa:2007ep,Satow:2010ia,HaraNemo:2008,Qin:2010pc,Nakkagawa:2012}.

In the present work,  we investigate
in a gauge-invariant way the spectral properties
of neutrinos at weak coupling,
without restricting ourselves to their dispersion relations,
in the full energy-momentum plane
at finite $T$ around the electroweak-scale.
We show that the neutrino develops
an ultrasoft excitation mode as well as
the normal and plasmino ones
without recourse to the HTL approximation.
We shall employ the $R_{\xi}$ gauge~\cite{Fujikawa:1972fe}
for examining the possible gauge-fixing dependence
of the spectral density in both broken and symmetric phases.
Furthermore, we also discuss
the implications of the resulting neutrino spectral properties
to the cosmology by considering
the electroweak-scale resonant leptogenesis.

This paper is organized as follows.
In the next section,
we introduce the Lagrangian of the electroweak theory
in the $R_{\xi}$ gauge~\cite{Fujikawa:1972fe} and
the basic quantities  to
evaluate the spectral properties of neutrinos
in field theory at finite temperature.
In Sec.~\ref{sec:Res},
we  show the numerical results for the self-energy
and the spectral density
of neutrinos in the broken phase.
The results will be
compared with the unitary-gauge HTL results
\cite{Boyanovsky:2005hk}.
In Sec.~\ref{sec:SDens_M0},
we  extend our study into the symmetric phase,
and investigate the fate of collective modes for neutrinos.
Then in Sec.~\ref{sec:discuss},
we discuss possible implications of
the neutrino spectral density obtained in the present work
in the context of resonant leptogenesis.
Finally, we make concluding remarks in Sec.~\ref{sec:sum}.
Appendices \ref{app:EW_params}--\ref{App:SEne_M0}
are devoted to explaining technical details.

\section{Preliminaries}\label{sec:EWT_SDens}

\subsection{Electroweak theory at finite temperature in $R_{\xi}$ gauge}
\label{subsec:EWPT_Rxi}
The Lagrangian to be considered is composed of
the lepton $L$,
the Higgs $\Phi$,
and the gauge boson
$\mathcal{G}_{\mu}=\mathcal{A}_{\mu},\mathcal{B}_{\mu}$ sectors,
\begin{align}
&\mathcal{L}[\Phi,L,\mathcal{G}_{\mu}]
=
\mathcal{L}_{\sub{L}}[L,\mathcal{G}_{\mu}]
+ \mathcal{L}_{\sub{H}}[\Phi,\mathcal{G}_{\mu}]
+ \mathcal{L}_{\sub{G}}[\mathcal{G}_{\mu}]
\ ,\label{eq:L_tot}
\end{align}
up to gauge fixing and
Faddeev-Popov ghost terms.
The Higgs sector is given by
\begin{align}
\mathcal{L}_{\sub{H}}
&=\bigl(\mathcal{D}_{\mu}\Phi\bigr)^{\dagger}
\bigl(\mathcal{D}^{\mu}\Phi\bigr)
+ \mu^2_0\Phi^{\dagger}\Phi
- \lambda\bigl(\Phi^{\dagger}\Phi\bigr)^2\ ,\label{eq:LH}\\
\mathcal{D}_{\mu}\Phi
&=      \biggl(
		\partial_{\mu}\mathbf{1}
		- ig\frac{\sigma^a\mathcal{A}_{\mu}^a}{2}
		- \frac{ig^{\prime}}{2}\mathcal{B}_{\mu}\mathbf{1}
	\biggr)\Phi \ ,
\end{align}
where $\sigma^{a=1,2,3}$ are Pauli matrices,
and $\mathcal{A}_{\mu}$ and
$\mathcal{B}_{\mu}$ are the gauge fields
of $\mathrm{SU}_\sub{W}(2)$ and $\mathrm{U}_\sub{Y}(1)$ groups,
respectively.
The field $\Phi$ represents the weak doublet scalar,
which can be parametrized as
\begin{align}
\Phi(x)
=	\frac{1}{\sqrt{2}}
\begin{bmatrix}
\phi^{2}(x)+i\phi^{1}(x)\\
v+h(x)-i\phi^3(x)
\end{bmatrix}
\ .
\end{align}
Here, the constant $v$
represents the vacuum expectation value (VEV).
The fluctuations
of the Higgs field and the Nambu-Goldstone modes
are denoted by $h(x)$ and $\phi^a(x)$, respectively,
the latter of which is to be eaten by the gauge fields.

The gauge-fixing term in the $R_{\xi}$ gauge reads
\begin{align}
\mathcal{L}_{\sub{GF}}
&=
-\frac{1}{2\xi}
\Bigl(
\bigl(F^a_{\sub{A}}(x)\bigr)^2
+ \bigl(F_{\sub{B}}(x)\bigr)^2
\Bigr)\ ,\label{eq:L_G_GF}
\end{align}
where
\begin{align}
&F^a_{\sub{A}}(x)
= \partial_{\mu}\mathcal{A}^{\mu,a}(x)
+ \xi m_{\sub{A}} \phi^a(x)\ ,\label{eq:GFA}\\
&F_{\sub{B}}(x)
= \partial_{\mu}\mathcal{B}^{\mu}(x)
- \xi m_{\sub{B}}\phi^3(x)\ ,\label{eq:GFB}
\end{align}
with
$(m_{\sub{A}},m_{\sub{B}})
=(g\mu_0/(2\sqrt{\lambda}),g^{\prime}\mu_0/(2\sqrt{\lambda}))$.
The gauge-fixing conditions
are specified by the parameter $\xi$.

From the gauge-boson sector $\mathcal{L}_{\sub{G}}$
and the gauge-fixing term $\mathcal{L}_{\sub{GF}}$,
the bare propagators of the gauge bosons
are obtained as a sum of the unitary gauge
and the $\xi$-dependent parts,
\begin{align}
G_{\mu\nu}(q)
&=
G_{\mu\nu}^{\sub{U}}(q) + G_{\mu\nu}^{\xi}(q)
\ ,\label{eq:G}\\
G_{\mu\nu}^{\sub{U}}(q)
&=
-\sum_{t=\pm}
\frac{t\bigl(g_{\mu\nu}-q_{\mu}q_{\nu}/M^2(T)\bigr)}
{2E_{\mathbf{q}}(M)\bigl(q_0-tE_{\mathbf{q}}(M)\bigr)}
\ ,\label{eq:GU}\\
G_{\mu\nu}^{\xi}(q)
&=
\sum_{t=\pm}
\frac{t\cdot q_{\mu}q_{\nu}/M^2(T)}
{2E_{\mathbf{q}}^{\xi}(M)
\bigl(q_0-tE_{\mathbf{q}}^{\xi}(M)\bigr)}
\ ,\label{eq:Gxi}
\end{align}
where
\begin{align}
E_{\mathbf{q}}(M)\equiv \sqrt{\abs{q}^2+M^2(T)}\ ,
E_{\mathbf{q}}^{\xi}(M)\equiv E_{\mathbf{q}}(\sqrt{\xi}M)
\ ,\label{eq:E}
\end{align}
and $M(T)$ represents
the $W$ boson, $Z$ boson or photon mass
at finite $T$:
$M(T)=M_{\sub{W}}(T),~M_{\sub{Z}}(T),~M_{\mathrm{ph}}(T)$.
In the weak coupling,
 the weak-boson masses
$M_{\sub{W,Z}}(T)$ at finite temperature
are given by simply replacing the VEV
at the vacuum ($v_0=\mu_0/\sqrt{\lambda}=246$ GeV)
with that at  finite temperature $v(T)$:
\begin{align}
&\bigl(
M_{\sub{W}}(T),~
M_{\sub{Z}}(T),~
M_{\mathrm{ph}}(T)
\bigr)\nonumber\\
&=
\Bigl(
\frac{gv(T)}{2},~
\frac{\sqrt{g^2 + g^{\prime 2}}~v(T)}{2},~
0
\Bigr)\ .\label{eq:MGT}
\end{align}
In fact,
the weak-boson masses (\ref{eq:MGT}) acquire
thermal corrections of order $gT$ from the one-loop self-energy,
as was shown by C. Manuel \cite{Manuel:1998vg}
under the condition $M_{\sub{W,Z}}(T) \ll T$ and $\lambda \to\infty$
in the Stueckelberg formalism.
However, our main focus is on the region satisfying
$M_{\sub{W,Z}}\sim gv(T)\gg gT$, or equivalently $v(T)\gg T$,
which means that the thermal corrections to
the masses are of higher order at weak coupling,
and can be neglected in the region of our interest.

Next, we discuss how to choose
the effective potential in the Higgs sector.
The lattice Monte Carlo simulations
have shown that
the electroweak symmetry breaking or restoration
is a smooth crossover \cite{Lat_EWPT},
for a Higgs mass $m_{\sub{H}}$ larger than about $\sim 70$ GeV
in the standard model.
Assuming that the new boson with the mass around $125$--$127$ GeV
observed at the LHC
\cite{:2012gu,:2012gk,Aaltonen:2012if,Abazov:2012qya}
is the Higgs particle,
the possibility of
a strong first-order electroweak transition is excluded
within the standard model.
Note that a strong first-order electroweak transition
is not necessary in the thermal leptogenesis scenario,
unlike some of other scenarios
of baryon number creation~\cite{Rubakov:1996vz}.
As pointed out in Ref.~\cite{Boyanovsky:2005hk},
the absence of the first-order transition
indicates that the weak-boson masses in electroweak plasma
$M_{\sub{W,Z}}(T)$ smoothly go to zero.
Therefore, there should exist a temperature regime
satisfying $M_{\sub{W,Z}}(T)\sim T$,
where we expect a nontrivial
spectral property of the standard model particles.
In order to implement this feature with a smooth transition,
we use the following Higgs effective potential~\cite{Kapusta}:
\begin{align}
V_{\mathrm{eff}}
&=
-\frac{1}{2}\mu^2(T)v^2(T)
+ \frac{\lambda}{4}v^4(T)
\ ,\label{eq:Veff}\\
\mu^2(T)
&=
\mu_0^2
\Bigl(
1-\frac{T^2}{T_c^2}
\Bigr)\ ,\\
T_{c}^2
&=
\frac{4\mu_0^2}{2\lambda + 3g^2/4 + g^{\prime~2}/4}\ ,
\end{align}
where $\mu_0^2=m_{\sub{H}}^2/2$
and $\lambda=m_{\sub{H}}^2/(2v_0^2)$
are the mass and coupling
parameters in the Higgs Lagrangian (\ref{eq:LH}).
This effective potential has been derived in the $R_{\xi}$ gauge
by taking account of
the leading order of the high-$T$ expansion
for thermal one-loop effects of the Higgs and gauge bosons
in addition to the tree-level Higgs potential.
The $\xi$ dependencies cancel out
between the Nambu-Goldstone bosons and the ghost contributions.
The effective potential leads to the second-order phase transition,
and the VEV at finite $T$ is obtained as
\begin{align}
v^2(T) = v_0^2
\Bigl(1-\frac{T^2}{T_c^2}\Bigr)\ ,\quad
T\leq T_c
\ .\label{eq:VEV}
\end{align}
Figure \ref{Fig:VEV} shows
the $v(T)$ and the weak-boson masses as a function of temperature.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=7.5cm]{VEV_T_SM.eps}
\caption{The VEV and gauge-boson masses
in thermal background as a function of temperature $T$.
All quantities are normalized by the vacuum VEV $v_0=246$ GeV.}
\label{Fig:VEV}
\end{center}
\end{figure}

\subsection{Self-energy and spectral density
for left-handed neutrinos at finite $T$}
\label{subsec:SEne_lepton}
In the broken phase of
the electroweak symmetry,
the Lagrangian for the lepton sector takes the following form:
\begin{align}
\mathcal{L}_{\sub{L}}
&=
\sum_{i=e,\mu,\tau}\Bigl[
(\bar{\nu}^i,\bar{l}_{\sub{L}}^i)
i\sla{-1.0}{-0.5}{\partial}
\begin{pmatrix}
\nu^i \\
l_{\sub{L}}^i
\end{pmatrix}
+ \bar{l}_{\sub{R}}^i
i\sla{-1.0}{-0.5}{\partial}
l_{\sub{R}}^i
\Bigr]\nonumber\\
&\quad +
\Bigl(
W^{\dagger}_{\mu}J^{\mu}_{\sub{W}}
+J^{\dagger}_{\sub{W}~\mu}W^{\mu}
+Z_{\mu}J^{\mu}_{\sub{Z}}
+A_{\mu}^{\sub{EM}}J^{\mu}_{\sub{EM}}
\Bigr)\ ,\label{eq:L_Lep}
\end{align}
where
$\nu^i=\mathcal{P}_{\sub{L}}\nu^i_{\sub{D}}$
and
$l_{\sub{L/R}}^i=\mathcal{P}_{\sub{L/R}}l^i$
represent the left-handed neutrinos
and left-handed (right-handed) charged leptons,
respectively.
Since we are interested in the temperature region comparable
to the weak-boson masses $T\sim M_{\sub{W,Z}}(T)$,
where the lepton masses
are much smaller than $M_{\sub{W,Z}}(T)$,
we have neglected such  tiny masses of the leptons.
It is expected that the massless approximation will not
affect the basic spectral properties of the neutrinos, as is
demonstrated in the case of light
quarks in QGP~\cite{Kitazawa:2007ep,Hidaka:2011rz}.

The gauge currents in the Lagrangian (\ref{eq:L_Lep}) are found to be
\begin{align}
J_{\sub{W}}^{\mu}
&=
\frac{g}{\sqrt{2}}\sum_{i=e,\mu,\tau}
\bar{l}^i\gamma^{\mu}\mathcal{P}_{\sub{L}}
\nu^i_{\sub{D}}
\ ,\label{eq:JW}\\
J_{\sub{Z}}^{\mu}
&=
\sum_{i=e,\mu,\tau}\Bigl[
\frac{g\sin^2\theta_{w}}{\cos\theta_{w}}
\bar{l}^i\gamma^{\mu}l^i
\nonumber\\
&\quad +
\frac{g}{\cos\theta_{w}}
(\bar{\nu}^i_{\sub{D}},\bar{l}^i)
\mathcal{P}_{\sub{R}}\gamma^\mu\mathcal{P}_{\sub{L}}
\frac{\sigma^3}{2}
\begin{pmatrix}
\nu^i_{\sub{D}}\\
l^i
\end{pmatrix}
\Bigr]
\ ,\label{eq:JZ}\\
J_{\sub{EM}}^{\mu}
&=
-e\sum_{i=e,\mu,\tau}\bar{l}^i\gamma^{\mu}l^i
\ ,\label{eq:JEM}
\end{align}
where the couplings $g$, $g^{\prime}$, $e$
and the Weinberg angle $\theta_w$
are given in Table \ref{tab:params} in Appendix \ref{app:EW_params}.
The Feynman diagram for the neutrino self-energy
is shown in Fig.~\ref{Fig:FD_SEneNeu}.
We note that a neutral-current tadpole diagram with
a lepton loop does not appear
in the $CP$-symmetric thermal background, which we assume
because we are motivated by the scenario of thermal leptogenesis,
where the $CP$ violation is considered to arise from
the decay of right-handed neutrinos
rather than the thermal background itself.
For the massless neutrinos,
it is sufficient to consider the single-generation case,
and thus we drop the generation index ``$i$.''

\begin{figure*}[ht]
\begin{center}
\includegraphics[width=15.0cm]{FD_SEneNeu.eps}
\caption{
One-loop Feynman diagrams
of the neutrino self-energies with the weak-boson exchanges.
The symbols ``$l^i$'' and ``$\nu_{\sub{D}}^i$''
represent a charged lepton and
a Dirac neutrino in the ``$i$th'' generation, respectively.
The oriented lines represent a fermion propagator.
The chirality is managed by the projection
operators $\mathcal{P}_{\sub L/R}=(1\mp \gamma_5)/2$
on the vertex.
In the right panel, $c=\cos\theta_w$.
}\label{Fig:FD_SEneNeu}
\end{center}
\end{figure*}

The Feynman diagrams listed in Fig.~\ref{Fig:FD_SEneNeu}
include the following one-loop integral as a common factor,
\begin{align}
&\sigma(\mathbf{p},i\omega_m;T,M(T))\nonumber\\
&\quad=T\sum_n\int\frac{d^3k}{(2\pi)^3}
(-i\gamma^{\mu})
G_{\mathrm{F}}(\mathbf{k},\omega_n)
(-i\gamma^{\nu})\nonumber\\
&\qquad\times G_{\mu\nu}(\mathbf{p-k},i\omega_m-i\omega_n)
\ ,\label{eq:SEne_common}
\end{align}
where $G_{\mathrm{F}}$ represents an imaginary-time
propagator for a lepton.
Self-energy corrections
in $G_{\mathrm{F}}$
are of order $g^2T$ and can be neglected  at weak coupling.
Accordingly we can use the massless propagator as $G_{\mathrm{F}}$,
\begin{align}
\label{eq:GF}
G_{\mathrm{F}}(\mathbf{k},\omega_n)
&=
\frac{i}{i\omega_n\gamma^0-\mathbf{k}\cdot\boldsymbol{\gamma}}
= \sum_{s=\pm}\frac{\Lambda_{\mathbf{k},s}\gamma^0}{i\omega_n-s\abs{k}},
\end{align}
with
\begin{align}
\Lambda_{\pm,\mathbf{k}}
&=
\frac{1\pm\gamma^0\hat{\mathbf{k}}\cdot\boldsymbol{\gamma}}{2}
\ ,\quad\quad
\hat{\mathbf{k}}=\mathbf{k}/\abs{k} \ .
\label{eq:Lambda}
\end{align}
The weak-boson propagator $G_{\mu\nu}$
is given by Eq.~(\ref{eq:G})
with the momentum identification
$q_{\mu} = (i\omega_m - i\omega_n,\mathbf{p} - \mathbf{k})$,
and the mass $M(T)$ is interpreted to be
the $W$ or $Z$ mass shown in Eq.~(\ref{eq:MGT}).
We note that
the projection operator $\mathcal{P}_{\sub{L/R}}$
has not been included in Eq.~(\ref{eq:SEne_common}),
and will be taken account of later.

The nontrivial thermal effect
emerges from the one-loop momentum integral
in the self-energy (\ref{eq:SEne_common}),
and we evaluate it
in the imaginary-time formalism.
Carrying out the summation over the Matsubara frequency $n$
and applying the analytic continuation
$i\omega_m\to \omega+i\eta$ in Eq.~(\ref{eq:SEne_common}),
we obtain the retarded self-energy as
\begin{align}
\sigma^{\mathrm{ret}}(\mathbf{p},\omega;T,M)
= \sigma^{\mathrm{ret}}_{\sub{U}}(\mathbf{p},\omega;T,M)
+ \sigma^{\mathrm{ret}}_{\xi}(\mathbf{p},\omega;T,M)
\ ,\label{eq:sig_ret}
\end{align}
where
\begin{align}
&\sigma^{\mathrm{ret}}_{\sub{U}}(\mathbf{p},\omega;T,M)\nonumber\\
&\quad= -\sum_{s,t = \pm}\int\frac{d^3k}{(2\pi)^3}
\frac{t\gamma^{\mu}\Lambda_{s,\mathbf{k}}\gamma^0\gamma^{\nu}}
{2E_{\mathbf{q}}}
\biggl(
g_{\mu\nu}-\frac{q_{\mu}q_{\nu}}{M^2(T)}
\biggr)\nonumber\\
&\qquad \times
\frac{N_{\mathrm{F}}(s\abs{k}/T) + N_{\mathrm{B}}(-tE_{\mathbf{q}}/T)}
{\omega - s\abs{k} -tE_{\mathbf{q}}(M) + i\eta}
\ ,\label{eq:sig_ret_U}\\
&\sigma^{\mathrm{ret}}_{\xi}(\mathbf{p},\omega;T,M)\nonumber\\
&\quad= -\sum_{s,t = \pm}\int\frac{d^3k}{(2\pi)^3}
\frac{t\gamma^{\mu}\Lambda_{s,\mathbf{k}}\gamma^0\gamma^{\nu}}
{2E_{\mathbf{q}}^{\xi}(M)}\frac{q_{\mu}q_{\nu}}{M^2(T)}\nonumber\\
&\qquad \times
\frac{N_{\mathrm{F}}(s\abs{k}/T) + N_{\mathrm{B}}(-tE_{\mathbf{q}}^{\xi}/T)}
{\omega - s\abs{k} -t E_{\mathbf{q}}^{\xi}(M) + i\eta}
\ .\label{eq:sig_ret_xi}
\end{align}
Here, the indices ``$U$'' and ``$\xi$"
are associated with the unitary gauge and
the $\xi$-dependent gauge-boson propagators
defined in Eqs.~(\ref{eq:G})--(\ref{eq:Gxi}).
The gauge-boson energy
$E_{\mathbf{q}}$ is defined in Eq.~(\ref{eq:E}),
and $N_{\mathrm{F,B}}$ represent
the Fermi-Dirac and Bose-Einstein distribution functions, respectively,
\begin{align}
N_{\mathrm{F}}(x)
= \frac{1}{e^x + 1}
\ ,\quad
N_{\mathrm{B}}(x)
= \frac{1}{e^x - 1}
\ .\label{eq:Dist}
\end{align}

Similarly to the propagator $G_{\mathrm{F}}$,
we decompose
the retarded self-energy $\sigma^{\mathrm{ret}}$ as
\begin{align}
\sigma^{\mathrm{ret}}(\mathbf{p},\omega;T,M(T))
= \sum_{s=\pm}\bigl[\Lambda_{s,\mathbf{p}}\gamma^0\bigr]\
\sigma_s(\mathbf{p},\omega;T,M(T))
\ ,\label{eq:chi_decomp}
\end{align}
where the coefficients
\begin{align}
\sigma_{\pm}(\mathbf{p},\omega;T,M(T))
=
\frac{1}{2}\mathrm{tr}_{\mathrm{spin}}
\Bigl[
\Sigma^{\mathrm{ret}}(\mathbf{p},\omega;T)
\Lambda_{\pm,\mathbf{p}}\gamma^0
\Bigr]
 \label{eq:sig_pm}
\end{align}
are responsible for finite-$T$ effects.
By using $\sigma_{\pm}$,
the retarded self-energy for
the left-handed neutrinos
$\Sigma_{\mathrm{ret}}^{(\nu)}$
is now constructed as
\begin{align}
\Sigma_{\mathrm{ret}}^{(\nu)}(\mathbf{p},\omega;T)
&=
\sum_{s=\pm}
\bigl[\mathcal{P}_{\sub{R}}
\Lambda_{s,\mathbf{p}}\gamma^0
\mathcal{P}_{\sub{L}}\bigr]\
\Sigma_{s}^{(\nu)}(\abs{p},\omega;T)
\ ,\label{eq:Sig_nu_ret}
\end{align}
where
\begin{align}
&\Sigma_{\pm}^{(\nu)}(\abs{p},\omega;T)
=\biggl(\frac{g}{\sqrt{2}}\biggr)^2
\sigma_{\pm}(\abs{p},\omega;T,M_{\sub{W}}(T))\nonumber\\
&\quad +
\biggl(\frac{g}{2\cos\theta_{w}}\biggr)^2
\sigma_{\pm}(\abs{p},\omega;T,M_{\sub{Z}}(T))
\ ,\label{eq:sig_nu_pm}
\end{align}
as read off from
the structure of the chirality and the gauge interaction given
by Eqs.~(\ref{eq:L_Lep})--(\ref{eq:JEM})
(and Fig.~\ref{Fig:FD_SEneNeu}).
In Eq.~(\ref{eq:Sig_nu_ret}),
the left-handed nature of massless neutrinos
is taken care of by the spinor matrix
$
\mathcal{P}_{\sub{L}}
\Lambda_{\pm,\mathbf{p}}\gamma^0
\mathcal{P}_{\sub{R}}\
$.

We shall now investigate
the neutrino spectral density,
which is defined by the imaginary part
of the retarded Green function,
\begin{align}
\rho^{(\nu)}(\mathbf{p},\omega;T)
&=
\sum_{s=\pm}
\bigl[\mathcal{P}_{\sub{R}}
\Lambda_{s,\mathbf{p}}\gamma^0
\mathcal{P}_{\sub{L}}\bigr]\
\rho_{s}^{(\nu)}(\abs{p},\omega;T)\nonumber\\
&\equiv
-\frac{1}{\pi}\mathrm{Im}\
G^{(\nu)}_{\mathrm{ret}}(\mathbf{p},\omega;T)
\ ,\label{eq:rho_def}
\end{align}
where
\begin{align}
G^{(\nu)}_{\mathrm{ret}}(\mathbf{p},\omega;T)
&=
\sum_{s=\pm}
\frac{\mathcal{P}_{\sub{R}}
\Lambda_{s,\mathbf{p}}\gamma^0
\mathcal{P}_{\sub{L}}}
{(\omega+i\eta)-s\abs{p}
-\Sigma^{(\nu)}_{s}(\abs{p},\omega;T)}
\ ,\label{eq:G_nu_ret}
\end{align}
or equivalently,
\begin{align}
&\rho_{\pm}^{(\nu)}(\abs{p},\omega;T)\nonumber\\
&= \frac{-\mathrm{Im}~\Sigma_{\pm}^{(\nu)}(\abs{p},\omega;T,\xi)/\pi
}{\{\mathcal{D}_{\pm}^{(\nu)}(\abs{p},\omega;T,\xi)\}^2
+\{\mathrm{Im}~\Sigma_{\pm}^{(\nu)}(\abs{p},\omega;T,\xi)\}^2}
\ ,\label{eq:rho_nu_pm}\\
&\mathcal{D}_{\pm}^{(\nu)}(\abs{p},\omega;T,\xi)
= \omega - \abs{p}
\mp \mathrm{Re}~\Sigma_{\pm}^{(\nu)}(\abs{p},\omega;T,\xi)
\ .\label{eq:Disp_p_nu}
\end{align}
The calculational procedure of
$\mathrm{Im}~\Sigma^{(\nu)}_{\pm}$
and
$\mathrm{Re}~\Sigma^{(\nu)}_{\pm}$
is summarized in Appendix \ref{App:SEne_sum}.

Note that finite-$T$ effects
on the spectral density $\rho^{(\nu)}$
of the left-handed neutrino are solely encoded in
the spectral densities $\rho_{\pm}^{(\nu)}(\abs{p},\omega;T)$
appearing as coefficients
in the decomposition (\ref{eq:rho_def}).
Thus, our focus will be put on
the spectral densities
$\rho_{\pm}^{(\nu)}(\abs{p},\omega;T)$
defined in Eq.~(\ref{eq:rho_nu_pm}).
Similarly, the thermal effects for
the spectral density of the massless charged lepton
are calculable without the complication of
the Dirac spinor structure.

Equations~(\ref{eq:sig_nu_pm}) and (\ref{eq:rho_nu_pm})
show that the computation of the neutrino spectral density
$\rho_{\pm}^{(\nu)}(\abs{p},\omega;T)$
is reduced to that of the self-energy $\sigma_{\pm}$
defined in Eq.~(\ref{eq:sig_pm}).
Fortunately,
this task is essentially the same as that of the
self-energy of the quark coupled with massive bosonic excitations
in a QGP, and this been carried out in a series of papers
\cite{3peakQGP,Kitazawa:2007ep,Satow:2010ia}.
Therefore, we here omit the technical details
but recapitulate them
with adequate modifications for the present context in Appendices
\ref{App:Landau_Damp} and \ref{App:SEne_sum}.

\section{Neutrino spectral density:
Numerical results}\label{sec:Res}
In this section, we show our main results
for the self-energy and the spectral density
of the massless left-handed neutrinos.
In Sec. \ref{subsec:Neu_SDens_pT},
we present $\rho_+^{(\nu)}$ defined in Eq.~(\ref{eq:rho_nu_pm})
in the $\omega$-$\abs{p}$ plane at various temperatures
in {}'t Hooft-Feynman gauge ($\xi=1$), and show that
the neutrino spectral density
has a three-peak structure in the low-energy and
low-momentum region when
the gauge-boson mass $M_{\sub{W,Z}}$ is comparable to $T$.
Next, in Sec.~\ref{subsec:Neu_SDens_3peak},
we analyze the three-peak structure in detail and the novel
low-lying collective mode for $\xi = 1$.
Finally, we examine the $\xi$ dependence
of the collective modes in Sec.~\ref{subsec:xi_dep}.
We clarify the relation between
the present results and those obtained in the previous work
\cite{Boyanovsky:2005hk}, where
the HTL approximation {is used} and the unitary gauge {is} adopted.
All results have been obtained by using the values
in Table \ref{tab:params} in Appendix \ref{app:EW_params}.

\subsection{Neutrino spectral density
in the $\omega$-$\abs{p}$ plane
}\label{subsec:Neu_SDens_pT}

In Fig.~\ref{Fig:SDens_SM221_1},
we show the neutrino spectral density
$\rho_{+}^{(\nu)}(\abs{p},\omega;T)$
in the $\omega$-$\abs{p}$ plane
at temperatures $T/v_{0}=0.2,~0.5$, and $0.8$.
Throughout this paper,
we do not analyze $\rho_-^{(\nu)}$
because that quantity can be calculated from $\rho_+^{(\nu)}$
due to the particle-antiparticle symmetry~\cite{3peakQGP}.
In the $T/v_0=0.2$ case (left panel),
the spectral density has sharp and narrow peaks
{almost} on the light cone, $\omega = \abs{p}$.
As shown in Fig.~\ref{Fig:VEV},
the weak-boson masses at $T/v_0=0.2$
are still larger than the temperature,
$M_{\sub{W}}/T \simeq 1.59$,
for which the effects of thermal excitations
Eqs.~(\ref{eq:app_IDist_II}) and (\ref{eq:app_IDist_II_x})
are exponentially suppressed.
As a result, the {general} property of the spectral density
at this temperature region
is similar to that of a massless particle at zero temperature.

This feature drastically changes
when $T$ reaches around $T/v_0\sim 0.35$,
where the weak-boson masses become
comparable to $T$, as is seen in Fig.~\ref{Fig:VEV}.
At this intermediate temperature regime
($T\gtrsim M_{\sub{W,Z}}$),
the weak bosons become thermally excited with
a considerable probability.
The middle panel of Fig.~\ref{Fig:SDens_SM221_1}
shows the spectral density $\rho_+^{(\nu)}$ at $T/v_0=0.5$,
a typical example at intermediate temperature.
We see that a three-peak structure is realized
in the low-energy and low-momentum region:
the three peaks consist of two peaks in
the positive- and negative-energy regions
and the sharp peak around the origin,
whose dispersion relations are quite different from that for the free massless
particle and are interpreted as collective excitations.
As the momentum $\abs{p}$ increases,
the peak in the positive-energy region
becomes sharper,
while the others are attenuated.
We should remark that
a quite similar result has been obtained for the quark spectral
density in a QGP \cite{3peakQGP,Satow:2010ia},
where the quarks are
coupled with massive scalar or vector-bosonic modes in the plasma.

In order to elucidate
the physical meanings of the observed peaks at
intermediate temperature $T/v_0 = 0.5$,
we compare in Fig.~\ref{Fig:SDensP_SM221_p_05}
the peak positions of $\rho_+^{(\nu)}$
seen in the middle panel of Fig.~\ref{Fig:SDens_SM221_1}
with the massless-HTL~\cite{HTL}
dispersion relation
$\omega_{\sub{HTL}}(\abs{p})$, which is defined as
a solution of
\begin{align}
\omega_{\sub{HTL}} - \abs{p}
- \mathrm{Re}~\Sigma_{+,\sub{HTL}}^{(\nu)}
(\abs{p},\omega_{\sub{HTL}};T) = 0
\ .\label{eq:Disp_HTL}
\end{align}
Here, the real part of the self-energy
$\mathrm{Re}~\Sigma_{+,\sub{HTL}}^{(\nu)}$
is given by taking the limit
$T \gg \omega,~\abs{p},\ M_{\sub{W,Z}}$
in the full one-loop result
as detailed in Appendix \ref{App:HTL}.
Figure \ref{Fig:SDensP_SM221_p_05} shows that
the positive (negative) branch in $\rho_+^{(\nu)}$
corresponds to the quasiparticle (antiplasmino) mode,
which is the familiar solution appearing in the massless-HTL approximation.

Although Eq.~(\ref{eq:Disp_HTL}) has a solution
in the ultrasoft region
$\omega_{\sub{HTL}}(\abs{p})\sim 0$
with $\abs{p}$ being finite,
the imaginary part in the massless-HTL approximation
$\mathrm{Im}~\Sigma_{\pm,\sub{HTL}}$ becomes huge,
and kills the peak structure at the ultrasoft region
$\omega_{\sub{HTL}}(\abs{p})\sim 0$.
Thus, the emergence of the collective modes
near the origin (ultrasoft mode) is characteristic of
the full spectral density $\rho_+^{(\nu)}$.

In higher-temperature region $T\gg M_{\sub{W,Z}}$,
the two peaks in the positive- and negative-energy regions
become sharper while the central peak is greatly attenuated
[see the right panel in Fig.~\ref{Fig:SDens_SM221_1} ($T/v_0 = 0.8$)].
As shown in Fig.~\ref{Fig:SDens_SM221_w_002_08_1},
{these two} peaks approximately stay
at the same position as the quasiparticle (antiplasmino) mode
of the massless-HTL approximation.
Thus, the {general} property of the spectral density
becomes closer to that of the massless-HTL approximation
in the high-temperature region $T\gg M_{\sub{W,Z}}$.
We note that
the high-temperature region does not satisfy the condition $T\ll v(T)$;
therefore, higher-loop corrections may be non-negligible.
In fact, it is known that a resummation of
higher-loop diagrams is necessary
for the ultrasoft-momentum region
$\omega, \abs{p}\lesssim g^2T$~in QED and QCD \cite{Hidaka:2011rz},
where a resummation scheme is developed for obtaining  sensible results.
It would be interesting
to develop such a resummation scheme in the electroweak theory
and investigate the spectral properties
in the vicinity of the electroweak phase transition, which is,
however, beyond the scope of the present work and left as a future task.

\begin{figure*}[ht]
\begin{center}
\includegraphics[width=5.8cm]{SDens_SM221_w_xx_02_1.eps}
\includegraphics[width=5.8cm]{SDens_SM221_w_xx_05_1.eps}
\includegraphics[width=5.8cm]{SDens_SM221_w_xx_08_1.eps}
\caption{The momentum evolution of
the spectral density $\rho_{+}^{(\nu)}v_0$
at $T/v_0=0.2$ (left), $0.5$ (middle),
and $0.8$ (right).
}\label{Fig:SDens_SM221_1}
\end{center}
\end{figure*}

\begin{figure}[ht]
\begin{center}
\includegraphics[width=8.0cm]{SDensP_SM221_p_05.eps}
\caption{The comparison between
the peak position of the spectral density $\rho_{+}^{(\nu)}$
shown in the middle panel of
Fig.~\protect\ref{Fig:SDens_SM221_1}
and the HTL dispersions given
by Eq.~(\protect\ref{eq:Disp_HTL}) at $T/v_0=0.5$
in the $\omega$-$\abs{p}$ plane.}
\label{Fig:SDensP_SM221_p_05}
\end{center}
\end{figure}

\begin{figure}[ht]
\begin{center}
\includegraphics[width=8.0cm]{SDens_SM221_w_002_08_1.eps}
\caption{The spectral density $\rho_{+}^{(\nu)}$ (solid red line)
as a function of $\omega$
at ($\abs{p}/v_0,T/v_0$) = ($0.02,0.8$)
compared with the spectral density obtained by using
the massless-HTL approximation (dashed blue line)
for the same conditions.}\label{Fig:SDens_SM221_w_002_08_1}
\end{center}
\end{figure}

\subsection{Emergence of the three-peak structure}
\label{subsec:Neu_SDens_3peak}

In this subsection, we investigate
the properties of the three-peak structure in detail.

In the upper panels of
Fig.~\ref{Fig:SDens_SEne_S_WZ_w_002_05_10},
we show the imaginary part of the self-energy,
$\mathrm{Im}~\Sigma_{\pm}^{(\nu)}$
given by Eq.~(\ref{eq:Im_sig_nu_pm})
as a function of $\omega/v_0$ at
$(\abs{p}/v_0,T/v_0)=(0.02,0.5)$.
Because of the growing statistical factors
Eqs.~(\ref{eq:app_IDist_II}) and (\ref{eq:app_IDist_II_x})
with increasing temperature,
the Landau damping effect becomes significant,
and the imaginary part of the self-energy
$\mathrm{Im}~\Sigma^{(\nu)}_{+}$
drastically grows in the space-like region.
A remarkable point is that
$\mathrm{Im}~\Sigma^{(\nu)}_+$
stays relatively small in the vicinity
of $\omega\sim 0$, and
has two peaks in the space-like region
($\omega^2 \le \abs{p}^2$).
By utilizing the Kramers-Kronig-like dispersion relation (\ref{eq:Re_sig_pm_T})
(which we call the Kramers-Kronig relation to avoid the confusion
with the dispersion relation as the momentum dependence of energy),
the real part
inevitably shows an oscillatory behavior
as displayed in the middle panel
of Fig.~\ref{Fig:SDens_SEne_S_WZ_w_002_05_10}.
The crossing points with the straight dashed line
representing $y=\omega-\abs{p}$
correspond to the solutions of the equation,
\begin{align}
\mathcal{D}^{(\nu)}_+(\abs{p},\omega;T,\xi) = 0
\ .\label{eq:Disp_nu}
\end{align}
Here, $\mathcal{D}^{(\nu)}_+$
is defined in Eq.~(\ref{eq:Disp_p_nu}).
Two of the solutions lead to
the two peaks of the imaginary part,
and hence do not lead to
a peak in the spectral density.
The remaining three solutions contribute
to the peaks in the spectral density
(lower panel of Fig.~\ref{Fig:SDens_SEne_S_WZ_w_002_05_10}).
We note that, in the previous QGP studies \cite{3peakQGP,Satow:2010ia},
similar results for the real and imaginary parts of
the self-energy have been obtained,
and the mechanism of the emergence of the three peaks in the spectral density
was interpreted in terms of a resonant scattering~\cite{3peakQGP}.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=7.5cm]{ImSEne_SM221_w_002_05_1.eps}
\includegraphics[width=7.5cm]{ReSEne_SM221_w_002_05_1.eps}
\includegraphics[width=7.5cm]{SDens_SM221_w_002_05_1.eps}
\caption{
The self-energies
$\mathrm{Im}~\Sigma_{+}^{(\nu)}/v_0$ (upper),
$\mathrm{Re}~\Sigma_{+}^{(\nu)}/v_0$ (middle),
and the spectral density $\rho_+^{(\nu)}v_0$
at $(\abs{p}/v_0,T/v_0)=(0.02,0.5)$.}
\label{Fig:SDens_SEne_S_WZ_w_002_05_10}
\end{center}
\end{figure}

In order to have a more complete overview,
we show the solution of Eq.~(\ref{eq:Disp_nu})
in the full $\omega$-$\abs{p}$ plane
for $T/v_0=0.35$, $0.42$, and $0.5$ cases
in Fig.~\ref{Fig:Disp_S_WZ_p_xx_10}; the resultant
function $\omega(\abs{p})$ is called the quasidispersion relation.
The upper panel ($T/v_0=0.35$)
corresponds to the dispersion relation
at a characteristic temperature
satisfying $T/M_{\sub{W,Z}}\sim 1$,
where the Landau damping effects start to dominate.
The quasidispersion curve
deviates from the light cone, and
shows a rapid increase for
$0.03 \leq \abs{p}/v_0 \leq 0.04$.
As $T$ is increased, this steep-rise behavior is evolved to
a ``back-bending'' shape as shown in
the middle panel of Fig.~\ref{Fig:Disp_S_WZ_p_xx_10} ($T/v_0=0.42$).
Eventually, the branch with the back-bending part
crosses the $\omega$ axis
around $T/v_0\simeq 0.475$,
at which point the additional two branches appear
in the negative-$\omega$ region; see the lower panel of Fig.~\ref{Fig:Disp_S_WZ_p_xx_10}.
Thus, we get five branches in the low-momentum region.
At $\abs{p}/v_0=0.02$,
these branches correspond to the five crossing points
in the middle panel of
Fig.~\ref{Fig:SDens_SEne_S_WZ_w_002_05_10}.

We should mention that a similar branch of the
neutrino dispersion relation was found in the HTL approximation
with the unitary gauge in Ref.~\cite{Boyanovsky:2005hk},
and the branch that touches the $\omega$ axis is
called a {\em pitchfork bifurcation}.
In Ref.~\cite{Boyanovsky:2005hk},
the pitchfork bifurcation is found to start
when the $T$-dependent mass scale
$\Delta=2M_{\sub{W}}^2(T)/(gT^2)$ exceeds
$\Delta_c\simeq 1.275$.
In the present study, the pitchfork bifurcation
starts from $T/v_0\simeq 0.475$, corresponding to
$\Delta_c\simeq 1.28$, which is consistent
with the finding in the previous
work \cite{Boyanovsky:2005hk},
and suggests the gauge independence of
the temperature at which the bifurcation starts.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=7.5cm]{Disp_SM221_p_035.eps}
\includegraphics[width=7.5cm]{Disp_SM221_p_042.eps}
\includegraphics[width=7.5cm]{Disp_SM221_p_05.eps}
\caption{The dispersion relations
at $T/v_0=0.35$ (upper),
$0.42$ (middle), and $0.5$ (lower).
The symbol $\times$
corresponds to a solution of
Eq.~(\protect\ref{eq:Disp_nu})
that gives a peak to the spectral density.
The symbol $\blacktriangle$
is a solution of
Eq.~(\protect\ref{eq:Disp_nu})
with a large imaginary part.}
\label{Fig:Disp_S_WZ_p_xx_10}
\end{center}
\end{figure}

\subsection{Gauge parameter $\xi$ dependence
of the three-peak structure}\label{subsec:xi_dep}
So far, we have investigated
the spectral density $\rho_{+}^{(\nu)}$
in the 't Hooft-Feynman gauge ($\xi = 1$),
and found the three-peak structure that consists of
the ultrasoft mode
as well as the quasiparticle and antiplasmino modes.
In this subsection, we investigate
the $\xi$ (in)dependence of the three-peak structure.

In Fig.~\ref{SEne_SM_221_w_002_05_xx},
we show the neutrino self-energies
for the gauge parameter $\xi = 0.1,\ 1,\ 10$.
We also display
the unitary-gauge result
which has been obtained
by utilizing the Proca formalism~\cite{3peakQGP},
which corresponds to $\xi\to \infty$.
The momentum and temperature are set to be the same as
in Fig.~\ref{Fig:SDens_SEne_S_WZ_w_002_05_10},
$(\abs{p}/v_0,T/v_0) = (0.02,0.5)$.
As shown in the figure,
the $\xi$ dependence
becomes significant with increasing $\omega$.
However, in the low-energy region $\omega/v_0\leq 0.15$
where the three-peak structure emerges
in the neutrino spectral density,
the $\xi$ dependence is found to be negligible
in both the imaginary part (upper panel)
and the real part (lower panel).
This indicates that
both the position and the width of the three peaks
are independent of the gauge parameter $\xi$.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=8.0cm]{ImSEne_SM221_w_002_05_xx.eps}
\includegraphics[width=8.0cm]{ReSEne_SM221_w_002_05_xx.eps}
\caption{The gauge parameter ($\xi$) dependence
of the imaginary (upper) and real (lower) parts of neutrino self-energies
at $(\abs{p}/v_0,T/v_0)=(0.02,0.5)$.
As shown in Fig.~\protect\ref{Fig:SDens_SEne_S_WZ_w_002_05_10},
the three-peak structure appears in
the low-energy region, $|\omega/v_0|\leq 0.15$,
where the $\xi$ dependence is negligible.}
\label{SEne_SM_221_w_002_05_xx}
\end{center}
\end{figure}

For the three peaks of the spectral density,
the two conditions
$M_{\sub{W,Z}}\gg \omega,\abs{p}$
and
$T\gg \omega,\abs{p}$
are satisfied.
Due to the former condition,
the unitary-gauge part of the self-energy dominates
[see Eqs.~(\ref{eq:Im_sig_pm_U})--(\ref{eq:Im_sig_pm_xi})],
which accounts for the $\xi$ independence of the three-peak structure.
By using both two conditions,
we can define {an extension of the HTL approximation to the case
where the boson mass is finite---which we call the {\em massive-HTL}
expansion---for the neutrino self-energy} as detailed in Appendix \ref{App:MHTL}.
The leading-order terms
reproduce the self-energy derived in Ref.~\cite{Boyanovsky:2005hk}.
The temperature and energy region for the pitchfork bifurcation
explained in the previous subsection is in the validity region
of this approximation scheme, which accounts for the gauge invariance
of $\Delta_c$ and the agreement of its values in the present and
the previous study \cite{Boyanovsky:2005hk}.

\section{Neutrino spectral density at $T \geq T_c$}
\label{sec:SDens_M0}
In the previous sections,
we have investigated the neutrino spectral properties
in the broken phase.
In this section, we discuss their properties
in the symmetric phase $T\geq T_c$.

In the symmetric phase,
the weak bosons become massless as is clear from Eq.(\ref{eq:MGT}).
Although higher-loop corrections for the weak-boson masses
as well as the neutrino self-energy
may become non-negligible in the symmetric phase,
in particular, for $\omega,\abs{p}\ll g^2T$,
we dare to analyze
the neutrino self-energy within the present approximation
and  show an advantageous property of  $R_\xi$ gauge:
in the unitary gauge,
the massless limit $M\to 0$
leads to the divergence in the self-energy
due to the prefactor proportional to $1/M^2$
in the imaginary part [Eq.~(\ref{eq:Im_sig_pm_U})], and
hence the massless limit cannot be well defined
within the unitary gauge.
This problem is solved {in $R_\xi$ gauge} by
the additional $\xi$-dependent part Eq.~(\ref{eq:Im_sig_pm_xi}):
All $\mathcal{O}(M^{-2})$ terms beautifully cancel out between
the unitary-gauge part [Eq.~(\ref{eq:Im_sig_pm_U})] and
the $\xi$-dependent part Eq.~(\ref{eq:Im_sig_pm_xi}),
and we find the finite imaginary part of the self-energy
in the vanishing gauge-boson masses
($T ,\omega, \abs{p} \gg M\to 0$) as
\begin{align}
&\mathrm{Im}~\sigma_{\pm}(\abs{p},\omega;T)|_{\sub{M}\to 0}
= \frac{\pm 1}{32\pi\abs{p}^2}\nonumber\\
&\quad\times\biggl[
-(1 - \xi)\abs{p}p^2
\bigl[
N_{\mathrm{F}}(X_{\mp})
+ N_{\mathrm{B}}(Y_{\mp})
\bigr]\nonumber\\
&\quad +
\bigl[
(1-\xi)(\omega \pm \abs{p})^2 + 2p^2
\bigr]\cdot
TI_{\sub{ST}}(X_+,X_-;\omega/T)\nonumber\\
&\quad -
4(\omega \mp \abs{p})\cdot
T^2J_{\sub{ST}}(X_+,X_-;\omega/T)
\biggr]\ ,\label{eq:Im_sig_pm_M0}
\end{align}
where $I_{\sub{ST}}$ and $J_{\sub{ST}}$ have been defined
in Eqs.~(\ref{eq:app_IDist_II}) and (\ref{eq:app_IDist_II_x}),
and their arguments $X_{\pm},Y_{\pm}$ are defined by
\begin{align}
&(X_{\pm},Y_{\pm})
\equiv\biggl(
\frac{\omega \mp \abs{p}}{2T},
\frac{-\omega \mp \abs{p}}{2T}
\biggr)\ .\label{eq:XY}
\end{align}
The technical details to derive Eq.~(\ref{eq:Im_sig_pm_M0})
are provided in Appendix \ref{App:SEne_M0}.

Furthermore,
the cancellations of $\mathcal{O}(M^{-2})$ terms
are also found
in the imaginary part of the vacuum self-energy
[$\mathrm{Im}~\sigma_{\pm,0}\equiv$
Eqs.~(\ref{eq:Im_sig_pm_U_0}) + (\ref{eq:Im_sig_pm_xi_0})],
which reads
\begin{align}
\mathrm{Im}~\sigma_{\pm,0}
(\abs{p},\omega)|_{\sub{M}\to 0}
= \frac{-\mathrm{sgn}(\omega)\xi}{16\pi}
\theta(p^2)(\omega \mp \abs{p})
\ .\label{eq:Im_sig_pm_T0M0}
\end{align}

In Fig.~\ref{Fig:SDens_M0221_w_002_1_xx},
we show the neutrino spectral density at $\abs{p}/v_0=0.02$
in the symmetric phase $T/v_0=1.0> T_c$ for $\xi = 1,~2,~10$.
We find the quasiparticle and the antiplasmino peaks,
whose positions are almost independent of $\xi$
and consistent with the HTL results~\cite{quasi_particle}.
One sees that the width seems to have a  significant $\xi$ dependence.
It should be noted, however, that
it was shown in Ref.~\cite{Satow:2010ia} that at one-loop order
the condition $\xi\lesssim 1/g$ guarantees
the smallness of the gauge dependence.
Since $g\simeq 0.7$, the upper limit of $\xi$ seems to be $\simeq 1.4$.
This estimate shows that the gauge dependence seen
in Fig.~\ref{Fig:SDens_M0221_w_002_1_xx} is consistent
with the analysis in Ref.~\cite{Satow:2010ia}
and tells us that we should restrict ourselves
to $\xi\lesssim 2$ to get a sensible result.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=8.0cm]{SDens_M0221_w_002_1_xx.eps}
\caption{The gauge parameter ($\xi$) dependence
of the neutrino spectral density
at $\abs{p}/v_0=0.02$
in the symmetric phase $T/v_0=1.0> T_c$.}
\label{Fig:SDens_M0221_w_002_1_xx}
\end{center}
\end{figure}

\section{Discussion: Toward application to resonant leptogenesis}
\label{sec:discuss}
\begin{figure*}[ht]
\begin{center}
\includegraphics[width=14.0cm]{NR_decay_all.eps}
\caption{
The diagram of the $N_{\sub{R}}^i$ decay to a
left-handed lepton $L^i=(\nu^i,l_{\sub{L}}^i)^T$ and
a Higgs doublet $\Phi$.
Note that the lepton propagators in all the diagrams
are given in terms of the spectral densities
$\rho^{(\nu)}$ and $\rho^{(l_{\sub{L}})}$.}
\label{Fig:NR_decay_all}
\end{center}
\end{figure*}
There is a growing interest
in the collective nature of the fermion quasiparticles
in the scenario of thermal leptogenesis
\cite{Drewes:2013gca,Anisimov:2010aq,Anisimov:2010gy,Kiessig:2010pr,Besak:2012qm,Garbrecht:2013,Laine:2013vpa,Drewes:2012ma}
because the novel collective fermion modes
may significantly modify lepton number creation.
In this section,
we argue that the neutrino collective modes with
the three peaks in the spectral function obtained in the previous sections
could possibly affect the leptogenesis and hence the BAU.

In the thermal leptogenesis,
the Lagrangian is extended to include
the right-handed neutrinos ($N_{\sub{R}}^i$, $i=1,2,3$),
\begin{align}
\delta\mathcal{L}
=\bar{N}^i_{\sub{R}} i\sla{-1.0}{-0.8}\partial N_{\sub{R}}^i
+Y_{ij}\bar{N}_{\sub{R}}^i\Phi^{\dagger}L^j
-\frac{M^{ij}_{\sub{R}}}{2}\bar{N}_{\sub{R}}^i N_{\sub{R}}^j
+ \text{H.c.}\ ,\label{eq:dL}
\end{align}
where $L^i$ and $\Phi$ represent the standard model left-handed lepton
$(\nu^i,l_{\sub{L}}^i)^T$ and the standard model Higgs doublet, respectively.
The Yukawa interaction ($Y_{ij}$) term in Eq.~(\ref{eq:dL})
gives rise to  the $N_{\sub{R}}$ decay shown in Fig.~\ref{Fig:NR_decay_all},
which causes lepton number creation.
The leading effects on the finite-$T$ interaction rate
are given by the left diagram
in Fig.~\ref{Fig:NR_decay_all},
and are expressed as
\begin{align}
\Gamma_{\sub{N}}^{i}(p)
&=
-\frac{1}{2p_0}\mathrm{tr}_{\mathrm{spin}}
\Bigl[
(\sla{0.0}{-1.0}{p} + M^i_{\sub{R}})~
\mathrm{Im}\Sigma_{\sub{N}}(p)
\Bigr]\ ,\label{eq:int_rate}\\
\Sigma_{\sub{N}}(p)
&=
-4(Y^{\dagger}Y)_{11}\nonumber\\
&\times
T\sum_{n}\int\frac{d^3k}{(2\pi)^3}
\tilde{G}_{\mathrm{F}}(k)
\tilde{G}_{\mathrm{S}}(p-k)
\ ,\label{eq:sig_N}
\end{align}
where we have adopted the mass basis
$M^{ij}_{\sub{R}}=\delta^{ij}M^i_{\sub{R}}$,
and $\tilde{G}_{\mathrm{F}}$ and $\tilde{G}_{\mathrm{S}}$
represent the full finite-$T$ propagators of a lepton
and the Higgs particle, respectively.
The $\tilde{G}_{\mathrm{F}}$ is expressed in terms of
the spectral density $\rho^{(\nu, l_{\sub{L}})}(\mathbf{k},\omega;T)$
of the left-handed neutrino $\nu$
(or the charged lepton $l_{\sub{L}}$), as follows:
\begin{align}
\tilde{G}_{\mathrm{F}}(\mathbf{k},i\omega_n;T)
=
\int_{-\infty}^{\infty}d\omega\
\frac{\rho^{(\nu, l_{\sub{L}})}(\mathbf{k},\omega;T)}
{\omega + i\omega_n}
\ .\label{eq:GF_rho}
\end{align}

At the electroweak-scale,
the spectral density $\rho^{(\nu)}$  in Eq.~(\ref{eq:GF_rho}) can have
the three-peak structure as was shown in the previous section.
Then---assuming resonant leptogenesis
at this scale~\cite{Pilaftsis:2005rv}---these collective
modes corresponding the peaks would modify
the creation rate of the lepton number $\Gamma_{\sub{N}}^{i}$.
Moreover, the spectral density having the three peaks
is involved in both the leading and subleading diagrams
shown in Fig.~\ref{Fig:NR_decay_all},
and modifies their interference effects
leading to the $CP$ asymmetry.

Here, a natural question is whether
the $N_{\sub{R}}$ decay into $\nu$ with the
collective nature which leads to leptogenesis
can take place before the sphaleron freeze-out.
If not, the three-peak structure would not have any relation to the BAU.
In order to study this problem,
we shall estimate the sphaleron freeze-out temperature $T_*$,
and investigate the spectral density of $\nu$ in the vicinity of $T_*$.

The net baryon number $N_{\mathrm{b}}$
is produced in the sphaleron process
when the changing rate of $N_{\mathrm{b}}$
is larger than the expanding rate of the universe $H(T)$,
\begin{align}
\Bigl|
\frac{1}{N_{\mathrm{b}}}\frac{d N_{\mathrm{b}}}{dt}
\Bigr|
\geq H(T)
\ .\label{eq:freeze_sph}
\end{align}
The freeze-out temperature $T_*$ is obtained from
the equality in Eq.~(\ref{eq:freeze_sph}).
The expanding rate $H(T)$ is given by
the Hubble parameter in the early universe at temperature $T$,
\begin{align}
H(T)
=1.66\sqrt{N_{\mathrm{dof}}}\frac{T^2}{M_{\mathrm{PL}}}
\simeq
T^2\times 1.41\times 10^{-18}\ (\mathrm{GeV})
\ ,\label{eq:Hubble}
\end{align}
with radiation degrees of freedom for the SM --
$N_{\mathrm{dof}}\simeq 106.75$,
and the Planck mass $M_{\mathrm{PL}}\simeq 1.22\times 10^{19}$ (GeV).

We evaluate the changing rate of the net baryon number \cite{Kapusta}
\begin{align}
\frac{1}{N_{\mathrm{b}}}\frac{d N_{\mathrm{b}}}{dt}
&=
-1100~
K(\lambda/g^2)~
g^7v(T)\nonumber\\
&\times\exp
\biggl[
-\frac{E_{\mathrm{sph}}(g,\lambda,v(T))}{T}
\biggr]\ ,\label{eq:Nb_evolv}
\end{align}
by using the same electroweak parameter set
and Higgs potential as those used
in the study of the neutrino spectral density
(see, Table \ref{tab:params}).
Under the static sphaleron background,
its free energy is known to have the expression,
\begin{align}
E_{\mathrm{sph}}=
\frac{4\pi v(T)}{g}
F_0(\lambda/g^2)\ ,\label{eq:E_sph}
\end{align}
where the dimensionless function $F_0(\lambda/g^2)$
is estimated to be
\begin{align}
F_0(\lambda/g^2)
=F_0(m_{\sub{H}}^2/(2g^2v_0^2))
\simeq F_0(0.309)\simeq 1.89\ .\label{eq:F0}
\end{align}
In the second relation --
we have used the values
listed in Table \ref{tab:params}
for $m_{\sub{H}},~v_0,~g$.
In the final relation,
we have taken
the results $F_0(0.1)=1.83$ and $F_0(1.0)=2.10$
from Table II in Ref.~\cite{Klinkhamer:1984di},
and performed a linear interpolation
to estimate $F_0(0.309)$.

The function $K(\lambda/g^2)$
is responsible for quantum fluctuations
around the sphaleron vacuum, and is well approximated
by using the expression \cite{kappa},
\begin{align}
K\Bigl(\frac{\lambda}{g^2}\Bigr)
&=
\exp\biggl[
-0.09\Bigl(\frac{\lambda}{g^2}-0.4\Bigr)^2
-0.13\Bigl(\frac{g^2}{\lambda}-2.5\Bigr)^2
\biggr]\nonumber\\
&\simeq 0.93\ ,\label{eq:kappa}
\end{align}
where again,
we have used the values
listed in Table \ref{tab:params}.

Substituting
Eqs.~(\ref{eq:E_sph})--(\ref{eq:kappa}) into Eq.~(\ref{eq:Nb_evolv}),
the changing rate of the net baryon number is obtained as
\begin{align}
\Big|
\frac{1}{N_{\mathrm{b}}}\frac{d N_{\mathrm{b}}}{dt}
\Big|
\simeq
51.02~v(T)~
\exp\biggl[
-36.51\frac{v(T)}{T}
\biggr]\ .\label{eq:Nb_evolv_num}
\end{align}
By using Eqs.~(\ref{eq:Nb_evolv_num}) and (\ref{eq:Hubble})
as well as Eq.~(\ref{eq:VEV}),
the inequality (\ref{eq:freeze_sph})
reduces to the following condition
\begin{align}
T\geq T_*\simeq 160~\mathrm{ GeV}
\ .\label{eq:freeze_T}
\end{align}

The leptons created at a temperature
satisfying $T/v_0\geq T_*/v_0\sim 0.65$ can contribute
to the BAU via the sphaleron process.
Figure~\ref{Fig:SDens_SM221_w_0007_xx_1}
shows the neutrino spectral density $\rho_{+}^{(\nu)}v_0$
for various temperatures around $T_*$
at a fixed low momentum $p/v_0=0.007$,
for which the sharp peaks are observed
in the ultrasoft region $\omega\sim 0$.
Thus, the three-peak structure is still observed in this temperature region
for small momenta, although the strength of the ultrasoft mode
may not be so strong.
This indicates that the three-peak collective modes of $\nu$
could affect leptogenesis before the freeze-out.
It is interesting that such a possibility
is shown in the definite setup.
We note that the condition $T/v_0 \sim 0.65$
corresponds to $T\sim v(T)$;
see Fig.~\ref{Fig:VEV}.
As explained in Sec. \ref{subsec:SEne_lepton},
higher-loop corrections
start to be non-negligible around this temperature,
and should be
taken into account to get more sensible results.
We leave this task as a future work.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=7.5cm]{SDens_SM221_w_0007_xx_1.eps}
\caption{The neutrino spectral density $\rho_{+}^{(\nu)}v_0$
in the vicinity of a sphaleron freeze-out temperature
$T_*/v_0\sim 0.65$
at a fixed low momentum $p/v_0=0.007$.
}\label{Fig:SDens_SM221_w_0007_xx_1}
\end{center}
\end{figure}

\section{Summary and concluding remarks}\label{sec:sum}
We have investigated the spectral properties
of standard model left-handed neutrinos
without restricting {ourselves} to the HTL approximation,
in the full energy-momentum plane
at finite $T$ around the electroweak-scale.
This analysis is motivated by the scenario of
the electroweak-scale resonant leptogenesis
\cite{Pilaftsis,Pilaftsis:2005rv},
where the spectral property of left-handed neutrinos
can affect lepton number creation through
the decay process of right-handed neutrinos $N_{\sub{R}}$,
as shown in Fig.~\ref{Fig:NR_decay}.
We have employed the $R_{\xi}$ gauge
in order to investigate
the possible gauge-fixing dependence of the spectral density.

In the intermediate-temperature region
$M_{\sub{W,Z}}\lesssim T$
($M_{\sub{W,Z}}$ is the weak-boson mass at finite $T$),
the neutrino spectral density involves
an ultrasoft mode as well as
normal and antiplasmino modes,
and shows a three-peak structure.
The three-peak structure
emerges independently of the gauge parameter, although a detailed
structure of the spectral density may have a slight gauge dependence.
Its emergence is accompanied
by a pitchfork bifurcation of the neutrino dispersion relation,
consistent with the previous finding in Ref.~\cite{Boyanovsky:2005hk},
where {the} HTL approximation was adopted solely in the unitary gauge.
We have mentioned that the mechanism of
the emergence of the collective excitations and the three-peak
structure is similar to what was discussed in Ref.~\cite{3peakQGP} in
a different context.

We have examined the neutrino spectral densities
in the symmetric phase by taking
the limit of vanishing gauge-boson masses ($M_{\sub{W,Z}}\to 0$).
In this limit,
the self-energy in the unitary gauge shows a divergence,
which completely cancels out in the $R_{\xi}$ gauge.
Thus, the $R_{\xi}$ gauge fixing has allowed us to
investigate the symmetric phase.
The neutrino spectral properties in the symmetric phase
are found to be similar to that in the HTL approximation.
We have found that the possible gauge dependence of the width
of the peaks in the spectral function is
controllable, and the gauge parameter $\xi$
should be restricted to $\xi\lesssim 2$ to have a sensible result.

We have also discussed a possible implication
of the present study for particle cosmology,
in particular in the resonant leptogenesis scenario,
where it makes sense to consider a thermal leptogenesis
at electroweak-scale temperature.
We have pointed out that
the collective modes of left-handed neutrinos
provide a novel decay channel
in the decay processes of the right-handed neutrinos,
and could modify lepton number creation.
We have estimated
the sphaleron freeze-out temperature $T_*$
and investigated
the spectral density in the vicinity of $T_*$.
Within the present setup with
the second-order electroweak phase transition,
the ultrasoft as well as (anti)plasmino modes
can appear at a temperature comparable to $T_*$
though the strength of ultrasoft modes might not be so strong.
Thus, the novel three-peak collective modes
could affect leptogenesis at $T\gtrsim T_*$,
and, therefore, the baryogenesis.

There are several subjects to be studied in the future:
First, it is desirable to estimate
how large the effects of two-loop or higher-order diagrams are
on the neutrino spectral density.
Second, the present formulation should be
extended to include the bare-fermion-mass effects
\cite{Kitazawa:2007ep}.
Finally,
it would be interesting to evaluate lepton number creation
with respect to the nontrivial spectral properties
of the neutrinos as well as the charged leptons
by adopting the explicit model of resonant leptogenesis.

\section*{Acknowledgements}\label{sec:ack}
We thank Chee Sheng Fong
for fruitful discussions on resonant leptogenesis.
We thank Marco Drewes for useful comments on the quantum effects
in the thermal leptogenesis and information on some references.
This work was supported by JSPS KAKENHI Grant Numbers
20540265, 23340067, 24$\cdot$56384, and 24740184.
T.K. was partially supported
by the Yukawa International Program for Quark-Hadron Sciences,
and by a Grant-in-Aid for the global COE program
``The Next Generation of Physics,
Spun from Universality and Emergence'' from MEXT.
D. S. was supported by
JSPS Strategic Young Researcher Overseas Visits Program
for Accelerating Brain Circulation (No. R2411).