\section{Introduction}

What is the best model to predict the nonlinear matter power spectrum ?
In this paper, we show that the standard perturbation (SPT) theory with two-loop corrections
is still one of the best models, even though various modified perturbation theories have been proposed in the last 5-10 yr.
In doing so, we give the relation between the SPT and the regularized multi-propagator method.

One of the key quantities in modern cosmology is the matter power spectrum because it contains
a lot of important information on evolution of the universe as well as on structure formation.
Recent observation of the baryon acoustic oscillations (BAO) in the power spectrum
provides a new method to precisely restrict cosmological parameters~\citep{Eisenstein/etal:2005}.

Recent progress on cosmological observation greatly motivates
various theoretical studies of accurate calculation of the nonlinear matter power spectrum.
At present, one of the most successful models is given by the multi-point propagator method
($\Gamma$-expansion method) with the regularized treatment of propagators
presented by \citet{Bernardeau:2008fa,Bernardeau:2011dp}, called ``Reg PT'' in~\citet{Taruya:2012ut}.
In particular, \citet{Taruya:2012ut} computed the solution of Reg PT with two-loop corrections,
giving theoretical predictions of the power spectrum which agree well
with $N$-body simulations at BAO scales.

Various modified perturbation theories partially sum up high-order terms in the SPT,
called ``resummation theories''.
Since SPT has exact but formal solutions at any order in perturbation theory,
any resummation theory should be represented in the context of SPT.
The coefficients of the $\Gamma$-expansion can easily be represented by the kernel function in SPT.
This is why we consider the relation between SPT and the $\Gamma$-expansion method.

We also proposed a method called the ``Wiener Hermite (WH) expansion``
which gives an accurate power spectrum over relatively wide range of wavenumber~\citep{Sugiyama:2012pc}.
There, we established the relation between the WH expansion and SPT, and showed the equivalence between
the WH expansion and the $\Gamma$-expansion method~\citep{Bernardeau:2008fa,Bernardeau:2011dp,Taruya:2012ut}.
Furthermore, by using an approximation of the kernel functions in SPT,
we derived the exponentially damping behavior of the power spectrum at each order of the WH expansion ($\Gamma$-expansion)
which has been known in the renormalized perturbation theory (RPT; \citet{Crocce:2005xz,Crocce:2005xy,Crocce:2007dt}).

We extend the approximation method of the kernel functions introduced in our previous work~\citep{Sugiyama:2012pc}.
As a result, we give an alternative explanation for Reg PT, and propose a natural extension of Reg PT.
The extended Reg PT gives the SPT solution with negligible correction terms.
Thus, we find that the Reg PT solution does not ever have more effective information on the evolution of dark matter than the SPT solution.
In addition, we also propose a modified version of the SPT solution which avoids the divergent behavior of the SPT solution
at small scales.

The $N$-body simulation results used in this paper were presented by \citet{Valageas:2010yw}.
These results and initial conditions at $z_{\rm ini} = 99$
were created by the public $N$-body codes {\it GADGET2} and {\it 2LPT} code, respectively~\citep{Springel:2005mi,Crocce:2006ve}.
These $N$-body simulations contain $2048^3$ particles
and were computed by combining the results with different box sizes $2048 h^{-1}$ ${\rm Mpc}$ and $4096$ $h^{-1}{\rm Mpc}$, called
$L11$-$N11$ and $L12$-$N11$.
The cosmological parameters we used were presented by
the {\it Wilkinson Microwave Anisotropy Probe} five year release~(\citet{Komatsu:2008hk},
$\Omega_m = 0.279$, $\Omega_{\Lambda} = 0.721$, $\Omega_b = 0.046$, $h = 0.701$, $n_s = 0.96$ and $\sigma_8=0.817$).
We used the program which is available on Taruya's homepage to compute the predicted power spectra
\footnote{ \url{ http://www-utap.phys.s.u-tokyo.ac.jp/~ataruya/}}.

This paper is organized as follows.
In Section~\ref{Review}, the relation between SPT and the $\Gamma$-expansion is reviewed.
Section~\ref{Ap_Kernel} proves the approximation of the kernel functions.
Using the approximated kernel functions, the alternative explanation for Reg PT is given in Section~\ref{Proof}.
In Section~\ref{Extension_of_RegPT}, it is proposed that the natural extension of Reg PT
is equivalent to the solution of SPT with some correction terms.
We also present the modified version of the SPT solution in Section~\ref{Ex_SPT}.
There, we reconfirm that the two-loop SPT solutions are better than those of the two-loop Reg PT and the closure theory,
and show that the divergent behavior of the solution in SPT is indeed removed by our modification.
In Section~\ref{Conclusion}, we summarize and discuss our results.

\section{Review of SPT and the $\Gamma$-expansion method}
\label{Review}

We shall briefly review SPT.
In SPT, we choose a cosmological model with $f=\Omega_m^{1/2}$
where $f \equiv d \ln D/ d \ln a $ is the linear growth rate with $D$ and $a$ being the linear growth factor and the scale factor.
As shown by \citet{1991MNRAS.251..128L},
this model is a good approximation in practice.
In this situation, the density perturbation and the velocity divergence of dark matter are expanded as~\citep{Bernardeau:2001qr},
\begin{equation}
		\delta(z,\kk) = \sum_{n=1}^{\infty} D^{n} \delta_{n}(\kk), \qquad
		\theta(z,\kk) = -aHf\sum_{n=1}^{\infty} D^n \theta_n(\kk),
\end{equation}
where $H$ is the Hubble parameter, and $\delta_n$ and $\theta_n$ are time-independent quantities with a Fourier mode $\kk$.
The velocity divergence is defined as $\theta \equiv \nabla \cdot \vv$, where $\vv$ is the velocity of dark matter.
The $n$th-order solutions in SPT are given by,
\begin{eqnarray}
		\delta_n(\kk) &=&  \int \frac{d^3p_1}{(2\pi)^3} \cdots \int \frac{d^3p_n}{(2\pi)^3}
		(2\pi)^3 \delta_{\rm D}(\kk-\pp_{[1,n]}) F_{n}([\pp_1,\pp_n]) \delta_{\rm L}(\pp_1)\cdots \delta_{\rm L}(\pp_n), \nonumber \\
		\theta_n(\kk) &=&  \int \frac{d^3p_1}{(2\pi)^3} \cdots \int \frac{d^3p_n}{(2\pi)^3}
		(2\pi)^3 \delta_{\rm D}(\kk-\pp_{[1,n]}) G_{n}([\pp_1,\pp_n]) \delta_{\rm L}(\pp_1)\cdots \delta_{\rm L}(\pp_n),
\label{sol_FG}
\end{eqnarray}
where $\pp_{[1,n]}\equiv\pp_1+\cdots+\pp_n$, $F_n(\pp_1,\dots,\pp_n) \equiv F_n([\pp_1,\pp_n])$, and
$\delta_{\rm D}$ is the Dirac delta function.
Linearized quantities, such as $\delta_{\rm L}$ and $\theta_{\rm L}$, are denoted by the subscript $L$.
The kernel functions $F$ and $G$ are constructed from the mode coupling functions
$\alpha(\kk_1,\kk_2) \equiv (\kk_1+\kk_2)\cdot\kk_2/k_1^2$ and $\beta(\kk_1,\kk_2)\equiv |\kk_1+\kk_2|^2 (\kk_1\cdot\kk_2)/2k_1^2k_2^2$
according to the following recursion relation:
\begin{eqnarray}
		 n F_{n}([\pp_1,\pp_{n}]) - G_{n}([\pp_1,\pp_{n}])
		&=& \sum_{m=1}^{n-1}G_m([\qq_1,\qq_m])\alpha(\qq_{[1,m]},\qq_{[m+1,n]})F_{n-m}([\qq_{m+1},\qq_{n}]), \nonumber \\
		-\frac{3}{2}F_{n}([\pp_1,\pp_{n}]) +\frac{2n+1}{2}G_{n}([\pp_1,\pp_{n}])
		&=& \sum_{m=1}^{n-1}G_m([\qq_1,\qq_m])\beta(\qq_{[1,m]},\qq_{[m+1,n]})G_{n-m}([\qq_{m+1},\qq_{n}]), \nonumber \\
		\label{FG}
\end{eqnarray}
where $F_1 = G_1 = 1$, and
the wavenumber vectors $\{\qq_1,\dots,\qq_n\} = \{\pp_1,\dots,\pp_n\}$ in the right hand side are completely-symmetrized.

The relation between the $r$th-order coefficient in the $\Gamma$-expansion and the kernel function is given by
\citep{Bernardeau:2008fa,Sugiyama:2012pc}
\begin{eqnarray}
		\Gamma^{(r)}(z,[\kk_1,\kk_r])
		\equiv 	D^r\Gamma^{(r)}_{\rm tree}([\kk_1,\kk_r])
		+ \sum_{n=1}^{\infty} D^{r+2n} \Gamma_{\rm n\mathchar`-loop}^{(r)}([\kk_1,\kk_r]),
\end{eqnarray}
with
\begin{eqnarray}
		&&\Gamma_{\rm tree}^{(r)}([\kk_1,\kk_r])\equiv F_r([\kk_1,\kk_r]), \nonumber \\
		&&\Gamma^{(r)}_{\rm n\mathchar`-loop}([\kk_1,\kk_r]) \nonumber \\
		&&\equiv
       \frac{1}{r!}\frac{(r+2n)!}{2^n n!}
          \int \frac{d^3p_1}{(2\pi)^3} \cdots  \int \frac{d^3p_n}{(2\pi)^3}
		F_{r+2n}([\kk_1,\kk_r],\pp_1,-\pp_1,\dots,\pp_n,-\pp_n)P_{\rm L}(p_1)\cdots P_{\rm L}(p_n), \nonumber \\
		\label{G-r}
\end{eqnarray}
where $P_{\rm L}$ is the present linear power spectrum.
The $n$-loop correction to the $r$th-order coefficient of the $\Gamma$-expansion
$\Gamma_{\rm n\mathchar`-loop}^{(r)}$ is of ${\cal O}( (P_{\rm L})^n)$,
and involves $n$-dimensional integrals.
The full nonlinear matter power spectrum is given by
\begin{eqnarray}
		P(z,k) = \sum_{r=1}^{\infty} P_{\rm \Gamma}^{(r)}(z,k),
\end{eqnarray}
where $P_{\rm \Gamma}$ is the $r$th-order contribution to the power spectrum in the $\Gamma$-expansion, defined as
\begin{equation}
		P_{\rm \Gamma}^{(r)}(z,k) \equiv  r! \int \frac{d^3k_1}{(2\pi)^3} \cdots  \int \frac{d^3k_r}{(2\pi)^3}
		(2\pi)^3 \delta_D(\kk-\kk_{[1,r]}) \left[\Gamma^{(r)}(z,[\kk_1,\kk_r])  \right]^2P_{\rm L}(k_1) \cdots P_{\rm L}(k_r).
	\label{power-r}
\end{equation}
Note that $P_{\rm \Gamma}$ which comprises $\Gamma_{\rm n\mathchar`-loop}^{(r)}\times \Gamma_{\rm m\mathchar`-loop}^{(r)}$
is of ${\cal O}\left( (P_{\rm L})^{r+n+m} \right)$,
and has $(r+n+m-1)$-dimensional integrals, also called the $(r+n+m-1)$-loop correction term.

The velocity divergence $\theta$ satisfies the same relation as the above by replacing $F$ with $G$.

\section{Approximation of Kernel Functions}
\label{Ap_Kernel}

Since the $\Gamma$-expansion method is rewritten by the kernel functions in SPT,
the problem to solve the nonlinear evolution of dark matter eventually reduces to compute the kernel functions.
Therefore, we develop an approximation of the kernel functions.

We shall impose the condition that the amplitude of a Fourier mode $\pp$ in $F_n$ and $G_n$ is much smaller than that of the others,
\begin{equation}
		|\pp_i| \gg |\pp| \to 0, \quad {\rm for} \ \{ i = 1,\dots,n-1\}.
\end{equation}
In this condition, we prove the following approximation of the kernel functions:
\begin{eqnarray}
		F_n([\pp_1,\pp_{n-1}],\pp) &\to& \frac{1}{n}\left(\frac{\pp_{[1,n-1]}\cdot \pp}{p^2} \right) F_{n-1}([\pp_1,\pp_{n-1}]), \nonumber \\
		G_n([\pp_1,\pp_{n-1}],\pp) &\to& \frac{1}{n}\left(\frac{\pp_{[1,n-1]}\cdot \pp}{p^2} \right) G_{n-1}([\pp_1,\pp_{n-1}]).
		\label{ap_FG}
\end{eqnarray}
We provide proof of this by the recursion relation in Equation~(\ref{FG}) and induction in $n$.
Note that these expressions are slightly different from
and more generalized than those presented in our previous work~\citep{Sugiyama:2012pc}.

First, the kernel functions $F_2$ and $G_2$ clearly satisfy Equation~(\ref{ap_FG}) for $|\pp_2| \gg |\pp_1| \to0$:
\begin{eqnarray}
		F_2(\pp_1,\pp_2)\big|_{p_1\to0}
		&\equiv& \frac{5}{7} + \frac{1}{2}\frac{\pp_1 \cdot \pp_2}{p_1p_2}\left( \frac{p_1}{p_2} + \frac{p_2}{p_1} \right)
		+ \frac{2}{7}\frac{(\pp_1\cdot\pp_2)^2}{p_1^2p_2^2}
		\to \frac{1}{2} \left( \frac{\pp_1\cdot\pp_2}{p_1^2} \right)F_1, \nonumber \\
        G_2(\pp_1,\pp_2)\big|_{p_1\to0}
		&\equiv& \frac{3}{7} + \frac{1}{2}\frac{\pp_1 \cdot \pp_2}{p_1p_2}\left( \frac{p_1}{p_2} + \frac{p_2}{p_1} \right)
		+ \frac{4}{7}\frac{(\pp_1\cdot\pp_2)^2}{p_1^2p_2^2}
		\to \frac{1}{2} \left( \frac{\pp_1\cdot\pp_2}{p_1^2} \right)G_1.
		\label{F_2G_2}
\end{eqnarray}

The next step is to prove the following relation for $|\pp|\to0$:
\begin{eqnarray}
		F_{n+1}([\pp_1,\pp_{n}],\pp) &\to& \frac{1}{n+1}\left(\frac{\pp_{[1,n]}\cdot \pp}{p^2} \right) F_{n}([\pp_1,\pp_{n}]), \nonumber \\
		G_{n+1}([\pp_1,\pp_{n}],\pp) &\to& \frac{1}{n+1}\left(\frac{\pp_{[1,n]}\cdot \pp}{p^2} \right) G_{n}([\pp_1,\pp_{n}]),
		\label{F_n+1G_n+1}
\end{eqnarray}
with the assumption that $F_i$ and $G_i$ for $\{i = 1,\dots,n\}$ satisfy Equation~(\ref{ap_FG}).
For this purpose, we rewrite the first line in Equation~(\ref{FG}) as follows:
\begin{eqnarray}
		&& (n+1) F_{n+1}([\pp_1,\pp_{n}],\pp) - G_{n+1}([\pp_1,\pp_{n}],\pp) \nonumber \\
		&=&\sum_{m=1}^{n-1}  \left( \frac{m+1}{n+1} \right)
		G_{m+1}([\qq_1,\qq_{m}],\pp)\alpha(\qq_{[1,m]}+\pp,\qq_{[m+1,n]})F_{n-m}([\qq_{m+1},\qq_{n}]) \nonumber \\
		&& + \sum_{m=1}^{n-1} \left( \frac{n+1-m}{n+1}  \right)
		G_m([\qq_1,\qq_{m}])\alpha(\qq_{[1,m]},\qq_{[m+1,n]}+\pp)F_{n+1-m}([\qq_{m+1},\qq_{n}],\pp) \nonumber \\
		&& + \frac{1}{n+1} \alpha(\pp,\pp_{[1,n]}) F_n([\pp_1,\pp_n])
		   +  \frac{1}{n+1}\alpha(\pp_{[1,n]},\pp) G_n([\pp_1,\pp_n]).
		\label{FG2}
\end{eqnarray}
This equation is approximated as
\begin{eqnarray}
		&& (n+1) F_{n+1}([\pp_1,\pp_{n}],\pp) - G_{n+1}([\pp_1,\pp_{n}],\pp) \nonumber \\
		& \to& \frac{1}{n+1} \left( \frac{\pp_{[1,n]}\cdot\pp}{p^2} \right)
		\sum_{m=1}^{n-1}G_{m}([\qq_1,\qq_{m}])\alpha(\qq_{[1,m]},\qq_{[m+1,n]})F_{n-m}([\qq_{m+1},\qq_{n}]) \nonumber \\
		&& + \frac{1}{n+1} \left( \frac{\pp_{[1,n]}\cdot\pp}{p^2} \right) F_n([\pp_1,\pp_n])
		   +  \frac{1}{n+1} G_n([\pp_1,\pp_n]),
		   \label{FG3}
\end{eqnarray}
where we used the following relations for $|\pp|\to0$:
\begin{eqnarray}
		G_{m+1}([\qq_1,\qq_m],\pp) &\to& \frac{1}{m+1} \left( \frac{\qq_{[1,m]}\cdot\pp}{p^2} \right)G_m([\qq_1,\qq_m]), \nonumber \\
		F_{n+1-m}([\qq_{m+1},\qq_n],\pp) &\to& \frac{1}{n+1-m} \left( \frac{\qq_{[m+1,n]}\cdot\pp}{p^2} \right),
		F_{n-m}([\qq_m,\qq_n]),\nonumber \\
	    \alpha(\qq_{[1,m]}+\pp,\qq_{[m+1,n]}) &\to&  \alpha(\qq_{[1,m]},\qq_{[m+1,n]}), \nonumber \\
        \alpha(\qq_{[1,m]},\qq_{[m+1,n]}+\pp) &\to&   \alpha(\qq_{[1,m]},\qq_{[m+1,n]}), \nonumber \\
		\alpha(\pp_{[1,n]},\pp) &\to& 1.
\end{eqnarray}
Using the first line in Equation~(\ref{FG}) and ignoring the last term in Equation~(\ref{FG3}),
we find
\begin{eqnarray}
		&& (n+1) F_{n+1}([\pp_1,\pp_{n}],\pp) - G_{n+1}([\pp_1,\pp_{n}],\pp)  \nonumber \\
		&&\to \frac{1}{n+1} \left( \frac{\pp_{[1,n]}\cdot\pp}{p^2} \right)
		     \left[ nF_n([\pp_1,\pp_n]) - G_n([\pp_1,\pp_n]) \right]
			 + \frac{1}{n+1} \left( \frac{\pp_{[1,n]}\cdot\pp}{p^2} \right) F_n([\pp_1,\pp_n]) \nonumber \\
			 && = \frac{1}{n+1}\left( \frac{\pp_{[1,n]}\cdot\pp}{p^2} \right)
		     \left[ (n+1) F_n([\pp_1,\pp_n]) - G_n([\pp_1,\pp_n])\right].
			 \label{ap_FG_delta}
\end{eqnarray}
For the second line in Equation~(\ref{FG}), similar calculations lead to
\begin{eqnarray}
		&& -\frac{3}{2}F_{n+1}([\pp_1,\pp_{n}],\pp) +\frac{2n+3}{2}G_{n+1}([\pp_1,\pp_{n}],\pp)  \nonumber \\
		&& \to  \frac{1}{n+1}\left( \frac{\pp_{[1,n]}\cdot\pp}{p^2} \right)
		\left[ -\frac{3}{2}F_{n}([\pp_1,\pp_{n}]) +\frac{2n+3}{2}G_{n}([\pp_1,\pp_{n}])  \right],
		\label{ap_FG_theta}
\end{eqnarray}
where we used
\begin{equation}
		\beta(\pp_{[1,n]},\pp) \to \frac{\pp_{[1,n]}\cdot\pp}{2p^2}, \ \ \mbox{for $|\pp|\to0$}.
\end{equation}
Combining Equations~(\ref{ap_FG_delta}) and (\ref{ap_FG_theta}), we finally obtain Equation~(\ref{F_n+1G_n+1}).
This concludes the proof.

For $p_{m+1},\cdots,p_n\to0$, Equation~(\ref{ap_FG}) is revised as
\begin{eqnarray}
		F_n([\pp_1,\pp_{n}])\big|_{p_{m+1},\dots,p_n\to0} &\to& \frac{m!}{n!}\left(\frac{\pp_{[1,m]}\cdot \pp_{m+1}}{p^2_{m+1}} \right)
     	   \cdots \left(\frac{\pp_{[1,m]}\cdot \pp_{n}}{p^2_{n}} \right) F_{m}([\pp_{1},\pp_{m}]),
\end{eqnarray}
where $n > m$.
Thus, we find that $F_n$ is approximated using $F_{m}$ at a lower order $m$ than $n$.
The case of $m=1$ leads to the Zel'dovich approximation in the limit that the total momentum is $p_1$:
\begin{eqnarray}
		F_n(\pp_1,\pp_{n}])|_{p_2,\dots,p_n\to0} \to
           \frac{1}{n!}\left(\frac{\pp_{1}\cdot \pp_{2}}{p^2_{2}} \right) \cdots \left(\frac{\pp_{1}\cdot \pp_{n}}{p^2_{n}} \right) .
\end{eqnarray}
Therefore, we interpret the approximation of Equation~(\ref{ap_FG}) as an extension of the Zel'dovich approximation.

\section{Reg PT}
\label{Proof}
Now, we are ready to derive Reg PT \citep{Bernardeau:2008fa,Bernardeau:2011dp,Taruya:2012ut}.
We give an alternative explanation for Reg PT by the relation between SPT and the $\Gamma$-expansion [Equation~(\ref{G-r})]
and the approximation of the kernel functions [Equation~(\ref{ap_FG})].

Before we proceed, we show that the $r$th-order coefficient of the $\Gamma$-expansion at the $n$-loop level
$\Gamma_{\rm n\mathchar`-loop}^{(r)}$ is approximately represented by $\Gamma_{\rm m\mathchar`-loop}^{(r)}$ at a lower $m$-loop level
than the $n$-loop level:
\begin{eqnarray}
		\Gamma_{\rm n\mathchar`-loop}^{(r)}([\kk_1,\kk_r])\big|_{p_{m+1},\cdots,p_n\to0}
		\to \frac{m!}{n!} \left( -\frac{k^2\sigma_v^2}{2} \right)^{n-m} \Gamma_{\rm m\mathchar`-loop}^{(r)}([\kk_1,\kk_r]),
		\label{G:n-mloop}
\end{eqnarray}
where $\sigma_v^2$ is the velocity dispersion of dark matter: $\sigma_v^2\equiv \int dp P_L(p)/6\pi $.
This is shown by substituting the following approximated kernel functions into Equation~(\ref{G-r}):
\begin{eqnarray}
		&&F_{r+2n}([\kk_1,\kk_r],\pp_1,-\pp_1,\cdots,\pp_n,-\pp_n)\big|_{p_{m+1},\cdots,p_n \to0} \nonumber \\
		&\to& \frac{(r+2m)!}{(r+2n)!} (-1)^{n-m}
		\left( \frac{\kk_{[1,r]}\cdot \pp_{m+1}}{p_{m+1}} \right)^2 \cdots \left( \frac{\kk_{[1,r]}\cdot \pp_n}{p_n^2} \right)^2
		F_{r+2m}\left( [\kk_1,\kk_r],\pp_1,-\pp_1,\cdots,\pp_m,-\pp_m \right). \nonumber \\
\end{eqnarray}

To lead Reg PT at the one-loop level, we need the cases of $m=0$ and $m=1$ in Equation~(\ref{G:n-mloop}):
\begin{eqnarray}
		\Gamma_{\rm n\mathchar`-loop}^{(r)}([\kk_1,\kk_r])\big|_{p_2,\cdots,p_n\to0}
		&=& \frac{1}{n!}\left( -\frac{k^2\sigma_v^2}{2} \right)^{n-1} \Gamma_{\rm 1\mathchar`-loop}^{(r)}([\kk_1,\kk_r]), \nonumber \\
		\Gamma_{\rm n\mathchar`-loop}^{(r)}([\kk_1,\kk_r])\big|_{p_1,\cdots,p_n\to0}
		&=& \frac{1}{n!}\left( -\frac{k^2\sigma_v^2}{2} \right)^{n} \Gamma_{\rm tree}^{(r)}([\kk_1,\kk_r]).
		\label{eq:1}
\end{eqnarray}
From Equation~(\ref{eq:1}), we derive an approximated expression of $\Gamma_{\rm n-loop}^{(r)}$ as follows:
\begin{eqnarray}
		\Gamma_{\rm n\mathchar`-loop}^{(r)}([\kk_1,\kk_r]) &\to& n\Gamma_{\rm n\mathchar`-loop}^{(r)}([\kk_1,\kk_r])\big|_{p_2,\dots,p_n \to0}
		 - (n-1) \Gamma_{\rm n\mathchar`-loop}^{(r)}([\kk_1,\kk_r])\big|_{p_1,\cdots,p_n \to0} \nonumber \\
		 &=& \frac{n}{n!}\left( -\frac{k^2 \sigma_v^2}{2} \right)^{n-1}\Gamma_{\rm 1\mathchar`-loop}^{(r)}([\kk_1,\kk_r])
		- \frac{n-1}{n!}\left( -\frac{k^2 \sigma_v^2}{2} \right)^{n}\Gamma_{\rm tree}^{(r)}([\kk_1,\kk_r]), \nonumber \\
		\label{del_r+2n}
\end{eqnarray}
where we multiply the first term in the right hand side by $n$ because
we choose a Fourier mode $\pp_1$ from $n$ Fourier modes $\pp_i$ $\{i=1,\cdots,n\}$.
Furthermore, we need to subtract $(n-1)\Gamma_{\rm n-loop}^{(r)}|_{p_1,\cdots,p_n \to0}$
because the first term $n\Gamma_{\rm n\mathchar`-loop}^{(r)}|_{p_2,\dots,p_n\to0}$
integrates the same region defined by $p_1,\dots,p_n\to0$ $n$ times in the multiple integral in Equation~(\ref{G-r}).
This expression is satisfied even in the case of $n=0$: $\Gamma_{\rm 0\mathchar`-loop}^{(r)} = \Gamma_{\rm tree}^{(r)}$.
Substituting Equation~(\ref{del_r+2n}) into Equation~(\ref{G-r}),
we find Reg PT with the one-loop corrections:
\begin{eqnarray}
		&&\Gamma^{(r)}(z,[\kk_1,\kk_r]) \nonumber \\
		&&\to \sum_{n=0}^{\infty} D^{r+2n}
		\left[ \frac{n}{n!}\left( -\frac{k^2 \sigma_v^2}{2} \right)^{n-1}\Gamma_{\rm 1\mathchar`-loop}^{(r)}([\kk_1,\kk_r])
		- \frac{n-1}{n!}\left( -\frac{k^2 \sigma_v^2}{2} \right)^{n}\Gamma_{\rm tree}^{(r)}([\kk_1,\kk_r])\right]
		\nonumber \\
		&=& \exp\left(   -\frac{k^2 D^2\sigma_v^2}{2} \right)
		D^r\left[ \Gamma_{\rm tree}^{(r)}([\kk_1,\kk_r]) + \left(D^2\Gamma_{\rm 1\mathchar`-loop}^{(r)}[\kk_1,\kk_r]
		+ \frac{k^2 D^2 \sigma_v^2}{2} \Gamma_{\rm tree}^{(r)}([\kk_1,\kk_r])   \right) \right]. \nonumber \\
		\label{Reg1}
\end{eqnarray}

As in the case of Reg PT at the one-loop level, we are further able to show the following expression:
\begin{eqnarray}
		&&\Gamma_{\rm n\mathchar`-loop}^{(r)}([\kk_1,\kk_r])  \nonumber \\
		&\to&
		 \frac{1}{(n-2)!}\left( -\frac{k^2 \sigma_v^2}{2} \right)^{n-2}
		 \Bigg[\Gamma_{\rm 2\mathchar`-loop}^{(r)}([\kk_1,\kk_r]) + \left( \frac{k^2\sigma_v^2}{2} \right)
		 \Gamma_{\rm 1\mathchar`-loop}^{(r)}([\kk_1,\kk_r])
		 + \frac{1}{2}\left( \frac{k^2\sigma_v^2}{2} \right)^2 \Gamma_{\rm tree}^{(r)}([\kk_1,\kk_r]) \Bigg] \nonumber \\
		&& +  \frac{1}{(n-1)!}\left( -\frac{k^2 \sigma_v^2}{2} \right)^{n-1}
		\Bigg[\Gamma_{\rm 1\mathchar`-loop}^{(r)}([\kk_1,\kk_r])
		+ \left( \frac{k^2\sigma_v^2}{2} \right) \Gamma_{\rm tree}^{(r)}([\kk_1,\kk_r]) \Bigg]
		\nonumber \\
		&& +  \frac{1}{n!}\left( -\frac{k^2 \sigma_v^2}{2} \right)^{n} \Gamma_{\rm tree}^{(r)}([\kk_1,\kk_r]).
\end{eqnarray}
This results in Reg PT with the two-loop corrections:
\begin{eqnarray}
	&&	\Gamma^{(r)}(z,[\kk_1,\kk_r]) \nonumber \\
	&&	\to \exp\left(  -\frac{k^2 D^2\sigma_v^2}{2} \right)
		D^r\Bigg[ \Gamma_{\rm tree}^{(r)}([\kk_1,\kk_r]) +
		\left(  D^2\Gamma_{\rm 1\mathchar`-loop}^{(r)}([\kk_1,\kk_r])
		+ \frac{k^2 D^2\sigma_v^2}{2}  \Gamma_{\rm tree}^{(r)}([\kk_1,\kk_r]) \right)
		\nonumber \\
		&& + \left( D^4\Gamma_{\rm 2\mathchar`-loop}^{(r)}([\kk_1,\kk_r])
		+ \left( \frac{k^2 D^2\sigma_v^2}{2}  \right)\Gamma_{\rm 1\mathchar`-loop}^{(r)}([\kk_1,\kk_r])
		+ \frac{1}{2} \left(\frac{k^2 D^2\sigma_v^2}{2}  \right)^2 \Gamma_{\rm tree}^{(r)}([\kk_1,\kk_r])  \right)	  \Bigg]. \nonumber \\
	\label{Reg2}
\end{eqnarray}

\section{Extension of Reg PT}
\label{Extension_of_RegPT}

In this section, we propose a natural extended version of Reg PT.
In Section~\ref{Proof},
we only applied the approximated kernel functions to multiple integrals involved in the coefficients of the $\Gamma$-expansion,
such as the integrals of $\pp_1$, \dots, $\pp_n$ in Equation~(\ref{G-r}).
Here, we further use the approximated kernel functions to calculate the multiple integrals
in the $r$th-order contribution to the power spectrum in the $\Gamma$-expansion,
such as the integrals of $\kk_1$, \dots, $\kk_r$ in Equation~(\ref{power-r}).
In other words, we apply our kernel approximation method to compute high-order terms of the $\Gamma$-expansion with Reg PT.

The authors in \cite{Taruya:2012ut} computed $P_{\rm \Gamma}^{(r)}$ up to the third order.
The terms at the first, second, and third order in the $\Gamma$-expansion
were calculated using Reg PT at the two-loop, one-loop (Equations~(\ref{Reg2}) and (\ref{Reg1})), and tree level, respectively
\citep[see][Equations~(23)-(26)]{Taruya:2012ut}.
On the other hand, we evaluate any order of the $\Gamma$-expansion using either Equation~(\ref{Reg1}) or Equation~(\ref{Reg2})
to keep consistency of the approximation.

The important relation to extend Reg PT is
that the $r$th-order coefficient in the $\Gamma$-expansion
is approximated by one at a lower order $m$ than $r$:
\begin{eqnarray}
		\Gamma_{\rm n\mathchar`-loop}^{(r)}([\kk_1,\kk_r])\big|_{k_{m+1},\dots,k_r\to0}
		\to \frac{m!}{r!}\left( \frac{\kk_{[1,m]}\cdot \kk_{m+1}}{k_{m+1}^2} \right)
		\cdots\left( \frac{\kk_{[1,m]}\cdot \kk_{m}}{k_{m}^2} \right) \Gamma_{\rm n\mathchar`-loop}^{(m)}([\kk_1,\kk_m]),
		\label{ap_G}
\end{eqnarray}
where we used the following approximated kernel function,
\begin{eqnarray}
		&&F_{r+2n}([\kk_1,\kk_r],\pp_1,-\pp_1,\dots,\pp_n,-\pp_n)\big|_{k_{m+1},\dots,k_n\to0} \nonumber \\
		&=&  \frac{(m+2n)!}{(r+2n)!}\left( \frac{\kk_{[1,m]}\cdot \kk_{m+1}}{k_{m+1}^2} \right)
		\cdots \cdots\left( \frac{\kk_{[1,m]}\cdot \kk_{m}}{k_{m}^2} \right)F_{m+2n}([\kk_1,\kk_m]).
\end{eqnarray}

\subsection{Extension of Reg PT: One-loop Level}

First, combining Equations~(\ref{power-r}) and (\ref{Reg1}),
the first-order contribution to the power spectrum in the $\Gamma$-expansion with Reg PT is given by
\begin{eqnarray}
		P_{\rm \Gamma}^{(1)}(z,k)=\left[ \Gamma^{(1)}(z,k) \right]^2P_{\rm L}(k) \to
		e^{-k^2 D^2\sigma_v^2}	\left[ 1 +  D^2 \Gamma_{\rm 1\mathchar`-loop}^{(1)}(k) + \frac{k^2 D^2 \sigma_v^2}{2}  \right]^2  D^2 P_{\rm L}(k).
		\label{WH1:1loop}
\end{eqnarray}

Next, we consider the second-order contribution calculated from Equation~(\ref{Reg1}):
\begin{eqnarray}
		P_{\rm \Gamma}^{(2)}(z,k)	&=& 2!\int \frac{d^3k_1}{(2\pi)^3}\int \frac{d^3k_2}{(2\pi)^3} (2\pi)^3 \delta_{D}(\kk-\kk_{[1,r]})
		\left[ \Gamma^{(2)}(z,\kk_1,\kk_2) \right]^2 P_{\rm L}(k_1) P_{\rm L}(k_2) \nonumber \\
		&\to&
		e^{-k^2 D^2\sigma_v^2}
	2! \int \frac{d^3k_1}{(2\pi)^3}\frac{d^3k_2}{(2\pi)^3}
	(2\pi)^3 \delta_D(\kk-\kk_{[1,2]}) D^2 P_{\rm L}(k_1)D^2 P_{\rm L}(k_2)\nonumber \\
	&& \times \Bigg[ \left( \Gamma_{\rm tree}^{(2)}(\kk_1,\kk_2) \right)^2
	+ 2 \Gamma_{\rm tree}^{(2)}(\kk_1,\kk_2)
	\left( D^2\Gamma_{\rm 1\mathchar`-loop}^{(2)}(\kk_1,\kk_2)
	+ \frac{k^2 D^2 \sigma_v^2}{2} \Gamma_{\rm tree}^{(2)}(\kk_1,\kk_2) \right)  \nonumber \\
	&& \hspace{5cm} + \left( D^2\Gamma_{\rm 1\mathchar`-loop}^{(2)}(\kk_1,\kk_2)
	+ \frac{k^2 D^2 \sigma_v^2}{2} \Gamma_{\rm tree}^{(2)}(\kk_1,\kk_2) \right)^2 \Bigg]. \nonumber \\
	\label{P_WH^2:1loop}
\end{eqnarray}
The first term in the square bracket leads to
\begin{equation}
		e^{-k^2D^2 \sigma_v^2} 2! \int \frac{d^3k_1}{(2\pi)^3}\frac{d^3k_2}{(2\pi)^3}
		(2\pi)^3 \delta_D(\kk-\kk_{[1,2]})
		\left[ F_2(\kk_1,\kk_2) \right]^2 D^2P_{\rm L}(k_1)D^2P_{\rm L}(k_2)= e^{-k^2D^2 \sigma_v^2}  D^4P_{22}(k).
		\label{WH2:1loop:part}
\end{equation}
If we only calculate Equations~(\ref{WH1:1loop}) and (\ref{WH2:1loop:part}),
we obtain a result called the one-loop Reg PT solution in \cite{Taruya:2012ut}:
\begin{eqnarray}
		P_{\rm Reg, 1\mathchar`-loop}(z,k) = e^{-k^2 D^2\sigma_v^2}
		\left[1 + D^2 \Gamma_{\rm 1\mathchar`-loop}^{(1)}(k) + \frac{k^2 D^2 \sigma_v^2}{2} \right]^2 D^2 P_{\rm L}(k)
		 + e^{-k^2D^2 \sigma_v^2}  D^4P_{22}(k).
		 \label{RegPT:1loop}
\end{eqnarray}
However, we further calculate the other terms in Equation~(\ref{P_WH^2:1loop}).
Although we need higher order terms in SPT than the one-loop level to strictly compute these terms,
we here approximate them using $P_{13}\equiv 2P_{\rm L} \Gamma_{\rm 1\mathchar`-loop}^{(1)}$ and $P_{22}$, which are at the one-loop level.
The second and third terms in the square bracket in Equation~(\ref{P_WH^2:1loop}) are approximated for $k_1\to0$ as
\begin{eqnarray}
	&& e^{-k^2 D^2\sigma_v^2}
	 2! \int \frac{d^3k_1}{(2\pi)^3}\frac{d^3k_2}{(2\pi)^3}
	 (2\pi)^3 \delta_D(\kk-\kk_1-\kk_2)\big|_{k_1\to0} D^2P_{\rm L}(k_1) D^2 P_{\rm L}(k_2)\nonumber \\
	&& \times \Bigg[ 2 \Gamma_{\rm tree}^{(2)}(\kk_1,\kk_2)
	\left( D^2\Gamma_{\rm 1\mathchar`-loop}^{(2)}(\kk_1,\kk_2)
	+ \frac{k^2 D^2 \sigma_v^2}{2} \Gamma_{\rm tree}^{(2)}(\kk_1,\kk_2) \right)\Bigg|_{k_1\to0}  \nonumber \\
	&& \hspace{2cm} + \left( D^2\Gamma_{\rm 1\mathchar`-loop}^{(2)}(\kk_1,\kk_2)
	+ \frac{k^2 D^2 \sigma_v^2}{2} \Gamma_{\rm tree}^{(2)}(\kk_1,\kk_2) \right)^2 \Bigg|_{k_1\to0} \Bigg]\nonumber \\
	&=& e^{-k^2 D^2\sigma_v^2}
	 2! \int \frac{d^3k_1}{(2\pi)^3}\frac{d^3k_2}{(2\pi)^3}
	(2\pi)^3 \delta_D(\kk-\kk_2) D^2P_{\rm L}(k_1) D^2 P_{\rm L}(k_2)\nonumber \\
	&& \times \frac{1}{4}\left( \frac{\kk_1\cdot\kk_2}{k_1^2} \right)^2
	\Bigg[ 	2 \Gamma_{\rm tree}^{(1)}(k_2)
	\left( D^2\Gamma_{\rm 1\mathchar`-loop}^{(1)}(k_2)
	+ \frac{k^2 D^2 \sigma_v^2}{2} \Gamma_{\rm tree}^{(1)}(k_2) \right)  \nonumber \\
	&& \hspace{5cm} + \left( D^2\Gamma_{\rm 1\mathchar`-loop}^{(1)}(k_2)
	+ \frac{k^2 D^2 \sigma_v^2}{2} \Gamma_{\rm tree}^{(1)}(k_2) \right)^2 \Bigg]\nonumber \\
	&=&  e^{-k^2 D^2\sigma_v^2}\left( \frac{k^2 D^2 \sigma_v^2}{2} \right)
	\left[ \left(1 + D^2 \Gamma_{\rm 1\mathchar`-loop}^{(1)}(k) + \frac{k^2 D^2 \sigma_v^2}{2} \right)^2 D^2 P_{\rm L}(k)
	- D^2 P_{\rm L}(k)\right],
	\label{PWH2:eq}
\end{eqnarray}
where we used the approximations for $\Gamma_{\rm 1\mathchar`-loop}^{(2)}$ and $\Gamma_{\rm tree}^{(2)}$ in Equation~(\ref{ap_G})
and $\delta_{\rm D}(\kk-\kk_{[1,2]})|_{k_1\to0} =\delta_{\rm D}(\kk-\kk_2)$.
Since we also need to consider the contribution from the region of $k_2 \to0$, we multiply Equation~(\ref{PWH2:eq}) by a factor of two.
Equation~(\ref{P_WH^2:1loop}) then becomes
\begin{eqnarray}
		P_{\rm \Gamma}^{(2)}(z,k)&\to& e^{-k^2D^2\sigma_v^2} D^4\left(P_{22}(k) - k^2 \sigma_v^2P_{\rm L}(k)  \right)  \nonumber \\
		&& + e^{-k^2D^2\sigma_v^2}(k^2 D^2 \sigma_v^2)
		\left(1 + D^2 \Gamma_{\rm 1\mathchar`-loop}^{(1)}(k) + \frac{k^2 D^2 \sigma_v^2}{2} \right)^2 D^2 P_{\rm L}(k).
	\label{WH2:1loop}
\end{eqnarray}

We apply the same analysis as above to an arbitrary order coefficient of the $\Gamma$-expansion.
The $r$th-order contribution to the power spectrum in the $\Gamma$-expansion is given from Equations~(\ref{power-r}) and (\ref{Reg1}) by
\begin{eqnarray}
	P_{\rm \Gamma}^{(r)}(z,k)&\to&e^{-k^2 D^2\sigma_v^2}
	r! \int \frac{d^3k_1}{(2\pi)^3} \cdots \frac{d^3k_r}{(2\pi)^3}
	(2\pi)^3 \delta_D(\kk-\kk_{[1,r]})D^2P_{\rm L}(k_1) \cdots D^r P_{\rm L}(k_r) \nonumber \\
	&& \times \Bigg[ \left[ \Gamma_{\rm tree}^{(r)}([\kk_1,\kk_r]) \right]^2
	+ 2 \Gamma_{\rm tree}^{(r)}([\kk_1,\kk_r])
	\left( D^{2}\Gamma_{\rm 1\mathchar`-loop}^{(r)}([\kk_1,\kk_r])
	+ \frac{k^2 D^2 \sigma_v^2}{2} \Gamma_{\rm tree}^{(r)}([\kk_1,\kk_r]) \right)  \nonumber \\
	&& \hspace{4cm}
	+ \left( D^{2}\Gamma_{\rm 1\mathchar`-loop}^{(r)}([\kk_1,\kk_r])
	+ \frac{k^2 D^2 \sigma_v^2}{2} \Gamma_{\rm tree}^{(r)}([\kk_1,\kk_r]) \right)^2 \Bigg].
	\label{P_WH^r:1loop}
\end{eqnarray}
The first term in the square bracket is approximated as
\begin{eqnarray}
	&&\to e^{-k^2 D^2\sigma_v^2}
	r! \int \frac{d^3k_1}{(2\pi)^3} \cdots \frac{d^3k_r}{(2\pi)^3} (2\pi)^3 \delta_D(\kk-\kk_{[1,r]})
	D^2 P_{\rm L}(k_1) \cdots D^2P_{\rm L}(k_r) \nonumber \\
	&&\times \left[  \frac{r(r-1)}{2} \left[\Gamma_{\rm tree}^{(r)}([\kk_1,\kk_r]) \right]^2  \Big|_{k_3,\dots,k_r\to0}
	+ \left( -r(r-1)  + r\right) \left[\Gamma_{\rm tree}^{(r)}([\kk_1,\kk_r]) \right]^2 \Big|_{k_2,\dots,k_r\to0}  \right]\nonumber \\
	&& = e^{-k^2D^2\sigma_v^2}\frac{(k^2D^2\sigma_v^2)^{r-2}}{(r-2)!} D^4\left(P_{22}(k) - k^2 \sigma_v^2 P_{\rm L}(k)  \right)
	 + e^{-k^2D^2\sigma_v^2}\frac{(k^2D^2\sigma_v^2)^{r-1}}{(r-1)!} D^2 P_{\rm L}(k),
\end{eqnarray}
where we multiply the first term by a factor of $r(r-1)/2$ because
we choose the two Fourier modes $\kk_1,\kk_2$ from the $r$ Fourier modes, $\kk_i$ $\{i=1,\cdots,r\}$.
We need the second term to exclude the redundant integrated regions defined by $k_2,\dots,k_r\to0$.
We used Equation~(\ref{ap_G})
and the approximation of the delta function: $\delta_{\rm D}(\kk-\kk_{[1,r]})|_{k_3,\dots,k_r\to0} = \delta_{\rm D}(\kk-\kk_{[1,2]})$
and $\delta_{\rm D}(\kk-\kk_{[1,r]})|_{k_2,\dots,k_r\to0} = \delta_{\rm D}(\kk-\kk_1)$.
The approximated solutions of the other terms in Equation~(\ref{P_WH^r:1loop})
are also calculated by the same derivation used in the case of $P_{\rm \Gamma}^{(2)}$:
\begin{eqnarray}
	e^{-k^2D^2\sigma_v^2}	\frac{(k^2D^2\sigma_v^2)^{r-1}}{(r-1)!}
	\left[\left( 1 + D^2 \Gamma_{\rm 1\mathchar`-loop}^{(1)}(k) + \frac{k^2 D^2 \sigma_v^2}{2} \right)^2 D^2P_{\rm L}(k) - D^2 P_{\rm L}(k)\right].
\end{eqnarray}
Therefore, we arrive at the approximated expression of $P_{\rm \Gamma}^{(r)}$:
\begin{eqnarray}
		P_{\rm \Gamma}^{(r)}(z,k)&\to &
		  e^{-k^2D^2\sigma_v^2}
       \frac{(k^2D^2\sigma_v^2)^{r-1}}{(r-1)!}
	   \left( 1 + D^2 \Gamma_{\rm 1\mathchar`-loop}^{(1)}(k) + \frac{k^2 D^2 \sigma_v^2}{2} \right)^2 D^2 P_{\rm L}(k) \nonumber \\
        && +e^{-k^2D^2\sigma_v^2}
		\frac{(k^2D^2\sigma_v^2)^{r-2}}{(r-2)!} D^4\left(P_{22}(k) - k^2 \sigma_v^2P_{\rm L}(k)  \right) .
		\label{EXREG1}
		\end{eqnarray}
We propose this expression as an extended version of the one-loop Reg PT solution.
This certainly recovers the results in the cases of $r=1$ and $r=2$.

Note that if we sum up all orders of the $\Gamma$-expansion,
we derive the following power spectrum from the extended Reg PT at the one-loop level:
\begin{eqnarray}
		P(z,k) = \sum_{r=1}^{\infty} P_{\rm \Gamma}^{(r)}(z,k)
		= D^2 P_{\rm L}(k) + D^4P_{\rm 1\mathchar`-loop}(k)
		+ \left( D^2 \Gamma_{\rm 1\mathchar`-loop}^{(1)}(k) + \frac{k^2 D^2 \sigma_v^2}{2}  \right)^2 D^2 P_{\rm L}(k),
		\label{Ex_SPT:1loop}
\end{eqnarray}
where $P_{\rm 1\mathchar`-loop}$ is the correction term at the one-loop level in SPT, defined as $P_{\rm 1\mathchar`-loop}\equiv P_{22} +P_{13}$.
\subsection{Extension of Reg PT: Two-loop Level}
Similar to the derivation used in Equation~(\ref{EXREG1}),
we derive the following approximated $P_{\rm \Gamma}^{(r)}$ from Equation~(\ref{Reg2}):
\begin{eqnarray}
	P_{\rm \Gamma}^{(r)}(z,k)&\to&e^{-k^2D^2\sigma_v^2}\frac{(k^2D^2\sigma_v^2)^{r-1}}{(r-1)!}
	\Bigg[ 1 + \left( D^2 \Gamma_{\rm 1\mathchar`-loop}^{(1)}(k) + \frac{k^2D^2\sigma_v^2}{2} \right)  \nonumber \\
	&& \hspace{3cm}+\left( D^4\Gamma_{\rm 2\mathchar`-loop}^{(1)}(k) + \frac{k^2D^2\sigma_v^2}{2} D^2\Gamma_{\rm 1\mathchar`-loop}^{(1)}(k)
	+ \frac{1}{2}\left( \frac{k^2D^2\sigma_v^2}{2} \right)^2	\right) \Bigg]^2D^2P_{\rm L}(k) \nonumber \\
		&& + e^{-k^2D^2\sigma_v^2}\frac{(k^2D^2\sigma_v^2)^{r-2}}{(r-2)!}
		D^4\left[ P_{22}(k) - k^2 \sigma_v^2 P_{\rm L}(k)\right] \nonumber \\
		&& + e^{-k^2D^2\sigma_v^2} \frac{(k^2D^2\sigma_v^2)^{r-2}}{(r-2)!}
		D^6\left[ P_{24}(k) + k^2 \sigma_v^2 P_{22}(k) - k^2 \sigma_v^2 \left( P_{13}(k) + k^2 \sigma_v^2 P_{\rm L}(k) \right) \right] \nonumber \\
		&&+  e^{-k^2D^2\sigma_v^2}\frac{(k^2D^2\sigma_v^2)^{r-2}}{(r-2)!}
          D^8\Bigg[ P_{44a}(k) + \frac{k^2 \sigma_v^2}{2} P_{24}(k) + \frac{(k^2 \sigma_v^2)^2}{4} P_{22}(k) \nonumber \\
		  &&\hspace{5cm}
		  - k^2 \sigma_v^2 \left( \Gamma_{\rm 1\mathchar`-loop}^{(1)}(k) + \frac{k^2  \sigma_v^2}{2}  \right)^2 P_{\rm L}(k)\Bigg] \nonumber \\
		&& + e^{-k^2D^2\sigma_v^2}\frac{(k^2D^2\sigma_v^2)^{r-3}}{(r-3)!}
		D^6\left[ P_{33b}(k) - k^2 \sigma_v^2 P_{22}(k) + \frac{(k^2 \sigma_v^2)^2}{2} P_{\rm L}(k) \right],
		\label{ExtendedRegPT:2loop}
\end{eqnarray}
where $P_{24}$, $P_{33b}$, and $P_{44a}$ are defined as
\begin{eqnarray}
		P_{24}(k)  &\equiv&  24	\int \frac{d^3k_1}{(2\pi)^3}\frac{d^3k_2}{(2\pi)^3}\frac{d^3p}{(2\pi)^3}
		(2\pi)^3 \delta_{D}(\kk-\kk_{[1,2]}) F_2(\kk_1,\kk_2) F_4(\kk_1,\kk_2,\pp,-\pp)P_L(p) P_{\rm L}(k_1) P_{\rm L}(k_2),\nonumber \\
		P_{33b}(k) &\equiv& 6	\int \frac{d^3k_1}{(2\pi)^3}\frac{d^3k_2}{(2\pi)^3}\frac{d^3k_3}{(2\pi)^3}
		(2\pi)^3 \delta_D(\kk-\kk_{[1,3]}) \left[  F_3(\kk_1,\kk_2,\kk_3)\right]^2 P_{\rm L}(k_1) P_{\rm L}(k_2) P_{\rm L}(k_3), \nonumber \\
		P_{44a}(k) &\equiv&  72 \int \frac{d^3k_1}{(2\pi)^3}\frac{d^3k_2}{(2\pi)^3} \frac{d^3p_1}{(2\pi)^3}\frac{d^3p_2}{(2\pi)^3}
		(2\pi)^3 \delta_D(\kk-\kk_{[1,2]}) \nonumber \\
		&& \times F_4(\kk_1,\kk_2,\pp_1,-\pp_1)F_4(\kk_1,\kk_2,\pp_2,-\pp_2) P_{\rm L}(k_1)P_{\rm L}(k_2) P_{\rm L}(p_1) P_{\rm L}(p_2).
		\label{corrections:2loop}
\end{eqnarray}

Equation~(\ref{ExtendedRegPT:2loop}) recovers the two-loop Reg PT solution which is given by \citep{Taruya:2012ut}
\begin{eqnarray}
		 P_{\rm Reg,2\mathchar`-loop}(z,k)
		&=&  e^{-k^2D^2\sigma_v^2}
        \Bigg[ 1 + \left( D^2 \Gamma_{\rm 1\mathchar`-loop}^{(1)}(k) + \frac{k^2D^2\sigma_v^2}{2} \right)  \nonumber \\
		&&\hspace{2cm} +\left( D^4\Gamma_{\rm 2\mathchar`-loop}^{(1)}(k) + \frac{k^2D^2\sigma_v^2}{2}
		D^2\Gamma_{\rm 1\mathchar`-loop}^{(1)}(k)
	       + \frac{1}{2}\left( \frac{k^2D^2\sigma_v^2}{2} \right)^2	\right) \Bigg]^2D^2P_{\rm L}(k) \nonumber \\
		&&+ e^{-k^2D^2\sigma_v^2}
		\Bigg[  D^4P_{22}(k) + D^6\left( P_{24}(k) + k^2 \sigma_v^2 P_{22}(k)\right)  \nonumber \\
		&&\hspace{3cm}
		+ D^8\left( P_{44a}(k) + \frac{k^2 \sigma_v^2}{2} P_{24}(k) + \frac{(k^2 \sigma_v^2)^2}{4}P_{22}(k) \right)\Bigg]  \nonumber \\
		&&+ e^{-k^2D^2\sigma_v^2} P_{33b}(k).
		\label{RegPT:2loop}
\end{eqnarray}
In the case of $r=1$, Equation~(\ref{ExtendedRegPT:2loop}) coincides with the first term in Equation~(\ref{RegPT:2loop}).
For $r=2$ and $r=3$, we have additional terms compared to the second and third terms in Equation~(\ref{RegPT:2loop}).
This is because
Equation~(\ref{ExtendedRegPT:2loop}) is computed using Reg PT at the two-loop level (Equation~(\ref{Reg2})),
although the second and third term in Equation~(\ref{RegPT:2loop}) are calculated
by Reg PT at the one-loop level (Equation~(\ref{Reg1})) and the tree level, respectively.
Furthermore, we can also derive higher order terms than the third order in the $\Gamma$-expansion from Equation~(\ref{ExtendedRegPT:2loop}).

The power spectrum including all orders of the $\Gamma$-expansion is given by
\begin{eqnarray}
		P(z,k) &=&  D^2 P_L(k) + D^4P_{\rm 1\mathchar`-loop}(k) + D^6 P_{\rm 2\mathchar`-loop}(k) \nonumber \\
		&&	+ 2D^8\left( \Gamma_{\rm 1\mathchar`-loop}^{(1)}(k) + \frac{k^2\sigma_v^2}{2}\right)
		\left( \Gamma_{\rm 2\mathchar`-loop}^{(1)}(k) + \frac{k^2\sigma_v^2}{2} \Gamma_{\rm 1\mathchar`-loop}^{(1)}(k)
		+ \frac{1}{2}\left( \frac{k^2\sigma_v^2}{2} \right)^2\right) P_{\rm L}(k) \nonumber \\
    &&+ D^8\left[ P_{44a}(k) + \frac{k^2 \sigma_v^2}{2} P_{24}(k)+ \frac{(k^2 \sigma_v^2)^2}{4} P_{22}(k)
	- k^2 \sigma_v^2 \left( \Gamma_{\rm 1\mathchar`-loop}^{(1)}(k) + \frac{k^2  \sigma_v^2}{2}  \right)^2D^2P_{\rm L}(k)\right] \nonumber \\
	&& + D^{10}\left( \Gamma_{\rm 2\mathchar`-loop}^{(1)}(k) + \frac{k^2\sigma_v^2}{2} \Gamma_{\rm 1\mathchar`-loop}^{(1)} (k)
	+ \frac{1}{2}\left( \frac{k^2\sigma_v^2}{2} \right)^2\right)^2  P_{\rm L}(k),
	\label{SPT2loop+}
\end{eqnarray}
where $P_{\rm 2\mathchar`-loop}$ is the correction term at the two-loop level in SPT, defined as
\begin{eqnarray}
		P_{\rm 2\mathchar`-loop} \equiv P_{15}(k) + P_{24}(k) + P_{33a}(k) + P_{33b}(k),
\end{eqnarray}
with $P_{15}\equiv 2\Gamma_{\rm 2\mathchar`-loop}^{(1)} P_{\rm L}$ and $P_{33a}\equiv [\Gamma_{\rm 1\mathchar`-loop}^{(1)}]^2 P_{\rm L}$.
As in the case of the extended 1-loop Reg PT,
we obtain the two-loop SPT solution with the additional correction terms.

\section{Modified SPT}
\label{Ex_SPT}

\begin{figure}[t]
				\begin{center}
						\epsscale{0.7}
						\plotone{f1.eps}
				\end{center}
		\caption{Ratios of the predicted power spectra $P$ to the smoothed linear power spectrum without BAO $P_{\rm nw}$
		presented by \cite{Eisenstein:1997ik} are plotted at $z=1.0$.
		The predicted power spectra are computed by
		SPT (red solid), Reg PT (blue solid, Equations~(\ref{RegPT:1loop}) and (\ref{RegPT:2loop})),
		the extended Reg PT (red dashed, Equations~(\ref{Ex_SPT:1loop}) and (\ref{SPT2loop+})), and $N$-body simulations (green symbols).
		}
		\label{fig:SG}
\end{figure}

We investigate the relation between the results from Reg PT (Equations~(\ref{RegPT:1loop}) and~(\ref{RegPT:2loop})),
the extended Reg PT (Equations~(\ref{Ex_SPT:1loop}) and~(\ref{SPT2loop+})), and SPT.
One might think that the results of the extended RegPT
give a better prediction for the nonlinear matter power spectra than those of SPT,
because the extended RegPT has the correction terms as well as the SPT solution.
However, this is not true.
Figure~\ref{fig:SG} shows that the additional terms in Equations~(\ref{Ex_SPT:1loop}) and~(\ref{SPT2loop+})
hardly contribute to the predicted nonlinear power spectra (see the red solid and dashed lines in Figure~\ref{fig:SG}).
Therefore, we do not need to consider the partial correction terms at higher order than the two-loop level in SPT,
such as $P_{44a}$, $\Gamma_{\rm 1\mathchar`-loop}^{(1)}\Gamma_{\rm 2\mathchar`-loop}^{(1)}$, and
$\left[ \Gamma_{\rm 2\mathchar`-loop}^{(1)} \right]^2$ in  Equation~(\ref{SPT2loop+}).
Moreover, we clearly find that the predictions from Reg PT (blue lines)
lose information on the nonlinear evolution of dark matter compared to the SPT predictions (red lines)
due to the truncation of the $\Gamma$-expansion at second or third order.

\begin{figure}[t]
		\epsscale{0.80}
		\plotone{f2.eps}
		\caption{Ratios of the predicted power spectra $P$ to the smoothed linear power spectrum $P_{\rm nw}$
		are plotted at the redshifts of $z=3.0$ 1.0, and 0.35.
		The predicted power spectra are computed by
		the one-loop SPT (purple solid), the two-loop SPT (red solid), the two-loop Reg PT in Equation~(\ref{RegPT:2loop}) (blue solid),
		the closure theory (orange solid), the $N$-body simulations (green symbols),
		and the modified two-loop SPT (Equation~(\ref{ExSPT:2loop}))
		where the finite truncated orders of the $\Gamma$-expansion go from 1 to 8
		(black dashed: $r_k=1\mathchar`-5$, 7, and 8 from left to right in order; black solid: $r_k=6$).
		Furthermore, the fractional differences, defined as ${\rm Diff[\%]} \equiv (P_{\rm Nbody}-P)*100/P_{\rm Nbody}$
		where $P_{\rm Nbody}$ is the predicted power spectrum by the $N$-body simulations,
		are plotted at each redshift.
		There, the plotted models are the one-loop SPT (purple symbols),
		the two-loop SPT (red symbols), the two-loop Reg PT (blue symbols), the closure theory (orange symbols),
		and the modified SPT truncated at the sixth order of the $\Gamma$-expansion (black symbols).
		}
		\label{fig:power}
\end{figure}

Now, we present the following modified SPT solution at the one- and two-loop level:
\begin{eqnarray}
		P_{\rm SPT,1\mathchar`-loop}(z,k) &=&	 e^{-k^2D^2\sigma_v^2}
        \sum_{r=1}^{r_k} \frac{(k^2D^2\sigma_v^2)^{r-1}}{(r-1)!}
	   \left( D^2 P_{\rm L}(k) + D^{4}\left( P_{13}(k) + k^2 \sigma_v^2 P_{\rm L}(k) \right) \right) \nonumber \\
	   && + e^{-k^2D^2\sigma_v^2}
		\sum_{r=2}^{r_k}\frac{(k^2D^2\sigma_v^2)^{r-2}}{(r-2)!} D^4\left(P_{22}(k) - k^2 \sigma_v^2P_{\rm L}(k)  \right)
		\label{ExSPT:1loop}
\end{eqnarray}
\begin{eqnarray}
		&&P_{\rm SPT,2\mathchar`-loop}(z,k) \nonumber \\
		&&=e^{-k^2D^2\sigma_v^2}  \sum_{r=1}^{r_k}\frac{(k^2D^2\sigma_v^2)^{r-1}}{(r-1)!}
		\Bigg[ D^2 P_{\rm L}(k) + D^4\left( P_{13}(k) + k^2 \sigma_v^2 P_L(k)  \right) \nonumber \\
		&&\hspace{5cm} + D^6\left( P_{33a}(k) + P_{15}(k) +	k^2 \sigma_v^2 P_{13}(k) + \frac{(k^2 \sigma_v^2)^2}{2} P_{\rm L}(k)\right) \Bigg]
		\nonumber \\
		&& + e^{-k^2D^2\sigma_v^2}  \sum_{r=2}^{r_k} \frac{(k^2D^2\sigma_v^2)^{r-2}}{(r-2)!}
		\Bigg[D^4\left( P_{22}(k) - k^2 \sigma_v^2 P_{\rm L}(k)\right) \nonumber \\
		&&\hspace{5cm}
		D^6\left( P_{24}(k) + k^2 \sigma_v^2 P_{22} - k^2 \sigma_v^2 \left( P_{13}(k) + k^2 \sigma_v^2 P_{\rm L}(k) \right)  \right) \Bigg]
		\nonumber \\
		&& + e^{-k^2D^2\sigma_v^2}  \sum_{r=3}^{r_k}\frac{(k^2D^2\sigma_v^2)^{r-3}}{(r-3)!}
		D^6\left[ P_{33b}(k) - k^2 \sigma_v^2 P_{22}(k) + \frac{(k^2 \sigma_v^2)^2}{2} P_{\rm L}(k) \right].
		\label{ExSPT:2loop}
\end{eqnarray}
These expressions are obtained by removing the additional terms at higher orders than the one- and two-loop level in SPT
from Equations~(\ref{EXREG1}) and (\ref{ExtendedRegPT:2loop}), respectively, and
cutting off the $\Gamma$-expansion at a finite order $r_k$.
For $r_k\to\infty$, the above solutions reduce to the usual SPT solutions.
Note that these solutions avoid the divergent behavior of the power spectrum at small scales
by considering the finite order $r_k$ one needs.

In Figure~\ref{fig:power},
we show a comparison between the predicted power spectra from
the modified SPT solution at the two-loop level (Equation~(\ref{ExSPT:2loop}))
with the finite truncated orders of the $\Gamma$-expansion going from $1$ to $8$ (black dashed),
the one-loop SPT (purple solid), the two-loop SPT (red solid),
the two-loop Reg PT (blue solid), the closure theory \citep{Taruya:2009ir} (orange solid), and the $N$-body simulations (green symbols).
First, the predictions of the two-loop SPT agree better with the $N$-body simulations
than those of the two-loop Reg PT and the closure theory at the redshifts ($z=3.0$, 1.0, and 0.35).
Next, the modified SPT solutions truncated at the third order of the $\Gamma$-expansion give almost the same results as the two-loop Reg PT.
As previously mentioned, this is because the additional correction terms in the two-loop Reg PT
have no effective information on the nonlinear evolution of dark matter.
Third, the divergent behavior in the two-loop SPT at small scales is indeed modified by the truncation of the $\Gamma$-expansion
(see the black dashed lines).
We emphasized the sixth-order solutions of the $\Gamma$-expansion in the modified SPT using the black solid lines.
The sixth-order solutions accidentally agree with the $N$-body results better than the two-loop SPT solutions
due to the damping factor from the truncated $\Gamma$-expansion
(see the red and black symbols in the fractional differences in Figure~\ref{fig:power}).
We suggest that this solution should be used instead of the two-loop SPT.
The solution guarantees accuracy comparable to the two-loop SPT and has no divergent behavior at small scales.

\section{Conclusion}
\label{Conclusion}

Based on the idea that any resummation theory should be described and solved through the kernel function in SPT,
we established an approximation of the kernel function (Equation~(\ref{ap_FG})).

This approximation explains the existing Reg PT model, and gives its natural extension (Equation~(\ref{ExtendedRegPT:2loop})).
However, the extended Reg PT is equivalent to SPT with negligibly small correction terms,
and gives almost the same result as for the SPT (see Figure~\ref{fig:SG}).
Therefore, we do not need to consider these additional terms.
Moreover, since the Reg PT solutions (Equations~(\ref{RegPT:1loop}) and (\ref{RegPT:2loop}))
truncate the $\Gamma$-expansion at the second and third order,
they do not contain more effective information on the nonlinear dark matter evolution than the SPT solutions.
In other words, while the original Reg PT solution becomes zero beyond BAO scales due to exponential damping,
the extended Reg PT solution allows us to predict the power spectrum at the high-$k$ region,
because the exponential damping factor is completely canceled out.
In fact, the nonlinear power spectra computed in the two-loop SPT agree better with the $N$-body results
than that of the two-loop Reg PT at redshifts of $z=3.0$, 1.0, and 0.35, especially at high-$k$ region.
Hence, the two-loop SPT is one of the best models to predict the nonlinear matter power spectrum
even at present (see Figure~\ref{fig:power}).

In the spirit of the $\Gamma$-expansion,
each loop corrections are concentrated at a narrow range of scales, and the reliable range of the approximation becomes much more obvious.
Therefore, the Reg PT two-loop solution, which truncates the $\Gamma$-expansion at the third order,
guarantees the validity of the predicted power spectrum at small scale regions.
In this paper, we showed that the Reg PT one- and two-loop solutions are a part of the SPT one- and two-loop solutions plus negligible terms
(Equation~(\ref{SPT2loop+}))
and extract the information at small scale regions in the SPT solutions.
Thereby, the predicted power spectra in the Reg PT model have the same behavior as those in the SPT model at the BAO scales
(see Figure~\ref{fig:power}).
Furthermore, we interpret the truncation of the $\Gamma$-expansion as the loss of the information at small scale regions.
In this sense the Reg PT solution does not have more effective information on the nonlinear power spectrum than the SPT solution.

We presented modified versions of the SPT solutions at
the one- and two-loop level (Equations~(\ref{ExSPT:1loop}) and (\ref{ExSPT:2loop})),
which are the SPT solutions described in the context of the $\Gamma$-expansion.
These solutions avoid the divergent behavior of the SPT solutions at small scales
by the truncation of the $\Gamma$-expansion at the finite order one needs.
We suggest that one should use the modified SPT solution at the sixth order of the $\Gamma$-expansion instead of the two-loop SPT solution,
because this solution guarantees accuracy comparable to the two-loop SPT and has no divergent behavior.

Finally, we shall mention some applications of our result.
So far the correlation function has not been computed from SPT because of the divergent behavior.
However, the modified SPT solution would allow us to do it.
Similarly, we occasionally
encounter the difficulty that we have to compute the integral of the linear or nonlinear power spectrum but the integrand does not converge,
such as the calculation of back reaction from the matter perturbations.
Our result would be useful for this case.
However, we leave these applications as our future work.

\acknowledgments
We thank T. Nishimichi and A. Taruya for providing numerical simulation results and useful comments.
We also thank L. Mercoli for checking this paper.
This work is supported in part by the GCOE Program  ``Weaving Science Web beyond Particle-Matter Hierarchy''
at Tohoku University and by a Grant-in-Aid for Scientific Research from JSPS (No. 24-3849 for N.S.S. and No. 20540245 for T.F.).


\section{Introduction}
 The 2D nature of the cuprate oxide superconductors and the fact
 that their normal state properties do not exhibit leading FL
behavior have lead to the suggestion that the ground state is not a
FL. In a FL the resistivity should follow a $T^2$ dependence as the
temperature $T$ goes to zero in the absence of impurities and it has
 been argued that the absence of this characteristic
FL behavior in several experiments for the cuprate
superconductors\cite{Iye,Takagi,Ando,Timusk} is evidence
that there is a qualitative difference between
the normal state of cuprates and that in the other metallic system.
Anderson\cite{Anderson} has argued that the ground
state of cuprates is
close to the one-dimensional system in which the elementary
excitations are collective modes, spinons and holons, with spin and
charge degrees of freedom decoupled as in the 1D Luttinger liquid.
Varma \emph{et al}.\cite{Varma} have put forward a model in which
the
weight in the quasiparticle pole vanishes logarithmically at
$p=p_f$. This is similar to case of the Luttinger liquid although in
 that case the weight is the quasiparticle poles at the Fermi
surface vanishes as $z_F\sim (p-p_f)^\alpha, 1>\alpha>0$\cite{Voit}.
The stability of the FL ground state has been investigated in
perturbation theory\cite{Engelbrecht,Fukuyama,Serene,Coffey}, by
renormalization group calculations\cite{Shanka} and bosonization of
the fermions\cite{Houghton,Castro}. These approaches have not
revealed any sign of FL breakdown other than due to the
instabilities of FL ground state familiar from three-dimensional
(3D) case : BCS, CDW or SDW. Metzner \emph{et al.} have reviewed
work on the role
of strong forward scattering in determining the ground state of Fermi
systems in different dimensions.\cite{Metzner}
As they point out for short interactions or the Coulomb interaction
the properties of Fermi systems in dimensions greater than one
are those of usual Fermi liquid.
 More recently Castellani \emph{et
al}.\cite{Castellani} have
suggested that the anomalous normal state properties of the cuprates
 are due to quantum critical points associated with
antiferromagnetism and charge density wave instabilities in
different doping ranges. In the region of a quantum critical point
temperature provides the only energy scale which it is argued is
consistent with data on the cuprates.

 An alternative explanation is that, although the ground state
develops from FL at $T=0$, the characteristic FL temperature
dependence is restricted to low temperatures compared to the high
superconducting transition temperature, $T_c$, or compared to small
energy scales given by large variations in the density of states
due to bandstructure effects such as a Van Hove
singularity\cite{Aristov}.

 The energy and temperature dependent characteristics of a FL arise
from quasiparticle interactions. Corrections
to these dependences also
 arise from the quasiparticle interactions themselves at energy and
temperature scales which depend on the nature of interactions. The
corrections limit the characteristic FL dependences to the vicinity
of the Fermi surface and to low temperature even in the absence of
bandstructure and other effects. We investigate the
corrections to these leading FL energy and temperature dependences
using a parabolic band in order to isolate
the effects of bandstructure. It is useful to
investigate the leading corrections for the FL with a simplest
model where non-essential complications are absent. Here we
calculate $\Sigma'(p,E)$ and $\Sigma''(p,E)$ for the paramagnon
model\cite{Doniach} and the electron gas. The paramagnon model
describes the physical system close to
a ferromagnetic instability where the self-energy contribution comes
 from incoherent long-wavelength spin fluctuations through particle-
hole($ph$) channel. The closer the system is to the instability the
stronger are the leading corrections to FL behavior
since they come from the
long wavelength limit of the effective interaction. This makes the
paramagnon model ideal for investigating corrections and the regions
of energies and temperatures over which they can characterize
calculated quantities.
Since the 2D electron liquid has recently been used to investigate
the corrections to FL behavior, we also calculate these corrections
for the Coulomb interaction. In these calculations $\Sigma(p,E)$ is
approximated by the $ph$ channel contribution.
 Unlike 3D this channel remains
important at low densities in 2D so that our results can also be
applied to low densities. The difference between 2D and 3D
is that the density of states for a parabolic band
at the Fermi surface is independent of
density in 2D. Consequently  repeated scattering in the $ph$
channel remains important at low density in 2D\cite{Kim}.

 In the section II we introduce the paramagnon model and in the
section III discuss the results for the single-particle self-energy
and thermodynamic properties.  In the section IV we
discuss the same kind of corrections for the 2D electron gas. We
give our conclusions in section V.

\section{Model}

 To calculate the leading corrections to the single-particle self-
energy  we first use a short range interaction between fermions.
The Hamiltonian is
\begin{equation}
H=\sum_{{\mathbf p}, \sigma} \xi_{{\mathbf p}}
c^\dagger_{{\mathbf p},\sigma}c_{{\mathbf p},\sigma}+
\sum_{{\mathbf p},{\mathbf q},\sigma, \sigma\prime}I({\mathbf q})
 c^\dagger_{{\mathbf p},\sigma}c_{{\mathbf p} ',\sigma\prime}
c^\dagger_{{\mathbf p} ' -{\mathbf q},\sigma\prime}c_{{\mathbf p}
 +{\mathbf q},\sigma}
\end{equation}
where $\xi_p=(p^2 -p_f^2)/2m$, $m$ is the mass of the fermions, $p_f$
 is the Fermi momentum and $I$ is the
strength of the interaction with a cutoff, $q_c$. This model,
called the paramagnon model, was used by Engelsberg \emph{et al}.
\cite{Doniach} to calculate the corrections to the linear
temperature dependences in specific heat of normal liquid $^3He$.
 Within this model the leading corrections to FL behavior come from
low energy long wavelength paramagnons.
The importance of paramagnons
in liquid $^3He$
was suggested by the enhancement of the observed static paramagnetic
 susceptibility and were shown to provide an explanation of the size
of corrections to the linear temperature dependence of the specific
heat. The single-particle self-energy is given by
the repeated scattering of $ph$ pairs which leads to an effective
interaction, $V^{eff}(\bf q, \omega)$.
\begin{equation}
\Sigma({\mathbf p}, \imath E_n)=-T\sum_{{\mathbf q},
\omega_l}G({\mathbf p} - {\mathbf q}, \imath E_n -\imath \omega_l)
{V}^{eff}({\mathbf q}, \omega_l)
\end{equation}
where $G({\mathbf p}, \imath E_n)$ is the unperturbed temperature
Green function and $\omega_l$ is Bose Matsubara frequencies.
$V^{eff}(\bf q,\omega)$ has two independent channels, the symmetric
(s) and the
antisymmetric (a) channels corresponding to spin exchanges of 0 or 1
 and is given by
\begin{equation}
{V}^{eff}({{\mathbf q}},\omega)={1\over 2}{V^2_s\chi
({\mathbf q}, \omega)\over {1-V_s\chi ({\mathbf q}, \omega)}}+
{3\over 2}{V^2_a\chi
({\mathbf q}, \omega)\over {1-V_a\chi ({\mathbf q}, \omega)}}
\end{equation}
where $V_s=I$ and $V_a=-I$. $\chi({\mathbf q},\omega)$ is the
polarization function for a 2D parabolic band\cite{Stern}.
 At $T=0$ the self-energy contributions
from the real and imaginary part on shell ($E=\xi_p$) is
\begin{eqnarray}
\Sigma({\mathbf p},\xi_{\mathbf p})&=& \sum_{\mathbf q}[\Theta(\xi_{
\mathbf p}-\xi_{{\mathbf p}-{\mathbf q}})-\Theta(-\xi_{{\mathbf p}-
{\mathbf q}})]V^{eff}({\mathbf q},\xi_{{\mathbf p}}-\xi_{{\mathbf p}
-{\mathbf q}})
\\ \nonumber \cr
&+&\int^{\infty}_{-\infty}{d\eta \over {2\pi}}\sum_{\mathbf q}
G({\mathbf p}-{\mathbf q},\xi_{\bf p} -\imath \eta)V^{eff}
({\mathbf q},\imath \eta)
\end{eqnarray}
with $\chi({\mathbf q},\omega)=\chi'({\mathbf q},\omega)+\imath\chi''
({\mathbf q},\omega)$ in $V_{eff}({\mathbf q},\omega)$.
 The leading corrections in $\Sigma(p,E)$ of interest here comes from
the  long-wavelength limit and are contained in the first term since
 the second term, the line integral along the imaginary frequency
axis, $\omega=\imath \eta$, does not contribute to the energy
dependence in the long wavelength limit.  The line integral in the
equation (4) does not contribute to the $\Sigma''(p,E)$ since the
imaginary part of the integrand is odd in $\eta$ and
gives contributions to $\Sigma'(p,\xi_p)$ which are proportional to
$q_c^2$.

\section{Results}
\subsection{Self-Energy At Zero Temperature}

 The leading dependence on $\xi_p$ in
the imaginary part of self-energy in
$\Sigma({\bf p},E)=\Sigma'({\bf p},E)+
\imath\Sigma''({\bf p},E)$ on shell is
\begin{equation}
\Sigma''({\bf p},\xi_{{\bf p}})=\beta
\xi^2_{\bf p} \ln |{\xi_{{\bf p}} \over \xi _{0} }|
+O(\xi_{\bf p}^4)
\end{equation}
where $\beta$ is independent of the direction for a parabolic band
and given by
\begin{eqnarray}
\beta&=&-\int^1_0 {d\mu \over \sqrt{1-\mu^2}}
({1 \over 2}{A^3_s \over [(1-\mu^2)+(A_s\mu)^2]}+
{3 \over 2}{A^3_a \over [(1-\mu^2)+(A_a\mu)^2]})
\\ \nonumber \cr
&\approx&-{(1+{\bar{I}}+{\bar{I}}^2)\over {(1-{\bar{I}}^2)^2}}
{{\bar{I}}^2 \over{4 E_f}},
\end{eqnarray}
$A_s={\bar I \over 1+\bar I}$, $A_a={\bar I \over 1-\bar I}$ and
$\bar I=N(0)I$ where $N(0)=m/2\pi\hbar$ is the density of states at
the Fermi surface for a 2D parabolic band for up or down spins.
 The real part of the self-energy for a parabolic band is
\begin{equation}
\Sigma'({\bf p},\xi_{\bf p})=-\alpha \xi_{{\bf p}}
-{\pi \over 2}\beta \xi_{{\bf p}}|\xi_{\bf p}|
-\gamma |{\xi}_{\bf p}|^{5/2}+O(\xi_p^3)
\end{equation}
where
for the paramagnon model,
\begin{mathletters}
\begin{equation}
\alpha \approx {(2+\bar{I}) \over {(1-{\bar{I}}^2})}
{{\bar{I}}^2 min[q_{c},2p_f] \over {\pi p_f}}
\end{equation}
\begin{equation}
\gamma \approx {\bar{I} \over {4\pi E^{3/2}_f}}
\end{equation}
\end{mathletters}
where $E_f=p_f^2/2m$  and $\xi_0 \propto p_f$ for $q_c>2p_f$
and $\xi_0 \propto q_c$ otherwise \cite{Kim}. The logarithmic
behavior is restricted to $|\xi_p|<\xi_0$.
 In the RPA approximation the largest contribution comes from
the antisymmetric channel since the contribution of the channel is
enhanced by a $1/(1-\bar I)^2$ factor in the case of a repulsive
interaction. The generic FL behavior in $\Sigma''(p,\xi_p)$ is shown
 in the figure\ \ref{seon} with a log-fit. The figure\ \ref{selnon}
shows an interaction $\bar{I}$ dependence in the logarithmic scale
of $\xi_p$ and $q_c$ dependence. The cut-off energy $\xi_0$ depends
only on $q_c$ and the slope $\beta$ depends on $\bar I$. The q-sum
in the equation (4) has a limit of $min[q_c,2p_f]$, so that the
log-fit for $q_c=3.0 p_f$ and $5.0 p_f$ has the same cut off in the
log dependence. The log fit is good to within 10 percent up to
$\xi_p=0.2E_f$ which, however, depends on $q_c$. The $\xi_p|\xi_p|$
term in $\Sigma'(p,\xi_p)$ is related to the
$\xi_p^2 \ln |\xi_p|$ term in the imaginary part through the
Kramers-Kronig relation\cite{Coffey} and also depends only on long
wavelength properties. In the figure\ \ref{ser2} the
${\xi_p}|{\xi_p}|$ term has been isolated in the logarithmic scale,
so that the slope 2 indicates ${\xi_p}^2$ in
$\ln|\Sigma'+\alpha {\xi_p}|$. This contribution comes from the
dependence of $V^{eff}(q,\omega)$ on the variable
$s=\omega/q v_f$\cite{footnote}
where $s$ is the variable which
appears in $\chi(\bf q,\omega)$ in the long-wavelength limit
\begin{equation}
\lim_{{\bf q} \rightarrow 0}\chi(q,\omega)
=\chi(s)=N(0)[-1+{s \over \sqrt{s^2-1}} \Theta(|s|-1)
+\imath {s \over \sqrt{1-s^2}} \Theta(1-|s|)].
\end{equation}
 As in the
$\Sigma''(p,\xi_p)$, the coefficient is independent of $q_c$ and
the log fit survives up to $\xi_p \approx \xi_0$. The term is present
 both for $\xi_p>0$ and $\xi_p<0$. The intercept point of vertical
axis shows the coefficient ${\pi \over 2}\beta$ and the curve for
$\xi_p <0$ ends at $\xi=-E_f$ in $\ln|\xi_p|$.
 The $|\xi_p|^{5/2}$ term is the leading zero sound contribution.
This term has a corresponding $(\xi-\xi_{th})^{3/2}$
term in $\Sigma''(p,\xi_p)$\cite{Kim}, where $\xi_{th}=(v_{zs}-v_f)$,
 determined by the velocity of the zero-sound mode, $v_{zs}=[
(1+\bar I)/\sqrt{1+2\bar I}]v_f$.

 In 3D the leading dependence in $\Sigma''(p,\xi_p)$ is $\xi_p^2$
and the leading correction to this , $|\xi_p|^3$,
corresponds to the term
$\xi_p^2 \ln \xi_p$ in 2D. The $|\xi_p|^3$ depends
only on long wavelength behavior\cite{Baym} and has a corresponding
$\xi_p^3 \ln |{\xi_p \over \xi_0}|$ term in $\Sigma'(p,\xi_p)$.
 The cut-off in the log, $\xi_0$, depends on both $\bar I=N(0)I$ and
 $q_c$ because of the stronger $q$ dependence in real part of
$\chi^{3D}(q,\omega)$. The magnitude
of these $\xi_p^2 \ln \xi_p$ and $\xi_p |\xi_p|$ terms in 2D and the
$|\xi_p^3|$ and $\xi_p^3 \ln \xi_p$ terms in 3D are enhanced by
repeated scattering of $ph$ pairs. The strength of the repeated
scattering depends on the coupling constant,
$\bar I$, which is independent of $p_f$ in 2D but goes to zero
 in 3D as $p_f \rightarrow 0$. As a result unlike the case in 3D,
where the $ph$ channel is not important for low densities, the
enhancement is independent of density in 2D and the results derived
here for 2D remain the leading corrections even in low densities.
 However both in 2D and 3D $\Sigma(p,E) \rightarrow 0$ as the phase
space for holes vanishes. The leading correction term of 2D FL is
limited to a small energy region which vanishes as $p_f$ decreases
in the low density limit.

 The dependences of $\Sigma(p,E)$ on the variables, $p$ and $E$,
are very different. For instance $\Sigma''(p,E)$ has a
$E^2\ln {|E| \over E_\pm}$ behavior only for
$p=p_f$, where $E_+=E_-=\xi_0$ are
cutoffs for $E>0$ and $E<0$ respectively. In the
figure\ \ref{seioff} and the figure\ \ref{seroff} we
show $\Sigma'(p,E)$ and $\Sigma''(p,E)$
vs. E for $p={p_f}$ and ${1.1p_f}$. As $p$ goes away from $p_f$, the
structure of the self-energy vanishes inside the Fermi surface
(for $p<p_f$ the structure vanishes outside the surface) due to
the step function in the zero temperature self-energy expression in
the equation (4) and so there is a threshold in the self-energy for
$|E|<|\xi_{{\bf p}-{\bf q}}|_{min}$. The sharp threshold effect is absent for
interaction without a cutoff, $q_c$. However if $V^{eff}(q,\omega)$
falls off with increasing $q$ then there is an effective $q_c$ and
associated with that there
is an effective cutoff in $E$ beyond which $\Sigma(p,E)$ is reduced in
magnitude. The most direct probe of the $E$ dependence
 of $\Sigma(p,E)$ is in angle resolved photoemission spectroscopy in
which the spectral density is measured. The quasi-particle peak in
the data depends on the $E$ dependence of $\Sigma(p,E)$ and has been
analyzed for the cuprate superconductors in both the normal and
superconducting state\cite{Shen1}. It has been found that the
linewidth of quasi-particle peak does not vary as $E^2$ which has
been pointed as not being the FL dependence\cite{Shen2}.
The strong $p$ and $E$ dependence of $\Sigma(p,E)$ found here
for a simple parabolic band shows that deviations from a simple
$E^2$ dependence are to be expected. This is especially the case
for cuprates with strong bandstructure effects and quasi-particle
interactions giving rise to low energy spin
fluctuations\cite{Kouznetsov}.

\subsection{Self-Energy at Finite Temperature}

 At finite temperature $\Sigma''(p,\xi_p)$ is
\begin{equation}
\Sigma''({\bf p},\xi_{\bf p})=
-\sum_{\bf q}[f(-\xi_{{\bf p}-{\bf q}})
+n(\xi_{\bf p}-\xi_{{\bf p}-{\bf q}})]
Im V^{eff}({\bf q},\xi_{\bf p}-\xi_{{\bf p}-{\bf q}})
\end{equation}
where $n$ and $f$ are Bose and Fermi distribution functions
respectively.
 The temperature dependence of
$\Sigma''(p_f,\xi_{p_f})$ is plotted for
different values of $q_c$ in the
figure\ \ref{tselnt}. $\Sigma''(p,\xi_p)$ calculated with
$\chi(q,\omega)$ evaluated at zero temperature is the solid lines
and with the temperature dependent $\chi''(q,\omega,T)$ are the
circles. As can be seen from figure\ \ref{tselnt}, the temperature
dependence in the $\chi''(q,\omega,T)$ does not affect the cut-off
$T_0$ since the $\chi(q,\omega,T)$ does not vary much respect to $T$
 for $T \ll E_f$ in the long wave length limit.  The leading
dependence of $\Sigma''(p,\xi_p)$ on $p$ and $T$ is given by
\begin{equation}
\Sigma''(p,\xi_p)=\beta(\xi_p^2+\pi^2 T^2) \ln(max[\xi_p,T]/T_0)
\end{equation}
which is shown on the figure\ \ref{tseln}.
The zero temperature and finite temperature cutoffs, $\xi_0$ and
$T_0$, are shown in figure\ \ref{cutoff}. They have similar values
and increase for $q_c \leq 2p_f$ and are independent of $q_c$ for
larger values of $q_c$. Although the self-energy at finite
temperature does not have explicit step functions as in zero
temperature case, the finite temperature cutoffs are independent of
$q_c \geq 2p_f$ since the log behavior comes from low temperatures.
 Since both $\xi_0$ and $T_0$ are proportional to $p_f$ at low
densities the leading FL behavior in the temperature case is also
limited to a region which depends on the density of the system.

 As $T$ increases beyond the degenerate temperature region, the
Bose distribution term dominates the contribution from the Fermi
Dirac distribution term in the equation (10). In the high
temperature limit $\Sigma''(p_f,\xi_{p_f})$ is approximated by
\begin{equation}
\Sigma''(p_f,\xi_{p_f}) \approx -\beta {8 min[q_c,2p_f] \over p_f}T
+O({1 \over T})
\end{equation}
 where the linear term in T comes from the Bose contribution.
 $\Sigma''(p_f,\xi_{p_f})$ has been graphed in two ways,
$\Sigma''(p_f,\xi_{p_f})/T^2$ vs. $\ln T$ and
$\ln[\Sigma''(p_f,\xi_{p_f})]$
vs. $\ln T$ in the figure\ \ref{tstl} for $q_c=0.1p_f$ and
$q_c=1.0p_f$.
 The dashed lines are the logarithmic and linear
temperature dependence determined from equations (11) and (12).
 The size of the crossover region from the low
temperature limit to the high temperature limit is small as can be
seen in the figure and is from $T=0.02E_f/k_B$ to $T=0.03E_f/k_B$.
 This is estimated by considering 10 percent
deviation from the fits. Although the linear $T$ dependence
comes from the high temperature limit, as soon as the temperature
dependence behavior leaves the $T^2\ln T$ behavior it tends to reach
 the linear temperature dependence rapidly. Since $T_0$ is
proportional to density the
leading FL behavior survives in a low temperature region which
shrinks with decreasing density and the linear high temperature
behavior is followed at lower and lower temperatures. The size of
 crossover also decreases with density. The same behavior is seen in
3D with similar crossover properties. Consequently this behavior is
 not determined by band-structure or low
dimensionality and it may be misleading to refer to it as
Luttinger-like\cite{Aristov}.

\subsection{Thermodynamic Properties}

 The contribution from quasiparticle interactions to the
thermodynamic potential, $\Delta \Omega$,
can be calculated by linked cluster expansion\cite{Mahan}. After
analytic continuation to the real $\omega$ axis $\Delta \Omega$ can
be split up into
$\Delta \Omega_{qp}$ from a quasiparticle contribution and
$\Delta \Omega_{cm}$ from a collective modes;
\begin{eqnarray}
\Delta \Omega(T)&=&-T\sum_{\imath \omega_n} \sum_{\bf q} \int^1_0
{d \eta \over \eta} \chi(\eta,q,\imath \omega_n)V^{eff}
(\eta,q,\imath \omega_n)
\\ \nonumber \cr
&=&\Delta \Omega_{qp}+\Delta \Omega_{cm}.
\end{eqnarray}
First, $\Delta \Omega_{qp}$ is
\begin{equation}
\Delta \Omega_{qp}(T)=\sum_{|\bf q|<q_c} \int^\infty_0
{d\omega \over \pi}
n(\omega)[F({\bf q},\omega)+2I^2\chi''({\bf q},\omega)],
\end{equation}
where
\begin{equation}
F(q,\omega)=\sum_{\lambda=s,a} \nu_\lambda
tan^{-1}[{-V_\lambda \chi''(q,\omega) \over 1-V_\lambda \chi'
(q, \omega)}].
\end{equation}
 By taking a derivative $\Delta \Omega_{qp}$ respect to $T$, the
shift in the entropy, $\Delta S_{qp}$, is
\begin{eqnarray}
\Delta S_{qp}(T)=&-&\sum_{|\bf q|<q_c}\int^\infty_0{\partial
n(\omega)
\over \partial T} [F({\bf q},\omega)+2I^2\chi''(\bf q,\omega)]
\\ \nonumber \cr
&-&\sum_{|\bf q|<q_c}\int^\infty_0 {d\omega \over 2\pi}
n(\omega)[{\partial F({\bf q},\omega) \over \partial T}
+2I^2{\partial \chi''({\bf q},\omega) \over \partial T}]
\end{eqnarray}
 The second term is negligible since, as shown in the previous
section, the
temperature dependence in $\chi(\bf q,\omega)$ is weak in the
long-wavelength limit. The temperature dependence in entropy mainly
comes from the first term. The entropy from quasiparticle
contribution is
\begin{equation}
\Delta S_{qp}(T)=\Gamma_1 T +\Gamma_2 T^2 +O(T^3),
\end{equation}
\begin{mathletters}
\begin{equation}
\Gamma_1={\pi \over 6T_f}(A_s+A_a) {q_c \over p_f}
\end{equation}
\begin{equation}
\Gamma_2=-{6 \zeta (3)n \over \pi T^2_F} \beta
\end{equation}
\end{mathletters}
where $n$ is density of particles and $\zeta$ is the Riemann zeta
function\cite{Coffey}.
 The figure\ \ref{entln} shows $T^2$ dependence for $q_c=0.1p_f$
and $q_c=p_f$. The $q_c$ dependent linear $T$ term has
been subtracted. As in the figure\ \ref{ser2}, the slope of the term
does not depend on $q_c$. This term can also be calculated by
including $\Sigma'(p,\xi_p)$ in the quasiparticle spectrum and using
 the expression for the entropy of non-interacting gas. The
quasiparticle spectrum has the $\xi_p|\xi_p|$ term which is
directly connected to $T^2$ term in the entropy\cite{Coffey}.
 This $T^2$ describes the corrections to the $\Gamma_1 T$ dependence
 in $\Delta S_{qp}$ to better than 10 percent up to a temperature
$T_S$ for entropy in the figure\ \ref{entln} which follows the same
trend as $T_0(\sim \xi_0)$ in the figure\ \ref{cutoff} but
 is smaller.

 The collective mode contribution, $\Delta \Omega_{cm}(T)$, is given
 by the zero-sound pole in the
symmetric channel in $V^{eff}({\bf q},\omega)$. Taking derivative
respect to $T$, one has a $T^2$ dependence in $\Delta S_{cm}$. In
the figure\ \ref{entln} $\Delta S_{cm}$ is compared with
$\Delta S_{qp}$.  The contribution to $T^2$ from the collective mode
 is about a percent of the quasi-particle contribution.

 The calculations discussed so far using the paramagnon model have
shown that FL behavior is limited to regions of low energy and
temperature which shrink with decreasing density. The cutoffs,
$\xi_0$ and $T_0$, are determined by the finite momentum dependence
of the quasiparticle interaction. We now turn to the 2D electron gas
 which has recently been of experimental and theoretical interest.

\section{2D Electron Gas}

 Murphy \emph{et al}.\cite{Murphy} have compared their data on the
tunneling conductance in 2D quantum well systems to the expression
for the single particle life time expression derived by Giuliani and
Quinn\cite{Giuliani} using long wavelength approximation and
found quantitative agreement if they multiplied this expression by
6.3. Jungwirth \emph{et al}.\cite{Jungwirth} have investigated the
same calculation with an effective interaction suggested by
MacDonald \emph{et al}.\cite{MacDonald} between fermions with a
static long wavelength limit in $V^{eff}$ and they obtained good
fits to the data. We calculate the imaginary part of self-energy for
 the quasi-particles in n-type doped GaAs using the RPA.

 Instead of applying static approximation calculated in RPA with
local corrections in $V^{eff}(q,\omega)$, we
 use the frequency dependent effective interaction,
\begin{equation}
V_q^{eff}({\bf q},\omega)={I_q^2 \chi({\bf q},\omega)
\over 1-I_q \chi({\bf q},\omega)}
\end{equation}
where $I_q=2\pi e^2/\epsilon_s q$ or $\bar I_q=2\pi e^2 N(0)
/\epsilon_s q=0.02$\AA$^{-1}/q$\cite{Murphy}($N(0)$ is density
 of states at Fermi level). For GaAs the static dielectric constant,
 $\epsilon_s$, is about 10 and $m^*/m=0.067$. The only free
 parameter is $p_f$. As pointed out in the discussion of the
contact interaction the coefficient of the
$(\xi_p^2+\pi^2 T^2)\ln(max[\xi_p,T]/T_0)$ dependence in
$\Sigma''(p,\xi_p)$ is determined
by the long wavelength properties of $V^{eff}(q,\omega)$.

 There are two contributions to $\Sigma(p,E)$. One comes from the
interaction of the quasi-particles with incoherent \emph{ph} pairs.
This contribution gives the leading FL corrections and was discussed
 in detail above for the contact interaction. The second contribution
 comes from the interaction of quasi-particles with the plasmon mode.
 Although  it does not contribute to the leading $p$
and $E$ dependence of $\Sigma''(p,E)$, it is comparable in
magnitude to that from the incoherent \emph{ph} pairs
at finite $E$. These contributions and
$\Sigma''(p,E)$ are plotted in figure\ \ref{plse} for $p=p_f$ and
$p=1.1p_f$ at $T=0$. $\Sigma''(p_f,E)$ is symmetric and leads to a
spectral density which is symmetric in the figure\ \ref{specq}. For
$p \ne p_f$ $\Sigma''(p,E)$ is no longer symmetric in $E$ as was
pointed out in the discussion of the contact interaction.

 Using the parameters for quasi-particles in n-type doped GaAs
quantum
 wells temperature dependence of $\Sigma''(p,E=\xi_p)$ is shown in
figure\ \ref{qseit} for $n=1.6$x$10^{-11}cm^{-2}$ which gives
$p_f=0.01$\AA$^{-1}$ and $E_f={p_f^2 \over 2m^*}=5.7 meV$.
 As in the contact interaction case there is a cross
over from $T^2\ln T$ to linear $T$ behavior which depends on
density. $T^2\ln T$ dependence is followed for $T_1<8 K$ and $T$
dependence for $T_2 > 156 K$ for $p_f=0.01$\AA$^{-1}$ system.
These values of $T_1$
and $T_2$ were determined by requiring the $\Sigma''(p_f,\xi_{p_f})$
 to fit the $T^2\ln T$ and $T$ dependences to 10 percent. The
$T^2\ln T$ dependence in the
Coulomb interaction is also limited by low density limit as in
contact interaction case. The cross over region ($T_1$ to $T_2$) has
 been estimated for different densities and is shown in table 1. The
 size of this region decreases with decreasing density system as in
the contact interaction case but is larger.

 Quasiparticle interactions of an isotropic electron gas do not
contribute to the transport relaxation time\cite{Mahan}. However the
results of the calculations discussed here suggest that it is
important to take into account the $p$ and $E$ dependences of
$\Sigma''(p,E)$ and the plasmon contribution when comparing
calculations of single particle properties in the 2D electron gas
with experiment data.

\section{Conclusion and Summary}

We calculated the corrections to FL behavior in the single particle
self-energies and in entropy using a contact and Coulomb
interaction. We demonstrated that these corrections take the forms,
 $\xi_p|\xi_p|$ in $\Sigma'(p,\xi_p)$ and $\xi_p^2 \ln|\xi_p|$ in
$\Sigma''(p,\xi_p)$, in a region close to the Fermi
surface which shrinks with decreasing density and increasing
temperature. In particular we have shown that at finite temperatures
 $\Sigma''(p_f,\xi_{p_f})$ rapidly goes over to
a linear $T$ dependence
even for a parabolic band. This type of linear
temperature leads to a
resistivity $\propto T$ which has been taken as evidence against a
FL ground state in the cuprate superconductors. Our results suggest
that an alternative explanation may be that the temperature below
which FL behavior is seen is below the superconducting transition
temperature. This has also been suggested by Aristov \emph{et al.
}\cite{Aristov} on the basis of band-structure effects in 2D Hubbard
Hamiltonian.

 This point has been addressed by Boebinger
\emph{et al}.\cite{Boebinger} who have applied magnetic fields to
suppress superconductivity in transport measurements on the cuprates
 at low temperatures. It is claimed that this  reveals the
normal state. They find that the non-FL behaviors persists. However
recent work by Chan \emph{et al}.\cite{Chan} suggests that single
particle properties of 2D electron systems are very sensitive to
modest applied magnetic fields. They found that a pseudogap appears
in the density of states in applied fields. A similar resistivity
was also found in transport properties of 2D electron system in
semiconductor hetero-structures by Simonian
\emph{et al.}\cite{Simonian}.
We will address the
influence of magnetic fields on 2D FL in a forthcoming paper.

This work was supported by the New York State Institute for
Superconductvity(NYSIS).