text:
\section{Introduction}
The Higgs boson discovered at the LHC~\cite{Aad:2012tfa,Chatrchyan:2012ufa} beautifully fits into the Standard Model (SM) predictions so far~\cite{CMS:yva}. The determination of its mass~\cite{Agashe:2014kda}
\al{
M_H
	&=	125.7\pm0.4\ensuremath{\,\text{GeV} }
}
completes the list of the SM parameters, among which the ones in the Higgs potential,
\al{
V	&=	m^2\ab{H}^2+\lambda\ab{H}^4,
}
have turned out to be $m^2\sim-\paren{90\ensuremath{\,\text{GeV} }}^2$ and $\lambda\simeq0.13$, depending on the precise values of the top and Higgs masses; see e.g.\ Ref.~\cite{Buttazzo:2013uya}.

We have not seen any hint of a new physics beyond the SM at the LHC, and it is important to guess at what scale it appears, as we know for sure that it must be somewhere in order to account for {\color{black} the tiny neutrino masses, dark matter, baryogenesis, inflation, etc.} In this work, we assume that the Higgs sector is not altered up to a very high scale,\footnote{
See e.g.\ Refs.~\cite{Haba:2013lga,Haba:2014zda,Hamada:2014xka,Haba:2014zja,Haba:2014sia,Bhattacharya:2014gva,Kawana:2014zxa} for a possible minimal extension of the SM with {\color{black} the dark matter and right-handed neutrinos}.
}
in accordance with the following indications:
The renormalization group (RG) running of the quartic coupling $\lambda$ revealed that it takes the minimum value at around the Planck scale $\sim10^{18}\ensuremath{\,\text{GeV} }$ and that the minimum value can be zero depending on the precise value of the top quark mass~\cite{Holthausen:2011aa,Bezrukov:2012sa,Degrassi:2012ry,Alekhin:2012py,Masina:2012tz,Hamada:2012bp,Jegerlehner:2013cta,Jegerlehner:2013nna,Hamada:2013cta,Buttazzo:2013uya,Branchina:2013jra,Kobakhidze:2014xda,Spencer-Smith:2014woa,Branchina:2014usa,Branchina:2014rva,Jegerlehner:2014xxa}.
We have also found that the bare Higgs mass can vanish at the Planck scale as well~\cite{Hamada:2012bp,Jegerlehner:2013cta,Jegerlehner:2013nna,Bian:2013xra,Hamada:2013cta,Masina:2013wja,Alsarhi:1991ji,Jones:2013aua}.\footnote{
See also Refs.~\cite{Jegerlehner:2014mua,Cherchiglia:2014gna,Bai:2014lea} for discussion of quadratic divergences.
}
That is, the Veltman condition~\cite{Veltman:1980mj} can be met at the Planck scale. In fact he speculates, {\it``This mass-relation, implying a certain cancellation between bosonic and fermionic effects, would in this view be due to an underlying supersymmetry.''} To summarize, it turned out that there is a triple coincidence: $\lambda$, its running, and the bare Higgs mass can all be accidentally small at around the Planck scale.

This is a direct hint for Planck scale physics in the context of  superstring theory.
The vanishing bare Higgs mass implies that the supersymmetry is restored at the Planck scale and that the Higgs field resides in a massless string state.
{\color{black} The} smallness of {\color{black} both} $\lambda$ and its beta function
{\color{black} is consistent with the Higgs potential being very flat around the string scale; see Fig.~\ref{phenomenological Higgs potential}.}\footnote{
\color{black} This is indeed suggested
by the multiple point criticality principle (MPP)~\cite{Froggatt:1995rt,Froggatt:2001pa,Nielsen:2012pu}, the classical conformality~\cite{Meissner:2007xv,Foot:2007iy,Meissner:2007xv,Iso:2009ss,Iso:2009nw,Hur:2011sv,Iso:2012jn,Chankowski:2014fva,Kobakhidze:2014afa,Gorsky:2014una,Kubo:2014ova,Foot:2014ifa,Kawana:2015tka}, the asymptotic safety~\cite{Shaposhnikov:2009pv}, the hidden duality and symmetry~\cite{Kawamura:2013kua,Kawamura:2013xwa}, and the maximum entropy principle~\cite{Kawai:2011qb,Kawai:2013wwa,Hamada:2014ofa,Kawana:2014vra,Hamada:2014xra}.
}
Such a flat potential opens up the possibility that the Higgs field plays the role of inflaton {\color{black} in the} early universe~\cite{Bezrukov:2007ep,Hamada:2013mya,Hamada:2014iga,Bezrukov:2014bra,Ko:2014eia,Xianyu:2014eba,Hamada:2014wna,Ibanez:2014swa,Bezrukov:2014ipa}.\footnote{
There are different models of the Higgs inflation involving higher dimensional operators~\cite{Germani:2010gm,Kamada:2010qe,Kamada:2012se,Nakayama:2014koa,Lee:2014spa}.
}
\begin{figure}[t]
\begin{center}
\hfill
\includegraphics[width=.5\textwidth]{phenomenological_potential.pdf}
\hfill\mbox{}
\caption{
{\color{black}
The SM Higgs potential $V$ as a function of the Higgs field $\varphi$.
Here we take $M_H=125\ensuremath{\,\text{GeV} }$ and tune the top mass in such a way that the potential becomes flat; see e.g.\ Ref.~\cite{Hamada:2014wna}.
}
}\label{phenomenological Higgs potential}
\end{center}
\end{figure}
{\color{black}
To understand the whole structure of the potential, it is crucial to investigate its behavior beyond the Planck scale.
The calculation based on field theory cannot be trusted in this region.
Although it is hard to reproduce the SM completely as a low energy effective theory of superstring, we can explore generic trans-Planckian structure of the Higgs field, under the assumption that the SM is close to a non-supersymmetric perturbative vacuum of superstring theory.
}

{\color{black} In four dimensions, string theory has {\color{black} many more} tachyon-free non-super\-sym\-metric vacua than the supersymmetric ones.
The latest LHC results suggest the possibility of the absence of the low energy supersymmetry, and the research based on the non-supersymmetric vacua is becoming more and more important~\cite{Blaszczyk:2014qoa,Angelantonj:2014dia,Hamada:2014eia,Nibbelink:2015ena,Abel:2015oxa}.}

{\color{black}
In such non-supersymmetric vacua,}
{\color{black} almost all the moduli are lifted up perturbatively,
contrary to the supersymmetric ones which typically possess tens or even hundreds of flat directions that cannot be raised purturbatively.}
{\color{black}
However, there remains a problem of instability in the non-supersymmetric models: The perturbative corrections generate tadpoles for the dilaton and other moduli such as the radii of toroidal compactifications.}
{\color{black}
The dilaton can be stabilized within the perturbation series when $g_s\sim 1$~\cite{Dine:1985he}, or else by the balance between the one-loop and the non-perturbative potentials when $g_s$ is small~\cite{Abel:2015oxa}. In this paper, we assume that the dilaton is already stabilized.
}{\color{black}
We will discuss other instabilities than the dilaton direction in Sections~2 and 3.
}

We start from the tachyon-free non-supersymmetric vacua of the heterotic string theory.
We assume that the Higgs comes from a closed string and that its emission vertex at the zero momentum can be decomposed into a product of operators whose conformal dimensions are $(1,0)$ and $(0,1)$.
This is realized in the following cases for example:
\begin{itemize}
\item The Higgs comes from an extra dimensional component of a gauge field~\cite{Manton:1979kb,Fairlie:1979at,Salam:1981xd,RandjbarDaemi:1982hi,Hosotani:1983xw,Hosotani:1983vn,Hosotani:1988bm}.
\item The Higgs is the only one doublet in generic fermionic constructions~\cite{Kawai:1986va,Kawai:1986ah,Lerche:1986cx,Antoniadis:1986rn}.
\item The Higgs comes from an untwisted sector in the orbifold construction~\cite{Dixon:1985jw,Dixon:1986jc}; see e.g.\ Ref.~\cite{Blaszczyk:2014qoa} {\color{black} for a recent model-building example}.\footnote{
In Ref.~\cite{Blaszczyk:2014qoa} the SM-like one Higgs doublet model is constructed,
in which Higgs is realized as an extra dimensional gauge field.
For example, the model under the $\mathbb{Z}_{6-I}$ orbifold compactification of $SO(16)\times SO(16)$ heterotic string with the shift vector
\als{
V=\paren{-1/2,\,-1/2,\,1/6,\,1/2,\,-2/3,\,-1/2,\,0,\,1/6}\paren{-2/3,\,-1/2,\,0,\,-1/2,\,-1/2,\,0,\,1/6,\,2/3}
}
and Wilson lines
\als{
A_5
	&=	\paren{1/2,\,1/2,\,-1/2,\,5/6,\,-1/6,\,1/2,\,1/6,\,-1/2}\paren{1/2,\,-1/6,\,-5/6,\,7/6,\,1/6,\,5/6,\,1/2,\,-1/6} \\
A_6	&=	\paren{1/2,\,1/2,\,-1/2,\,5/6,\,-1/6,\,1/2,\,1/6,\,-1/2}\paren{1/2,\,-1/6,\,-5/6,\,7/6,\,1/6,\,5/6,\,1/2,\,-1/6}
}
fits in all the three criteria.
We thank the authors of Ref.~\cite{Blaszczyk:2014qoa} on this point.
}
\end{itemize}
Then we consider multiple insertions of such emission vertices to evaluate the effective potential.
It is very important to understand the whole shape of the Higgs potential in order to discuss the initial condition of the Higgs inflation, as well as to examine whether the MPP is realized or not.
We will show that, in the large field region, the Higgs potential is connected to a runaway vacuum
with vanishing energy, which corresponds to opening up an extra dimension.
We find that such potential can realize an eternal inflation.

This paper is organized as follows.
In Sec.~\ref{classification}, we show that the potential in the large field limit with fixed radius can be classified into the above three categories.
In Sec.~\ref{SO(16)}, we compute the one-loop partition function as a function of a background field in $SO(16)\times SO(16)$ non-supersymmetric heterotic string on $\mathbb{R}^{1,8}\times S^1$, as a concrete toy model~\cite{Dixon:1986iz,AlvarezGaume:1986jb,Ginsparg:1986wr,Itoyama:1986ei,Itoyama:1987rc}. We explicitly check that the limiting behavior of the potential fits into the three categories mentioned above. We argue that physically this corresponds to opening up a multi degrees of freedom space above the Planck scale and that the runaway vacuum is a direction in this space.
In Sec.~\ref{eternal section}, we point out a possibility that the Higgs inflation is preceded by an eternal inflation, which occurs either in a domain wall or in a false vacuum.
In Sec.~\ref{solution to cosmological constant}, we show a possible explanation for the vanishing cosmological constant in terms of the MPP, and consider a possible mechanism to yield the observed value of the order of $\paren{\text{meV}}^4$.
In Sec.~\ref{summary}, we summarize our results.
In Appendix~\ref{notation}, we summarize our notation for several mathematical functions. In Appendix~\ref{fermionic construction}, we review the fermionic construction that we use for the heterotic superstring theory. The computation of the partition function is also outlined. In Appendix~\ref{T-duality section}, we review the T-duality that we use in this work. In Appendix~\ref{MPP review}, we review the MPP.

\section{Higgs potential in string theory}\label{classification}
In this section, we show how to treat the large constant background of a massless mode in closed string theory. In general, we start from a worldsheet action, say,
\al{
S_0={1\over 2\pi\alpha'}\int \text{d}^2z\,G_{MN}\,\partial X^M  \bar{\partial} X^N+\cdots,
}
where $G_{MN}$ is the target space metric, $M,N,\dots$ run from $0$ to $D-1${\color{black} , and $\alpha'$ is the string tension}.
In general, a genetic massless string state has the emission vertex
\al{
\mathcal{O}\fn{z,\bar{z}}e^{ik\cdot X},
}
where $k^2=0$ and $\mathcal{O}\fn{z,\bar{z}}$ has conformal dimensions $(1,1)$ to preserve the conformal symmetry on the worldsheet.

\begin{figure}[tn]
\begin{center}
\hfill
\includegraphics[width=.9\textwidth]{string_diagram.pdf}
\hfill\mbox{}
\caption{
Partition function under the presence of the background $\phi$. Summing up all the possible insertions of $\phi$, it exponentiates to yield Eq.~\eqref{shifted action}. This picture shows the one-loop case.
}\label{string diagram}
\end{center}
\end{figure}

As said in Introduction, we assume in this paper that the emission vertex at the zero momentum of the physical Higgs can be decomposed into a product of the $(1,0)$ operator $\mathcal O_L\fn{z}$ and the $(0,1)$ operator $\mathcal O_R\fn{\bar z}$:
\al{
\mathcal O\fn{z,\bar z}
	&=	\mathcal O_L\fn{z}\,\mathcal O_R\fn{\bar z}.
}
An operator of this form is exactly marginal: Insertions of the operator $\phi\,\mathcal O\fn{z,\bar z}$ can be exponentiated without renormalization, and hence the deformation of the worldsheet action
\al{
S	&=	S_0
		+\phi\int \text{d}^2 z\,\mathcal{O} (z,\bar{z})
		\label{shifted action}
}
keeps the theory conformally invariant; see Fig.~\ref{string diagram}.

We want to know the effective potential for the background: $V\fn{\phi}$.
At the tree-level, the potential vanishes
\al{
V_\text{tree}(\phi)=0.
}
This is because the one-point function of any emission vertex, especially that of the graviton, vanishes on the sphere as it has non-zero conformal dimension. At the one-loop level and higher, we have non-zero effective potential.\footnote{
On the whole plane that is mapped from the sphere, an operator $\mathcal O$ with the scale dimension $d_s$ satisfies $\langle\mathcal O\fn{\lambda z}\rangle=\langle\mathcal O\fn{z}\rangle\lambda^{-d_s}$ and the translational invariance reads $\langle\mathcal O\fn{\lambda z}\rangle=\langle\mathcal O\fn{z}\rangle$. Hence we get $\langle\mathcal O\fn{z}\rangle=0$ for $d_s\neq0$. On the other hand, for torus and surfaces with higher genera, we cannot define the scale transformation, unlike the plane.
}

The $D$-dimensional energy density is given by
\al{
V_\text{$g$-loop}
	&=	-{Z_{g}\over \mathcal V_D},
		\label{energy density}
}
where $\mathcal V_D$ is the volume of $D$-dimensional spacetime and $Z_{g}$ is the partition function on the worldsheet with genus $g$ after moduli integration.
We note that the potential~\eqref{energy density} is given in the Jordan frame that does not yet make the gravitational action canonical; we will come back to this point in Secs.~\ref{graviton background} and \ref{S1 with Wilson}.

We emphasize that in string theory, the partition function $Z_g$ can be obtained even for the field value larger than the Planck scale, unlike the ordinary quantum field theory where infinite number of Planck-suppressed operators become relevant and uncontrollable.

Before generalizing to arbitrary compactification, we first analyze two simple examples to build intuition:
In Sec.~\ref{graviton background}, we study the large field limit of the radion, namely an extra dimensional component of the graviton under the toroidal compactification. This limit corresponds to the large radius limit of the compactified dimension.
In Sec.~\ref{momentum boost}, we further turn on the Wilson line and the anti-symmetric tensor field. We can analyze this setup by considering the corresponding boost in the momentum space~\cite{Narain:1985jj,Narain:1986am}.
From the analysis of the spectrum of these modes, we argue that the effective potential in the large field limit can be classified into three categories, namely, runaway, periodic, and chaotic.
(In Sec.~\ref{SO(16)}, we will confirm it by a concrete computation for the toroidal compactification of the $SO(16)\times SO(16)$ heterotic string theory.)

In Sec.~\ref{generalization}, we discuss more general compactifications, and show that the same classification holds.

\subsection{Radion potential}\label{graviton background}
As said above, we start from the toroidal compactification of the $(D-1)$th direction: $X^{D-1}\sim X^{D-1}+2\pi R$.
The emission vertex of the radion, $G_{D-1\,D-1}$, is
\al{
\partial X^{D-1}\bar{\partial} X^{D-1}\,e^{i k\cdot X}.
}
Its constant background is given by setting the momentum $k=0$.

We want the partition function with the radion background $\phi$:
\al{
S_\text{worldsheet}={1\over 2\pi\alpha'}\int \text{d}^2z \paren{1+\phi}\partial X^{D-1}  \bar{\partial} X^{D-1}+\cdots.
}
In this case, we can transform the action into the original form with $\phi=0$ by the field redefinition
\al{
X'^{D-1}=\sqrt{1+\phi}\,X^{D-1},
}
which however changes the periodicity as
\al{
X'^{D-1}\sim X'^{D-1}+2\pi\sqrt{1+\phi}\,R.
}
That is, the radion background changes the radius of $S^1$ to
\al{
R'	&:=	\sqrt{1+\phi}\,R.
}
Therefore if the compactification radius $R'$ is large, the effective action is proportional to it, and the $(D-1)$-dimensional effective action for large $\phi$ becomes
\al{
S_\text{eff}
	&\sim
		\int \text{d}^{D-1}x \sqrt{-g}\,R'\left(\mathcal{R}-C-{2\over R'^2}\paren{\partial R'}^2\right)\nonumber\\
	&=	\int \text{d}^{D-1}x \sqrt{-g}\sqrt{1+\phi}\,R\left(\mathcal{R}-C-{1\over2\paren{1+\phi}^2}\paren{\partial\phi}^2\right)
		\label{potential in lower dimensions}
}
up to an overall numerical coefficient, where we have taken the $\alpha'=1$ units, $\mathcal R$ is the Ricci scalar in $(D-1)$-dimensions, and $C$ is a $\phi$-independent constant that is generated from loop corrections in the non-supersymmetric string theory.
$C$ can be viewed as the $D$-dimensional cosmological constant.

This can be confirmed at the one-loop level as follows. The radius dependent part of the one-loop partition function before the moduli integration is
\al{\label{radius dependence}
\sum_{n,w=-\infty}^\infty \exp\left[
2\pi i \tau_1 n w-\pi \tau_2 \alpha'\left(\left({n\over R'}\right)^2+\left({R' w\over \alpha'}\right)^2\right)
\right],
}
where $n$ and $w$ are the Kaluza-Klein (KK) and winding numbers, respectively, and $\tau=\tau_1+i\tau_2$ is the moduli of the worldsheet torus.
In the large radius limit $R'\gg\sqrt{\alpha'}$, we can rewrite Eq.~\eqref{radius dependence} by the Poisson resummation formula:
\al{
{R'\over\sqrt{\pi\tau_2\alpha'}}\sum_{m,w}
\exp\left[
-{\pi R'^2\over \alpha'\tau_2}|m-w\tau|^2
\right].
	\label{prop to R after Poisson}
}
We see that the partition function becomes indeed proportional to $R'$ in the large $R'$ limit. Note that in the large $R'$ limit, only the $w=0$ modes contribute, and hence that the winding modes are not important here.

We then rewrite the action~\eqref{potential in lower dimensions} in the Einstein frame.
In $(D-1)$-dimensions, the field redefinition by the Weyl transformation, $g_{\mu\nu}^\text{E}=e^{2\omega} g_{\mu\nu}$, gives us the volume element and the Ricci scalar in the Einstein frame as
\al{
\sqrt{-g^\text{E}}
	&=	e^{\paren{D-1}\omega}\sqrt{-g}, \\
\mathcal{R}^\text{E}
	&=	e^{-2\omega}\sqbr{
			\mathcal{R}
			-2\paren{D-2}\nabla^2\omega
			-\paren{D-3}\paren{D-2}g^{\mu\nu}\partial_\mu\omega\partial_\nu\omega
			},\label{curvature}
}
respectively.
By choosing $e^{\paren{D-3}\omega}=R'$, we get the Einstein frame action:
\al{
S_\text{eff}
	&=	\int \text{d}^{D-1}x \sqrt{-g^\text{E}}\paren{
			\mathcal{R}^\text{E}
			+\paren{D-3}\paren{D-2}g^{\text{E}\mu\nu}\,\partial_\mu\omega\,\partial_\nu\omega
			-e^{-2\omega}C
			-{2\over R'^2}\paren{\partial R'}^2
			}\nonumber\\
	&=	\int \text{d}^{D-1}x \sqrt{-g^\text{E}}\paren{
			\mathcal{R}^\text{E}
			-{D-4\over D-3}{g^{\text{E}\mu\nu}\over R'^2}\partial_\mu R'\,\partial_\nu R'
			-{C\over R'^{2/\paren{D-3}}}
			}\nonumber\\
	&=	\int \text{d}^{D-1}x \sqrt{-g^\text{E}}\paren{
			\mathcal{R}^\text{E}
			-{g^{\text{E}\mu\nu}\over2}\partial_\mu \chi\,\partial_\nu \chi
			-C\exp\fn{-\sqrt{2}\chi\over\sqrt{\paren{D-3}\paren{D-4}}}
			},
			\label{to Einstein}
}
where the second term in Eq.~\eqref{curvature} has become a total derivative and we have defined $R'=:\exp\fn{{\chi\over\sqrt{2}}\sqrt{D-3\over D-4}}$. When $D>4$ and $C>0$, we see that the last term, the potential, becomes runaway for large $R'$ or $\chi$.\footnote{
The small radius limit $R'\ll\sqrt{\alpha'}$ is the same {\color{black} as the large radius limit} due to the T-duality: {\color{black} $R'\longleftrightarrow \alpha'/R'$}~\cite{Kikkawa:1984cp,Sakai:1985cs,Maharana:1992my}.
}

To summarize, the large field limit of the radion $\phi$, the extra dimensional component of the graviton, leads to the decompactification of the corresponding dimension.
{\color{black} This decompactified vacuum corresponds to the runaway potential if the cosmological constant is positive~\cite{Appelquist:1982zs,Appelquist:1983vs}.}
{\color{black} Since the large radius limit is equivalent to the weak coupling limit, the runaway vacuum corresponds to a free theory. Therefore this runaway nature is not altered by the higher order corrections. We will see in Section~\ref{summary} that this argument also applies to the dilaton background.}

\subsection{Boost on momentum lattice}\label{momentum boost}
As the second example, we turn on the backgrounds for graviton, gauge, and anti-symmetric tensor fields.
Let $p$ and $q$ be the numbers of the compactified dimensions in the left and right moving sectors of the closed string, other than our four dimensions. We take $p\geq q$ without loss of generality.
The spectrum of $(p+q)$-dimensional momenta $(\vec k_\text{L},\vec k_\text{R})$ of the non-oscillatory mode
is restricted to form an (even self-dual) momentum lattice, due to the modular invariance~\cite{Narain:1985jj,Narain:1986am}; see Appendix~\ref{bosonic part}. Different lattices that are related by the $SO(p,q)$ rotation of $(\vec k_\text{L},\vec k_\text{R})$ correspond to different compactifications, up to the $SO(p)\times SO(q)$ rotation that leaves $\vec k_\text{L}^2$ and $\vec k_\text{R}^2$ invariant. Therefore the compactifications are classified by the transformation
\al{\label{Narain}
{SO(p,q)\over SO(p)\times SO(q)}.
}
This is the moduli space of the theory at the tree level, which is lifted up by the loop corrections in non-supersymmetric string theory.

The boost in the momentum space corresponds to putting constant backgrounds for the degrees of freedom that are massless at the tree-level~\cite{Narain:1985jj,Narain:1986am}:
\al{
C_{ij}\,\partial X^i_\text{L}\,\bar\partial X_\text{R}^{\bar j},
	\label{C_ij}
}
where $i$ and $\bar j$ run for $1,\dots,p$ and $1,\dots,q$, respectively.
In terms of $q$-dimensional fields, they can be interpreted as the symmetric tensor (metric), antisymmetric tensor, and ${U(1)}^{p-q}$ gauge fields (Wilson lines), whose total number is
\al{
{q\paren{q+1}\over 2}+{q\paren{q-1}\over2}+q\paren{p-q}=pq.
}
Indeed, this agrees with the number of degrees of freedom of the coset space~\eqref{Narain}:
\al{
{\paren{p+q}\paren{p+q-1}\over 2}-{p\paren{p-1}\over2}-{q\paren{q-1}\over2}=pq.
}

\begin{figure}[tn]
\begin{center}
\hfill
\includegraphics[width=.44\textwidth]{lattice_decomp_deco.pdf}
\hfill
\includegraphics[width=.45\textwidth]{lattice_chaotic_deco.pdf}
\hfill\mbox{}
\caption{Schematic picture of the momentum boost in the $k_\text{R}$ vs $k_\text{L}$ plane. The light cone in the momentum space is depicted by the dashed diagonal lines. The sets of lighter (magenta) and black dots represent the initial momentum lattice and the one after the boost, respectively. Left: There exists a point of the initial lattice on the light cone. Then there exist infinite amount of its integer multiplications on the light cone. In the infinite boost limit, they are contracted to form a decompactified dimension, which is represented by the black dots. Right: There is no initial point on the light cone, and such a decompactification does not occur.
}\label{momentum lattice}
\end{center}
\end{figure}
We are interested in switching on the background of a single field. If the emission vertex of the field is given by $c_{i\bar j}\,\partial X^i_\text{L}\,\bar\partial X_\text{R}^{\bar j}$, this corresponds to adding
\al{
\lambda\,c_{ij}\,\partial X^i_\text{L}\,\bar\partial X_\text{R}^{\bar j}
}
to the worldsheet action, where $\lambda$ represents the strength of the background.
In general, the $SO(p)\times SO(q)$ rotation can make $c_{i\bar j}$ into the diagonal form
\al{
c_{i\bar j}	&\to
	\bmat{
		*	&		& 	&  \\
			&	*	&		&  \\
			&		&\ddots &  \\
			&	 & &* \\
			&	 &  &\\
			&  &  &
		},\label{c_ij}
}
where the blank slots stand for zero.
This background corresponds to the combination of $q$ boosts in the $1$-$\bar 1$, \dots, $q$-$\bar q$ planes.
That is, the $(p+q)$-dimensional vector
\al{
k=\paren{k_\text{L}^1,\dots,k_\text{L}^p;k_\text{R}^{\bar 1},\dots,k_\text{R}^{\bar q}}
}
is transformed by
\al{
\bmat{k_\text{L}'^i\\ k_\text{R}'^{\bar i}}
	&=	\bmat{\cosh\eta_i&\sinh\eta_i\\ \sinh\eta_i&\cosh\eta_i}
			\bmat{k_\text{L}^i\\ k_\text{R}^{\bar i}},	\nonumber\\
k_\text{L}'^j
	&=	k_\text{L}^j,
}
for $i=1,\dots,q$ and $j=q+1,\,\dots,\,p$.

Let us first consider the effect of a boost in a single plane:
\al{
\bmat{k_\text{L}'\\ k_\text{R}'}
	&=	\bmat{\cosh\eta&\sinh\eta\\ \sinh\eta&\cosh\eta}
			\bmat{k_\text{L}\\ k_\text{R}}.
}
Then one of $k_\text{L}\pm k_\text{R}$ is contracted and the other expanded:
\al{
k'_\text{L}+k'_\text{R}&=e^\eta\paren{k_\text{L}+k_\text{R}},\nonumber\\
k'_\text{L}-k'_\text{R}&=e^{-\eta}\paren{k_\text{L}-k_\text{R}}.
}
The effective potential in the large $\eta$ limit depends on whether or not there exists a lattice point on the light cone in this plane, as is illustrated schematically in Fig.~\ref{momentum lattice}.
There are two possibilities in the infinite boost limit:
\begin{itemize}
\item If a point in the initial momentum lattice sits on the light cone as in the left panel in Fig.~\ref{momentum lattice}, infinite amount of its integer multiplications on the light cone are contracted to form a continuous spectrum.
This behavior is the same as that of the KK momenta in the large radius limit discussed in Sec.~\ref{graviton background}.
The resultant partition function becomes proportional to the radius $R$.
The same argument as Sec.~\ref{graviton background} gives us the runaway potential.

\item If no point sits on the light cone in the initial momentum lattice, as in the right panel in Fig.~\ref{momentum lattice}, then the continuum is not formed by the infinite boost.
For a given amount of boost, the closest point to the origin contributes the most to the partition function. Then the potential becomes either periodic or chaotic for larger and larger boost.
\end{itemize}
The fate of the large field limit depends on whether or not a lattice point sits on the light cone of the boost plane in the momentum space.

In the case of the multiple boosts~\eqref{c_ij}, the boost in each plane is independent from the others. However, if there are several degenerate massless states as in Eq.~\eqref{C_ij}, we should better consider all of them simultaneously. As we will see in Sec.~\ref{SO(16)} in a concrete model, the asymptotic behavior of the potential remains essentially the same.

\subsection{General compactifications}\label{generalization}
We discuss the large field limit in more general setup including compactification on a curved space, possibly involving orbifolding etc., or even the case without having a geometrical interpretation.
We will show that the classification still holds: runaway, periodic, and chaotic.

\begin{figure}
\begin{center}
\includegraphics[width=.7\textwidth]{graviton.pdf}
\caption{Left: The Higgs emission vertex $\partial Y\,\bar\partial Z$ is single-valued around the graviton emission vertex because $Y$ and $Z$ are independent of the spacetime coordinate $X^\mu$. Right: Exponential mapping around the graviton emission vertex. $\partial Y$ and $\bar\partial Z$ are periodic around the cylinder, e.g.\ around the (red) circle.}\label{graviton}
\end{center}
\end{figure}
As said above, the emission vertex of a massless field must be written in terms of a $(1,1)$ operator, and we assume that this operator separates into the holomorphic and anti-holomorphic parts,
\al{
\mathcal{O}_{(1,1)}=\mathcal{O}_{(1,0)}\times \mathcal{O}_{(0,1)},
}
on the worldsheet.
Then we can write at least locally,
\al{
&\mathcal{O}_{(1,0)}=\partial Y,
&\mathcal{O}_{(0,1)}=\bar{\partial} Z,
}
where $Y$ and $Z$ are free worldsheet scalars.
If we further assume that the Higgs field is uniquely identified, i.e., that it does not mix with other massless states at the tree level, then it suffices to consider a single background as in Eq.~\eqref{shifted action}. In this case we may not need to consider the multi-field potential discussed above.

We can show that $\partial Y$ and $\bar\partial Z$ are periodic at least in one sector:
In fact, if we insert the graviton emission vertex $\partial X^\mu\,\bar\partial X^\nu e^{ik\cdot X}$ near the Higgs emission vertex, the latter is single valued in the neighborhood of the former. This is because $Y$ and $Z$ are independent of the spacetime coordinates $X^\mu$. Therefore, $\partial Y$ and $\bar\partial Z$ are periodic in the graviton sector; see Fig.~\ref{graviton}.

In such a sector, we can mode-expand $\partial Y$ and $\bar\partial Z$.
Let us consider the simultaneous eigenvalues $\paren{p_Y,p_Z}$ of the constant modes of $\partial Y$ and $\bar\partial Z$.
The set of the pairs of eigenvalues form a momentum lattice $\Gamma_P=\Set{\paren{p_Y,p_Z}}$: If there exist states $s_1$ and $s_2$ with momenta $\paren{p_{Y1},p_{Z1}}$ and $\paren{p_{Y2},p_{Z2}}$, respectively, there is a state with the momentum $\paren{p_{Y1}+p_{Y2},\,p_{Z1}+p_{Z2}}$; such a state appears when $s_1$ and $s_2$ merge.
If $\Gamma_P$ contains a non-zero vector, it forms the momentum lattice.
Then the same argument applies as in Sec.~\ref{momentum boost}.
Putting a constant background for $\mathcal{O}_{(1,1)}$ corresponds to the momentum boost.
If there is a point on the light cone with $p_Y/p_Z$ being a rational number, then a runaway direction emerges in the infinite boost limit. If not, namely if there is no such point, then the potential becomes chaotic.

\section{$SO(16)\times SO(16)$ heterotic string}\label{SO(16)}
We verify the argument in the previous section in the concrete model: the $SO(16)\times SO(16)$ heterotic string theory~\cite{Dixon:1986iz,AlvarezGaume:1986jb}. This model breaks supersymmetry at the string scale but, unlike the bosonic string theory in 26 dimensions, the tachyonic modes are projected out as in the ordinary heterotic superstring theories.
In the fermionic construction, the modular invariance of the partition function restricts the allowed set of the fermion numbers in Neveu-Schwarz (NS) and Ramond (R) sectors.
The classification of the ten dimensional string theories is completed in Ref.~\cite{Kawai:1986vd}.
The $SO(16)\times SO(16)$ model~\cite{Dixon:1986iz,AlvarezGaume:1986jb} is the only one that has neither a tachyon nor a supersymmetry in ten dimensions.

We write the uncompactified dimensions $X^\mu$ ($\mu=0,\dots,9$), and the compactified ones $X_L^I$ ($I=1,\dots,16$) for the left movers.
We then compactify this model on $S^1$~\cite{Ginsparg:1986wr}:
\al{
X^9	&\sim X^9+2\pi R.
}
We further turn on a Wilson line for the gauge field $A_{\mu=9}^{I=1}$, and compute the one-loop partition function.

In Appendix~\ref{fermionic construction}, we spell out the construction of the model and the computation of the partition function; the notations for the theta functions are put in Appendix~\ref{notation}.
In Sec.~\ref{SO(16)xSO(16) partition function}, we review the partition function in the $SO(16)\times SO(16)$ heterotic string theory in 10 dimensions. In Sec.~\ref{S1 with Wilson} we compute the one-loop partition function of this model for the case described above.

\subsection{Partition function of $SO(16)\times SO(16)$ string}
\label{SO(16)xSO(16) partition function}
We first review the computation of the partition function in the $SO(16)\times SO(16)$ non-supersymmetric heterotic string~\cite{Dixon:1986iz,AlvarezGaume:1986jb}. Here we have chosen a non-super\-sym\-metric string as a toy model because, as discussed in Introduction, the low energy data at the electroweak scale suggests via the Veltman condition that the supersymmetry is broken at the Planck scale. In such a non-supersymmetric theory, the flat direction of the effective potential is raised perturbatively. Detailed procedure of the fermionic construction of the model is explained in Appendix~\ref{formalism section} and \ref{concrete heterotic models}.

Let us write down the contribution from the momentum lattice after the bosonization in each $\alpha\vec w$ sector:
\al{
\hat Z_{T^2,\alpha\vec w}
	&=	\Tr_\text{$\alpha\vec w$} e^{2\pi i\tau_1\paren{L_0-\bar L_0}-2\pi\tau_2\paren{L_0+\bar L_0}}\bigg|_\text{momentum lattice}.
}
In our case, they are
\al{
\hat Z_{T^2,\vec{0}}&=\phantom{-}{1\over8}\paren{\paren{\bar{\vartheta}_{00}}^4-\paren{\bar{\vartheta}_{01}}^4}\paren{\paren{\vartheta_{00}}^8+\paren{\vartheta_{01}}^8}^2,\nonumber\\
\hat Z_{T^2,\vec w_0}&=-{1\over8}\paren{\bar{\vartheta}_{10}}^4\paren{\vartheta_{10}}^{16},\nonumber\\
\hat Z_{T^2,\vec w_1}&=\phantom{-}{1\over8}\paren{\paren{\bar{\vartheta}_{00}}^4-\paren{\bar{\vartheta}_{01}}^4}\paren{\vartheta_{10}}^{16},\nonumber\\
\hat Z_{T^2,\vec w_2}&=\phantom{-}{1\over8}\paren{\paren{\bar{\vartheta}_{00}}^4+\paren{\bar{\vartheta}_{01}}^4}\paren{\vartheta_{10}}^8\paren{\paren{\vartheta_{00}}^8-\paren{\vartheta_{01}}^8},\nonumber\\
\hat Z_{T^2,\vec w_0+\vec w_1}&=-{1\over8}\paren{\bar{\vartheta}_{10}}^4\paren{\paren{\vartheta_{00}}^8-\paren{\vartheta_{01}}^8}^2,\nonumber\\
\hat Z_{T^2,\vec w_0+\vec w_2}&=-{1\over8}\paren{\bar{\vartheta}_{10}}^4\paren{\paren{\vartheta_{00}}^8+\paren{\vartheta_{01}}^8}\paren{\vartheta_{10}}^8,\nonumber\\
\hat Z_{T^2,\vec w_1+\vec w_2}&=\phantom{-}{1\over8}\paren{\paren{\bar{\vartheta}_{00}}^4+\paren{\bar{\vartheta}_{01}}^4}\paren{\vartheta_{10}}^8\paren{\paren{\vartheta_{00}}^8-\paren{\vartheta_{01}}^8},\nonumber\\
\hat Z_{T^2,\vec w_0+\vec w_1+\vec w_2}&=-{1\over8}\paren{\bar{\vartheta}_{10}}^4\paren{\paren{\vartheta_{00}}^8+\paren{\vartheta_{01}}^8}\paren{\vartheta_{10}}^8,
		\label{SO(16) partition function}
}
where $\vec 0$ and $\vec{w}_i$ are basis vectors for the boundary conditions on the fermions; see Appendix~\ref{fermion one-loop} for details.

Let us sum up all the above contributions, multiplied by those from the oscillator modes in the bosonization.
Including also the spacetime momentum and oscillator modes from the bosonic $X^m$ ($m=2,\dots,9$), we get the one-loop vacuum amplitude~\cite{Dixon:1986iz,AlvarezGaume:1986jb}:
\al{
Z_{T^2}&={V_{10}\over\alpha'^5}{1\over2\paren{2\pi}^{10}}\int_F {\text{d}\tau_1\,\text{d}\tau_2\over \tau_2^6}{1\over\ab{\eta(\tau)}^{16}{\eta(\tau)}^{16}\,{\bar{\eta}(\bar{\tau})}^4}
\sum_\text{sector $\alpha\vec w$}\hat Z_{T^2,\alpha\vec w}\nonumber\\
&=
{V_{10}\over\alpha'^5}{1\over4 (2\pi)^{10}}\int_F {\text{d}\tau_1\,\text{d}\tau_2\over \tau_2^6}{1\over\ab{\eta(\tau)}^{16}{\eta(\tau)}^{16}\,{\bar{\eta}(\bar{\tau})}^4}\nonumber\\
&\quad
\times
\left[
\paren{\bar{\vartheta}_{01}}^4\paren{\vartheta_{10}}^8\paren{\paren{\vartheta_{00}}^8-\paren{\vartheta_{01}}^8}
+\paren{\bar{\vartheta}_{10}}^4\paren{\vartheta_{01}}^8\paren{\paren{\vartheta_{00}}^8-\paren{\vartheta_{10}}^8}
\right],
	\label{partition function for non-SUSY}
}
where $F$ represents the fundamental region,
\al{
F	&:=	\Set{\paren{\tau_1,\tau_2}|-1/2\leq\tau_1\leq1/2,\ \ab{\tau}=\ab{\tau_1+i\tau_2}\geq 1},
}
and we have used the Jacobi's identity:
\al{
\paren{\bar{\vartheta}_{00}}^4-\paren{\bar{\vartheta}_{01}}^4-\paren{\bar{\vartheta}_{10}}^4=0.
}
We can see from this identity that the contributions between $\vec{w}_0$ and $\vec{w}_1$ cancel.
By the numerical calculation, we obtain~\cite{Dixon:1986iz,AlvarezGaume:1986jb}
\al{
\rho_{10}
	&=	-{Z_{T^2}\over V_{10}}\simeq \paren{3.9\times 10^{-6}} {1\over\alpha'^5}.
}
\subsection{$S^1$ compactification with Wilson line}
\label{S1 with Wilson}
Now we compactify the $m=9$ direction on $S^1$ with radius $R$: $X^9\sim X^9+2\pi R$~\cite{Ginsparg:1986wr}.
Here we consider a large field limit of an extra $m=9$ dimensional component of the gauge field, $A_{m=9}^{I=1}$.
We will find three possible large field limits discussed in the previous section.

The emission vertex for the gauge field with the polarization and momentum $\epsilon^m$ and $k$, respectively, is
\al{
\epsilon_m\left(i\bar{\partial}X^m+{\alpha'\over 2} \paren{k\cdot\psi_R} \psi^{m}_R\right)\partial X^I_L\,e^{ik\cdot X},
	\label{gauge vertex op}
}
where indices run such that $m=2,\dots,9$ and $I=1,\dots,16$.
We see by putting $k=0$ in Eq.~\eqref{gauge vertex op} that a constant background $A_m^I$ corresponds to adding
\al{
A_m^I\int \text{d}^2z\,\bar{\partial}X^{m}\,\partial X^I_{L},
	\label{vertex op of A}
}
to the worldsheet action.\footnote{
In obtaining the constant background by putting $k=0$, it is again important that the $A$ is massless at the tree-level.
}
In particular, we switch on the component of $I=1$ and $m=9$, and write $A:=A_9^1$:
\al{
A\int \text{d}^2z\,\bar{\partial}X^{9}\,\partial X^1_{L}.
	\label{vertex op of A comp}
}

Turning on the Wilson line background $A$ does not affect the oscillator modes since Eq.~\eqref{vertex op of A} is a total derivative in the worldsheet action; only the momentum lattice of the center-of-mass mode is changed by $A$.

Let $l_L$ be the momentum of $X^{I=1}_L$. After fermionization, we have
\al{
l_L&=\sqrt{{2\over\alpha'}}m,
}
where $m\in\mathbb{Z}$ and $\mathbb{Z}+1/2$ for the NS (anti-periodic) and R (periodic) boundary conditions, respectively.
Let $p_L$ and $p_R$ be the spacetime momenta of the $S^1$-compactified direction $X^{m=9}$ for the left and right movers, respectively:
\al{
p_L&={n\over R}+{R w\over\alpha'},\nonumber\\
p_R&={n\over R}-{R w\over\alpha'},
	\label{internal momentum}
}
where $n\in\mathbb Z$ and $w\in\mathbb Z$ are the KK and winding numbers, respectively.

Turning on the background $A$ corresponds to the boost on the momentum lattice~\cite{Narain:1986am}:
\al{
\bmat{
l_L'\\
p_R'
}
=
\bmat{
\cosh\eta&\sinh\eta\\
\sinh\eta&\cosh\eta
}
\bmat{
l_L\\
p_R
},\label{lL pR mixing}
}
since there appears only $l_L$ and $p_R$ in Eq.~\eqref{vertex op of A comp}.
This boost necessarily changes the radius of the compactification too.

we will see that the identification
\al{
A	&=	\sinh\eta,
	\label{A-id}
}
gives the correct answer below.
Let us define $r$ by
\al{\label{r-id}
r	&:=	{R\over\cosh\eta},
}
which will turn out to be the compactification radius in the presence of $A$.
Note that in the language of Sec.~\ref{momentum boost}, we have 17 left-moving and 1 right-moving internal dimensions ($p=17$ and $q=1$).
The non-trivial transformations on the compactified space are
\al{
{SO(17,1)\over SO(17)}.
	\label{tree moduli space}
}
Among them, we have chosen the boost between the left $I=1$ and right $m=9$ dimensions with the momenta $l_L$ and $p_R$, respectively.
The left momentum of the $m=9$ dimension, $p_L$, is untouched.
We will soon use the rotation between $l_L$ and $p_L$ that belongs to $SO(17)$.

We now show the validity of the identification~\eqref{A-id}.
In terms of $A$ and $r$, we have
\al{\label{pR''}
p_R'
	&=	p_R\cosh\eta +l_L\sinh\eta &
	&=	{n\over r}-{rw\over\alpha'}\paren{1+A^2}+\sqrt{{2\over\alpha'}}m A,\\
l_L'
	&=	l_L\cosh\eta +p_R\sinh\eta	&
	&=	\sqrt{{2\over\alpha'}}m \sqrt{1+A^2}+{n\over r}{A\over\sqrt{1+A^2}}-{rw\over\alpha'}A\sqrt{1+A^2},\\
p_L'
	&=	p_L	&
	&=	{n\over r}{1\over\sqrt{1+A^2}}+{r w\over\alpha'}\sqrt{1+A^2}.
}
We further rotate by a part of $SO(17)$ in Eq.~\eqref{tree moduli space},
\al{
\bmat{
l_L''\\
p_L''
}
&=
\bmat{
\cos\theta&\sin\theta\\
-\sin\theta&\cos\theta
}
\bmat{
l_L'\\
p_L'
},	\label{lL pL mixing}\\
p_R''
	&=	p_R', \nonumber
}
with
\al{
\cos\theta	&=	{1\over\sqrt{1+A^2}},	&
\sin\theta	&=	-{A\over\sqrt{1+A^2}},
}
to get
\al{
l_L''&=\sqrt{{2\over\alpha'}}m-2{rw\over\alpha'}A,\\
p_L''&={n\over r}+{rw\over\alpha'}\paren{1-A^2}+\sqrt{{2\over\alpha'}}mA.
\label{pL''}
}
The spectrum becomes
\al{
\sum_\text{all modes}\paren{l_L''^2+p_L''^2+p_R'^2}
	&=	\sum_{m,n,w}\Bigg[
			\paren{\sqrt{{2\over\alpha'}}m-2{rw\over\alpha'}A}^2
			+\paren{{n\over r}+{rw\over\alpha'}\paren{1-A^2}+\sqrt{{2\over\alpha'}}mA}^2\nonumber\\
	&\phantom{\mbox{}=	\sum_{m,n,l}\Bigg[}
			+\paren{{n\over r}-{rw\over\alpha'}\paren{1+A^2}+\sqrt{{2\over\alpha'}}m A}^2
			\Bigg].
			\label{spectrum with Wilson line}
}
As promised, this result~\eqref{spectrum with Wilson line} correctly reproduces that in Ref.~\cite{Narain:1986am,Ginsparg:1986wr}, which is obtained from the quantization of the scalar field under constraints.
Furthermore, from Eq.~\eqref{spectrum with Wilson line}, we see
\al{
\left.\paren{l_L''^2+p_L''^2}\right|_{m=w=0}
	=	\left.p_R''^2\right|_{m=w=0}
	=	{n^2\over r^2},
		\label{large r limit summation}
}
which indicates that $r$ is the physical radius of $S^1$.

Now let us discuss the T-dual transformations that can be read off from the above result.
\begin{itemize}
\item We can see that the shift
\al{
A	&\rightarrow
		A+{\sqrt{2\alpha'}\over r}
		\label{A shift}
}
leaves the spectrum~\eqref{spectrum with Wilson line} unchanged.\footnote{
After the shift of $A$, redefine the mode numbers by $n'=n+2m-2w$, $w'=w$, and $m'=m-2w$.
}
\item From Eq.~\eqref{internal momentum}, we see that the spectrum is invariant under the T-dual transformation~\cite{Kikkawa:1984cp,Sakai:1985cs,Maharana:1992my}
\al{
R	&\to {\alpha'\over R},
	\label{T-dual tf basic}
}
or in terms of $r$ and $A$, $r\to \alpha'/\paren{1+A^2}r$.
\end{itemize}
By defining
\al{\label{tautilde}
\tilde\tau
	&=	\tilde\tau_1+i\tilde\tau_2
	:=	{r A\over\sqrt{\alpha'}}+i{r\over\sqrt{\alpha'}},
}
we can write down the enlarged T-dual transformation:\footnote{
The $S$-transformation is the transformation~\eqref{T-dual tf basic} composed with $A\to-A$, while the $T$ is Eq.~\eqref{A shift}.
}
\al{
&S:\quad \tilde\tau\rightarrow-{1\over\tilde\tau}\nonumber\\
&T:\quad \tilde\tau\rightarrow\tilde\tau+\sqrt{2}.
	\label{T-duality transformation}
}
The general form of the T-dual transformation is
\al{\label{general transformation}
&\tilde\tau'={a\tilde\tau+b\over c\tilde\tau+d},
}
where $ad-bc=1$ and $a$, $b$, $c$, and $d$ are either
\al{
a	&\in	\mathbb Z,	&
b	&\in	\sqrt{2}\mathbb Z,	&
c	&\in	\sqrt{2}\mathbb Z,	&
d	&\in	\mathbb Z,
}
or
\al{
a	&\in	\sqrt{2}\mathbb Z,	&
b	&\in	\mathbb Z,	&
c	&\in	\mathbb Z,	&
d	&\in	\sqrt{2}\mathbb Z.
}
The fundamental region is $-1/\sqrt{2}\leq \tilde\tau_1\leq1/\sqrt{2},\, \ab{\tilde\tau}\geq 1$.
More details can be found in Appendix~\ref{T-duality section}.

\subsection{Effective potential under Wilson line}
Let us write down the contribution from the momentum lattice after the bosonization
in each sector $\alpha\vec w$; this time we include the momentum \eqref{internal momentum} of the $S^1$-compactified $X^{m=9}$ which is modified by the Wilson line $A$ as in Eqs.~\eqref{pR''} and \eqref{pL''}:
\al{
\tilde Z_{T^2,\alpha\vec w}
	&=	\Tr_{\alpha\vec w} e^{2\pi i\tau_1\paren{L_0-\bar L_0}-2\pi\tau_2\paren{L_0+\bar L_0}}\bigg|_{\text{momentum lattice}}.
}
Concretely,
\al{
\tilde Z_{T^2,\vec{0}}&=\phantom{-}{1\over8}\paren{\paren{\bar{\vartheta}_{00}}^4-\paren{\bar{\vartheta}_{01}}^4}\sum_{m\in\mathbb Z} g_m\fn{\eta,R}
\paren{\paren{\vartheta_{00}}^7+\paren{-1}^m\paren{\vartheta_{01}}^7}\paren{\paren{\vartheta_{00}}^8+\paren{\vartheta_{01}}^8},
\nonumber\\
\tilde Z_{T^2,\vec w_0}&=-{1\over8}
\paren{\bar{\vartheta}_{10}}^4
\sum_{m\in\mathbb Z+1/2}g_m\fn{\eta,R}
\paren{\vartheta_{10}}^{15},
\nonumber\\
\tilde Z_{T^2,\vec w_1}&=\phantom{-}{1\over8}\paren{\paren{\bar{\vartheta}_{00}}^4-\paren{\bar{\vartheta}_{01}}^4}
\sum_{m\in\mathbb Z+1/2}g_m\fn{\eta,R}
\paren{\vartheta_{10}}^{15},
\nonumber\\
\tilde Z_{T^2,\vec w_2}&=\phantom{-}{1\over8}\paren{\paren{\bar{\vartheta}_{00}}^4+\paren{\bar{\vartheta}_{01}}^4}
\sum_{m\in\mathbb Z+1/2}g_m\fn{\eta,R}
\paren{\vartheta_{10}}^7\paren{\paren{\vartheta_{00}}^8-\paren{\vartheta_{01}}^8},
\nonumber\\
\tilde Z_{T^2,\vec w_0+\vec w_1}&=-{1\over8}\paren{\bar{\vartheta}_{10}}^4
\sum_{m\in\mathbb Z}g_m\fn{\eta,R}
\paren{\paren{\vartheta_{00}}^7-\paren{-1}^m\paren{\vartheta_{01}}^7}
\paren{\paren{\vartheta_{00}}^8-\paren{\vartheta_{01}}^8},
\nonumber\\
\tilde Z_{T^2,\vec w_0+\vec w_2}&=-{1\over8}\paren{\bar{\vartheta}_{10}}^4
\sum_{m\in\mathbb Z}g_m\fn{\eta,R}
\paren{\paren{\vartheta_{00}}^7+\paren{-1}^m\paren{\vartheta_{01}}^7}
\paren{\vartheta_{10}}^8,
\nonumber\\
\tilde Z_{T^2,\vec w_1+\vec w_2}&=\phantom{-}{1\over8}\paren{\paren{\bar{\vartheta}_{00}}^4+\paren{\bar{\vartheta}_{01}}^4}
\sum_{m\in\mathbb Z}g_m\fn{\eta,R}
\paren{\paren{\vartheta_{00}}^7-\paren{-1}^m\paren{\vartheta_{01}}^7}\paren{\vartheta_{10}}^8,
\nonumber\\
\tilde Z_{T^2,\vec w_0+\vec w_1+\vec w_2}&=-{1\over8}\paren{\bar{\vartheta}_{10}}^4
\sum_{m\in\mathbb Z+1/2}g_m\fn{\eta,R}
\paren{\vartheta_{10}}^7
\paren{\paren{\vartheta_{00}}^8+\paren{\vartheta_{01}}^8},
	\label{partition function under A}
}
where
\al{
g_m\fn{\eta,R}
&=
\sum_{n,w=-\infty}^\infty \exp\left[
	\pi i \alpha'{\tau_1\over2}\paren{l_L''^2+p_L''^2-p_R''^2}
	-{\pi\over 2}\tau_2\alpha'\paren{l_L''^2+p_L''^2+p_R''^2}
\right]
\nonumber\\
&=\sum_{n,w=-\infty}^\infty \exp\left[
\pi i \tau_1\paren{m^2+2 n w}-{\pi\over 4}\tau_2\alpha'
	\paren{
	e^{2\eta}\paren{l_L+p_R}^2+e^{-2\eta}\paren{l_L-p_R}^2+2p_L^2
	}
\right]\label{Wilson-line-dependent part}
}
contains the information of the Wilson line.
We can check that the $\eta\to 0$ limit reduces Eq.~\eqref{partition function under A} to Eq.~\eqref{SO(16) partition function}, multiplied by the contribution from the compactified dimension shown in Appendix.~\ref{bosonic part}.

Including the oscillator modes and the spacetime coordinates $X^m$ ($m=2,\dots,9$), we get
\al{
Z_{T^2}
&={V_{9}\over\alpha'^{9/2}}{1\over2 (2\pi)^9}\int_F {d\tau_1 d\tau_2\over \tau_2^{11/2}}{1\over\ab{\eta(\tau)}^{16}{\eta(\tau)}^{16}{\bar{\eta}(\bar{\tau})}^4}
\sum_\text{sector $\alpha\vec w$}\tilde{Z}_{T^2,\alpha \vec{w}}\nonumber\\
&={V_{9}\over\alpha'^{9/2}}{1\over8 (2\pi)^9}\int_F {d\tau_1 d\tau_2\over \tau_2^{11/2}}{1\over\ab{\eta(\tau)}^{16}{\eta(\tau)}^{16}{\bar{\eta}(\bar{\tau})}^4}\nonumber\\
&\quad\times\Bigg(
\sum_{m\in\mathbb Z+1/2}g_m\fn{\eta,R}\paren{\vartheta_{10}}^7
	\paren{
		\paren{\bar{\vartheta}_{01}}^4\paren{\vartheta_{00}}^8
		-\paren{\bar{\vartheta}_{00}}^4\paren{\vartheta_{01}}^8
		}\nonumber\\
&\phantom{\quad\times\Bigg(}
	+\sum_{m\in\mathbb Z}g_m\fn{\eta,R}\bigg[
\paren{\vartheta_{00}}^7\paren{\paren{\bar{\vartheta}_{10}}^4\paren{\vartheta_{01}}^8+\paren{\bar{\vartheta}_{01}}^4\paren{\vartheta_{10}}^8}\nonumber\\
&\phantom{\quad\times\Bigg(+\sum_{m\in\mathbb Z}g_m\fn{\eta,R}\bigg[}
+\paren{-1}^m \paren{\vartheta_{01}}^7
\paren{\paren{\bar{\vartheta}_{10}}^4\paren{\vartheta_{00}}^8-\paren{\bar{\vartheta}_{00}}^4\paren{\vartheta_{10}}^8}
\bigg]
\Bigg).\label{Wilsonline partition function}
}
The 9 dimensional energy density in the Jordan frame is given by
\al{
\rho_9
	&=	-{Z_{T^2}\over V_9};
}
see Eq.~\eqref{energy density}.

\begin{figure}[t]
\begin{center}
\hfill
\includegraphics[width=.4\textwidth]{partition_function_r1.pdf}
\hfill
\includegraphics[width=.4\textwidth]{partition_function_rSqrt2.pdf}
\hfill\mbox{}
\caption{
The potential $\rho_9$ in Jordan frame as a function of $A$ with $r=\sqrt{\alpha'}$ (left) and $\sqrt{2\alpha'}$ (right), all in units of $\alpha'=1$.
We can see the periodicity $A\to A+\sqrt{2\alpha'}/r$, up to the distortions due to numerical errors.
}\label{periodicity}
\end{center}
\end{figure}
In Fig.~\ref{periodicity}, we plot $\rho_9$ as a function of $A$ for $r=\sqrt{\alpha'}$ (left) and $\sqrt{2\alpha'}$ (right), all in units of $\alpha'=1$.
The summation over $n$ and $m$ in Eqs.~\eqref{Wilson-line-dependent part} and \eqref{Wilsonline partition function} are truncated by $\ab{n},\ab{m}\leq10$ and the numerical integration is performed within $\tau_2\leq4$.
We can see the periodicity $A\to A+\sqrt{2\alpha'}/r$.

\begin{figure}[t]
\begin{center}
\hfill
\includegraphics[width=.35\textwidth]{contour.pdf}
\hfill
\includegraphics[width=.6\textwidth]{3Dplot.pdf}
\hfill\mbox{}
\caption{
Contour and 3D plots are shown in the left and right panels, respectively, for the energy density $\rho_9$ in the Jordan frame as a function of $\tilde\tau_1=rA/\sqrt{\alpha'}$ and $\tilde\tau_2=r/\sqrt{\alpha'}$, with all their values being given in $\alpha'=1$ units. In the left, we shade the fundamental region for the T-dual transformation: $\ab{\tau_1}\leq1/\sqrt{2}$, $\ab{\tau}\geq 1$. We can see the shift-symmetry $\tilde\tau_1\to\tilde\tau_1+\sqrt{2}$, up to distortions due to numerical errors.
}\label{Jordan potential}
\end{center}
\end{figure}
For varying $A$ and $r$, we plot $\rho_9$ as a function of $\tilde\tau_1=rA/\sqrt{\alpha'}$ and $\tilde\tau_2=r/\sqrt{\alpha'}$ in Fig.~\ref{Jordan potential}.
Note that in the large $r$ ($=\sqrt{\alpha'}\tilde\tau_2$) limit, the Jordan frame potential becomes proportional to $r$. This can also be seen analytically from the fact that in the large $r$ limit, the contributing modes are as in Eq.~\eqref{large r limit summation}, which results in the same expression as Eq.~\eqref{prop to R after Poisson}.
To repeat, we have obtained both numerically and analytically that the Jordan frame potential is proportional to $r$ at the one-loop level. For large $r$ limit,
all the higher loop corrections have the same behavior since it comes from the fact that the energy is proportional to the volume of the compactified dimension.

\begin{figure}[t]
\begin{center}
\hfill
\includegraphics[width=.35\textwidth]{contour_Einstein.pdf}
\includegraphics[width=.6\textwidth]{Einstein_3Dplot.pdf}
\hfill\mbox{}
\caption{
Contour and 3D plots are shown in the left and right panels, respectively, for the energy density $V_\text{E}$ in the Einstein frame as a function of $\tilde\tau_1=rA/\sqrt{\alpha'}$ and $\tilde\tau_2=r/\sqrt{\alpha'}$, with all their values being given in $\alpha'=1$ units.
The shaded fundamental region and the existence of the $\sqrt{2}$-shift are the same as in Fig.~\ref{Jordan potential}. We see that the potential becomes runaway for the large radius limit $r\gg\sqrt{\alpha'}$.
}\label{Einstein potential}
\end{center}
\end{figure}
Now let us turn to the Einstein frame:
\al{
V_E(r)&=-{1\over \paren{{2\pi r}}^{2/7}}{Z_{T^2}\over2\pi r V_9},\nonumber\\
&=-{1\over\alpha'^{9/2}}{1\over2 \paren{2\pi}^{72/7}}{1\over r^{9/7}}\int_F {d\tau_1 d\tau_2\over \tau_2^{11/2}}{1\over\ab{\eta(\tau)}^{16}{\eta(\tau)}^{16}{\bar{\eta}(\bar{\tau})}^4}
\sum_\text{sector $\alpha\vec w$}\tilde Z_{T^2,\alpha\vec w};
}
see Eq.~\eqref{to Einstein}.
We plot this potential in Fig.~\ref{Einstein potential}.
Important fact is that the potential in the Einstein frame becomes runaway for the large radius limit $r\gg\sqrt{\alpha'}$. As discussed above, this behavior should not be altered by the higher loop corrections.

Note that this effective potential in the Einstein frame is reliable only for large $r\gg\sqrt{\alpha'}$ since the treatment in terms of the effective field theory~\eqref{potential in lower dimensions} becomes valid only in this limit; furthermore, we can regard $r$ as the physical radius only in this limit; see also the argument around Eq.~\eqref{large r limit summation}.

\subsection{Large boost limit}
We want to examine the behavior of the Higgs potential in the large field limit.
However, in this nine dimensional toy model, there are two flat directions {\color{black} at this level}, namely, $A$ and $R$.
If the Higgs comes from a similar mechanism to the gauge-Higgs unification,
the Higgs field should be identified with $A$.
Therefore, we check the large $A$ limit for a fixed $R$.
This limit is nothing but the large boost limit as is easily seen from Eq.~\eqref{A-id}: $\eta\rightarrow \infty$.
From Eqs.~\eqref{r-id} and \eqref{tautilde},
the trajectory in the $\tilde\tau_1$-$\tilde\tau_2$ plane is given by
\al{
&\tilde\tau_1={R\over\sqrt{\alpha'}}\tanh\eta,\nonumber\\
&\tilde\tau_2={R\over\sqrt{\alpha'}}{1\over\cosh\eta}.
	\label{trajectory of constant R}
}
Since $\tilde\tau_1^2+\tilde\tau_2^2=R^2/\alpha'$, this path starts from $\paren{0,\,R/\sqrt{\alpha'}}$ for $\eta=0$, and moves on the circle toward $\paren{R/\sqrt{\alpha'},\,0}$ as $\eta\to\infty$.
The question is what this trajectory is when mapped onto the fundamental region.
The large $\eta$ behavior depends on the value of $R/\sqrt{\alpha'}$:
\begin{itemize}
\item If $R/\sqrt{\alpha'}\in \sqrt{2}\mathbb Q$, then $\tilde\tau_2$ ($=r/\sqrt{\alpha'}$) goes to infinity in the large $\eta$ limit.
This can be seen as follows.
Since $\tilde\tau\to R/\sqrt{\alpha'}$ as $\eta\to\infty$, let us check to what point $R/\sqrt{\alpha'}$ is mapped in the fundamental region.
Let us write $R/\sqrt{\alpha'}=\sqrt{2}p/q$ with $p,q\in\mathbb Z$.
By an appropriate times of $\sqrt{2}$-shifts ($T$-transfomation in \eqref{T-duality transformation}), we can always make $\ab{p}<\ab{q}$.
Performing the inversion ($S$-transfomation in \eqref{T-duality transformation}), and again doing an appropriate times of $\sqrt{2}$-shifts, we can make the numerator $p$ smaller and smaller; eventually we get $p/q\to 0$.
This corresponds to the infinity $\tau_2\to\infty$ in the fundamental region.

This behavior is expected from the discussion of the general momentum boost in Sec.~\ref{momentum boost}.
In fact, if and only if $R/\sqrt{\alpha'}\in \sqrt{2}\mathbb Q$, we can have a lattice point on the light cone in the momentum space, that is, there exist $n,m,w\in\mathbb Z$ such that\footnote{
This can be proved as follows.
First we show that the conditions~\eqref{lL pR condition} and \eqref{pL condition} can be met for an arbitrary $R/\sqrt{\alpha'}\in\sqrt{2}\mathbb Q$ by an appropriate choice of $n,m,w$.
Let us write $R/\sqrt{\alpha'}=\sqrt{2}q_n/q_d$ with $q_n,q_d\in\mathbb Z$.
The condition~\eqref{pL condition} reads $nq_d/q_n+2wq_n/q_d=0$.
We can choose $n$ and $w$ such that $n=n'q_n$ and $w=w'q_d$ with $n',w'\in\mathbb Z$, resulting in the condition $n'q_d+2w'q_n=0$. This can be satisfied by setting $w'=q_d$ and $n'=-2q_n$.
Then the condition~\eqref{lL pR condition} reads
$0	\stackrel{!}{=}
		m+{1\over2}\paren{n'q_d-2q_nw'}
	=	m+n'q_d$, which can be satisfied by choosing $m=-n'q_d$.

Next we show that if $R/\sqrt{\alpha'}\slashed\in\sqrt{2}\mathbb Q$, there is no set of $n,m,w\in\mathbb Z$ that satisfies Eqs.~\eqref{lL pR condition} and \eqref{pL condition}.
By putting Eq.~\eqref{pL condition} into Eq.~\eqref{lL pR condition}, we get the condition
$m\pm \sqrt{2}n{\sqrt{\alpha'}\over R}=0$. Therefore, it is necessary that $R/\sqrt{\alpha'}=\mp\sqrt{2}n/m$ with $n,m\in\mathbb Z$.
}
\al{
l_L^2-p_R^
&={2\over\alpha'}
\sqbr{m+{1\over\sqrt{2}}\left({n\over R/\sqrt{\alpha'}}-{R\over\sqrt{\alpha'}}w\right)}
\sqbr{m-{1\over\sqrt{2}}\left({n\over R/\sqrt{\alpha'}}-{R\over\sqrt{\alpha'}}w\right)}
=0,	\label{lL pR condition}\\
p_L^2&={1\over\alpha'}\left({n\over R/\sqrt{\alpha'}}+{R\over\sqrt{\alpha'}}w\right)^2=0.
	\label{pL condition}
}
For $R/\sqrt{\alpha'}\in\sqrt{2}\mathbb Q$, there is a point on the light cone in the momentum space. Following the argument of Sec.~\ref{momentum boost}, the Lorentz boost between $l_L$ and $p_R$ opens up a new dimension.

\item
If $R/\sqrt{\alpha'}\slashed\in \sqrt{2}\mathbb Q$, the potential becomes either periodic or chaotic.
Let us check in what case we get the periodic potential.
\begin{itemize}
\item
The periodic case is realized if, starting from a point $\tilde{\tau}$~\eqref{trajectory of constant R} with the boost $\eta$, we get another point on the trajectory with the boost $\eta+\eta_c$,
\al{
\tilde\tau_1'	&=	{R\over\sqrt{\alpha'}}\tanh\fn{\eta+\eta_c}, \nonumber\\
\tilde\tau_2'	&=	{R\over\sqrt{\alpha'}}{1\over\cosh\fn{\eta+\eta_c}},
}
which can be mapped from $\tilde\tau$ by an appropriate T-dual transformation~\eqref{general transformation}.

In general, the transformation of $\tilde\tau_2$ is as shown in Eq.~\eqref{explicit T-dual}, and we get
\al{
\tilde\tau_2'
&={\tilde\tau_2\over\ab{c\tilde\tau+d}^2}={\tilde\tau_2\over c^2{R^2\over\alpha'}+d^2+2cd\tilde\tau_1}
={R\over\sqrt{\alpha'}}{1\over \paren{c^2{R^2\over\alpha'}+d^2}\cosh\eta+2cd{R\over\sqrt{\alpha'}}\sinh\eta}\nonumber\\
&={R\over\sqrt{\alpha'}}{1\over\cosh\fn{\eta-\eta_2}},
}
where we have defined $\eta_2$ by
\al{
\tanh\eta_2:=-{2cd{R\over\sqrt{\alpha'}}\over c^2{R^2\over\alpha'}+d^2}.
}
On the other hand,
the same transformation maps $\tilde{\tau}_1$ to
\al{
\tilde\tau_1'&={ac\ab{\tilde\tau}^2+\paren{ad+bc}\tilde\tau_1+bd\over \ab{c\tilde\tau+d}^2}
={ac{R^2\over\alpha'}+bd+\paren{ad+bc}\tau_1\over\cosh\fn{\eta-\eta_2}/\cosh\eta}\nonumber\\
&={\paren{ac{R^2\over\alpha'}+bd}\cosh\eta+{R\over\sqrt{\alpha'}}\paren{ad+bc}\sinh\eta\over\cosh\fn{\eta-\eta_2}}\nonumber\\
&=
{R\over\sqrt{\alpha'}}{\sinh\fn{\eta-\eta_1}\over\cosh\fn{\eta-\eta_2}},
}
where we have defined $\eta_1$ by
\al{
\tanh\eta_1=-{ac{R^2\over\alpha'}+bd\over {R\over\sqrt{\alpha'}}\paren{ad+bc}}.
}
The trajectory becomes periodic if and only if $\eta_1=\eta_2$, that is,
\al{
{2cd{R\over\sqrt{\alpha'}}\over c^2{R^2\over\alpha'}+d^2}
	&=	{ac{R^2\over\alpha'}+bd\over {R\over\sqrt{\alpha'}}\paren{ad+bc}},
}
or
\al{
\paren{d^2-c^2{R^2\over\alpha'}}\paren{bd-ac{R^2\over\alpha'}}
	&=	0.
}
Vanishing first factor means $\eta_2=\infty$, and the finite period is obtained when and only when the last factor becomes zero:
\al{
bd-ac{R^2\over\alpha'}
	&=	0.
}
Therefore, the partition function becomes periodic if and only if $R^2/\alpha'$ can be written as
\al{
&
{R\over \sqrt{2\alpha'}} \slashed\in \mathbb{Q}, &
&{R^2\over \alpha'}={bd\over ac},
		\label{periodicity condition}
}
where $ad-bc=1$ and either $a,d\in\sqrt{2}\mathbb Z$, $b,c\in\mathbb Z$ or $a,d\in\mathbb Z$, $b,c\in\sqrt{2}\mathbb Z$.

\item In particular, if $R^2/\alpha'$ is an irrational number then the condition~\eqref{periodicity condition} cannot be met (unless $ac=0$ that leads to the trivial $\eta_1=0$), and the partition function becomes non-periodic, namely chaotic.
\end{itemize}
\end{itemize}
\begin{figure}[t]
\begin{center}
\hfill
\includegraphics[width=.32\textwidth]{Rsqrt2gray.pdf}
\hfill
\includegraphics[width=.32\textwidth]{R2gray.pdf}
\hfill
\includegraphics[width=.32\textwidth]{R2_13gray.pdf}
\hfill\mbox{}
\caption{The trajectory that starts from $\eta=0$ at $\paren{\tilde\tau_1,\tilde\tau_2}=\paren{0,R/\sqrt{\alpha'}}$ for a fixed value of $R/\sqrt{\alpha'}$ being $\sqrt{2}$, 2, and $2^{1/3}$ in the left, center, and right panels, respectively, showing the runaway, periodic, and chaotic limits.
We have shaded the fundamental region for the T-dual transformations.
}\label{T-duality}
\end{center}
\end{figure}
As a check, we show the numerical results for $R/\sqrt{\alpha'}=\sqrt{2}$, $2$, and $2^{1/3}$ in Fig.~\ref{T-duality}.
We see that they show the runaway, periodic, and chaotic limits, respectively.
The result presented in this section provides a concrete example of the general argument presented in Sec.~\ref{classification}.
It is plausible that the large Higgs field limit in string theory fits into either one of these three.

Note that our computation is based on the one-loop effective potential and that the higher order corrections are significant around the region $A,R^{-1}\sim M_\text{s}$ ($=1/\sqrt{\alpha'}$). Therefore, the result so far should be interpreted as an effort to guess what is the physical large field limit along a potential valley after including all the higher order corrections.
In Fig.~\ref{T-duality}, we have checked the large $A$ limit for a fixed $R$. Is this a physical limit, and if not, what should it be?
Comparing Figs.~\ref{Jordan potential} and \ref{Einstein potential}, we see that it is a generic feature that there is a runaway vacuum no matter what the structure is around $A,R^{-1}\sim M_\text{s}$. It seems plausible that if the physical large $A$ limit is not the one with fixed $\tilde\tau_2$, then large $A$ limit goes into the runaway vacuum after all. However, we consider all the three limits, runaway, periodic, and chaotic in order not to loose generality.

\begin{figure}[t]
\begin{center}
\hfill
\includegraphics[width=.5\textwidth]{potential.pdf}
\hfill\mbox{}
\caption{
Schematic figure for the Higgs potential. Low energy side is determined phenomenologically. High energy side represents a runaway direction in the multi degrees of freedom space.
}\label{potential}
\end{center}
\end{figure}

As said above, the extrapolation from the low energy data has revealed that there is the quasi-flat direction of the Higgs potential in the SM.
We are interested in the potential for the large field values.
Beyond the string or Planck scale, there opens up several quasi-flat directions in general. Therefore we need to consider a multi-dimensional field space. In the example examined in this section, it corresponds to the $A$-$R$ (or $\tilde\tau_1$-$\tilde\tau_2$) plane. As we have seen in this section, generally there is at least one runaway direction in this space that corresponds to opening up an extra dimension; see Fig.~\ref{potential}. We will discuss its physical implications in the subsequent sections.

\section{Eternal Higgs inflation}\label{eternal section}

As shown in Introduction, the Higgs potential $V\sim\lambda_\text{eff}\ab{H}^4$ in the SM shows a quite peculiar behavior when extrapolated to very large field values: all of the $\lambda_\text{eff}$, its running, and the bare Higgs mass can be accidentally small.
In Ref.~\cite{Hamada:2014iga}, we have proposed a possibility that this behavior, so to say the criticality, is a consequence of the Planck scale physics and that the criticality is closely related to the cosmic inflation.

\begin{figure}[tn]
\begin{center}
\hfill
\includegraphics[width=.7\textwidth]{domainwall.pdf}
\hfill\mbox{}
\caption{Schematic figure for the maximum that yields the domain wall, which becomes the source for the eternal inflation.
}\label{domainwall}
\end{center}
\end{figure}
We have seen that the large field limit goes down to a runaway direction, which corresponds to opening up an extra dimension, in the multi degrees of freedom space, as shown in Fig.~\ref{potential}.
Therefore, there is at least one maximum of the potential around the Planck scale; see Fig.~\ref{domainwall}.
This maximum can be a source of an eternal inflation at the core of the domain wall~\cite{Hamada:2014raa} between the electroweak vacuum and the runaway vacuum, in which the fifth dimension is opened up. In order for this to work, the curvature of the potential at the maximum must be sufficiently small~\cite{Sakai:1995nh}:
\al{
\left.M_P^2{V_{\varphi\varphi}\over V}\right|_\text{maximum}\lesssim 1.4.
	\label{DW condition}
}
In our scenario, this can be naturally satisfied as follows.
The potential for the fifth dimension can be seen by putting $D=5$ in Eq.~\eqref{to Einstein}. In stringy language, the action for the fifth dimension $R'\gg M_\text{s}^{-1}$ is coming from the one-loop potential: In the Einstein frame, we get
\al{
S_\text{eff}
	&\sim	{M_\text{s}^2\over g_\text{s}^2}\int \text{d}^4x\sqrt{-g}
			\paren{
				\mathcal R-{\paren{\partial R'}^2\over R'^2}-g_\text{s}^2M_\text{s}^2{1\over R'}
				}.
}
Switching to the canonical field $R'=e^{g_\text{s}\chi/M_\text{s}}$, we get
\al{
S_\text{eff}
	&\sim	\int \text{d}^4x\sqrt{-g}\paren{
				M_\text{P}^2\mathcal R-\paren{\partial\chi}^2-e^{-\chi/M_\text{P}}M_\text{s}^4
				},
}
where $M_\text{P}=M_\text{s}/g_\text{s}$.
Therefore, the stringy potential also gives
\al{
V_{\chi\chi}
	&\sim	{V\over M_\text{P}^2}.
}
It is remarkable that the potential changes of order unity when we vary $\chi$ by $M_\text{P}$, not by $M_\text{s}$, for large $\chi$. On the other hand at low energies, the SM potential in the Einstein frame exhibits the same behavior if the non-minimal coupling $\xi$ is of order ten~\cite{Hamada:2014wna}.
Therefore, it is natural to conclude that the condition~\eqref{DW condition} is also met around the maximum.

We note that in the original version of the topological Higgs inflation~\cite{Hamada:2014raa}, $\xi$ is used to make the maximum of the potential, and hence that it cannot account for the observed fluctuation of the cosmic microwave background (CMB). On the other hand, the scenario proposed in this paper allows the Higgs to be the source for both the eternal topological inflation and for the one that accounts for the CMB fluctuation, simultaneously.

\begin{figure}[tn]
\begin{center}
\hfill
\includegraphics[width=.5\textwidth]{runaway.pdf}
\hfill\mbox{}
\caption{
Schematic figure for the Higgs potential smoothly connected to the runaway direction.
}\label{runaway figure}
\end{center}
\end{figure}
\begin{figure}[tn]
\begin{center}
\hfill
\includegraphics[width=.4\textwidth]{tunneling1.pdf}
\hfill
\includegraphics[width=.5\textwidth]{tunneling2.pdf}
\hfill\mbox{}
\caption{
Schematic figure for the Higgs potential. On the left, the false vacuum has higher energy than the quasi-flat potential in the SM, while on the right, it has lower energy.
}\label{false vacuum}
\end{center}
\end{figure}
There are two possibilities for the potential beyond the maximum:
\begin{itemize}
\item The potential smoothly becomes runaway as in Fig.~\ref{runaway figure}.
\item The potential has another local minimum as in Fig.~\ref{false vacuum}.
\end{itemize}
In the latter case, the false vacuum gives another mechanism of eternal inflation.
This situation is similar to some of the originators' idea of the inflation using a first order phase transition~\cite{Sato:1980yn,Guth:1980zm}. In the medium of the false vacuum, which is indicated by the (red) dot in Fig.~\ref{false vacuum}, there appears a bubble of the electroweak vacuum due to the tunneling, which is indicated by the dotted arrow.
This eternal inflation in the false vacuum had caused the so-called the graceful exit problem in the old inflation scenario~\cite{Linde:1981mu,Hawking:1982ga,Guth:1982pn}.
However in the left case in Fig.~\ref{false vacuum}, the space inside the bubble experiences the second stage of inflation~\cite{Hamada:2014iga,Hamada:2014wna}, after the dotted arrow in the figure, and hence this problem is ameliorated as we do not need bubbles to collide.
In the right case in Fig.~\ref{false vacuum}, we need another inflation to account for the observed CMB fluctuation such as the $B-L$ Higgs inflation.

\section{Cosmological constant}\label{solution to cosmological constant}
As is reviewed in detail in Appendix~\ref{MPP review}, the MPP requires  degenerate vacua at the field value of the order of the Planck scale~\cite{Froggatt:1995rt,Froggatt:2001pa,Nielsen:2012pu}.
The cosmological constant of the runaway vacuum is exactly zero.
Then the MPP tells us that our electroweak vacuum must have the zero cosmological constant too.
This is a new solution to the cosmological constant problem in terms of the MPP.\footnote{
See also Ref.~\cite{Nielsen:2012pu} in which the cosmological constant problem is discussed in a different perspective.
}

On the other hand,
the current universe is being dominated by the cosmological constan
~\cite{Ade:2013zuv}
\al{
\rho_\Lambda^\text{obs}

	\simeq
		\paren{2.2\,\text{meV}}^4,
		\label{cosmological constant observed}
}
and is entering the second inflationary stage.
This will eventually lead to the de Sitter space $dS_4$ with the length scale $H^{-1}$, where
\al{
H^2	=	{\rho_\Lambda^\text{obs}\over 3M_P^2}.

}

We will discuss the possibility that the existence of the finite cosmological constant is understood as a statistical fluctuation.

\begin{figure}[tn]
\begin{center}
\hfill
\includegraphics[width=.4\textwidth]{universe.pdf}
\hfill\mbox{}
\caption{
Universe is divided into parts that will eventually become causally disconnected to each other in the end of their histories.
}\label{universe}
\end{center}
\end{figure}
First we point out that our universe is a part of a large universe
whose cosmological constant is fixed to zero by the MPP.\footnote{
The argument in this section may also apply for the multiverse~\cite{Kawai:2011qb,Kawai:2013wwa,Hamada:2014ofa,Kawana:2014vra,Hamada:2014xra}.
}
The large universe can be divided into parts that will eventually become causally disconnected de Sitter spaces
in the end of their histories, as in Fig.~\ref{universe}.
After the Euclideanization, each de Sitter space becomes $S^4$ with radius $r_U=1/H$.

We consider one of the $S^4$'s and latticize it by the lattice spacing of the order of $\l_P=1/M_\text{P}$, and
let $S_i$ be the action on each site labeled by $i$.
The total action for the $S^4$ becomes the sum over positions:
\al{
S	&=	\sum_{i=1}^{r_U^4/l_P^4} S_i.
}
Assuming that $S_i$ are independent of each other,
the vanishing cosmological constant for the large universe leads to $\Braket{S_i}=0$ for each $i$ and in particular to $\Braket{S}=0$ for this part.
Therefore the value of $S$ fluctuates around zero and its variance can be evaluated as
\al{\label{fluctuation}
\Braket{S^2}\sim
	N:={r_U^4\over l_P^4},
}
where we have assumed that the variance of each $S_i$ is of order unity.

We interpret Eq.~\eqref{fluctuation} as the variance of the actions of the $S^4$'s in the large universe.
Then the typical amount of the energy density of one $S^4$ is estimated as
\al{
\rho_\Lambda
	\sim	{\sqrt{\Braket{S^2}}\over r_U^4}
	\sim	{1\over l_P^2 r_U^2}\sim\paren{\ensuremath{\,\text{meV} }}^4.
}
Thus, we have obtained the right amount of the cosmological constant as
the fluctuation from zero.
{\color{black} This result has been obtained in Ref.~\cite{Sorkin:2007bd} in the context of causal set theory.}

We note that the value of $H$ is not really a prediction in this argument. We have rather provided a consistent explanation of having a finite amount of the cosmological constant, even though it is fixed to be zero for the large universe.

\section{Summary \color{black} and discussions}\label{summary}
We have studied possible large field limits of the SM Higgs, assuming that it is coming from a massless state at the tree level in heterotic string theory with its supersymmetry broken at the string scale.
In the toroidal compactification, putting a background for such a massless state corresponds to a boost in the momentum lattice. We have classified the large boost limits with fixed radius into three categories: runaway, periodic, and chaotic.

As a concrete toy model, we have examined the ten-dimensional $SO(16)\times SO(16)$ non-supersymmetric heterotic string, with a dimension being compactified on $S^1$ with the radius $R$.
We have considered the large field limit of a Wilson line on the $S^1$ with fixed $R$, and reproduced these three limits.
We have argued that this behavior is universal if the zero momentum limit of the emission vertex of the Higgs is written as a product of holomorphic (1,0) and anti-holomorphic (0,1) operators, not only in the case of toroidal compactification.
In the known models of fermionic construction and of orbifolding, the emission vertex tends to be written as such a product, and our argument applies for these wide class of models.

Physically several degrees of freedom appears when the Higgs field value becomes larger than the Planck scale.
We have argued that there exists an runaway direction in this multi degrees of freedom space. This runaway vacuum corresponds to opening up an extra dimension.

It is noteworthy that this potential fits into the criteria of the MPP proposed by Froggatt and Nielsen.
The MPP requires that the electroweak vacuum is degenerate with this runaway vacuum, and hence that the cosmological constant of the electroweak vacuum is tuned to be zero in the large universe.
We have speculated that the observed amount of the cosmological constant can be understood as a fluctuation from zero in the framework of the MPP.

We may get the eternal inflation from this potential.
It is realized either as a topological inflation at the domain wall between the two vacua or as a decay from the false vacuum that traps the Higgs field.
In both cases, the Higgs field, which is rolling down the potential, may cause the succeeding inflation, which accounts for the observed CMB fluctuations, along the quasi-flat potential around the critical point.

It would be interesting to study the limit in more realistic SM-like model with the orbifolding, fermionic constructions, etc; see e.g.\ Ref.~\cite{Blaszczyk:2014qoa}.

{\color{black}
Finally we comment on the dilaton potential. Though we consider the general compactifications which may not even have a geometric interpretation, let us illustrate the situation starting from a conventional ten dimensional string theory. The low energy effective action in ten dimensions reads
\al{
S	&=	{M_\text{s}^8\over g_\text{s}^2}\int\text{d}^{10}x\sqrt{-g}\,e^{-2\Phi}
			\paren{\mathcal R+4\partial_\mu\Phi\,\partial^\mu\Phi+\cdots}\nonumber\\
	&\quad	+M_\text{s}^{10}\int\text{d}^{10}x\sqrt{-g}\paren{-C+\cdots}\nonumber\\
	&\quad	+\mathcal O\fn{g_s^2e^{2\Phi}},
}
where $\Phi$ is the dilaton field and $C$ is the dimensionless cosmological constant induced at the one-loop level. We note that in this string frame, $g_\text{s}$ and $\Phi$ always appear in the combination $g_\text{s}e^\Phi$. After the compactification,
\al{
S	&=	{M_\text{s}^2\over g_\text{s}^2}\paren{M_\text{s}^6V_6}\int\text{d}^4x\sqrt{-g_4}\,e^{-2\Phi}
			\paren{\mathcal R_4+4\partial_\mu\Phi\,\partial^\mu\Phi+\cdots}\nonumber\\
	&\quad	+M_\text{s}^4\paren{M_\text{s}^6V_6}\int\text{d}^4x\sqrt{-g_4}\paren{-C+\cdots}\nonumber\\
	&\quad	+\mathcal O\fn{g_s^2e^{2\Phi}},
		\label{string frame}
}
where $V_6$ is the compactification volume. Switching to the Einstein frame, we get
\al{
S	&=	{M_\text{s}^2\over g_\text{s}^2}\paren{M_\text{s}^6V_6}\int\text{d}^4x\sqrt{-g_E}
			\paren{\mathcal R_E-2\partial_\mu\Phi\,\partial^\mu\Phi+\cdots}\nonumber\\
	&\quad	+M_\text{s}^4\paren{M_\text{s}^6V_6}\int\text{d}^4x\sqrt{-g_E}\paren{-C\,e^{4\Phi}+\cdots}\nonumber\\
	&\quad	+\cdots.
}
We see from the second line that the dilaton has the runaway potential $e^{4\Phi}$ for $\Phi\to-\infty$ if the cosmological constant $C$ is positive. In this limit, the expansion parameter $g_se^\Phi$ in Eq.~\eqref{string frame} becomes small, and the theory is weakly coupled. Since all the higher-loop corrections come with this combination as well, the runaway behavior is not altered by taking them into account. Therefore, this direction $\Phi\to-\infty$ necessarily comprises one of the runaway directions~\cite{Dine:1985he} in Fig.~\ref{potential}, and hence the arguments in Sections~\ref{eternal section} and \ref{solution to cosmological constant} apply quite generally.
}

\subsection*{Acknowledgement}
We thank Michael Blaszczyk, Stefan Groot Nibbelink, Orestis Loukas, S\'aul Ramos-S\'anchez, and Toshifumi Yamashita for useful comments.
This work is in part supported by the Grant-in-Aid for Scientific Research Nos.\ 22540277 (HK), 23104009, 20244028, and 23740192 (KO). The work of Y. H. is supported by a Grant-in-Aid for Japan Society for the Promotion of Science (JSPS) Fellows No.25$\cdot$1107.

redpajama_set_name: RedPajamaArXiv


Mid January. The fire in my fireplace, reduced to red embers, would be extinguished for another night. The house slept, except for Coco, our Dog Of All Breeds, who had ventured outside before she would settle in on the blanket in our bedroom. Ready to join my family I went onto the patio to call Coco and, while I waited her for to become aware that her nemesis, the chipmunk, would not appear from under 3 feet of snow, I was hypnotized by the clear January night.
All of life was in its proper place. The fresh snow, not yet ruined by the treads of snowmobiles, extended in many directions. With the exception of a night light on the porch of Sunset Mountain Lodge the dark embraced me and the only sound that occasionally echoed was a dull rumble that issued forth from the ice as the surface of the lake expanded under the freezing process. Frost covered tree branches hung peacefully on this windless night.
Across the lake, in the direction of Pilgrim Camp, First Brother Mountain disappeared into the sky as earth and heaven joined as one. The peak of the mountain drew my glance into a sky with only a sliver of a moon, a new moon, that enabled the stars to shine even brighter on a backdrop of black. There they were, constellations, in the configuration that had prevailed since earliest times. The Milky Way. Big Dipper. Leo. Holding fast. Unchangeable, no matter what may have been happening on earth.
In fact, this cold January night offered a paradigm for a world where each particle was arranged in perfect harmony. All was right. All was in place.
Then, suddenly, the cosmos erupted. A shooting star raced across the sky and, within seconds, burned out in some mysterious place in the universe. I have always been fascinated by shooting stars which, scientifically, are not stars; instead they are particles from comets that break off in space, enter the atmosphere at awesome speed and, rubbing against the air particles, cause friction, overheat and send balls of fire earthwards. In a brief moment the shooting star, or meteor, had disappeared, the flaming tail quickly fading. Once, in the realm of myth, the shooting star was thought to be cast down to earth as a gift from the angels, although the recipients of these gifts remains unknown. Believers in the dark life maintain that the gods are displaying anger, aglow in their incendiary existence.
When I was young, on a clear January night I would search the sky for the wondrous phenomena. But now, advanced in years, the meteor, appearing for but a second, warns me that in many ways our lives are like that flash in space. We live on the continuum of eternity for a brief instant and then, vanish, our life as tenuous as the shooting star.
But this realization need not be disheartening. Instead, with eyes wide open to that startling shot across the sky, I resolved to grasp life, aware of the awesome journey of simply living, and ride the fiery tail until it is no more.


text:
Sonny Chiba Dies From COVID-19
August 20 | 2021
dynaimage.cdn.cnn.com
Japanese martial arts film star Shinichi "Sonny" Chiba passed away Thursday from COVID-19 complications at the age of 82. Born Sadaho Maeda, Chiba was a gymnastics champion who began studying karate while a university student. A pupil of the late kyokushin founder, Mas Oyama, Chiba achieved a black belt under Oyama and went on to study several other martial arts. He began acting in the 1960s eventually starring in more than 100 films as he carved out a niche for himself with his own rough-hewn brand of action.
Following in the wake of Bruce Lee, Chiba became one of the first Asian cinematic martial arts stars to achieve international success when his 1974 movie "Gekitotsu! Satsujin ken" was released overseas under the title "The Street Fighter." The success of the film lead to two sequels and made Chiba a cult favorite. Among his admirers was Hollywood director Quentin Tarantino who cast Chiba in "Kill Bill: Volume 1." Chiba would also influence the action movie genre throughout Japan with his formation of a popular school for martial arts stuntmen, the Japan Action Club (later called Japan Action Enterprise).
10 Things You Didn't Know About Sho Kosugi and Ninja Warriors ... ›
King Boxer, Master of the Flying Guillotine and Other Awesome ... ›
Top 20 Martial Arts Actors of All Time - Black Belt Magazine ›
10 Things You Didn't Know About Sho Kosugi and Ninja Warriors! - Black Belt Magazine ›
Shin'ichi Chiba - IMDb ›
Sonny Chiba, martial arts star of 'Kill Bill,' dies of Covid-19 ... ›
Sonny Chiba - Wikipedia ›

redpajama_set_name: RedPajamaCommonCrawl


text:
Home biography Steven Avery
Steven Avery Biography
Facts of Steven Avery
1962 , July-9
Quick Timeline of Steven Avery
1962 Steven Avery was born
1981 First Crime committed
1982 Steven was married
1982 Sent jail for the second time
1985 Trio Crime
1988 Steven was divorced
2005 Arrested for killing a photographer .
2016 Currently Steven Avery
Detail Timeline of Steven Avery
Steven Avery was born
Steven Avery was born on July 9 , 1962 in Manitowoc County, Wisconsin, United States . His parents were Allan Avery and Dolores Avery . Their family had a salvage yard which was 40-acre . He has two brothers and one sister .
First Crime committed
Steven was sent to jail as early as the age of 18 in 1981 , he was found guilty for burglarizing a bar with a friend and was sent to jail for 10 months in the Manitowoc County Jail, he was released on probation in order to a pay restitution.
Steven was married
Steven met his wife in the age of 19 . Avery found his wife Lori Dassey very pretty and humble he told the media . Shortly , after they met they married in 1982 . They also have four children , which includes twins also .
Sent jail for the second time
Steven Avery was sent to jail in the late 1982 for the second time . This time it was for animal cruelty . One man admitted that , he and Avery took Avery's cat and poured gas and oil and threw it util it died in a bonfire . He was jailed for this crime until August 1983
Trio Crime
In the year 1985 , his crimes were tripple i.e he had comminted three crimes in the same year which assaulting his cousin, a Manitowoc County sheriff's deputy; possessing a firearm as a felon; and for the rape of Penny Beernsten.He was put into jail for 24 years for commiting three crimes .
Steven was divorced
In 1988 , after being married for six years the couple decided to get split . It was mainly because his wife Lori was depressed and she had sent a letter telling "I just can't handle these kids anymore. Between the crying, yelling, talking back and fighting I just can't anymore."
Arrested for killing a photographer .
In 2005 , Avery had a meeting with photographer Teresa Halbach who went missing then after in August .Her body was found in November , 2005 near Steve's Home .In 2007 , he was found guilty and was sentenced to life in prison .
Currently Steven Avery
Currently , Steven Avery has a American documentary based on his life Making a Murderer on netflix . The series explores the story of Steven Avery, a man from Manitowoc County, Wisconsin, who served 18 years in prison for the wrongful conviction of sexual assault and attempted murder of Penny Beerntsen,
Steven Avery met his wife when he was 19 years old , he told the media she was pretty and shortly after their meting they got married in 1982 . They had total of four children within three years , one of them were twins .After Steven was arrested and taken to jail , Lori Dassey got very tired and frustrated and wrote a letter to Avery telling "I just can't handle these kids anymore. Between the crying, yelling, talking back and fighting I just can't anymore." They got divorced in 1988 .
Steven was sent to jail as early as the age of 18 in 1981 , he was found guilty for burglarizing a bar with a friend and was sent to jail for 10 months .In 1982 , not even after one year he was found guilty of animal cruelty .
In 2005 , Avery had a meeting with photographer Teresa Halbach who went missing then after .Her body was found in nov , 2005 near Steve's Home .In 2007 , he was found guilty and was sentenced to life in prison .
In August, Zellner filed a motion . The motion claims that investigators framed Avery and the defense is trying to prove this using sophisticated DNA testing that was not available at the time of the trial .
Related Posts on Steven Avery


text:
BMC Video
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Burnett Medical Center History
The first hospital in Grantsburg opened on November 19, 1930 - one year after the organization was founded. The cost: $25,700. Forty-seven local residents purchased initial shares in the corporation, and more shares were sold as time went on. Two physicians and one surgeon were on the medical staff.
Throughout the 1930s, 1940s, and 1950s the hospital had its ups and downs. In 1953 the loss of practicing physicians in Grantsburg put the hospital on a month-to-month operational basis with the definite possibility of closing. The hospital limped along until another doctor was found to meet area health care needs.
In 1956, stockholders resolved to launch Grantsburg Hospital Inc., a new not-for-profit corporation, to secure the benefits of a not-for-profit hospital for all area residents. Hopes of remodeling the old Community Hospital structure turned toward building a new 31-bed facility on St. George Avenue. Citizens throughout the surrounding area in Wisconsin and Minnesota banded together to raise a full quarter of the total construction cost.
New Healthcare Facility
On an early June day in 1961, the new healthcare facility on St. George Avenue in Grantsburg opened, and the hospital grew. In 1970 a 53-bed extended care facility was added to meet the growing needs of an aging population. In 1985 an expanded emergency room, exam rooms, ambulance garage, and supply area were added as well.
For many years in the 1960s, 70s, and 80s the area was under-served with too few practicing physicians. Residents increasingly looked elsewhere for their health care. In the years between 1988 and 1993, Burnett General Hospital's economic size dropped. In 1990 Burnett General faced a financial loss of $390,942.
Determined that the hospital would survive, a forward-looking board of directors turned to Brim Healthcare to manage the hospital. Through Brim, the medical center drew national strength while keeping local control and its focus on the special needs of the people it serves. Operating margins began to improve. A new era had begun.
By 1995 the hospital had evolved from an inpatient care facility to a modern health-care provider of inpatient and outpatient services. Reflecting the hospital's growth, the hospital became Burnett Medical Center.
Time for Additional Growth
Now that Burnett Medical Center was headed in the right direction, a determination was made to provide a more appealing environment to the people served and at the same time organize services offered. To meet these objectives, in 1996 and 1997 Burnett Medical Center underwent a complete internal makeover. Many departments were moved and/or redesigned to provide more efficient work areas. Departments that worked closely together were realigned to be in close proximity to each other. A 10,000 square foot wing was added to the east end of the hospital to accommodate a new operating room and procedure room, a larger recovery area, an expanded Radiology department with a CT scan room, a storeroom to house needed supplies, and a loading dock. Existing spaces within the hospital and extended care facility were redecorated to make them more appealing and inviting to patients, residents, and visitors. The building renovation was completed in late 1997 and was a welcome addition to the now thriving hospital and extended care facility. The extended care facility has subsequently been renamed the Continuing Care Center and on March 1, 2010, BMC revised the Continuing Care Center license from 53 beds to 50 beds (23 double occupancy semi-private rooms and 4 private rooms).
A Desire to Purchase the Local Clinic
Burnett Medical Center had always worked closely with its next-door neighbor, the Grantsburg family practice clinic. The clinic, owned by Allina, needed immediate attention if it was going to prosper and remain open. After repeated requests by Burnett Medical Center to purchase the clinic, Allina, in 2001, finally agreed to sell. In November 2001, clinic physical assets were purchased from Allina and the Burnett Medical Center Clinic was born. With increased effort by Brim management, the Clinic was reestablished as a viable enterprise and is running strong and smoothly today with adequate medical personnel who provide a wide range of services to the entire Burnett Medical Center complex.
BMC Expansion Project
In the spring of 2004, Burnett Medical Center's board of directors finalized a strategic plan that included the development of a Master Facility Plan. The impetus for the plan was to improve the patient access points: primary care clinic, specialty clinics, and emergency department, as well as to expand the facility in order to accommodate the ever-increasing patient care space needs of their medical staff and visiting specialists. It was also the intent to fully integrate all healthcare services offered into the main hospital, for the convenience of their patients.
In July of 2004, Burnett Medical Center's board hired the architectural firm of BWBR to assist them in developing BMC's expansion plan. Initially they were asked to assemble a plan that did not include a parcel of BMC's land that was utilized by Grantsburg Village Golf Course. After reviewing that option, the board recognized that in order to remain at its current location, BMC would need to be able to utilize all of its available property.
The plan approved by Burnett Medical Center's board of directors at their November 2005 meeting allowed for southward expansion of the current facility while maintaining flexibility for future growth. Additionally, a new main entrance was decided on, providing access and parking directly off highway 70. While revisions to the golf course that surrounds BMC on three sides was necessary, working with the Village of Grantsburg ensured that the completion of the building project would continue to allow for a nine-hole golf course.
Expansion Project Progress
Burnett Medical Center has taken on a new shape since its groundbreaking ceremony in April 2007 that began PHASE I of it's building expansion project. The new expansion, completed in May 2008, includes a new main entrance, parking and driveway off of highway 70, an emergency department, combined family practice and specialty clinic, centralized registration, health information department, and a portion of the rehabilitation services department. PHASE II which began in May 2008, includes renovating portions of the current facility to include a new business office and hospital pharmacy, expanded hospital lab and rehabilitation services departments, a new Chapel/Grief room, additional office and conference space, and a new dining room. Administrative offices that include Human Resources, Accounting, Administrative Assistant services, and Marketing were renovated as part of PHASE III completed in November 2008. Also completed as part of this phase were new Activities and Social Services offices.
Since completion of the building expansion which greatly enhanced the patient experience here at Burnett Medical Center, we set out to update and upgrade our medical technologies to further serve our patients more efficiently and effectively. Some of the recent technology upgrades include:
In July of 2009, Burnett Medical Center began a project to implement an Electronic Medical Record (EMR) system. An EMR system simply means that patient history and care information is stored in a computer system. There are many reasons why we have undertaken this important project. The foremost reason is to improve the quality of patient care. The EMR allows our providers to access a patient's information regardless of where the patient is seen within the facility. For example, an allergy that was documented during a patient's visit to our clinic will be easily accessible if that same patient presents to our emergency department.
Implementation of the EMR allows us to better track the care provided and ensure that charges are made correctly and in a timely manner. In addition, we are able to streamline the processes in our patient care departments to both control costs and improve the quality of care.
Only those employees who need access to a patient's information are allowed to access it. Ensuring the privacy of our patients is accomplished through the security and tracking features that are a part of the new EMR system, along with the policies and procedures that are already in place.
Omnicell
Also, in 2009, Burnett Medical Center acquired an Omnicell automated medication dispensing system for it's Pharmacy Department. The benefits of an Omnicell System include:
Overall, better quality of care
Faster access to first dose, emergency medications, and new medication orders
Complete tracking of dispensed medications within the facility
Fewer mistakes caused by manual steps and workarounds
Improved patient safety resulting in increased patient satisfaction
Reduced costs for the Pharmacy Department and the facility
The benefits are very clear in having this secure, accurate, and efficient medication management system here at Burnett Medical Center.
In early 2013, Burnett Medical Center acquired a new piece of equipment called a Vitros 5600 for the Laboratory Department. The Vitros 5600 is a state of the art, integrated analysis system that combines the workload of many laboratory instruments on to a single platform. This integration provides greater efficiency, productivity, and testing capacity within the facility. The analyzer can perform over 120 different tests, providing Burnett Medical Center physicians more diagnostic information in less time.
The addition of digital x-ray equipment in the winter of 2013 marked the last step in transforming Burnett Medical Center's Radiology Department into a completely digital department. Now with digital imaging, radiographic images are no longer on film but performed and stored digitally. Digital technology allows instant access to radiology images by the patient care team without duplication or transporting of films. Digital images provide better dynamic range and contrast, improving diagnostic accuracy in less time. The Radiology Department has been renamed Diagnostic Imaging to more accurately reflect the services provided within the department.
The Burnett Medical Center Foundation provided funding to purchase a new system for the anesthesia department at BMC which can be utilized during surgery or an emergency. The system, known as a GlideScope video laryngoscope, is an airway device with a video camera that aids in the placement of breathing tubes in patients with difficult and unpredictable airways. Because seconds count during anesthesia or emergencies, the GlideScope ensures greater patient safety and care.
Burnett Medical Center's ongoing efforts to expand and improve our facilities and services will help us facilitate our mission to provide compassionate, comprehensive, community healthcare that exceeds our customer's expectations.
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I truly love this bakery spot in La Marqueta. The owner is just as sweet as her treats. They have delicious Latin flavored treats and awesome coffee. Oh and their pineapple cake...yumm!
My wife and I were so pleased with our purchase! Not only did the cake taste so delicious but everyone kept talking about how nice the cake looked sprinkle splash will definitely receive more business from us! And guys don't think twice about ordering your next birthday cake from here, worth every penny!!!!


text:
SMAP 009 est le neuvième album studio sorti le  et le premier sans l'un des membres originaux, Katsuyuki Mori, qui a quitté le groupe en .

Détails de l'album
L'album sort le  en une seule édition sur le label Victor Entertainment après le précédent album SMAP 008 TACOMAX sorti en .

Il atteint la  du classement hebdomadaire des ventes de l'Oricon.

L'opus comprend au total 10 titres dont un single sortis quelques mois auparavant dont : Hadaka no Ōsama ~Shibutoku Tsuyoku~ (en ). L'un des membres, Katsuyuki Mori (qui a quitté le groupe en mai 1996, avant la sortie du disque), fait sa seule apparition sur le single et n'est cependant pas crédité sur l'album.

Formation
Membres crédités sur l'album :
 Masahiro Nakai : leader ; chœurs
 Takuya Kimura : chant principal (en solo sur la piste n°4)
 Goro Inagaki : chœurs
 Tsuyoshi Kusanagi : chœurs
 Shingo Katori : chœurs (en solo sur la piste n°6)

Membre autre:
 Katsuyuki Mori : chant principal (ne participe que sur la piste n°10) [non crédité]

Liste des titres

Liens externes
  Fiche de l'album sur le site de l'Oricon
  Fiche de l'album sur le site de Victor Entertainment

Album musical sorti en 1996
Album de SMAP
Album publié par Victor Entertainment

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NuGet Gallery | Packages matching Tags:"codegenerator"
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CodeGenHero is an amazing Visual Studio extension that enhances developer productivity via code generation. This package should be added to projects that are the source CodeGenHero templates (template generators).
CodeGenHero is an amazing Visual Studio extension that enhances developer productivity via code generation. This package should be added to projects that are the target of CodeGenHero templates implementing the repository pattern.
CodeGenHero is an amazing Visual Studio extension that enhances developer productivity via code generation. This package should be added to projects that are the target of CodeGenHero templates that use logging.