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\title{\Large Renormalization of Hamiltonian Field Theory; a non-perturbative and non-unitarity approach}
\bigskip
\author{Amir H. Rezaeian}
\email{Rezaeian@Theory.phy.umist.ac.uk}
\author{Niels R. Walet}
\email{Niels.Walet@umist.ac.uk}  
\affiliation{Department of Physics, University of Manchester\\ Institute of Science and Technology, PO Box 88, Manchester, M60 1QD, UK}
\date{\today}


\begin{abstract}
Renormalization of Hamiltonian field theory is usually a rather
painful algebraic or numerical exercise. By combining a method based on
the coupled cluster method, analysed in detail by Suzuki and Okamoto,
with a Wilsonian approach to renormalization, we show that a powerful
and elegant method exist to solve such problems. The method is in
principle non-perturbative, and is not necessarily
unitary. Nevertheless, we show that we can apply a loop expansion in
the method. As an illustrative example, we obtain results for the
one-loop renormalized extended Lee model.
\end{abstract}
\pacs{11.10 Gh, 11.10.Ef, 21.30.Fe}
\maketitle
\newpage
\section{Introduction}
Although many attempts have been made during the last few decades to
understand strongly interacting relativistic systems, a satisfactory
and definitive theory still does not exist. Strongly interacting
\emph{non-relativistic} systems can be successfully described by using
traditional many-body theories \cite{h}, but the relativistic bound
state is much more complicated. Two important areas where the relativistic
bound-state issue plays a role are in relativistic nuclear physics and
in quantum chromodynamic~(QCD).

In these areas, the concept of effective field theory (EFT) has also
drawn substantial attention in recent years (see, e.g.,
Refs.~\cite{1,2}). The basis of the EFT concept is the recognition of
the occurrence of different energy scales in nature, where each energy
scale has its own characteristic degrees of freedom. In strong
interactions the transition from the fundamental to effective level is
induced by a phase transition that takes place around
$\lambda_{QCD}\simeq 1 \text{GeV}$ via the spontaneous breaking of
chiral symmetry, which generates pseudoscalar Goldstone bosons.  This
coincides, of course, with the emergence of nuclei and nuclear mather,
as opposed to the quark-gluon plasma and quark matter expected to
occur at high temperature and high density. Therefore, at low energies
$(E<\lambda_{QCD})$, the relevant degrees of freedom are not quarks
and gluons, but pseudoscalar mesons and potentially other hadrons. The
resulting description is a chiral EFT, which has much in common with
traditional potential models. In particular, there might be an
intermediate regime where a non-relativistic model is inadequate but
where relatively few hadronic degrees of freedom can be used to
faithfully describe both nuclear structure and response. If this is
the case, one should be able to describe strongly interacting hadronic
systems in this effective model.


The power of Hamiltonian methods is well known from the study of
non-relativistic many-body systems and from strongly-interacting few
particle systems, even though a Lagrangian approach is usually chosen
for relativistic theories. Hamiltonian methods for
strongly-interacting systems are intrinsically non-perturbative and
usually contains a Tamm-Dancoff type approximation, in the sense that,
in practice, one has to limit the bound state as an expansion over
states containing a small number of particles. This truncation of the
Fock space give rise to a new class of non-perturbative divergences,
since the truncation does not allow us to take into account all
diagrams for any given order in perturbation theory. Therefore
renormalization issues have to be considered carefully. Two very
different remedies for this issue are the use of light-front
Tamm-Dancoff field theory (LFFT) 
\cite{3} and the application of the coupled cluster method (CCM) \cite{4,5}. 
In the LFFT the quantisation plane is chosen to coincide with the 
light front, therefore the divergences that plagued the original theories seem
to disappear \cite{6}, furthermore, not having to include interactions
in boost operators allows a renormalizable truncation scheme
\cite{7}. One of the most important difficulties in LFFT is the
complicated structure of the renormalization process \cite{8}. In the
standard form of CCM, on the other hand, the amplitudes obey a system of coupled
non-linear equations which contain some ill-defined terms because of
ultraviolet divergences. It has been shown \cite{9} that the
ill-defined amplitudes, which are also called critical topologies, can
be systematically removed, by exploiting the linked-cluster property
of the ground state. This can be done by introducing a mapping which
transfers them into a finite representations without making any
approximation such as a coupling expansion. Thus far this resummation
method has been restricted to supernormalizable theories due to its
complexity.


A natural question in the renormalization within the Hamiltonian
formalism arises, since one could also perform renormalization in a
Lagrangian framework and finally construct the corresponding
Hamiltonian by means of standard Legendre transformation.~[It should
be noticed that this is not generally applicable if the Lagrangian
contains higher-order time-derivatives.] The Hamiltonian formalism,
despite a certain lack of elegance, has the advantage that it is very
economical, and one can use all the know-how of quantum many-body
theory. In the last decade extensive attempts have been made to give a
prescription for renormalization within the Hamiltonian formalism
\cite{10}. Commonly unitary transformations are used to decouple the
high- and low-energy modes aiming at the partial diagonalization of
the Hamiltonian. In this paper we introduce a method for obtaining the
low energy effective operators in the framework of CCM. Neither
perturbation theory nor unitarity are essential for this method. The
method is non-perturbative, since there is no expansion in the coupling
constant, however, CCM can be conceived as a topological expansion in
the number of correlated excitations. We show that introducing a
double similarity transformation using linked-cluster amplitudes will
simplify the partial diagonization. However, a price must be paid: due
to the truncation the similarity transformations are not unitary, and
accordingly the hermiticity of the resultant effective Hamiltonian is
not manifest. This is related to the fact that we have a biorthogonal
representation of the given many-body problem. There is a long
tradition of such approaches. The first we are aware of are Dyson-type bosonization
schemes
\cite{11}. [Here one chooses to map the generators of a Lie algebra, such that the raising generators have a particularly simple representation.] The space of states is mapped onto a larger space where the physically realizable
states are obtained by constrained dynamics. This is closely related
to CCM formalism, where extended phase space is a complex manifold,
the constraint function performing this elimination has been shown to
be of second class and the physical shell itself was found to be a
K\"ahler manifold
\cite{12}. The second is the Suzuki-Lee method in the nuclear many-body 
problem (NMT) \cite{13,Ni}, which reduces the full many-body
problem to a problem in a small configuration space and introduces a
related effective interaction. The effective interaction is naturally
understood as the result of certain transformations which decouple a
model space from its complement. As is well know in the theory of effective
interactions, unitarity of the transformation used for decoupling or
diagonalization is not necessary. Actually, the advantage of a non-unitarity
approach is that it can give a very simple description for both
diagonalization and the ground state. This has been discused by many
authors \cite{14} and, although it might lead to a non-hermitian
effective Hamiltonian, it has been shown that hermiticity can be
recovered
\cite{12,15}. Nevertheless non-hermiticity is negligible if the model
space and its compliments are not strongly correlated \cite{16}. We
may hope that $S$ and $S'$ are small (in a somewhat ill-defined
sense), in other words, we may require that some of the coherence has
already been obtained by optimizing the model space. (This can be
done, for example, by a Hartree-Bogolubov transformation). Then the
correlations can be added via the CCM. The application of this
procedure to two-dimensional $\phi^{4}$ theory has been shown to give
a stable result outside the critical region \cite{17}. Therefore
defining a good model space can in principle control the accuracy of
CCM.

 To solve the relativistic bound state problem one needs to
 systematically and simultaneously decouple 1) the high-energy from
 low-energy modes and 2) the many-body from few-particle states. We
 emphasize in this paper that CCM can in principle be an adequate
 method to attack both these problems. Our hope is to fully utilize
 Wilsonian renormalization
\cite{18} within the CCM formalism. Here the high energy modes will be
integrated out leading to a modified low-energy Hamiltonian in an effective
many-body space. 


As an illustrative but necessarily simple example of our approach, we
employ the extended Lee model (ELM) which is a simple model connecting
elementary and composite particles
\cite{19}. The ELM involves a particular interaction
$V\leftrightarrows N,\theta$ and $N\leftrightarrows V,\bar{\theta}$
between two kinds of fermions $V,N$ and a boson $\theta$ and
anti-boson $\bar{\theta}$. It has been shown that the
composite-particle theory, the meson pair theory of Wentzel
\cite{20}, can be obtained as a strong-coupling limit of 
the ELM, in which limit the wave function renormalization constant of
the $V$ particle vanishes
\cite{19}.  For generality we keep the coupling of $NV\theta$ and
$NV\bar{\theta}$ interaction different. If we remove the anti-meson
field ($\bar{\theta}$) and accordingly crossing symmetry the resultant
Hamiltonian is the so-called Lee model. The renormalization of the Lee
model has been extensively studied \cite{21,23}. This is rather
trivial since analyticity of the Lee model is due to an inherent
exact Tamm-Dancoff approximation (which limits the number of mesons present
at any instant). This makes it simple to exhibit many
field-theoretical features. To our knowledge less has been done for
ELM. 

The organization of this paper is as follows. In section II, we
discuss our approach and it's foundation. In section III, as an illustrative example,
we apply our method to the ELM, to work out one-loop renormalized low-energy Hamiltonian. Finally we conclude and present a outlook in section IV.

\section{Formalism}
The discussion in this section is partially based on the work of
Suzuki and Okamoto \cite{Ni}.  Let us consider a system described by a
Hamiltonian $H(\Lambda)$ which has, at the very beginning, a large
cut-off $\Lambda$. We assume that the renormalized Hamiltonian up to
scale $\Lambda$ is expressible in terms of renormalized fields,
couplings and masses, where the effect of renormalization can be
computed by $Z(\Lambda)$ factors. Now imagine that we restrict the
Hamiltonian to a lower energy scale $(\mu)$, where we want to find an
effective Hamiltonian $H(\mu)$ which has the same energy spectrum as
the original Hamiltonian in the smaller space. Formally, we wish to
transform the Hamiltonian to a new basis, where the high-energy modes
$\mu<k<\Lambda$, decouple from the low-energy ones, while the
low-energy spectrum remains unchanged. We define two subspaces, one
intermediate-energy space $Q$ containing modes with $\mu<k<\Lambda$ and a
low-energy space $P$ with $\mu\leq k$. Our renormalization approach is
based on decoupling of the complement space $Q$ from the model space $P$. The
operators $P$ and $Q$ which project a state onto the model space and
its complement, satisfy $P^2=P $, $Q^2=Q$, $PQ=0$ and
\(P+Q=\openone\). We introduce an isometry operator $G$ which maps
states in the $P$- onto the $Q$- space,
\begin{equation}
|q\rangle=G|p\rangle  \hspace{1cm} (|q\rangle\in Q,|p\rangle\in P).
\end{equation} 
The operator $G$ is the basic ingredient in a family of
``integrating-out operators'', which passes information about the
correlations of the high energy modes to the low-energy space. The
operator $G$ obeys $ G=QGP$, $GQ=0$, $PG=0$ and $G^n =0$ for $
n\geqslant 2$. The rather surprising direction for $G$ to act in is
due to the definition Eq.~(\ref{eq4}) below (cf. the relation between
the active and passive view of rotations). In order to give a general
form of the effective low-energy Hamiltonian, we define another
operator $X(n,\mu,\Lambda)$, which maps states in the $P$-space onto
the full-space,
\begin{equation}
X(n,\mu,\Lambda)= (1+G)(1+G^{\dag}G+GG^{\dag})^{n}. \label{eq2}
\end{equation}
($n$ is a real number.) The inverse of $X(n, \mu, \Lambda)$ can be
obtained explicitly,
\begin{equation}
X^{-1}(n,\mu,\Lambda)= (1+G^{\dag}G+GG^{\dag})^{-n}(1-G). \label{eq3}
\end{equation}
The special case $n=0$ is equivalent to the transformation
introduced in Ref.~\cite{25} to relate the hermitian and non-hermitian effective operators in the energy-independent Suzuki-Lee approach. We now consider the transformation of $H(\Lambda)$ defined as
\begin{equation}
\overline{H}(n,\mu,\Lambda)=X^{-1}(n,\mu,\Lambda)H(\Lambda)X(n,\mu,\Lambda). \label{eq4}
\end{equation}
One can prove that if $\overline{H}(n,\mu, \Lambda)$ satisfies the desirable decoupling property,
\begin{equation}
Q \overline{H}(n,\mu,\Lambda)P=0, \label{eq5}
\end{equation} 
or more explicitly, by substituting the definition of $X(n,\mu,\Lambda)$ and $X^{-1}(n,\mu,\Lambda)$ from Eqs.~(\ref{eq2}-\ref{eq3}),
\begin{equation}
QH(\Lambda)P+QH(\Lambda)QG-GPH(\Lambda)P-GPH(\Lambda)QG=0, \label{eq6}
\end{equation}
that ${\cal H}(n,\mu)=P\overline{H}(n,\mu,\Lambda)P$ is an effective
Hamiltonian for the low energy degrees of freedom. In other words, it
should have the same low-energy eigenvalues as the original
Hamiltonian. The proof is as follows: 


Consider an eigenvalue equation in the $P$ space with $\{|\phi(k)\rangle \in P\}$,
\begin{eqnarray}
&&P\overline{H}(n,\mu,\Lambda)P|\phi(k)\rangle=E_{k}PX^{-1}(n,\mu,\Lambda)X(n,\mu,\Lambda)P|\phi(k)\rangle. \label{eq7}\
\end{eqnarray}
By multiplying both sides by $X(n,\mu,\Lambda)$ and making use of the decoupling property Eq.~(\ref{eq5}), we obtain 
\begin{equation}
H(\Lambda)X(n,\mu,\Lambda)P|\phi(k)\rangle=E_{k}X(n,\mu,\Lambda)P|\phi(k)\rangle. \label{eq8}
\end{equation}
This equation means that $E_{k}$ in Eq.~(\ref{eq7}) agrees with one of
the eigenvalue of $ H(\Lambda)$ and $
X(n,\mu,\Lambda)P|\phi(k)\rangle$ is the corresponding eigenstate. In
the same way we can also obtain the $Q$-space effective Hamiltonian,
from the definition of $\overline{H}(n,\mu,\Lambda)$. It can be
seen that if $G$ satisfies the requirement in Eq.~(\ref{eq6}), then
we have additional decoupling condition 
\begin{equation}
 P \overline{H}(n,\mu,\Lambda)Q=0. \label{decoupling}
\end{equation}
We will argue later that \emph{both} of the decoupling conditions
Eq.~(\ref{eq5}) and (\ref{decoupling}) are necessary in order to have
a sector-independent renormalization scheme. The word ``sector'' here
means the given truncated Fock space. Let us now clarify the meaning
of this concept. To maintain the generality of
the previous discussion, we use here the well known Bloch-Feshbach
formalism \cite{b,fesh}. The Bloch-Feshbach method exploits projection
operators in the Hilbert space in order to determine effective
operators in some restricted model space. This technique seems to be
more universal than Wilson's renormalization formulated in a Lagrangian
framework. This is due to the fact that in the Bloch-Feshbach formalism,
other irrelevant degrees of freedom (such as high angular momentum,
spin degrees of freedom, number of particles, etc.) can be
systematically eliminated in the same fashion. 


Assume that the full space Schr\"odinger equation is
$H(\Lambda)|\psi\rangle=E|\psi\rangle$ and for simplicity 
$|\psi\rangle$ has been normalized to one. We explicitly construct the
effective Hamiltonian in this formalism,
\begin{eqnarray}
H^{\text{eff}}&=&P\overline{H}P+P\overline{H}Q\frac{1}{E-Q\overline{H}Q}Q\overline{H}P,
\label{F1}\
\end{eqnarray}
where $\overline{H}$ can be a similarity transformed Hamiltonian. This
equation resembles Brueckner's reaction matrix (or ``G''-matrix) equation
in nuclear many-body theory (NMT). In the same way for arbitrary
operator $O$ (after a potential similarity transformation), we construct
the effective operator
\begin{eqnarray}
O^{\text{eff}}&=&P\overline{O}P+P\overline{H}Q\frac{1}{E-Q\overline{H}Q}Q\overline{O}P+P\overline{O}Q\frac{1}{E-Q\overline{H}Q}Q\overline{H}P\nonumber\\
&+&P\overline{H}Q\frac{1}{E-Q\overline{H}Q}
Q\overline{O}Q\frac{1}{E-Q\overline{H}Q} Q\overline{H}P.\label{F2}\
\end{eqnarray}
The $E$-dependence in Eqs.~(\ref{F1}) and (\ref{F2}) emerges from the
fact that the effective interaction in the reduced space is not
assumed to be decoupled from the excluded space. However, by using the
decoupling conditions introduced in Eq.~(\ref{eq5}) and
(\ref{decoupling}), we observe that energy dependence can be removed,
and the effective operators become
\begin{eqnarray}
H^{\text{eff}}&=&P\overline{H}P={\cal H}(n,\mu),\label{final}\nonumber\\
O^{\text{eff}}&=&P\overline{O}P={\cal O}(n,\mu).\
\end{eqnarray}
The decoupling property makes the operators in one sector independent
of the other sector. The effects of the excluded
sector is taken into account by imposing the decoupling
conditions. This is closely related to the folded diagram method in
NMT for removing energy-dependence \cite{folded}. (It is well-known in NMT that
$E$-dependence in the $G$-matrix emerges from non-folded diagrams which
can be systematically eliminated using the effective
interaction approach). The above argument was given without assuming
an explicit form for $X$ and thus the decoupling conditions are more
fundamental than the prescription used to derive these condition. We
now show that Lorentz covariance in a given sector does not hinge on
special forms of similarity transformation. We assume ten Poincar\'e
generators $L_{i}$ satisfying
\begin{equation}
[L_{i},L_{j}]=\sum a^{k}_{ij}L_{k},
\end{equation}
where the $a^{k}_{ij}$ are the known structure coefficients. One
can show that if the operators $L_{i}$ satisfy the decoupling
conditions~ $Q\bar{L}_{i}P=0$ and $P\bar{L}_{i}Q=0$ then it follows
that
\begin{equation}
[L_{i}^{\text{eff}},L_{j}^{\text{eff}}]=\sum
a^{k}_{ij}L_{k}^{\text{eff}}.
\end{equation}
This leads to a relativistic description even after simultaneously
integrating out the high-frequency modes and reducing the number of
particles.


Note that the solution to Eq.~(\ref{eq6}) is independent of the number
$n$. One can make use of Eq.~(\ref{eq6}) and its complex conjugate to
show that for any real number $n$, the following relation for the effective low-energy Hamiltonian 
\begin{equation}
 {\cal H}(n,\mu)= {\cal H}^{\dag}(-n-1,\mu). \label{eq10}
\end{equation}
The case $n=-1/2$ is special since the effective Hamiltonian is hermitian, which can be obtained as 
\begin{equation}
{\cal H}(-1/2,\mu)=(P+G^{\dag}G)^{1/2}H(\Lambda)(P+G)(P+G^{\dag}G)^{-1/2}. \label{hermi}
\end{equation}
Hermiticity can be verified from the relation \cite{26} 
\begin{equation}
e^{T}P=(1+G)(P+G^{\dag}G)^{-1/2},
\end{equation}
where,
\begin{equation}
T=\arctan(G-G^{\dag})=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{2n+1}\big(G(G^{\dag}G)^{n}-\text{h.c}.).
\end{equation}
Since the operator $T$ is anti-hermitian, $e^{T}$ is a unitary
operator. From the above expression Eq.~(\ref{hermi}) can be written
in the explicitly hermitian form
\begin{equation}
{\cal H}(-1/2,\mu)=Pe^{-T}H(\Lambda)e^{T}P.
\end{equation}
As was already emphasized renormalization based on unitary
transformations is more complicated and non-economical. Thus we will
explore a non-unitary approach. An interesting non-hermitian effective
low-energy Hamiltonian can be obtained for $n=0$,
\begin{equation}
{\cal H}(0,\mu,\Lambda)=PH(\Lambda)(P+QG). \label{eq11}
\end{equation}
This form resembles the Bloch and Horowitz type of effective
Hamiltonian as used in NMT \cite{b}, and in the context of the CCM,
this form leads to the folded diagram expansion well known in
many-body theory \cite{f}. It is of interest that various effective
low-energy Hamiltonians can be constructed according to
Eq.~(\ref{final}) by the use of the mapping operator $G$ which all
obey the decoupling property Eq.~(\ref{eq5}) and
Eq.~(\ref{decoupling}). Neither perturbation theory nor hermiticity is
essential for this large class of effective Hamiltonians. 


The CCM approach is, of course, just one of the ways of describing the
relevant spectrum by means of non-unitary transformations. According
to our prescription the model space is $P:\{|L\rangle
\bigotimes|0,b\rangle_{h}, L\leq \mu\}$, where $|0,b\rangle_{h}$ is a
bare high energy vacuum (the ground state of high-momentum of
free-Hamiltonian) which is annihilated by all the high frequency
annihilation operators $
\{C_{I}\}$, the set of indices $\{I\}$ therefore defines a subsystem, or cluster, within the full system of a given configuration. The
actual choice depends upon the particular system under
consideration. In general, the operator $\{C_{I}\}$ will be products
(or sums of products) of suitable single-particle operators. We assume
that the annihilation and its corresponding creation
$\{C^{\dag}_{I}\}$ subalgebras and the state $ |0,b\rangle_{h}$ are
cyclic, so that the linear combination of state
$\{C^{\dag}_{I}|0,b\rangle_{h}\}$ and $\{~_{h}\langle b, 0|C_{I}\}$
span the decoupled Hilbert space of the high-momentum modes,
$\{|H\rangle\}$, where $\mu<H<\Lambda$. It is also convenient, but not
necessary, to impose the orthogonality condition, \(\langle
0|C_{I}C^{\dag}_{J}|0\rangle=\delta(I,J)\), where $\delta(I,J)$ is a
Kronecker delta symbol. The complement space is $Q:\{|L\rangle
\bigotimes \left(|H\rangle-|0,b\rangle_{h}\right)\}$. Our main goal is
to decouple the $P$-space from the $Q$-space. This gives sense to the
partial diagonalization of the high-energy part of the
Hamiltonian. The states in full Hilbert space are constructed by
adding correlated clusters of high-energy modes onto the $P$-space, or
equivalently integrating out the high-energy modes from the Hamiltonian,
\begin{eqnarray}
&& |f\rangle=X(\mu,\Lambda)|p\rangle=e^{\hat{S}}e^{-\hat{S}'}|0,b\rangle_{h}\bigotimes|L\rangle=e^{\hat{S}}|0,b\rangle_{h}\bigotimes|L\rangle, \label{eq13} \\
&& \langle \widetilde{f}|= \langle L|\bigotimes ~_{h}\langle b,0|X^{-1}(\mu,\Lambda)=\langle L|\bigotimes ~_{h}\langle 0|e^{\hat{S}'}e^{-\hat{S}}, \label{eq14}
\end{eqnarray}
where the operators $X(\mu,\Lambda)$ and $X^{-1}(\mu,\Lambda)$ have been expanded in terms of independent coupled cluster excitations $I$,
\begin{eqnarray}
&&\hat{S}=\sum_{m=0}\hat{S}_{m}\left(\frac{\mu}{\Lambda}\right)^{m}, \hspace{2cm} \hat{S}_{m}=\sideset{}{'}\sum_{I}\hat{s}_{I}C^{\dag}_{I},\nonumber\\
&&\hat{S}'=\sum_{m=0}\hat{S}'_{m}\left(\frac{\mu}{\Lambda}\right)^{m}, \hspace{2cm} \hat{S}'_{m}=\sideset{}{'}\sum_{I}{}\hat{s}'_{I}C_{I}.\
\end{eqnarray}
Here the primed sum means that at least one fast particle is created
or destroyed $(I\neq 0)$, and momentum conservation in $P\bigoplus Q$
is included in $\hat{s}_{I}$ and
$\hat{s}'_{I}$. $\hat{S}_{m}(\hat{S}'_{m})$ are not generally
commutable in the low-energy Fock space, however by construction they
are commutable in the high-energy Fock space. It is immediately clear
that states in the interacting Hilbert space are normalized, $\langle
\widetilde{f}|f\rangle=~_{h}\langle b,0|0,b\rangle_{h}=1$. We have
two types of parameters in this procedure, one is the coupling
constant of the theory ($\lambda$) and the other is the ratio of cutoffs
($\mu/\Lambda$). The explicit power counting makes the degree of
divergence of each order smaller than the previous one. According to
our logic, Eq.~(\ref{eq11}) can be written as
\begin{equation}
\hat{{\cal H}}(\mu)=~_{h}\langle b,0|X^{-1}(\mu,\Lambda)H(\Lambda)X(\mu,\Lambda)|0,b\rangle_{h},  \label{eq15}
\end{equation}
with $X(\mu,\Lambda)$ and $X^{-1}(\mu,\Lambda)$ defined in
Eq.~(\ref{eq13}) and Eq.~(\ref{eq14}). It is well-know in many-body
applications that the exponential Ansatz Eq.~(\ref{eq13}) and
(\ref{eq14}) guarantees automatically proper size-exclusivity and
conformity with the Goldstone liked-cluster theorem to all level of
truncation. This parametrization does not manifestly preserve
hermitian conjugacy. However, it is compatible with the
Hellmann-Feynman theorem (HFT), in other words, demanding hermiticity
will violate this theorem at any level of truncation. On the other
hand, with this parametrization the phase space $ \{\hat{s}_{I},\hat{s}'_{I}\}$
for a given $m$ is a symplectic differentiable manifold. Thereby all
the geometrical properties of the configuration space can be precisely
defined.~(In a more ordinary language, the canonical equations of
motion with respect to phase space, define a set of trajectories,
which fill the whole dynamically allowed region of the phase space.) 
There is a deep connection between these three properties and we can
not give up one without losing something else as well \cite{27}. The
individual amplitudes for a given $m$, \(\{\hat{s}_{I}^{m},\hat{s}'^{m}_{I}\}
\equiv\{\hat{s}_{I},\hat{s}'_{I}\}_{m}\), can generally be functionals of the low-
and high-energy field operators and have to be fixed by the dynamics
of quantum system. This is a complicated sets of requirements.
However, we require less than that. Suppose that after a similar
transformation of Hamiltonian, $\overline{H}$, we obtain an effective
Hamiltonian of the form
\begin{equation}
\overline{H}= H(\text{low})+H_{\text{free}}(\text{high})+C^{\dag}_{I}V_{IJ}C_{J}, \label{eqH}
\end{equation}
where $V_{IJ}$ is an arbitrary operator in the low frequency space
. The $I$ and $J$ indices should be chosen such that the last term in
Eq.~(\ref{eqH}) contains at least one creation- operator and one
annihilation-operator of high frequency.  By using
Rayleigh-Schr\"odinger perturbation theory, it can be shown that the
free high-energy vacuum state of $H_{\text{free}}(\text{high})$ is
annihilated by Eq.~(\ref{eqH}) and remains without correction at
any order of perturbation theory. Having said that, we will now
consider how to find the individual amplitudes $
\{\hat{s}_{I},\hat{s}'_{I}\}_{m}$ that transfer the Hamiltonian into the form 
Eq.~(\ref{eqH}). We split the Hamiltonian in five parts:
\begin{equation}
H=H_{1}+H_{2}^{\text{free}}(\text{high})+V_{C}(C^{\dag}_{I})+V_{A}(C_{I})+V_{B}, \label{eq16}
\end{equation}
where $H_{1}$ contains only the low frequency modes with $k\leq \mu$,
$H_{2}$ is the free Hamiltonian for all modes with $\mu<k<\Lambda$,
$V_{C}$ contains low frequency operators and products of the high
frequency creation operators $C_{I}^{\dag}$ and $ V_{A}$ is the
hermitian conjugate of $V_{C}$. The remaining terms are contained in
$V_{B}$, these terms contain at least one annihilation and creation
operators of the high energy modes. Our goal is to eliminate $V_{C}$
and $V_{A}$ since as previously became clear $ V_{B}$ is already
partially diagonal and has less effect after acting on the vacuum. The
ket-state coefficients $\{\hat{s}_{I}\}_{m}$ are worked out via the
ket-state Schr\"odinger equation
\(H(\Lambda)|f\rangle=E|f \rangle\) written in the form
\begin{equation}
\langle 0|C_{I}e^{-\hat{S}}He^{\hat{S}}|0\rangle=0, \hspace{2cm} \forall I\neq 0. \label{eq17}
\end{equation}
The bra-state coefficient $\{\hat{s}_{I},\hat{s}'_{I}\}_{m}$ are obtained by
making use of the Schr\"odinger equation defined for the bra-state,~$\langle \widetilde{f}|H(\Lambda)=\langle \widetilde{f}|E$. Firstly we
project both sides on $C^{\dag}_{I}|0\rangle$, then we eliminate $E$ 
by making use of the ket-state equation projection with the state $\langle 0|e^{\hat{S}'}C^{\dag}_{I}$ to yield the equations
\begin{equation}
\langle 0|e^{\hat{S}'}e^{-\hat{S}}[H,C^{\dag}_{I}]e^{\hat{S}}e^{-\hat{S}'}|0\rangle=0,  \hspace{1cm}\forall I\neq 0. \label{eq18}
\end{equation}
Alternatively one can in unified way apply $ e^{\hat{S}}e^{-\hat{S}'}C^{\dag}_{I}|0\rangle$  on Schr\"odinger equation defined for the bra-state and obtain
\begin{equation}
\langle 0|e^{\hat{S}'}e^{-\hat{S}}He^{\hat{S}}e^{-\hat{S}'}C^{\dag}_{I}|0\rangle=0, 
\hspace{2cm}\forall I\neq 0. \label{eq19}
\end{equation}
Equation (\ref{eq17}) and Eqs.~(\ref{eq18}) or (\ref{eq19}) provide two
sets of formally exact, microscopic, operatorial coupled non-linear equations for
the ket and bra. One can solve the coupled equations in
Eq.~(\ref{eq17}) to work out $\{\hat{s}_{I}\}_{m}$ and then use them as an
input in Eq.~(\ref{eq18}) or (\ref{eq19}). 


It is important to notice that Eq.~(\ref{eq17}) and (\ref{eq18}) can
be also derived by requiring that the effective low-energy Hamiltonian
defined in Eq.~(\ref{eq15}), be stationary (i.e. $\delta
\hat{\mathcal{H}}(\mu)=0$) with respect to all variations in each of the
independent functional
\(\{\hat{s}_{I},\hat{s}'_{I}\}_{m}\). One can easily verify that the requirements \(
\delta \hat{\mathcal{H}}(\mu)/\delta \hat{s}_{I}=0\) and \(\delta
\hat{\mathcal{H}}(\mu)/\delta \hat{s}'_{I}=0\) yield  
Eq.~(\ref{eq17}) and Eq.~(\ref{eq18}). The combination of
Eqs.~(\ref{eq17}) and (\ref{eq18}) does not manifestly satisfy the
decoupling property as set out in Eqs.~(\ref{eq5}) and
(\ref{decoupling}). On the other hand Eqs.~(\ref{eq17}) and
(\ref{eq19}) satisfy these conditions. Equations (\ref{eq17}) and
(\ref{eq19}) imply that all interactions including creation and
annihilation of fast particles (``$I$'') are eliminated from the
transformed Hamiltonian $\mathcal{H(\mu)}$ in Eq.~(\ref{eq15}). In
other words, these are decoupling conditions leading to the
elimination of $V_{C}$ and $V_{A}$ from Eq.~(\ref{eq16}), which is, in
essence, a block-diagonalization. Therefore it makes sense for our
purpose to use Eq.~(\ref{eq17}) and Eq.~(\ref{eq19}) for obtaining the
unknown coefficients, losing some of the elegance of the CCM
elsewhere.


So far everything has been introduced rigorously without invoking any
approximation. In practice one needs to truncate both sets of
coefficients $\{\hat{s}_{I},\hat{s}'_{I}\}_{m}$ at a given order of
$m$. A consistent truncation scheme is the so-called SUB($n,m$)
scheme, where the $n$-body partition of the operator
$\{\hat{S},\hat{S}'\}$ is truncated so that one sets the higher
partition with $I>n$ to zero at a given accuracy $m$. Notice that,
Eq.~(\ref{eq18}) and (\ref{eq19}) provide two equivalent sets of
equations in the exact form, however after the truncation they can 
in principle be different. Eqs.~(\ref{eq17}) and (\ref{eq19}) are
compatible with the decoupling property at any level of the
truncation, whereas Eqs.~(\ref{eq17}) combined with (\ref{eq18}) are fully
consistent with HFT at any level of truncation. Thus the low-energy
effective form of an arbitrary operator can be computed according to
Eq.~(\ref{final}) in the same truncation scheme used for
the renormalization of the Hamiltonian. In particular, we will show that
only in the lowest order ($m=0$), equations (\ref{eq18}) and
(\ref{eq19}) are equivalent, independent of the physical system and
the truncation scheme. 


Although our method is non-perturbative, perturbation theory can be
recovered from it. In this way, its simple structure for loop
expansion will be obvious and we will observe that at lower order
hermiticity is preserved. Now we illustrate how this is realizable in
our approach. Assume that $V_{C}$ and $V_{A}$ are of order $\lambda$,
we will diagonalize the Hamiltonian, at leading order in $\lambda$ up
to the desired accuracy in $\mu/\Lambda$. We use the
commutator-expansion
\begin{equation}
e^{-S}He^{S}=H+[H,S]+\frac{1}{2!}\big[[H,S],S]+...~. \label{eq20}
\end{equation} 
Eq.~(\ref{eq17}) can be organized perturbatively in order of $m$, aiming at elimination of the high momenta degree of freedom up to the first order in the coupling constant, thus yields
\begin{eqnarray} 
&&m=0:\langle 0|C_{I}(V_{C}+[H_{2},S_{0}])|0\rangle=0,\nonumber\\
&&m=1:\langle 0|C_{I}([H_{1},S_{0}]+[H_{2},S_{1}]+[V_{A},S_{1}]+[V_{C},S_{1}])|0\rangle=0,\nonumber\\
&& \hspace{2cm}\vdots\nonumber\\
&&m=n:\langle 0|C_{I}([H_{1},S_{n-1}]+[H_{2},S_{n}]+[V_{A},S_{n}]+[V_{C},S_{n}])|0\rangle=0,\label{eq21}\
\end{eqnarray}
where $I\neq 0$. Notice that $S_{0}$ is chosen to cancel $V_{C}$ in
the effective Hamiltonian, hence it is at least of order of $\lambda$,
consequently it generates a new term $[H_{1},S_{0}]$ which is of
higher order in $ \mu/\Lambda$ and can be cancelled out on the next
orders by $S_{1}$. The logic for obtaining the equations above
is based on the fact that $S_{n}$ should be smaller than $S_{n-1}$
(for sake of convergence) and that the equations should be consistent with each
other. Since $H_{2}, V_{A}, V_{C}\approx \Lambda $ and $H_{1}\approx\mu$, from Eq.~(\ref{eq21}) we have the desired relation $S_{n}\approx\frac{\mu}{\Lambda}S_{n-1}$. The same procedure can be applied for Eq.~(\ref{eq19}) which
leads to the introduction of a new series of equations in order of
$m$,
\begin{eqnarray} 
&&m=0:\langle
0|(V_{A}-[H_{2},S'_{0}])C^{\dag}_{I}|0\rangle=0,\nonumber\\
&&m=1:\langle
0|([H_{1},S'_{0}]+[H_{2},S'_{1}]+[V_{C},S'_{1}]+[V_{A},S'_{1}]-[V_{A},S_{1}])C^{\dag}_{I}|0\rangle=0,
\nonumber\\ && \hspace{2cm}\vdots\nonumber\\ &&m=n:\langle
0|([H_{1},S'_{n-1}]+[H_{2},S'_{n}]+[V_{C},S'_{n}]+[V_{A},S'_{n}]-[V_{A},S_{n}])C^{\dag}_{I}|0\rangle=0.\label{eq23}\
\end{eqnarray}
Alternatively, we can use Eq.~(\ref{eq18}) to yield the equations
\begin{eqnarray} 
&&m=0:\langle
0|\big([V_{A},C^{\dag}_{I}]-\big[[H_{2},C^{\dag}_{I}],S'_{0}\big]\big)|0\rangle=0,\nonumber\\
&&m=1:\langle
0|\big(\big[[V_{A},C^{\dag}_{I}],S_{1}\big]-\big[[H_{2},C^{\dag}_{I}],S'_{1}\big]-\big[[V_{A},C^{\dag}_{I}],S'_{1}\big]\big)|0\rangle=0,
\nonumber\\ && \hspace{2cm}\vdots\nonumber\\ &&m=n:\langle
0|\big(\big[V_{A},C^{\dag}_{I}],S_{n}\big]-\big[[H_{2},C^{\dag}_{I}],S'_{n}\big]-\big[[V_{A},C^{\dag}_{I}],S'_{n}\big]\big)|0\rangle=0.\label{eq24}\
\end{eqnarray}
It is obvious that at  order $m=0$, Eq.~(\ref{eq23}) and (\ref{eq24})
are the same and $S'_{0}=S^{\dag}_{0}$, which indicates that the
similarity transformation at this level remains unitary.  It should be
noted that diagonalization at first order in the coupling constant
introduces a low-energy effective Hamiltonian in Eq.~(\ref{eq15}) which
is valid up to the order $\lambda^{3}$. In the same way,
diagonalization at second order in $\lambda$ modifies the Hamiltonian
at order $\lambda^{4}$ and leads generally to a non-unitarity
transformation. In this way one can proceed to diagonalize the
Hamiltonian at a given order in $\lambda$ with desired accuracy in
$\mu/\Lambda$ . Finally, the renormalization process is completed by
introducing the correct $Z(\Lambda)$ factors which redefine the
divergences emerging from Eq.~(\ref{eq15}).

\section{Renormalization of the extended Lee model}
 As an illustrative example, we will now apply the formalism introduced in the last chapter to determine
the effective Hamiltonian for ELM up to the one-loop order.
We define four kinds of particles, the $V$-particle and $N$-particle as two different fermions and the $\theta$ and $\bar{\theta}$ as a scalar boson and anti-boson respectively. Here $a(k)$, $a^{\dag}(k)$ and $b(k)$, $b^{\dag}(k)$ are the annihilation and creation operators which satisfy boson commutator rules. The $V(p)$, $V^{\dag}(p)$ and $N(p)$, $N^{\dag}(p)$ define the fermion sector and obey the usual anticommutator rules. The bare ELM Hamiltonian then reads
\begin{eqnarray}
H&=&H_{0}+H_{I},\label{eq25}\nonumber\\ 
H_{0}&=&\int{d^{3}p
~\omega_{V}(p)V^{\dag}(p)V(p)}+\int{d^{3}p
~\omega_{N}(p)N^{\dag}(p)N(p)}\nonumber\\ &+&\int{d^{3}k
~\omega_{\theta}(k)a^{\dag}(k)a(k)}+\int{d^{3}k
~\omega_{\bar{\theta}}(k)b^{\dag}(k)b(k)},\nonumber\\
H_{I}&=&\lambda_{1}(2\pi)^{-3/2}\int{\frac{d^{3}kd^{3}p}{(2\omega_{\theta}(k))^{1/2}}V^{\dag}(p)N(p-k)a(k)}\nonumber\\
&+&\lambda_{2}(2\pi)^{-3/2}\int{\frac{d^{3}kd^{3}p}{(2\omega_{\bar{\theta}}(k))^{1/2}}N^{\dag}(p)V(p-k)b(k)}+\text{h.c.}.\
\end{eqnarray}
 The kinetic energy generically is defined
 \(\omega_{O}(k)=\sqrt{k^{2}+m^{2}_{O}}\) where the indice $O$ can be either
 $(V,N,\theta,\bar{\theta})$. The interaction term in $H_{I}$
 describes the processes;
\begin{eqnarray}
&& V\rightleftarrows N+\theta,\label{eq26}\\
&& N\rightleftarrows V+\bar{\theta}. \label{eq27}\
\end{eqnarray}
The crossing symmetry become manifest if we take
$\lambda_{1}=\lambda_{2}$ and equal masses for boson and anti-boson. 
For sake of generality we will ignore crossing symmetry at the
moment. The Lee model can be recovered if we decouple the  anti-boson
$\bar{\theta}$, $\lambda_{2}\to 0$. In the Lee model the virtual
process Eq.~(\ref{eq27}) is not included and thus the $N$-particle state
remains unrenormalized and the model become exactly solvable. 


It is believed that the Lee model is asymptotically free for
space-time dimension $D$ less than four \cite{22}. With on-shell
renormalization one can show that the Lee model for $D>4$ (odd $D$) is
ultraviolet stable and not asymptotically free \cite{23}. It is well
known that such a model in four dimension exhibit a ghost state as the
cutoff is removed. The Hamiltonian Eq.~(\ref{eq25}) exhibits two
symmetries; it is straightforward to verify that following operators
commute with $H$
\begin{eqnarray}
B&=&\int{d^{3}p~V^{\dag}(p)V(p)}+\int{d^{3}p~N^{\dag}(p)N(p)},\label{sym}\nonumber\\
Q&=&\int{d^{3}p~N^{\dag}(p)N(p)}+\int{d^{3}k~b^{\dag}(p)b(p)}-\int{d^{3}k~a^{\dag}(p)a(p)}.\
\end{eqnarray}
Clearly $B$ is a baryon number operator and $Q$ is a charge
operator. We assign the charges $1,0,-1$ and $1$ to the
$N,V,\theta$ and $\bar{\theta}$, respectively. The sectors of the ELM
are labeled by the eigenvalue $(b,q)$ of the operators $(B,Q)$. According
to our formulation the ultraviolet-finite Hamiltonian is obtained
by introducing $Z$-factors, which depend on the UV cutoff $\Lambda$
and some arbitrary renormalization scale $M$ in such way that
effective Hamiltonian does not depend on $\Lambda$. The bare
Hamiltonian can be rewritten
\begin{eqnarray}
H&=&\int{d^{3}p~Z^{2}_{V}Z_{M_{V}}\omega_{V}(p)V^{\dag}(p)V(p)}+\int{d^{3}p~Z^{2}_{N}Z_{M_{N}}\omega_{N}(p)N^{\dag}(p)N(p)}\nonumber\\
&+&\int{d^{3}k~Z^{2}_{\theta}Z_{M_{\theta}}\omega_{\theta}(k)a^{\dag}(k)a(k)}+\int{d^{3}k~Z^{2}_{\bar{\theta}}Z_{M_{\bar{\theta}}}\omega_{\bar{\theta}}(k)b^{\dag}(k)b(k)}\nonumber\\
&+&\int{\frac{\lambda_{1}d^{3}kd^{3}p}{(2(2\pi)^{3}\omega_{\theta}(k))^{1/2}}}Z_{\lambda_{1}}Z_{V}Z_{N}Z_{\theta}V^{\dag}(p)N(p-k)a(k)\nonumber\\
&+&\int{\frac{\lambda_{2}d^{3}kd^{3}p}{(2(2\pi)^{3}\omega_{\bar{\theta}}(k))^{1/2}}}Z_{\lambda_{2}}Z_{V}Z_{N}Z_{\bar{\theta}} N^{\dag}(p)V(p-k)b(k)\label{eq28}\nonumber\\
&+&\text{h.c.}~.\
\end{eqnarray}
Although the factorization in the above equation is scheme-dependent,
the $\Lambda$ dependence itself is not. Each of $Z$-factors has an
expansion of the form.
\begin{equation}
Z=1+f_{1}(\Lambda)\lambda +f_{2}(\Lambda)\lambda^{2}+\hdots,
\end{equation}
where $\lambda$ is a generic coupling constant of theory and has been
defined at a given renormalization scale $M$. The functions $f_{n}$
will be obtained order-by-order, by summing up contributions of
the fast modes between $\mu$ and $\Lambda$, in the sense that $Z(\Lambda)\to Z(\mu)$ and $f_{n}(\Lambda)\to
f_{n}(\mu)$. This means that the low-energy correlation functions are
invariants of the renormalization group flow. One can therefore assume
that the $Z$'s are initially $1$ and choose the corresponding $f$'s
from the condition that the cut-off dependence be cancelled out after
computing the effective Hamiltonian in the desired loop order. We
split the original Hamiltonian in the form of Eq.~(\ref{eq16});
\begin{eqnarray}
H_{1}&=&H(\int^{\mu}_{0}),\nonumber\\
H_{2}&=&H_{0}(\int^{\Lambda}_{\mu}),\nonumber\\
V_{C}&=&\int^{\mu}_{0}\int^{\Lambda}_{\mu}\frac{d^{3}p'd^{3}k}{(2(2\pi)^{3}\omega_{\theta}(k))^{1/2}}\lambda_{1} N^{\dag}(p'-k)V(p')a^{\dag}(k)\nonumber\\
&+&\int^{\mu}_{0}\int^{\Lambda}_{\mu}\frac{d^{3}p'd^{3}k}{(2(2\pi)^{3}\omega_{\bar{\theta}}(k))^{1/2}}\lambda_{2}V^{\dag}(p'-k)N(p')b^{\dag}(k), \nonumber\\
V_{A}&=&V^{\dag}_{C},\nonumber\\
V_{B}&=&\int^{\mu}_{0}\int^{\Lambda}_{\mu}\frac{d^{3}pd^{3}k'}{(2(2\pi)^{3}\omega_{\theta}(k'))^{1/2}}\lambda_{1}V^{\dag}(p)N(p-k')a(k')\nonumber\\
&+&\int^{\mu}_{0}\int^{\Lambda}_{\mu}\frac{d^{3}pd^{3}k'}{(2(2\pi)^{3}\omega_{\bar{\theta}}(k'))^{1/2}}\lambda_{2}N^{\dag}(p)V(p-k')b(k')\nonumber\\
&+&\int^{\Lambda}_{\mu}\int^{\Lambda}_{\mu}\frac{d^{3}pd^{3}k}{(2(2\pi)^{3}\omega_{{\theta}}(k))^{1/2}}\lambda_{1}V^{\dag}(p)N(p-k)a(k)\nonumber\\
&+&\int^{\Lambda}_{\mu}\int^{\Lambda}_{\mu}\frac{d^{3}pd^{3}k}{(2(2\pi)^{3}\omega_{\bar{\theta}}(k))^{1/2}}\lambda_{2}N^{\dag}(p)V(p-k)b(k)+\text{h.c.}~ .\
\end{eqnarray}
Here $p'$ and $k'$ stand for low momenta ($p',k'<\mu$). If the
arguments of an operator are all low momenta ($p'$ or $k'$), this
indicates low momentum operators. The arguments in $H(\int^{\mu}_{0})$
and $H_{0}(\int^{\Lambda}_{\mu})$ means that all the momentum
integrations involved in Eq.~(\ref{eq25}) are running between
$0<p'<\mu$ for the former and $\mu<p<\Lambda$ for the latter,
respectively. The configuration space of the high momentum operators
are specified by \(\{C_{I}\to V^{n_{1}}N^{n_{2}}a^{n_{3}}b^{n_{4}},
C^{\dag}_{I}\to{C^{\dag}_{I}\to
(V^{\dag})^{n_{1}}(N^{\dag})^{n_{2}}(a^{\dag})^{n_{3}}(b^{\dag})^{n_{4}}}\}\)
with $ n_{1}+n_{2}+n_{3}+n_{4}=I $. Aiming at a one-loop expansion the
corresponding $S$ and $S'$ operators which preserve the symmetry
property Eq.~(\ref{sym}), can be chosen as 
\begin{eqnarray}
S_{m}&=&\int{d^{3}p'd^{3}k~S^{1}_{m}(p')V^{\dag}(p'-k)b^{\dag}(k)}+\int{d^{3}p'd^{3}k~S^{2}_{m}(p')N^{\dag}(p'-k)a^{\dag}(k)},\nonumber\\
S^{1}_{m}&=&S^{N}_{m}N(p')+S^{Vb}_{m}V(p'-k')b(k')+S^{Va}_{m}V(p'+k')a^{\dag}(k'),\nonumber\\
S^{2}_{m}&=&S^{V}_{m}V(p')+S^{Na}_{m}N(p'-k')a(k')+S^{Nb}_{m}N(p'+k')b^{\dag}(k').\label{eq29}\
\end{eqnarray}  
We have ignored the $I=1$ configuration, since there are no tadpole type
diagrams. The truncation of $S_{I}$ in configuration space should be
consistent with the loop expansion, for example in $\phi^{4}$
theories, the renormalization up to two-loops requires that we take
into account the expansion up to $I=4$ , this may allow to
eliminate pure terms like as $a_{k}a_{p}a_{q}a_{r}$ or $a^{\dag}_{k}a^{\dag}_{p}a^{\dag}_{q}a^{\dag}_{r}$. Here we confine our
attention to the elimination of the high-momentum degrees of freedom up to the
first order in coupling constant and second order in $\mu/\Lambda$. The
unknown coefficients in Eq. (\ref{eq29}) can be obtained by making use of
Eq. (\ref{eq21}),

\begin{eqnarray} 
S^{V}_{0}&=&\frac{
\lambda_{1}}{(2(2\pi)^{3}\omega_{\theta}(k))^{1/2}(\omega_{N}(p'-k)+\omega_{\theta}(k))},\nonumber\\
S^{N}_{0}&=&\frac{\lambda_{2}}{(2(2\pi)^{3}\omega_{\bar{\theta}}(k))^{1/2}(\omega_{V}(p'-k)+\omega_{\bar{\theta}}(k))},\nonumber\\
S^{Vb}_{0}&=&S^{Va}_{0}=0,\nonumber\\
S^{V}_{1}&=&\frac{\lambda_{1}\omega_{V}(p')}{(2(2\pi)^{3}\omega_{\theta}(k))^{1/2}(\omega_{N}(p'-k)+\omega_{\theta}(k))^{2}},\nonumber\\
S^{N}_{1}&=&\frac{\lambda_{2}\omega_{N}(p')}{(2(2\pi)^{3}\omega_{\bar{\theta}}(k))^{1/2}(\omega_{V}(p'-k)+\omega_{\bar{\theta}}(k))^{2}},\nonumber\\
S^{Va}_{1}&=&\frac{\lambda_{1}\lambda_{2}}{2(2\pi)^{3}(\omega_{\bar{\theta}}(k)\omega_{\theta}(k'))^{1/2}(\omega_{V}(p'-k)+\omega_{\bar{\theta}}(k))^{2}},
\nonumber\\
S^{Nb}_{1}&=&\frac{\lambda_{1}\lambda_{2}}{2(2\pi)^{3}(\omega_{\bar{\theta}}(k')\omega_{\theta}(k))^{1/2}(\omega_{N}(p'-k)+\omega_{\theta}(k))^{2}},\nonumber\\
S^{Na}_{1}&=&\frac{\lambda_{1}^{2}}{2(2\pi)^{3}(\omega_{\theta}(k')\omega_{\theta}(k))^{1/2}(\omega_{N}(p'-k)+\omega_{\theta}(k))^{2}},\nonumber\\
S^{Vb}_{1}&=&\frac{\lambda_{2}^{2}}{2(2\pi)^{3}(\omega_{\bar{\theta}}(k')\omega_{\bar{\theta}}(k))^{1/2}(\omega_{V}(p'-k)+\omega_{\bar{\theta}}(k))^{2}}.\
\end{eqnarray}
It is easy to observe that Eq.~(\ref{eq23}) will be satisfied if we
require $ S'_{m}=S^{\dag}_{m}$, since, up to
the first in $\lambda$ the similarity transformation introduced in
Eq.~(\ref{eq15}) remains unitary. Equally one could use Eq.~(\ref{eq24}) to
obtain $S'$, it is obtained that $S'_{0}=S_{0}^{\dag}$ and $S'_{1}=0$,
we will show that the renormalization feature of our model up to this
order will remain unchanged, however the effective low-energy
Hamiltonian will be different. As was already pointed out, this is
because Eq.~(\ref{eq24}) requires a different truncation scheme. The
effective Hamiltonian is now produced by plugging the $S$ and $S'$
defined in Eq.~(\ref{eq29}) into Eq.~(\ref{eq15}). With naive
power-counting one can identify the potentially divergent terms.  At
the lower order of expansion, the divergent term is $\langle
0|[V_{A},S_{0}]|0\rangle $, the divergence in this term arises from a
double contraction of high-energy fields.

At this step the
contributions of the terms $[V_{B},S_{0}(S'_{0})]$ and
$[H_{1},S_{0}(S'_{0})]$ are zero, after projection on to the
high-frequency vacuum. There is one other divergent term, $\langle
0|[V_{C},S'_{0}]|0\rangle $, but this is harmless and will be cancelled
out by $\langle 0|[\big[H_{2},S_{0}],S'_{0}\big]|0\rangle$. We thus
obtain
\begin{eqnarray} 
\delta H(\lambda)&=&-\frac{\lambda_{1}}{2(2\pi)^{3}}\int_{\mu}^{\Lambda} {\frac{d^{3}k}{\omega_{\theta}(k)(\omega_{N}(p'-k)+\omega_{\theta}(k))}}\big[\int d^{3}p' N^{\dag}(p')N(p')\big]\nonumber\\
&-&\frac{\lambda_{2}}{2(2\pi)^{3}}\int_{\mu}^{\Lambda}{ \frac{d^{3}k}{\omega_{\bar{\theta}}(k)(\omega_{V}(p'-k)+\omega_{\bar{\theta}}(k))}}\big[\int d^{3}p' V^{\dag}(p')V(p')\big].\
\end{eqnarray} 
From this expression one can immediately deduce the renormalization
factors $Z_{m_{V}}$ and $Z_{m_{N}}$, we take $\omega_{O}\simeq |k|$
for $\mu \gg m_{O}$, therefore
\begin{eqnarray} 
Z_{m_{V}}&=& 1+\frac{\lambda_{1}^{2}}{8\pi^{2}}(\Lambda-\mu),\nonumber\\
Z_{m_{N}}&=& 1+\frac{\lambda_{2}^{2}}{8\pi^{2}}(\Lambda-\mu).\
\end{eqnarray}
There is no mass renormalization for $\theta$ and $\bar{\theta}$ and
accordingly there are no vacuum polarization type diagrams. Thus $\theta
$ and $\bar{\theta}$ remain unrenormalized,
$Z_{\theta}=Z_{m_{\theta}}=Z_{\bar{\theta}}=Z_{m_{\bar{\theta}}}=1
$. The other contribution of $H^{\text{eff}}$ at one-loop which are
not zero after projecting on to vacuum come from
\begin{equation}
\delta H(\lambda)=-\langle 0|\big[[H_{1},S_{1}],S'_{0}\big]+\langle 0|\big[[H_{1},S_{1}],S'_{1}\big]|0\rangle. 
\end{equation} 
The divergent contribution emerges from the first terms, the leading
divergence of this expression is logarithmic which means that we can
neglect the difference between $p$ and $k-p'$ (for the divergent
contribution only). After evaluating a momentum integral we finally
get,
\begin{eqnarray} 
\delta H&=&-\frac{\lambda_{1}^{2}}{16\pi^{2}}\ln \left[\frac{\Lambda}{\mu}\right]\int \omega_{V}(p')V^{\dag}(p')V(p')-\frac{\lambda_{2}^{2}}{16\pi^{2}}\ln\left[\frac{\Lambda}{\mu}\right]\int \omega_{N}(p')N^{\dag}(p')N(p')\nonumber\\
&-&\frac{\lambda_{1}^{2}+\lambda_{2}^{2}}{32\pi^{2}}\ln\left[\frac{\Lambda}{\mu}\right]\int \frac{\lambda_{1}}{((2\pi)^{3}\omega_{\theta}(k'))}V^{\dag}(p')N(p'-k')a(k')\nonumber\\
&-&\frac{\lambda_{1}^{2}+\lambda_{2}^{2}}{32\pi^{2}}\ln\left[\frac{\Lambda}{\mu}\right]\int \frac{\lambda_{2}}{((2\pi)^{3}\omega_{\bar{\theta}}(k'))}N^{\dag}(p')V(p'-k')b(k').\
\end{eqnarray}
From this expression we deduce the renormalization factor $Z_{\lambda_{1}},Z_{\lambda_{2}},Z_{V} $ and $Z_{N}$:
\begin{eqnarray}
&&Z_{V}^{2}=1+\frac{\lambda_{1}^{2}}{16\pi^{2}}\ln\frac{\Lambda}{\mu},\nonumber\\
&&Z_{N}^{2}=1+\frac{\lambda_{2}^{2}}{16\pi^{2}}\ln\frac{\Lambda}{\mu},\nonumber\\
&&Z_{\lambda_{1}}=Z_{\lambda_{2}}=1+\frac{\lambda_{1}^{2}+\lambda_{2}^{2}}{32\pi^{2}}\ln\frac{\Lambda}{\mu}.\
\end{eqnarray}
It is obvious from the equations above that one can define renormalized coupling constants in terms of the bare couplings and wave function renormalization 
\begin{equation}
\lambda_{i}=\lambda_{i}^{0}/Z_{V}Z_{N}, \hspace{2cm}i=1,2. 
\end{equation}
This definition corresponds for $\lambda_{1}=\lambda_{2}$ with the renormalization introduced to compute the $T$-matrix for the $N-\theta$ interaction \cite{19,24}. The one-loop $\beta$-function and anomalous dimension $\gamma$ are
\begin{eqnarray}
&&\beta_{\lambda_{i}}(\lambda_{1},\lambda_{2})=\frac{\partial \lambda_{i}}{\partial \ln M}|_{\Lambda}=\frac{\lambda_{i}}{32\pi^{2}}(\lambda_{1}^{2}+\lambda_{2}^{2}),\hspace{.5cm} i=1,2.\nonumber\\
&&\gamma_{V}=1/2\frac{\partial \ln Z_{V}}{\partial \ln M}|_{\Lambda}=-\frac{\lambda_{1}^{2}}{32\pi^{2}},\nonumber\\
&&\gamma_{N}=1/2\frac{\partial \ln Z_{N}}{\partial \ln M}|_{\Lambda}=-\frac{\lambda_{2}^{2}}{32\pi^{2}}.\label{g}\
\end{eqnarray}
Since the fixed points of the theory are the zero solutions of the
$\beta$-function, one immediately identifies the trivial solution
$\lambda_{1}=\lambda_{2}=0$ (we ignore the nonphysical imaginary
solution). It is now obvious from Eq.~(\ref{g}) that
$\gamma_{V}=\gamma_{N}=0$ only at trivial solutions of the
$\beta$-functions. This result is in correspondence with property that
for real field theories the $\gamma$-function is not zero unless at
trivial fixed point of the theory \cite{c}. Now to investigate the
behaviour of the theory at high momentum, we must compute the
momentum-dependent effective coupling constant $\lambda_{1}(k)$ and $
\lambda_{2}(k)$ by
\begin{eqnarray}
\left\{\begin{array}{ll}
& k\frac{d\lambda_{1}(k)}{dk}=\beta_{\lambda_{1}}(\lambda_{1}(k)\lambda_{2}(k)) ,\hspace{.51cm} \lambda_{1}(k)|_{k=1}=\lambda^{ph}_{1}\\
&\hspace{8cm},\\
&k\frac{d\lambda_{2}(k)}{dk}=\beta_{\lambda_{2}}(\lambda_{1}(k),\lambda_{2}(k)) ,\hspace{.51cm} \lambda_{2}(k)|_{k=1}=\lambda^{ph}_{2}\label{cou}\\
\end{array} 
\right.
\end{eqnarray}
 where $\lambda_{1}^{ph}$ and $\lambda_{2}^{ph}$ are dimensionless
 physical renormalized coupling constants defined at the renormalization
 scale $k=1$. The coupled equations (\ref{cou}) can be solved by going to
 polar coordinates $r^{2}(k)=\lambda_{1}^{2}(k)+\lambda_{2}^{2}(k)$ and
 $ \theta(k)=\tan^{-1}\frac{\lambda_{1}(k)}{\lambda_{2}(k)}$, 
\begin{eqnarray}
\lambda_{1}(k)&=&\frac{\bar{r}}{\sqrt{1-(16\pi^{2})^{-1}\bar{r}^{2}\ln k}}
\sin {\bar{\theta}},\nonumber\\
\lambda_{2}(k)&=&\frac{\bar{r}}{\sqrt{1-(16\pi^{2})^{-1}\bar{r}^{2}\ln k}}
\cos{ \bar{\theta}}, \label{so}\
\end{eqnarray}
where $\bar{r}$ and $\bar{\theta}$ denote the value at the renormalization
scale. The behaviour of the ELM in the deep-Euclidean region is
obtained by allowing $ k\rightarrow \infty$. From Eq.~(\ref{so}) one
observes that $\lambda_{1}(k)$ and $\lambda_{2}(k)$ in this region are
imaginary. This means that the effective Hamiltonian is non-hermitian
and the theory generates ghost states when the cut-off is removed. The
ghost state appears as a pole in $ V$ and $N$-propagators. Since a
theory is said to exhibit asymptotic freedom if (i)
$\frac{d\beta}{d\lambda}|_{\lambda(\infty)}<0$ (ultraviolet stability
at the fixed point $\lambda(\infty)$ and (ii)
$\lambda(\infty)=\lim_{k\rightarrow \infty}\lambda(k)=0$,
Eqs.~(\ref{so}) indicate that the ELM can not exhibit asymptotic
freedom at $D=4$.

\section{Conclusion and outlook}
In this paper we have outlined a strategy to derive effective
renormalized operators in a Hamiltonian formulation of a field
theory. The effective low-frequency operator is obtained by the
condition that it should exhibit decoupling between the low- and
high-frequency degrees of freedom. We showed that the similarity
transformation approach to renormalization can be systematically
classified. The non-hermitian formulation gives a very simple
description of decoupling, leading to a partial diagonalization of the
high-energy part. The techniques proposed are known from the coupled
cluster many-body theory and invoke neither perturbation nor unitarity
transformation. We showed that our formalism can be solved
perturbatively. In this way, it was revealed that diagonalization at
first order in coupling constant defines a correct low-energy
effective Hamiltonian which is valid up to the order $\lambda^{3}$. As
an example we applied this method to the computation of the one-loop
renormalized Hamiltonian in ELM. We showed that up to this order the
low-energy effective Hamiltonian for the ELM is manifestly
hermitian. It was demonstrated that ELM can not exhibit asymptotic
freedom at $D=4$ and when the cut-off is removed, ghost states are
generated. The application of the method proposed here to more
realistic models is the subject of future publications. One of the key
features which has not yet been exploited is the non-perturbative aspect of
the method; it may well be able to obtain effective degrees of freedom
that are very different from the ones occurs at the high-energy
scale. This is a promising avenue for future work. 
\section{Acknowledgment}
One of the authors (AHR) acknowledges support from British Government ORS award and UMIST grant. The work of NRW is supported by the UK engineering and physical sciences research council under grant GR/N15672.
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