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\begin{document} YITP-01-36 
\hspace{10cm}
\today
\\
\vspace{3cm}
\thispagestyle{empty}
\begin{center} {\LARGE    Local
Casimir energy for solitons
   } 
\\  \vspace{2cm} {Alfred Scharff Goldhaber\footnote{e-mail:
goldhab@insti.physics.sunysb.edu},  Andrei
Litvintsev\footnote{e-mail:
litvint@insti.physics.sunysb.edu}  and Peter van
Nieuwenhuizen\footnote{e-mail: 
vannieu@insti.physics.sunysb.edu}\\
 { \it C.N.Yang Institute for Theoretical Physics, SUNY at
Stony Brook},
\\ {\it Stony Brook, NY 11794 } }
%\it
\abstract{ Direct calculation of the one-loop
contributions to the energy density of
bosonic and supersymmetric 
$\phi^4$ kinks exhibits:  (1) {\it
Local mode regularization}. 
Requiring the mode density in the soliton and trivial
sector to be equal at each point in space yields the
anomalous part of the energy density.  (2) 
{\it Phase space factorization}. A striking
position-momentum factorization for reflectionless
potentials gives the non-anomalous energy density a
simple relation to that for the bound state.  
For the supersymmetric kink, energy and central
charge densities agree as expected. }
\end{center}
\newpage










{\it Introduction.---}Quantum corrections to solitons were
of great interest in the
  1970's and 1980's, and again in the last few years, due to
the present  activity in quantum field theories with
dualities between extended objects and pointlike objects.  
Dashen, Hasslacher, and Neveu
\cite{dashen}, in a 1974 article that has become a classic, 
computed the one-loop corrections
 to the mass of the bosonic kink in
$\phi^4$ field theory and to the bosonic soliton in
sine-Gordon theory.  For the latter, there exist exact
analytical methods associated with the complete
integrability of the system, authenticating the perturbative
calculation.  Our work here uses general principles but
focuses on the kink, for which exact results are not
available.  Dashen et al. put the object (classical
background field corresponding to kink or to s-G soliton) in
a box of length
$L$ to discretize the
 continuous  spectrum, and used mode number regularization
(equal numbers of modes in the topological and trivial
sectors, including the zero mode in this counting) for the
ultraviolet divergences.  They imposed periodic boundary
conditions (PBC) on the meson field which describes the
fluctuations around the trivial or topological vacuum
solutions, and added a logarithmically divergent mass
counterterm whose finite part
 was fixed by requiring absence of tadpoles in the trivial
background. They found for the kink 
\begin{equation} M^{(1)} = 
\sum \frac{1}{2} \hbar \omega_n - \sum \frac{1}{2}
\hbar
\omega_n^{(0)} +
\Delta M = - \hbar m \left( \frac{3}{2 \pi} -
\frac{\sqrt{3}}{12}
\right) <0
\label{mass}  \ \ ,
\end{equation} where $m$ is the mass of the meson in the
trivial background. This result remains unchallenged. 

The supersymmetric (susy) case, as well as the general case
including fermions, proved more difficult. The action reads 
\begin{equation} {\cal L}=- \frac{1}{2} \left(
\partial_{\mu}\phi
\right)^{2}-  \frac{1}{2}
\overline{\psi}\!\!\!\not\!\partial\psi -
\frac{1}{2} U^{2} - c\frac{1}{2}
\frac{dU}{d\phi}\overline{\psi}\psi
\ \ ,\end{equation} where $- \frac{1}{2} U^{2}=
-\frac{\lambda}{4}(\phi^{2}-\mu^{2}/\lambda)^{2}
$,  the meson mass is 
$m=\mu\sqrt{2}$, and $c=1$ for supersymmetry. Dashen et al.
did not explicitly compute the fermionic corrections to the
soliton mass, stating ``The actual computation of [the
contribution to]
$M^{(1)}$ [due to fermions] can be carried out along the
lines of the Appendix. As the result is rather complicated
and not particularly illuminating we will not give it here''
(page 4137 of
\cite{dashen}). 





Several authors later computed $M^{(1)}$ for the susy kink
and found different answers. It became clear that the method
of Dashen et al. yielded results that
depended on the boundary conditions  for the fluctuating
fields. In fact,
repeating exactly the same steps for the susy kink as taken
by Dashen et al. for the bosonic kink (using PBC also for
the fermions,
$\psi_{\pm}(-L/2) =\psi_{\pm}(L/2) $), taking equal numbers
of modes in all four sectors, including one term with
$\omega \simeq  0
$ in the bosonic kink sector and one term with
$\omega=0$ in the fermionic kink sector (explicitly,
there are two real independent solutions with
$\omega=0$, one localized at the kink and 
one at the boundary, and the coefficient
of each satisfies
$c^2=\frac{1}{2}$ \cite{fred}) 
 gives 
 $M^{(1)}=\hbar m (\frac{1}{4} -
\frac{1}{2 \pi})$ \cite{rebhan},  which we now know is a
correct answer to an inappropriate question, because it
includes boundary energy.
Schonfeld
\cite{schonfeld} finessed the problem of a single kink with
its sensitivity to boundary conditions by considering the
kink-antikink system with PBC, and taking into account two
terms with
$\omega
\sim 0$ in the bosonic kink sector and one term with 
$\omega
\sim 0$ in the fermionic kink sector,  he obtained what we
now know to be the correct answer. The problem of boundary
contributions was circumvented by other methods in
\cite{misha} and
\cite{graham}. The fermionic contribution to
$M^{(1)}$ is given by \begin{equation} M^{(1)}_f=\hbar
m\left(\frac{1}{\pi}-\frac{\sqrt{3}}{12}\right)>0 \
\ ,
\end{equation} and the total one-loop correction is thus
\begin{equation} M^{(1)}_{susy}=-\frac{\hbar m}{2\pi}
\label{Ms}\ \  .
\end{equation}

 With attention restricted to the kink alone, it was shown
in \cite{fred} that  to eliminate boundary contributions
from the fermionic part of the energy one should average
over quartets of boundary  conditions for the fermionic
fluctuations -- periodic, antiperiodic, twisted periodic and
twisted antiperiodic, where twisting means interchange of
the upper and lower components of the fermion wave function
$\psi_{\pm}\to\psi_{\mp}$ \cite{misha}.  This averaging 
 is necessary to preserve a fermionic
$Z_2$ gauge invariance, and also preserves a chiral
$Z_2$ rigid symmetry. The results in
\cite{fred} give a complete, though  intricate,
understanding of the correct way to calculate
$M^{(1)}$ in terms of mode frequencies $\omega_n$.


In light of the complexities which boundary conditions
generate for the problem including fermions, the most
important advance since
\cite{dashen}  was the approach of Shifman, Vainshtein, and
Voloshin 
\cite{shifman},  who used higher  space-derivative
regularization (factors $(1-\partial_x^2/M^2)$ for the
kinetic terms but not the  interactions) to compute the central
charge densities of the susy sine-Gordon soliton and kink. 
Their  scheme is manifestly susy, canonical (no  higher time
derivatives), and  independent of  boundary
conditions  (because
it yields a local density).  They showed
the energy density is equal to the central charge
density, which they computed -- including an anomaly
recognizable as an
$M^2/M^2$  effect  (The presence of an anomaly was first 
suggested in
\cite{misha}.).  They  verified that the one-loop
correction 
$Z^{(1)}$ to the central charge of the kink is equal to
(\ref{Ms}).  

 A `pseudo-supersymmetry'
allows one to compute the energy density for the
bosonic sine-Gordon soliton as a central charge density
\cite{shifman}, and quite possibly similar  techniques would work
for the bosonic kink.  Our approach here is instead to attack the 
 Casimir energy
density directly, freeing the calculation from
dependence on supersymmetry.  In doing so, we have encountered a
principle which appears new in the literature, and yet has roots in
early quantum physics. {\it Local mode regularization}
(lmr), or mode density regularization, is the local
counterpart of the familiar global mode regularization or
mode number regularization.  Neither the local nor
the global version represents a complete regularization
scheme (for example,  we shall see for our problem that point
splitting  determines the central charge density, but  lmr
alone is insufficient). The principle  is that, when 
fluctuations are  expanded into modes
$\phi_n(x)$, the cut-off local mode density $\rho_N(x) =
\sum_{n=1}^N
\phi^*_n(x)\phi_n(x) $ should be background-independent,
i.e., the same in the trivial ($\rho^{(0)}$) and kink
($\rho$) sectors. For the bosonic kink this means that one
must truncate the sums at different upper bounds
\begin{equation}
\rho_N(x) = \rho_{N+\Delta N(x)}^{(0)} (x) \ \ .
\label{eq:eq3}
\end{equation} This equation determines $\Delta N(x)$ in
terms of
$N$, and clearly
$\Delta N(x) $ is 
$x$-dependent. For the susy kink one can begin by fixing the
mode number  cut-off  $N_b^{(0)}$ for the bosons and
$N_f^{(0)}$  for the fermions in the trivial sector such
that here the bosonic and fermionic mode densities
 are equal. From (\ref{eq:eq3}) one then obtains in  the
nontrivial sector the requirement
\begin{equation}
\rho_{N,b}(x) = \rho_{N+\Delta N(x),f} (x) \ \ ,
\end{equation} which again determines $\Delta N(x) $ in
terms of
$N$.  We use this principle to compute the anomalous energy
density of the bosonic kink, as well as of the
supersymmetric kink, which as mentioned was obtained already
in
\cite{shifman} through the equality of energy and central
charge densities.  The lmr principle appears necessary and
sufficient for regularization of Casimir energy density in
one space dimension, and at least necessary in
higher dimensions.  

For the non-anomalous contributions to both the bosonic and
susy kink densities, we find empirically another striking
regularity, {\it phase space factorization}.  The continuum
contribution to the Casimir energy density in phase space 
exhibits a remarkable  factorization, involving a few terms
each with simple
momentum-dependent factors multiplying functions related to the
bound-state energy density in coordinate space.  We believe this
factorization should hold for all reflectionless potentials, but
might not extend farther.  

We use point-splitting regularization to
obtain the anomaly in the central charge density
 near the kink, of course confirming \cite{shifman}.
Finally, we discuss the physical basis for lmr. 


{\it  Bosonic kink energy density.---}For the energy density
of the bosonic kink, one must evaluate sums $\sum \frac{1}{2}
\omega_n
\phi^*_n(x) \phi_n(x) $.  As these sums differ from the
density sums $\sum \phi_n^*(x) \phi_n(x)$,  one
expects in general a nonvanishing one-loop correction to the
energy density, and hence to the quantum mass.  Let us begin
with explicit expressions for the mode eigenfunctions,  so
that one may follow the argument in detail.  The wave
functions of the continuous spectrum (using  $ |\phi_n|^2(x)=1$
away from the kink to determine the normalization constant ${\cal
N}$) obey
\begin{equation}
\phi(k,x) = \frac{e^{ikx}}{\cal
N} \left[ -3 \tanh^2
\frac{mx}{2}+1+ 4
\left(
\frac{k}{m} \right)^2 + 6i \frac{k}{m} \tanh
\frac{mx}{2} \right] \ \ ,
\end{equation} with $\omega= \sqrt{k^2+m^2} $. The zero mode
with
$\omega_0=0$ is given by
\begin{equation}
\phi_0(x) = \sqrt{\frac{3m}{8}} \cosh^{-2} \left(
\frac{mx}{2}
\right) \ \ .
\end{equation} The bound state with $\omega_B =
\frac{\sqrt{3}}{2} m$ is given by
\begin{equation}
\phi_B(x) =\sqrt{\frac{3m}{4}} \frac{\sinh \left(
\frac{mx}{2}
\right)}{\cosh^2\left( \frac{mx}{2} \right)} \ \ .
\end{equation} The density of the continuous spectrum  can be
written as follows
$$ |\phi(k,x)|^2 = \frac{1}{{\cal
N}^2} \left[
\frac{4 \omega^2}{m^2} \left( \frac{4
\omega^2}{m^2}-3
\right) - 12 \frac{\omega^2}{m^2} \cosh^{-2}
\frac{mx}{2} + 9 \cosh^{-4}
\frac{mx}{2}
\right] 
$$
\begin{equation}  = 1 - \frac{m
\phi_B^2(x)}{\omega^2-\omega_B^2}- 
\frac{2 m \phi_0^2(x)}{\omega^2} \ \ 
\label{eq10m}
\end{equation} (with $ {\cal
N}^2= 16 \frac{\omega^2}{m^2} \left(
\frac{\omega^2}{m^2} - \frac{\omega_B^2}{m^2}
\right)$).
 As $ \int_{-\infty}^{\infty} \frac{dk}{2 \pi }
\frac{1}{\omega^2-\omega_B^2} = \frac{1}{m} $, while
$ 
\int_{-\infty}^{\infty} \frac{dk}{2 \pi }
\frac{1}{\omega^2} =
\frac{1}{2m} $, it is clear that the completeness relation 
\begin{equation}
\int_{-\infty}^{\infty} \frac{dk}{2 \pi} \left\{
|\phi(k,x)|^2-1 
\right\}+ \phi_0^2(x) + \phi_B^2(x) =0 
\label{comp}
\end{equation}  is satisfied.  Eq (\ref{eq10m}) may be
written in a remarkable formula perhaps true for all
reflectionless potentials,  showing factorization of the
difference in mode densities in phase space, where the
position dependence of each term is given by the
corresponding bound-state probability density.
\begin{equation} |\phi(k,x)|^2 - 1 = - \sum_j \phi_{j}^2(x) 
\frac{2
\sqrt{m^2-\omega_{j}^2}}{\omega^2-\omega^2_{j}} \ \ ,
\end{equation} satisfying the completeness relation,
as one may check by performing the integration over
$k$.  

Note that all the above expressions for the
density do not refer to any particular choice
of boundary conditions, which of course do
affect eigenenergies and the corresponding
wave  functions.  The reason 
is that the
choice of boundary conditions will contribute
to the density away from the boundary at most
terms of order $1/L$. In the large-$L$ or
continuum limit such terms could contribute a
finite amount to the total energy obtained by
integration over the entire interval between
the boundaries. However, for the integral just
over an interval around the kink,  these terms
may be neglected, and the kink energy and
energy density can be computed in
terms of the continuum, modified-plane-wave
solutions.


The requirement in (\ref{eq:eq3}) that the topological vacuum
density and the trivial vacuum density be equal leads via
(\ref{comp}) to 
$$
\frac{\Delta \Lambda(x)}{\pi} = 
%\phi_0^2(x) +
%\phi_B^2(x) +
\int_\Lambda^\infty \frac{dk}{\pi} \left( 1-|\phi(k,x)|^2
\right)
 =\int_\Lambda^\infty dk\frac{2}{\pi}\sum_j\frac{\sqrt{m^2-
\omega^2_{j}}}{\omega^2-
\omega^2_{j}}
\phi_{j}^2(x)$$
\begin{equation} = \frac{m}{\pi \Lambda} \left( \phi_B^2(x)
+ 2
\phi_0^2(x)
\right) +{\cal O} \left( \frac{1}{\Lambda^2}
\right)= \frac{3 m^2}{4 \pi \Lambda} \frac{1}{\cosh^2
\frac{mx}{2} } + {\cal O} \left( \frac{1}{\Lambda^2}
\right)\label{deltalambda} \ \ .
\end{equation}

With this result for $\Delta \Lambda(x)$ we can evaluate the
energy density
$\epsilon(x)$. Adding also the counter term $\Delta M(x) =
\sum_j 2\phi_{j}^2(x)\sqrt{m^2-\omega_{j}^2}
\int_0^{\Lambda}\frac{dk}{2\pi}\frac{1}{\omega}=
\frac{m^2}{4} \left(
\cosh^{-2} \frac{mx}{2} \right) \frac{3}{2 \pi}
\int_0^\Lambda 
\frac{dk }{\omega} $ and rewriting $\cosh^{-2}
\frac{mx}{2}$ as $ 
\frac{4}{3m} \phi_B^2(x) + \frac{8}{3m} \phi_0^2(x)$ yields
(setting from now on $\hbar=1$)
$$\epsilon_{\rm Cas}(x)=
\epsilon(x) - \epsilon^{(0)} (x) = \frac{1}{2}
\omega_B
\phi_B^2(x) +  2\int_0^\Lambda \frac{dk}{2 \pi} |\phi(k,x)|^2
\frac{1}{2} \omega - 2 \int_0^{\Lambda+\Delta
\Lambda(x)}
\frac{dk}{2
\pi}  \frac{1}{2} \omega + \Delta M(x)
 =$$
\begin{equation} \frac{1}{2} \omega_B \phi_B^2(x) -
\int_0^\Lambda
\frac{dk}{2 \pi}
\left(
\frac{m \phi_B^2(x)}{k^2+\frac{1}{4} m^2} + \frac{2 m
\phi_0^2(x)}{k^2+m^2}
\right)
 \omega
 - \frac{ \Delta \Lambda (x)}{2\pi} 
\Lambda + m \left(
\phi_B^2(x) + 2 \phi_0^2(x) \right) \int_0^\Lambda 
\frac{dk}{2 \pi}
\frac{1}{\omega} \ \ .
\end{equation} The two quadratic divergences proportional to
$\int_0^\Lambda dk \omega$ have canceled because we
subtracted the energy density of the trivial vacuum, while
the counter term cancels the remaining logarithmic
divergence.  Again, each term is proportional to a
bound-state probability density.


The result is finite and reads 
\begin{equation}
\epsilon_{\rm Cas}(x) = 
\frac{1}{2} 
\omega_B \phi_B^2(x) - m 
\int_0^\Lambda  
\frac{dk}{2 \pi} 
\left( 
\frac{\omega}{k^2+\frac{1}{4} m^2} -
\frac{1}{\omega} \right) \phi_B^2(x) - 
\frac{m}{2 \pi} \left(\phi_B^2(x)+2 \phi_0^2(x)
\right) \ \ .\label{bonomaly}
\end{equation} The last term is the contribution from the
anomaly (the term due to 
$\Delta \Lambda (x) $). Using the integral
\begin{equation}
\int_0^\infty \frac{dk}{2 \pi} \left(
\frac{1}{k^2+\frac{1}{4} m^2} -
\frac{1}{k^2+m^2}
\right) \omega = \frac{1}{2 \sqrt{3} } \ \ ,
\end{equation} we obtain 
\begin{equation}
\epsilon_{\rm Cas}(x) = \left(
\frac{1}{2} \omega_B - \frac{m}{2 \sqrt{3}} -
\frac{m}{2 \pi}
\right)
\phi_B^2(x) - \frac{m}{\pi} \phi_0^2(x) \ \ . 
\end{equation}  This formula can be rewritten as 
follows, 
\begin{equation}
\epsilon_{\rm Cas}(x)=
\sum_j \frac{1}{2} \left( 1 - \frac{2}{\pi} \arctan
\frac{\omega_j}{\sqrt{m^2- \omega_j^2}} \right)
\omega_j\phi^2_{j}(x)-
\sum_j\frac{1}{\pi}\sqrt{m^2-
\omega^2_{j}}
\phi_{j}^2(x) \ \ , \label{fad} \end{equation} where in the
first sum the contribution with 1 comes from the bound
states, and that with arctan comes from the continuum, while
the second sum is the anomaly contribution.  Such formulas
for the total mass can be found in
\cite{Faddeev}, though we are unaware of local versions in
the literature.  This kink example might be an
illustration of a general factorization rule, valid for a
wide class of reflectionless potentials.  While we
have not tested it for other cases, and do not
know how to prove it other than by explicit
computation, we believe that its simplicity
and elegance make the rule worthy of further
investigation. 

Integration of $\epsilon_{Cas}(x)$ over $x$ yields 
\begin{equation} M^{(1)}= \frac{1}{2}
\omega_B(1-\frac{2}{3})-
\frac{m}{2\pi} -\frac{m}{\pi} =
\frac{\sqrt{3} m}{12} -
\frac{3m}{2
\pi} \ \ ,
\end{equation} in agreement with (\ref{mass}).

Eq. (\ref{fad}) gives an elegant expression for local
energy density which certainly provides the correct total
quantum energy of the bosonic kink.  However, to obtain
the correct local Casimir energy density, one must start
with an expression for the energy density of each mode
including a quadratic term in the gradient of the boson
field $\frac{1}{2}(\partial_x\chi)^2$, whereas our formulae above
implicitly used the expression coming from the field equation 
$-\frac{1}{2}\chi\partial_x^2\chi$.  Therefore we need to add 
to (\ref{fad}) the
difference, a perfect differential of a function whch vanishes
far from the kink,
\begin{equation}
\Delta\epsilon_{\rm Cas} (x)=
\frac{1}{4}\partial_x^2\langle\chi^2(x)\rangle
=
\frac{1}{4}\partial_x^2\left(-\frac{3}{8\pi}\frac{1}{\cosh^4(mx/2)}
+\frac{1}{4\sqrt{3}}\frac{{\rm tanh}^2(mx/2)}{\cosh^2(mx/2)}\right)
\ \ , 
\end{equation}
where the propagator at equal times and positions
$\langle\chi^2(x)\rangle$ (excluding the zero mode
which solves the homogeneous equation) \cite{shifman} can be
obtained by integrating $\frac{|\phi|^2}{2\omega}$ in (\ref{eq10m})
w.r.t. $\frac{dk}{2\pi}$ and adding $\frac{\phi_B^2}{2\omega}$.

As observed in \cite{shifman}, besides the Casimir energy density
there is another consequence of the zero-point oscillations,
namely, a position-dependent shift $\phi_1$ in the classical 
background field.  This in turn implies a further term
in the local energy density, given by
\begin{equation}
\Delta \epsilon_{(
\phi_1)}(x)=\partial_x\phi_1\partial_x\phi_{\rm
kink}+(\frac{1}{2}U^2)'\phi_1=
\partial_x(\phi_1\partial_x\phi_{\rm kink})  \ \ ,\label{ephi}
\end{equation}
but of course no shift at
this order in the total energy, because the classical energy
is stationary with respect to arbitrary small           variations
of the  classical field about its equilibrium form.  Decomposing
the Heisenberg field
$\Phi(x,t)$ as $\phi_{\rm kink}(x)+\phi_1(x)+\chi(x,t)$, with
the quantum fluctuation field obeying $\langle\chi\rangle=0$,
and taking the expectation value of the $\Phi$ field 
equation $\langle -\partial_t^2\Phi
+\partial_x^2\Phi-(\frac{1}{2}U^2)'\rangle=0$ 
gives
\begin{equation}
\partial_x^2\phi_1-(\frac{1}{2}U^2)''\phi_1=
\frac{1}{2!}(\frac{1}{2}U^2)'''\langle
\chi^2\rangle-\frac{1}{2}\Delta
m^2\phi_{\rm kink}
\
\ ,
\end{equation}
from which one may verify
\begin{equation}
\phi_1=\frac
{\lambda}{m^2}
{\bigg [ } \left
(\frac{1}{2\sqrt{3}}+\frac{3}{4\pi}\right
)
\frac{1}{\cosh^2(mx/2)}
-\frac{\sqrt{3}}{4}(m\partial_m+2\lambda\partial_{\lambda}){\bigg
] }\phi_{\rm kink}
\
\ \label{phi1}.
\end{equation}
Through one-loop order  (as in the susy case 
\cite{shifman}), the effect of the second term is to replace
 the renormalized mass
$m$ and coupling $\lambda$ in $\phi_{\rm kink}$  with the pole
mass $\bar m=m(1-\sqrt{3}\frac{\lambda}{4m^2})$ given in
\cite{rebhan}, eq.(7) and the adjusted coupling $\bar \lambda =
\frac{\bar m^2} {m^2}\lambda$.  If we then rewrite the classical
energy in terms of $\bar m$ and $\bar \lambda$, the classical mass
is multiplied by a factor $1-\sqrt{3}\frac{\lambda}{4m^2}$.
As $\phi_1$ cannot shift the total mass, we know even without
explicit calculation that the classical energy density in terms of
the barred quantities must be renormalized by a compensating factor
$1+\sqrt{3}\frac{\lambda}{4m^2}$.  The first term in (\ref{phi1})
is sensitive only to bosonic fluctuations and hence unchanged
in the susy case \cite{shifman}; it contributes according to
(\ref{ephi}). The total one-loop bosonic energy density becomes
$${\cal E}(x) = U^2(\bar\lambda,\bar m,\phi_{kink}(\bar \lambda,
\bar m,x))(1+\sqrt{3}\frac{\bar\lambda}{4\bar m^2}) + 
\epsilon_{\rm
Cas}(x)+\Delta \epsilon_{\rm
Cas}(x) + $$
\begin{equation}\frac{\bar m}{8}\partial_x {\bigg [ } 
\frac{9}{4\pi}\frac{{\rm tanh}(\bar m x/2)}{\cosh^4(\bar m x/2)}
+\frac{1}{2\sqrt{3}}{\bigg (}\frac{3}{\cosh^2(\bar m
x/2)}+\frac{1}{\cosh^4 (\bar m x/2)}{\bigg ) }{\rm tanh} (\bar m
x/2) {\bigg ] } \ \  .
\end{equation}

{\it Susy kink energy density.---} For the susy kink we
choose the cut-offs in the trivial sector in such a way that
the bosonic and fermionic densities in the trivial sector
are equal. To make also the bosonic and fermionic densities
in the topological sector equal, we use a cut-off
$\Lambda$ for the bosons and $\Lambda+\Delta
\Lambda(x)$ for the fermions. The fermion is described by a
Majorana two-component spinor
$\psi={\psi_+
\choose \psi_-}$. As $\psi_+(k,x)$ is proportional to
$\phi(k,x)$ while
$\psi_-(k,x) =
\frac{i}{\omega} \left( \partial_x + m \tanh
\frac{mx}{2} \right)
\psi_+(k,x)
$  for solutions proportional to $\exp (-i \omega t)$ 
according to the Dirac equation \cite{rebhan}, one obtains
for the wave functions of the continuous fermionic spectrum
\begin{equation}
\psi_+(k,x) = \frac{1}{\sqrt{2}} \phi(k,x)
\end{equation}
\begin{equation}
\psi_-(k,x) = \frac{1}{\sqrt{2}} \frac{\omega}{{\cal
N} m}
\left( - 4
\frac{k}{m} - 2 i \tanh \frac{mx}{2} \right) e^{ikx}
\ \ .\end{equation} In the difference of the densities the
constant term of course cancels, giving
$$ |\phi(k,x)|^2 - |\psi_+(k,x)|^2 - |\psi_-(k,x)|^2 =
|\psi_+(k,x)|^2- |\psi_-(k,x)|^2 \
\ 
$$
\begin{equation} = \frac{1}{2 {\cal
N}^2} \left( 9 \cosh^{-4}
\frac{mx}{2} - 8 \left(
\frac{\omega}{m} \right)^2 \cosh^{-2}
\frac{mx}{2} \right) \ \ .  \label{asym}
\end{equation} In order that the bosonic and fermionic
densities agree one must satisfy
$$
\phi_0^2(x) + \phi_B^2(x) + 2 \int_0^\Lambda
\frac{dk}{2 \pi}  |\phi(k,x)|^2 
$$
\begin{equation} = \frac{1}{2} \phi_0^2(x)+ \left
(\frac{1}{2}
\phi_B^2(x) +
\frac{m}{8 \cosh^2 \frac{mx}{2} }\right ) + 2
\int_0^{\Lambda +
\Delta \Lambda} \frac{dk}{2\pi}\left\{ |\psi_+(k,x)|^2+
|\psi_-(k,x)|^2
\right\} \ \ .
\end{equation}  The factor $\frac{1}{2}$ in
$\frac{1}{2}\phi_0^2(x)$ comes from the mode expansion
$\psi_+(x,t)=c_0\phi_0(x,t)+{\ ...}$ with $\{ c_0,c_0\}=1$. 
This $\frac{1}{2}$ is the analogue for Majorana fermions of
the fractional fermion charge discovered by Jackiw and Rebbi
for Dirac fermions
\cite{JR}.
 The two terms in parentheses give the
$\psi_+$ and
$\psi_-$ contributions of the bound state: 
$\psi_{B+}^2=\frac{1}{2}\phi_B^2$ and
$|\psi_{B-}|^2=\frac{m}{8 \cosh^2 \frac{mx}{2} }$.  
  We obtain
$$
\frac{1}{2} \phi_0^2(x) +  \psi_{B+}^2(x) 
-|\psi_{B-}|^2(x)+ 2
\int_0^\Lambda
 \frac{dk}{2 \pi}  \left( |\psi_+(\Lambda ,x)|^2 -
|\psi_-(\Lambda , x)|^2 \right)
$$

\begin{equation}      =\frac{\Delta \Lambda(x)}{\pi} \left(
|\psi_+(k,x)|^2 + |\psi_-(k,x)|^2 \right) \ \ .
\end{equation} Using the completeness relation, and taking
the large $k$ limit $ |\psi_+(k,x)|^2 + |\psi_-(k,x)|^2
 \to 1$, one finds
\begin{equation}
\frac{\Delta \Lambda(x)}{\pi} = -2 
\int_\Lambda^\infty
 \frac{dk}{2 \pi}  \left( |\psi_+(k,x)|^2 - |\psi_-(k,x)|^2
\right) \ \ .
\end{equation} As we are interested only in the $1/\Lambda$
term, the calculation is easy.  From (\ref{asym}) we find
\begin{equation}
\frac{\Delta \Lambda (x) }{\pi} = \frac{m^2}{4 \pi
\Lambda}
\frac{1}{\cosh^2\frac{mx}{2}} \ \ .
\end{equation} 

With this result in hand, we compute the difference in
energy densities for the susy kink  
$$
\epsilon_{\rm b}(x) - \epsilon_{\rm f}(x) =
\frac{1}{2}
\omega_B \left(\phi_B^2(x)-\psi_B^{\dagger}(x)
\psi_B(x)
\right)
$$
\begin{equation} + 2 \int_0^\Lambda \frac{dk}{2 \pi}  \left(
|\psi_+(k,x)|^2 - |\psi_-(k,x)|^2 \right) \frac{1}{2}
\omega -
\frac{\Delta \Lambda(x)}{\pi} \frac{1}{2} \Lambda +
\Delta M(x)  \ \ .
\end{equation} The counter term in the susy case,
\begin{equation}
\Delta M(x) = \frac{m^2}{4} \frac{1}{\cosh^2
\frac{mx}{2} }
\int_0^\Lambda 
\frac{dk}{2 \pi} \frac{1}{\omega} \ \ ,
\end{equation}  is smaller (but nonvanishing).
  By the same process as in the bosonic case, this yields
\begin{equation}
\epsilon_{\rm Cas}=\epsilon_{\rm b}(x) - \epsilon_{\rm f}(x)=
(1-\frac{2}{3})\frac{1}{2}
\omega_B \left(\phi_B^2(x)-\psi_B^{\dagger}(x)
\psi_B(x)\right) - \frac{m^2}{8 \pi }
\frac{1}{\cosh^2\frac{mx}{2}} \ \ ,
\end{equation} where the last term, the contribution from the
anomaly, agrees with the central charge density anomaly
eq.(3.38) in
\cite{shifman}. Integration over $x$ yields the one-loop
correction to the mass of the susy kink 
\begin{equation} M^{(1)}_{susy} =  \frac{1}{6} \omega_B(1-1)
-
\frac{m}{2 \pi} =-
\frac{m}{2 \pi} \ \ ,
\end{equation}   which of course is the accepted answer. Note
that the non-anomalous contributions from the bosons and the
fermions do not cancel locally, but in the integral they do:
$\frac{1}{6}
\omega_B(1-1)=0$.  As in the
bosonic case we must add the missing term in the bosonic Casimir
energy density
$\Delta \epsilon_{\rm Cas}$, as well
as include the shift for the susy case $\phi_1(x)$ in the background
field.  We have computed this $\phi_1$, again using the
second-order field equation for $\Phi$, getting the same
result found in \cite{shifman} from a first-order differential
equation based on susy considerations (Iterating the susy relation
$\langle\partial_x\phi+U\rangle=0$, one finds agreement by using
the identity $(\frac{1}{2}\partial_x+U')\langle
\chi^2\rangle +\frac{1}{2}\langle\bar \psi\psi\rangle=0$, which
holds because the nonzero modes of $\chi$ and $(\partial_x+U')\chi$
satisfy the same field eqations as the components of $\psi$.). 



 For the susy energy density
we find full agreement, after restoring a missing factor of
$\frac{1}{2}$ in the first line of (5.21) of
\cite{shifman}, kindly pointed out to us by the authors.

{\it Central charge density.---}We now compute the anomalous
contribution to the
density
$\zeta (x)$. Before  regularization one has
${\cal H}(x)=\frac{1}{2}\dot{\phi}^2
+\frac{1}{2}\phi'^2+\frac{1}{2}U^2+
\frac{i}{2}\left (
\psi_+\psi'_+ +
\psi_-\psi'_-\right )-iU'\psi_+\psi_-$, and
$\zeta (x) = U
\partial_x \phi$  (note that
\cite{shifman} have the opposite sign convention for
$\zeta$).  Using the equal-time anticommutators of the
fermionic fields
$\psi_+(x)$ and $\psi_-(x)$ one obtains 
\begin{equation}
\{Q_{\pm}, j_{\pm}(y)\} = 2 {\cal H} (y)\pm 2 \zeta (y);
\ \ \ \  j_{\pm} = \dot{\phi} \psi_{\pm} + (
\phi^\prime
\pm U) \psi_{\mp};
\ \ \ \  Q_\pm = \int_{-L/2}^{L/2} j_{\pm} dx \ \ ;
\end{equation}
$$
\zeta(y) = \int dx \left[ \frac{1}{2} \left(
\{ \psi_+(x), \psi_+(y) \} - \{ \psi_-(x), \psi_-(y)
\}
\right) 
\left( \frac{1}{2} \dot{\phi} (x) \dot{\phi} (y) -
\frac{1}{2} 
\phi^\prime(x) \phi^\prime(y) + \frac{1}{2}U(x) U(y)
\right)
\right.
$$
\begin{equation} 
\left. + \frac{1}{2} \left(
\{ \psi_+(x), \psi_+(y) \} + \{ \psi_-(x), \psi_-(y)
\}
\right) 
\left(
\frac{1}{2} \phi^\prime(x) U(y) + \frac{1}{2} U(x)
\phi^\prime (y)
\right) \right]
\label{eq:A} \ \ .
\end{equation} Naively the anticommutators $
\{ \psi_+(x), \psi_-(y) \} 
$ and $
\{ \psi_+(y), \psi_-(x) \} 
$, as well as  terms involving bosonic commutators, all
vanish, and also the first line in $\zeta(y)$ vanishes,
while the second line gives
$$\zeta(y)=
\int_{-\infty}^{+\infty}  dx \delta (x-y) \left[
\frac{1}{2} \langle \eta(x) \eta(x) \rangle U^{\prime
\prime}
\phi^\prime (x) + \langle \eta^\prime(x)
\eta(x) \rangle U^\prime(x) \right]
$$
\begin{equation} = \int_{-\infty}^{+\infty} dx \delta(x-y)
\frac{\partial}{\partial x}
\left[
\frac{1}{2} \langle 
\eta(x) \eta(x) \rangle U^\prime(x) \right] \ \ ;
\end{equation}
\begin{equation}\int_{-\infty}^{\infty}dy\zeta(y) =
\frac{1}{2}
\langle
\eta(x)
\eta(x)
\rangle U^\prime(x) |_{-\infty}^\infty
\label{eq_B} \ \ .
\end{equation} This is the expression obtained in
\cite{rebhan}. Below we show that with proper regularization
this term yields the anomaly. The naive result in
(\ref{eq_B}) contains a free field propagator for
$\eta$, because at $x=
\pm
\infty$ the effects of the kink disappear, and adding the
counterterm to the central charge due to mass
renormalization, all  quantum corrections to the central
charge would seem to vanish. In the  approach of
\cite{shifman}, on the other hand, the central charge
contains  a naive term $\phi^\prime U$ and an explicit
correction term which is also a total derivative and
proportional to
$1/M^2$. Because their $\eta$ propagator contains an extra
regulating factor $(k^2+M^2)^{-1}$, the contribution in
(\ref{eq_B}) now cancels even after regularization, and the
extra term proportional to
$1/M^2$ yields the anomaly.

In our case we start from (\ref{eq:A}), but without extra
terms as in
\cite{shifman}. We regulate (\ref{eq:A}) and show that after
regularization the result is still a total derivative, but
instead of the total derivative in (\ref{eq_B}), rather a
total derivative with an extra term. The
$x$-integral of the first term vanishes as in
\cite{rebhan} but the second term yields the anomaly
\begin{equation}
\int_{-\infty}^\infty \zeta (x) dx = \left.
\frac{W^{\prime\prime}(\phi)}{4 \pi}
\right|_{-\infty}^{\infty} \ \ ,
\end{equation} where $W^\prime(\phi) =  U$. Hence
$M^{(1)}=-Z^{(1)}$ in agreement with the invariance of the
background under $Q_{+}$ (which corresponds to susy
transformation with parameter $\epsilon_-$).

The crucial identity which we need is
\begin{equation}
\delta(x-y) \langle \eta^\prime (x) \eta(y)  \rangle ( f(x)
- f(y) ) = - \frac{1}{2 \pi} \delta(x-y) f^\prime(x) \ \ ,
\label{eq37m}
\end{equation} where $f$ is any smooth function of $x$. The
proof of this identity follows from $\langle \eta(x)
\eta(y)
\rangle  = -\frac{1}{2 \pi}
\ln |x-y| + A(x,y)$, where $A$ is a smooth function. The
actual calculation of the anomaly is now very simple. 
Expanding $U^{\prime}(x)$ in terms of $x+y$ and $x-y$, the
latter contributes according to (\ref{eq37m}).
\begin{equation}
\int_{-\infty}^\infty  \zeta(y) dy =
\int_{-\infty}^\infty dxdy 
\left[  \frac{1}{4\pi }  \delta(x-y) U^{\prime
\prime}(\phi)
\phi^\prime(x) \right] =  \frac{m}{2 \pi } \ \ .
\end{equation} Here we used $U= \sqrt{\frac{\lambda}{2}}
\left(
\phi^2 -
\frac{\mu^2}{\lambda}
\right) $, $\phi = \frac{\mu}{\sqrt{\lambda}}
\tanh{
\frac{\mu x}{\sqrt{2}}}
$, and $m= \mu \sqrt{2}$.  Again we have the accepted
result, as well as further evidence that the anomaly in the
central charge density is more straightforward to compute
directly than is the  anomaly in the energy density.

Although we showed in a simple way that the term 
$\langle \eta^\prime(x) \eta(y) \rangle U^\prime(y)
$ produces the anomaly if one does not set $x=y$ too soon,
for completeness one should show that none of the other
terms in $\zeta(y)$ produces further similar contributions.
In \cite{shifman} a more complicated but also more powerful 
regularization scheme was used to prove this. Our
observation  pinpoints the place where naive methods missed
the anomaly.

{\it Foundations and conclusions.---}Finally we comment on
the physical basis for lmr. In
Planck's original formulation, the number of degrees of
freedom is defined by the available volume in phase space.
To fix the total number of modes while introducing a
background potential affecting the fluctuations is simply to
conserve the total phase space available. The work of
Einstein and Debye on crystal vibration contributions to
heat capacity introduced the notion of a local density of
degrees of freedom. As was true for their work, in a lattice
approach the number of degrees of freedom  per unit volume
evidently does not change when interactions are  introduced,
and the local mode density should be equal to this number of
degrees of freedom. 

Also point splitting methods clarify the meaning  of lmr.
Consider the bosonic local mode density regulated by point
splitting
\begin{equation}
\rho(k,x)=\int_{-\infty}^{\infty} dy \phi^*\left(
k,x-\frac{y}{2}\right)
\phi\left( k,x+\frac{y}{2}\right) f(y) \ \ ,\label{dens}
\end{equation} where $f(y)$ is a function sharply peaked
around
$y=0$, with $\int dyf(y)=1$. For large $k$,  the WKB
approximation for
$\phi(k,x)$ is
\begin{equation}
\phi(k,x)=e^{ikx}e^{-i\int^x dx^{\prime}V(x^{\prime})/2k} \
\ ,
\end{equation} where
$V(x)=U(\phi(x))U^{\prime\prime}(\phi(x)) +
(U^{\prime})^2(\phi(x))-(U^{\prime})^2(\phi(|x|\to\infty))$.
Substituting this expression into $\rho(k,x)$ one finds for
the integrand of (\ref{dens})
\begin{equation}
e^{iky}e^{-i\int_{x-\frac{y}{2}}^{x+\frac{y}{2}}
\frac{V(x^{\prime})}{2k}dx^{\prime}} f(y)
\simeq e^{i\left( k-\frac{V(x)}{2k}\right)y} f(y)
 \ \ .
\end{equation} In the trivial sector
$\rho(k,x)=\tilde{f}(k)$, where $\tilde{f}(k)$ is the
Fourier transform of
$f(x)$, but in the kink sector one finds a modification 
$\rho(k,x)=\tilde{f}\left(k-V(x)/2k\right)$. The energy 
density therefore contains a term
$\delta\epsilon(x)=\int\delta\rho(k,x)\frac{1}{2}\omega
\frac{dk}{2\pi}$, and expanding
$\tilde{f}$ we find
\begin{equation}
\delta\epsilon(x)=  -\int_{-\infty}^{\infty}\frac{dk}{2\pi}
\frac{V(x)}{2k}\frac{\omega}{2}
\frac{\partial}{\partial k}
\tilde{f}(k) \ \ .
\end{equation} This is the anomaly in (\ref{bonomaly}).

For the supersymmetric case, one gains insight into the
requirement of equal bosonic and fermionic mode densities by
considering the $N=2$ theory, where there is an abelian
charge density which should be invariant under
supersymmetry.  That requirement automatically imposes the
constraint represented by lmr.

While all of the above are appealing arguments, the accepted
criterion for determining the validity of a regulation
procedure is to insert regulators into the action in such a
way that all relevant symmetries are satisfied at the
regulator level, and then to deduce consequences for specific
quantities.  Thus in the present case a definitive check on
the validity of lmr would be to use, for example,
higher-derivative regulation, which obeys supersymmetry, and
check that  this scheme implies lmr.  This important
analysis remains to be done.

We have seen that lmr permits one to
isolate and then compute directly the anomalous contribution
to the energy density of the bosonic or susy kink.  Further,
we found remarkable phase space factorization identities
for the non-anomalous contributions to the energy density,
which might hold for all reflectionless potentials.  These
non-anomalous contributions are independent of the
regularization method (though sensitive to renormalization
conditions because they only are convergent after
subtraction of the mass counter term).   Elsewhere
\cite{future} we compute the divergent energy density at the
boundary of the kink with supersymmetric boundary
conditions, and obtain an analytic expression for the
anomaly near the boundary, which in the limit when the
regulator energy goes to infinity becomes a delta-function
contribution just at the boundary, in agreement with
expectations of \cite{shifman}.    It would be interesting
to explore further local mode regularization, comparing
with complete regularization schemes and studying solitons
in higher dimensions such as the magnetic monopole, and also
explore phase space factorization, seeking a theoretical
basis as well as additional examples.

We thank Kevin Cahill, Ludwig Faddeev, and
Stefan  Vandoren for discussions, and in particular Arkady
Vainshtein for useful detailed comments on a number of occasions.  
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\end{document}

