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\ifx\tocauthor\undefined\def\tocauthor#1{\relax}\fi
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\begin{document}

\title{Puzzling Aspects of the Adiabatic Effective Action in Chern Simons
  Theories}

\tocauthor{A. Guay, M. Leblanc, R. MacKenzie and M. B. Paranjape}
\author{ A. Guay, M. Leblanc, R. MacKenzie, M. B. Paranjape\\
\small\em Groupe de physique des particules, D\'epartement de Physique, 
\\
\small\em Universit\'e de Montr\'eal, C.P. 6128,\\
\small\em succ. centreville, Montr\'eal, 
Qu\'ebec, Canada, H3C 3J7}


\maketitle
\begin{abstract}
The parity odd part of the  adiabatic effective action in a 2+1
dimensional non-abelian Yang-Mills theory with a Chern-Simons term and
scalar fields contains the Chern-Simons term and several other pretendors or
``would be" Chern-Simons terms.  We compute all of these terms, and we find
that the renormalization of the coefficient of the Chern-Simons term is
altered,  whether or not the symmetry is spontaneously broken. The
renormalization appears to be that which would correspond to the (putative)
minimal residual unbroken gauge group.  There remain two puzzles.  Firstly, how
can this renormalization be altered when the symmetry is not spontaneously
broken?   Secondly, are the Chern-Simons term and the set of pretenders linearly
independent?
\end{abstract}

The Chern-Simons (CS) term is given by
\be
L_{CS}={\mu\over 2}\int d^3 x
\epsilon^{\mu\nu\lambda}\big(A_\mu^a\partial_\nu A_\lambda^a+ {1\over
3}g f_{abc}A_\mu^aA_\nu^bA_\lambda^c\big)\label{one}
\ee
where $f_{abc}$ are the structure of the gauge group and $g$ is the
coupling constant. It is not gauge invariant but it can be thought of as a 
Wess-Zumino term which is manifestly gauge invariant\cite{ll}:
\be
L_{CS}={\mu\over 2}\int d^4 x {tr (F\tilde F)\over 32\pi^2}.\label{two}
\ee
The integration is over a $3+1$ dimensional manifold with a boundary
corresponding to the $2+1$ dimensional space-time.  This expression is
gauge invariant but it is not unique, it depends on
the particular extension of the gauge fields that has been made into
the additional dimension.  To obtain a sensible theory in the case of such 
multi-valued functionals we have
to quantize the coefficient of  the
term in order that its (discrete) ambiguity is only a harmless integer multiple
of $2\pi$.  This yields the quantization condition
\be
{4\pi\mu\over g^2}=n\in Z.
\ee
Quantum corrections renormalize the value of $n$.  It was found by 
Pisarski and Rao \cite{pr}that for an $SU(N)$ theory, 
\be
n\rightarrow n+N. \label{three}
\ee
This
correction comes from a one loop, and there can be no further
contributions from higher loops.  The case with symmetry breaking was studied
for partial symmetry breaking, for the example
$SU(3)\rightarrow SU(2)$ with a scalar field in the fundamental
representation with $<\phi > = (0,0,v)$.  Calculating the renormalization of
the coefficient of the residual $SU(2)$ sub-group yields $n\rightarrow
n+2$ and not the usual renormalization appropriate for the $SU(3)$
original symmetry $n\rightarrow n+3$\cite{kmpp,cdhl}.  Hence there
seems to be a correction to this renormalization in equation (\ref{three}) 
due to the scalar
fields.  For the broken generators, the correction is a complicated
function of the vacuum expectation value\cite{ks}, which should be explainable
in terms of the effective action for the gauge fields, by terms which mix with 
the Chern-Simons terms.  

Consider a simpler model, with an $SU(2)$ spontaneously broken
by a fundamental scalar doublet $\phi $.  Terms in the effective
action\cite{ks}, which we will call ``would be'' Chern-Simons terms, 
\bea
&&\alpha (\phi^\dagger\phi )\epsilon^{\mu\nu\lambda}(\phi^\dagger T^a
{\stackrel{\leftrightarrow} {D}}_\mu \phi F_{\nu\lambda}^a)\nonumber\\
&+&\beta (\phi^\dagger\phi )
\epsilon^{\mu\nu\lambda}\epsilon_{abc}(\phi^\dagger T^a
{\stackrel{\leftrightarrow} {D}}_\mu\phi\phi^\dagger T^a{\stackrel{\leftrightarrow} {D}}_\mu\phi\phi^\dagger T^a
{\stackrel{\leftrightarrow} {D}}_\mu\phi)\label{four}\nonumber\\
\eea
become
\be
\alpha (\phi^\dagger\phi )\epsilon^{\mu\nu\lambda} v^2 A_\mu^a
F_{\nu\lambda}^a +\beta (\phi^\dagger\phi ) \epsilon^{\mu\nu\lambda}
\epsilon_{abc}v^6 A_\mu^aA_\nu^bA_\lambda^c .
\ee
These mix with the Chern-Simons term and hide its 
true renormalization. The problem is to disentangle the contributions of such 
terms from the 
true renormalization.  The background field effective action is the ideal
way to proceed.  

The explicit Lagrangian that we consider is
\bea
{\cal L} (\phi, A_\mu^a)&=& -{1\over 4}F_{\mu\nu}^aF^{a\mu\nu} -{\mu\over 2}
\epsilon^{\mu\nu\lambda}\big(A_\mu^a\partial_\nu A_\lambda^a+ {1\over
3}g \epsilon_{abc}A_\mu^aA_\nu^bA_\lambda^c\big)\nonumber\\
&+&(D_\mu\phi )^\dagger D^\mu\phi +m^2
\phi^\dagger\phi -{\lambda\over 4}(\phi^\dagger\phi )^2 +{\rm gauge}\,\,\
{\rm fixing}\nonumber\\
\eea
where $a,b,c$ take values from $1$ to $3$, and $\phi$ is a doublet.
The effective action computed in an adiabatic expansion, considering only the
parity odd part, has three possible terms, equation (\ref{one}) and the two 
``would be'' CS terms in equation
(\ref{four}), that are relevant for the renormalization of the CS term.  If we
define the coefficients $-C_1, C_2$ and $C_3$ respectively for the three terms,
then for $\varphi  = (0,v/\sqrt 2 )$ we get  
\be
\left( -C_1 +{gv^2\over 2}C_2 \left( {v^2\over 2}\right) \right) 
\epsilon^{\mu\nu\lambda}A_\mu^a\partial_\nu A_\lambda^a
\ee
while the three gauge field term is 
\be
\left( -{C_1\over 3} + {gv^2\over 4}C_2 \left( {v^2\over 2}\right)  +{g^2v^6\over 64}
C_3 \left({v^2\over 2}\right)  \right) 
\epsilon^{\mu\nu\lambda} \epsilon_{abc}g A_\mu^aA_\nu^bA_\lambda^c.
\ee
Hence and independent calculation of $C_2$ and $C_3$ needs to be done
in order to isolate the true renormalization of the CS term, which is contained
in $C_1$.  For example, computing the coefficient of the term
\be
\epsilon^{\mu\nu\lambda}(\varphi^\dagger iT^a\vec\partial_\mu\phi )
(\partial_\nu A^a_\lambda -\partial_\lambda A^a_\nu )
\ee
will give $C_2$.  
The  calculation has been done for this case, of $SU(2)$ 
completely broken by a scalar doublet.  We find that
all corrections to the CS term are absorbed into the ``would be'' CS terms.  
This means that in fact there is no renormalization of the coefficient of the
CS term here.  
\be
n\rightarrow n \,\, (\neq n+2)
\ee
It seems that each gauge boson that gets a Higg's mass does not contribute 
to the renormalization.  Correspondingly in the case considered in the 
beginning $SU(3)\rightarrow SU(2)$ we expect that the renormalization is
\be
n\rightarrow n+2 \,\, (\neq n+3)
\ee
The first puzzling aspect of the result is that it seems to be
independent of whether or not the symmetry is broken.  Since we shift
the scalar field in the background field method, we give all
putatively massive gauge bosons a Higg's mass, even though we may
remove this at the end if the symmetry is not broken, ie.  if the
minimum of the effective potential is at vanishing shift.  Putting the
shift to zero will not matter in the calculation of the
renormalization of the coefficient of the CS term.  If, as in our
case, we find no renormalization, $n\rightarrow n$, this result will
not change if we evaluate it at zero shift of the scalar field.  It
does not depend on the scalar field, it is just a constant (integer
valued) function.  A priori we expect that without symmetry breaking,
the scalar fields cannot modify the renormalization of the CS term 
in concordance with the Coleman-Hill theorem (which actually only applies for
abelian theories).  This lack of commutativity, between computing the
renormalization of the coefficient of the CS term with a shifted
scalar field followed by letting the shift vanish or computing it
directly without shifting the scalar field, warrants further
investigation.

The second puzzling aspect comes from the almost linear dependence of the 
Chern-Simons term and the appropriate parts of the ``would be" terms.  The
``would be" terms are constructed out of the current
\be
J^a_\mu =\phi^\dagger i\stackrel{\leftrightarrow}{D}_\mu T^a\phi .
\ee
If $\phi \ne 0$, we can write 
\be
\phi = {\cal U} \bar\phi
\ee
where
\be
\bar\phi =\left(\begin{array}{c}0\\v(x)\end{array}\right)
\ee
with $v(x)$ real.  Furthermore if we write
\be
A_\mu = \left(\bar A_\mu\right)^{\cal U}\label{fr}
\ee
Then
\be J^a_\mu (\bar A, \bar\phi )= g {v^2\over 2} \bar A_\mu^a ,
\ee
so that the ``would be" terms (which are of course absolutely gauge invariant)
become, for example
\be
\epsilon^{\mu\nu\lambda}J^a_\mu (\bar A, \bar\phi )F^a_{\nu\lambda} (A) =
 g {v^2\over 2} \bar A_\mu^aF^a_{\nu\lambda} (\bar A).
\ee
Thus the part in this ``would be" term that is proportional to $1/v^2$ is not
independent of the part of the Chern-Simons term
\be
\epsilon^{\mu\nu\lambda} \bar A_\mu^aF^a_{\nu\lambda} (\bar A) .
\ee
Correspondingly the term
\be
\epsilon^{\mu\nu\lambda}\epsilon_{abc}J^a_\mu (\bar A, \bar\phi )
J^b_\nu (\bar A, \bar\phi) J^c_\lambda (\bar A, \bar\phi )=
\left( g {v^2\over 2}\right)^3 \epsilon^{\mu\nu\lambda}\epsilon_{abc}
 \bar A_\mu^a \bar A_\nu^b \bar A_\lambda^c
\ee
which does not seem to be independent of the other part of the Chern-Simons
term.    Hence it seems that the relevant parts of the ``would be " terms are not
linearly independent of the Chern-Simons term.  

This would be catastrophic, and it is probably not true.  First we needed that
$\phi\ne 0$ which is not the case in general.  Moreover since under the field redefinition
in equation(\ref{fr}) the Chern-Simons term transforms as
\be
C.S.(A)\rightarrow C.S.({\bar A}) + {4\pi\mu\over g^2}{2\pi\over 24 \pi^2}\int 
tr ({\cal U}^\dagger d{\cal U} ),
\ee
that is
\be
\delta C.S. = 2\pi n \times {\rm Winding \# }\quad ({\cal U}),
\ee
hence the Chern-Simons term and the ``would be" terms are not linearly
independent, but only by a topological winding number.  

Thus an independent confirmation of the previous calculation is possible.  We
should simply calculate the variation of the effective action under a topologically
non-trivial gauge transformation.  The change under such a gauge
transformation is only sensitive to the Chern-Simons terms, the others being
explicitly gauge invariant.  Such a claculation has been done for the pure gauge
theory, but remains to be done with the presence of the scalar fields.  



\section*{Acknowledgements}
We thank NSERC of Canada and FCAR du Qu\'ebec for financial support.

\begin{thebibliography}{99}
\bibitem{ll}A. Guay, M. Leblanc, R. B. MacKenzie, M. B. Paranjape   
proceedings of the 1998 Lake Louise Winter Institute, to be published.
\bibitem{pr}R. Pisarski, S. Rao   \Journal{\PRD} {32} {2081}{1985}.
\bibitem{ch}S. Coleman, B. Hill \Journal{\PLB} {159} {184}{1985}.
\bibitem{kmpp}A. Khare, R. MacKenzie, P. Panigrahi, M. B. Paranjape  
\Journal{\PLB}{355} {236}{1995}.
\bibitem{cdhl}L. Chen, G. Dunne, K. Haller, E. Lim-Lombridas
\Journal{\PLB}{348} {468}{1995}.
\bibitem{ks}S. Khlebnikov, M. Shaposhnikov \Journal{\PLB}{254} {148}{1991}.
\end{thebibliography}

\end{document}

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