QUANTUM CORRECTIONS TO NEWTON'S LAW IN RESUMMED QUANTUM GRAVITY
\footnotesize B.F.L. WARD
We present the elements of resummed quantum gravity, a new approach to quantum gravity based on the work of Feynman using the simplest example of a scalar field as the representative matter. We show that we get a UV finite quantum correction to Newton's law. \keywords{Quantum Gravity; Newton; Resummation}
BU-HEPP-04-10.tex
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Trim Size: 9.75in x 6.5in
%% Text Area: 8in (include Runningheads) x 5in
%% ws-ijmpa.tex   :   28 July 2003
%% Tex file to use with ws-ijmpa.cls written in Latex2E. 
%% The content, structure, format and layout of this style file is the 
%% property of World Scientific Publishing Co. Pte. Ltd. 
%% Copyright 1995, 2002 by World Scientific Publishing Co. 
%% All rights are reserved.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%

\documentclass{ws-ijmpa}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{cite}
\begin{document}

\markboth{\footnotesize B.F.L. WARD}
{QUANTUM CORRECTIONS TO NEWTON'S LAW IN RESUMMED QUANTUM GRAVITY}

%%%%%%%%%%%%%%%%%%%%% Publisher's Area please ignore %%%%%%%%%%%%%%%
%
\catchline{}{}{}{}{}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\title{QUANTUM CORRECTIONS TO NEWTON'S LAW IN RESUMMED QUANTUM GRAVITY
}

\author{\footnotesize B.F.L. WARD}

\address{Department of Physics, Baylor University, One Bear Place \#97316\\
Waco, Texas 76798-7316,
USA
}

\maketitle

%\pub{Received (Day Month Year)}{Revised (Day Month Year)}

\begin{abstract}
We present the elements of resummed quantum gravity, a new
approach to quantum gravity based on the work of Feynman using the simplest
example of a scalar field as the representative matter.
We show that we get a UV finite quantum correction to Newton's law.

\keywords{Quantum Gravity; Newton; Resummation}
\end{abstract}

\section{Introduction}
Newton's law, one of the most basic laws in
physics, is a special case of the solutions of the
classical field equations of Albert Einstein's general theory of relativity.
% Specifically,
%$g_{00}=1+2\Phi_N \Rightarrow \nabla^2\Phi_N=4\pi G_N\rho$
%from
%$R^{\alpha\gamma}-\frac{1}{2}g^{\alpha\gamma}R  =-8\pi G_N T^{\alpha\gamma}$, etc., where he have now introduced the familiar metric of space-time
%$g_{\mu\nu}$, the Newtonian potential $\Phi_N$, Newton's constant
%$G_N$, the mass density $\rho$, the contracted Riemann 
%tensor $R^{\alpha\gamma}$,
%and the appropriate energy momentum tensor $T^{\alpha\gamma}$.
Successful tests of Einstein's theory in classical physics
are described in Refs.~\cite{mtw,sw1,abs}.
Quantum mechanics, as formulated by Heisenberg and Schroedinger, following Bohr,has explained, in 
the Standard Model(SM)~\cite{sm}, 
all empirically established quantum phenomena except the quantum treatment of Newton's law. This obtains
even with the tremendous progress in quantum field theory,
superstrings~\cite{gsw,jp}, loop quantum gravity~\cite{lqg}, etc.
In this paper, we address this issue by using
a new approach~\cite{bw1,bw2} to quantum gravity(QG), 
building on previous work by Feynman~\cite{f1,f2}, to get a minimal
union of Bohr's and Einstein's ideas. 
%basic idea quantum gravity is a point particle 
%quantum field theory
%and its apparent bad UV
%behavior is due to our naivete --
%nothing fundamental prevents
%the union of Bohr's and Einstein's ideas.

%The approaches to the to the attendant bad UV behavior
%have been summarized in Ref.~\cite{wein1}. 
%There are four approaches~\cite{wein1} to the attendant bad UV behavior 
%of quantum gravity (QG):
%extended theories of gravitation such as supersymmetric theories - superstrings
%and loop quantum gravity;
%resummation, a new version of which we discuss presently;
%composite gravitons; and,
%asymptotic safety -- fixed point theory, recently pursued with
%success in Refs.~\cite{laut,reuter2}.
From the view of the generic approaches~\cite{wein1}
to the attendant bad UV behavior of QG, 
our approach, based on YFS methods~\cite{yfs,yfs1},
is a new version of the resummation approach~\cite{wein1} and
allows us to make contact
with both the extended theory~\cite{wein1} 
and the asymptotic safety~\cite{wein1,laut,reuter2}
approaches and to address~\cite{bw1,bw2} issues in black hole physics, some of which relate to Hawking~\cite{hawk} radiation.\par

The description of our new resummed QG theory is already presented
in these {\it Proceedings} in Ref.~\cite{bw2}, to which we refer
the reader. Here, we go directly to the issue of the
quantum corrections to Newton's law.

\section{Quantum Corrections to Newton's Law in Resummed QG}

The model which we use~\cite{bw1,bw2} is gravity coupled to a 
scalar field as formulated in Refs.~\cite{f1,f2}. 
In our resummed QG theory, the graphs in Fig.~\ref{fig1}
\begin{figure}
\begin{center}
\epsfig{file=fig1cn.eps,width=77mm,height=38mm}
\end{center}
\caption{\baselineskip=7mm     The scalar one-loop contribution to the
graviton propagator. $q$ is the 4-momentum of the graviton.}
\label{fig1}
\end{figure}\noindent 
become finite as we explain in Refs.~\cite{bw1,bw2}.
In this way, we get a UV finite quantum correction to Newton's law
{\em without modifying Einstein's theory}. In Refs.~\cite{bw1},
we show that this UV finiteness holds for all orders in the loop expansion.\par

Specifically, introducing the YFS resummed propagators as derived
in Refs.~\cite{bw2,bw1} into Fig. 1 yields
, by the standard methods~\cite{bw1}, 
that the graviton propagator denominator,
$q^2 +\frac{1}{2}q^4\Sigma^{T(2)}+i\epsilon$, 
is specified by
\begin{equation} 
-\frac{1}{2}\Sigma^{T(2)} \cong \frac{c_2}{360\pi M_{Pl}^2}
\label{sigma}
\end{equation}
%(here, $k\rightarrow (ik^0,\vec k)$ by Wick rotation, and we work in the
%transverse-traceless space){\small
%\begin{equation}
%\begin{split}
%i\Sigma(q)^{1a}_{\bar\mu\bar\nu;\mu\nu}&=i\kappa^2\frac{\int d^4k}{2(2\pi)^4}\\
%&\frac{\left(k'_{\bar\mu}k_{\bar\nu}+k'_{\bar\nu}k_{\bar\mu}\right)e^{\frac{\kappa^2|{k'}^2|}{8\pi^2}\ln\left(\frac{m^2}{m^2+|{k'}^2|}\right)}}
%{\left({k'}^2-m^2+i\epsilon\right)}\\
%   &\qquad   \frac{
%\left(k'_{\mu}k_{\nu}+k'_{\nu}k_{\mu}\right)e^{\frac{\kappa^2|k^2|}{8\pi^2}\ln\left(\frac{m^2}{m^2+|k^2|}\right)}}
%{\left(k^2-m^2+i\epsilon\right)}.
%\end{split}
%\label{eq2p}
%\end{equation}}\noindent
%This integral is now convergent! So is that for Fig 1b.
%Continuing to work in the transverse-traceless space, 
%we get the graviton propagator denominator
%\begin{equation}
%q^2 +\frac{1}{2}q^4\Sigma^{T(2)}+i\epsilon
%\label{prop}
%\end{equation}
%where the transverse, traceless self-energy 
%function $\Sigma^T(q^2)$ 
%follows from eq.(\ref{eq2p}) for Fig. 1a and its analog for Fig. 1b
%by the standard methods.
%For the coefficient of $q^4$ in $\Sigma^T(q^2)$ for $|q^2|>>m^2$
%we get
%\begin{equation} 
%-\frac{1}{2}\Sigma^{T(2)} \cong \frac{c_2}{360\pi M_{Pl}^2}
%\label{sigma}
%\end{equation}
for
%\begin{equation} 
$c_2 = \int^{\infty}_{0}dx x^3(1+x)^{-4-\lambda_c x}\cong 72.1$ 
%\label{int1}
%\end{equation}
where $\lambda_c=\frac{2m^2}{\pi M_{Pl}^2}$.
This implies the Newton potential
\begin{equation}
\Phi_{N}(r)= -\frac{G_N M_1M_2}{r}(1-e^{-ar})
\label{newtn}
\end{equation}
where $a=1/\sqrt{-\frac{1}{2}\Sigma^{T(2)}}\simeq 3.96 M_{Pl}$
when for definiteness we set $m\cong 120$GeV~\cite{lewwg}.
We note that
%\begin{equation}
$c_2 \cong \ln\frac{1}{\lambda_c}-\ln\ln\frac{1}{\lambda_c}-\frac{\ln\ln\frac{1}{\lambda_c}}{\ln\frac{1}{\lambda_c}-\ln\ln\frac{1}{\lambda_c}}-\frac{11}{6}$.
%\label{anal1}
%\end{equation}
Without resummation, $\lambda_c=0$,
and $c_2$ is infinite
and, as this is
the coefficient of $q^4$ in the inverse propagator, 
{\bf no renormalization of the field and/or of the mass could remove
such an infinity}. In our new approach, 
this infinity is absent.\par

%%%%START HERE
%Our result in (\ref{sigma}) is gauge invariant, as our 
%approach involves the exact re-arrangement of the Feynman series
%%{\color{red}(see mpla17(2002)2371)}
%and the original series is gauge
%invariant. 
We can make a 
cross check of our gauge invariant~\cite{bw1} analysis 
with the gauge invariant analysis of
Ref.~\cite{thvelt1} where the complete
result of the one-loop divergences of our scalar field coupled
to Einstein's gravity have been computed. Since $c_2$ diverges
without our resummation, it follows~\cite{bw1} that
%Thus,
%the dominant terms which we are isolating in this talk
%are precisely those that
%are given in Ref.~\cite{thvelt1}, where 
we need to make the correspondence
between the poles in $n$, the dimension of space-time,
at $n=4$ calculated in Ref.~\cite{thvelt1}
and the leading 
log $\ln\frac{1}{\lambda_c}$. This implies~\cite{bw1}
%means we  
%set
%$c_2$ equal to its value when $\lambda_c = 0$ in $n$ dimensions
%and allow $n \rightarrow 4$.
%{\color{red}$\rightarrow$}
\begin{equation}
\frac{1}{(2-n/2)} \leftrightarrow c_2,
\label{crsp1}
\end{equation}
so that, if we look at the limit $q^2\rightarrow 0$,
we find that the coefficient of $q^4$ 
in the graviton propagator denominator above
is $3/(2-n/2)$ times the coefficient 
of $c_2$ on the right-hand side of (\ref{sigma}), in
complete agreement with the result that is implied by
eq.(3.40) in Ref.~\cite{thvelt1}
, for example. 
%The results in Ref.~\cite{thvelt1}
%are also gauge invariant.


%Our gauge invariant result for $\Sigma^{T(2)}$
% in (\ref{sigma})
%can be shown~\cite{bw1} to be consistent
%with the one-loop analysis of QG in Ref.~\cite{thvelt1}.
%CROSS CHECK WITH 't HOOFT AND VELTMAN, {\Color{Red}Ann. Inst. Henri Poincare {\bf XX}, 69 (1974)}, {\Color{Black}WHERE THE COMPLETE
%RESULT OF THE ONE-LOOP DIVERGENCES OF OUR SCALAR FIELD COUPLED
%TO EINSTEIN'S GRAVITY HAVE BEEN COMPUTED}. 
%\newpage


Sub-Planck scale physics is accessible to point particle field theory
so that current superstring theories may be
phenomenological models
for a more fundamental theory (TUT=The Ultimate Theory) just as the old string theory~\cite{schw1} is such a model for QCD. Other types of correspondences
are not excluded here~\cite{superichep02}.
Our deep Euclidean studies are complementary 
to the low energy studies
of Ref.~\cite{dono1}. The effective cut-off which we generate dynamically
is at $M_{Pl}$ so that, while renormalizable quantum field theory (QFT)  
below $M_{Pl}$
is unaffected, some non-renormalizable QFT's are given new 
life here -- other problems notwithstanding.

Some phenomenological implications of (\ref{newtn}) are presented in Ref.~\cite{bw1,bw2}. To sum up, it appears that we may have indeed realized a minimal union
of the ideas of Bohr and Einstein.\par

\section*{Acknowledgments}
We thank Prof. S. Jadach for useful discussions.
This work was partly supported by the US Department of Energy Contract  
DE-FG05-91ER40627
and by NATO Grants PST.CLG.977751,980342.

\begin{thebibliography}{0}
\bibitem{mtw} C. Misner, K.S. Thorne and J.A. Wheeler,
{\it Gravitation},(Freeman, San Francisco, 1973).
\bibitem{sw1} S. Weinberg, {\it Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity},( John Wiley, New York, 1972).
\bibitem{abs} R. Adler, M. Bazin and M. Schiffer, {\it Introduction to General
Relativity },( McGraw-Hill, New York, 1965 ).
\bibitem{sm}S.L. Glashow, {\it Nucl. Phys.} {\bf 22} (1961) 579; 
S. Weinberg, {\it Phys. Rev. Lett.} {\bf 19} (1967) 1264;
A. Salam, in {\it Elementary Particle Theory}, ed. N. Svartholm
(Almqvist and Wiksells, Stockholm, 1968), p. 367;
G.~'t Hooft and M.~Veltman, {\it Nucl. Phys.}B {\bf 44}, 189 (1972)  
and {\bf B50}, 318 (1972); 
G.~'t Hooft, {\it ibid.} {\bf 35}, 167 (1971); M.~Veltman, {\it ibid.} {\bf 7}, 637 (1968);
D. J. Gross and F. Wilczek, 
{\it Phys. Rev. Lett.} {\bf 30} (1973) 1343;
H. David Politzer, {\it ibid.}{\bf 30} (1973) 1346; see also
, for example, F. Wilczek, in {\it Proc. 16th International Symposium on Lepton and Photon Interactions, Ithaca, 1993}, eds. P. Drell and D.L. Rubin 
(AIP, NY, 1994) p. 593, and references therein.
\bibitem{gsw} See, for example, M. Green, J. Schwarz and E. Witten,
{\it Superstring Theory, v. 1 and v.2}, ( Cambridge Univ. 
Press, Cambridge, 1987 ) and references therein.
\bibitem{jp}
See, for example, J. Polchinski, {\it String Theory, v. 1 and v. 2},
(Cambridge Univ. Press, Cambridge, 1998), and references therein.
\bibitem{lqg}
See for example V.N. Melnikov, {\it Gravit. Cosmol.} {\bf 9}, 118 (2003);
L. Smolin, hep-th/0303185, and references therein.
\bibitem{bw1} B.F.L. Ward,{\it Mod. Phys. Lett.}A{\bf 17}, 2371 (2002); 
{\it ibid.}{\bf 19} (2004)143; {\it J. Cos. Astropart. Phys.}{\bf 0402}, 011 (2004).
\bibitem{bw2} B.F.L. Ward, in these {\it Proceedings}, in press.
\bibitem{f1} R. P. Feynman, {\it Acta Phys. Pol.} {\bf 24}, 697 (1963).
\bibitem{f2} R. P. Feynman, {\it Feynman Lectures on Gravitation},
eds. F.B. Moringo and W.G. Wagner (Caltech, Pasadena, 1971).
\bibitem{wein1}
S. Weinberg, in{\it General Relativity}, eds. S.W. Hawking
and W. Israel,( Cambridge Univ. Press, Cambridge, 1979) p.790.
\bibitem{yfs}D.~R.~Yennie, S.~C.~Frautschi, and H.~Suura, Ann. Phys. {\bf 13}, 379 (1961);
see also K.~T.~Mahanthappa, {\it Phys.~Rev.}~{\bf 126}, 329 (1962), for a related analysis.
\bibitem{yfs1} See S. Jadach {\it et al.},{\it Comput. Phys. Commun.} {\bf 102},229 (1997); S. Jadach, M. Skrzypek and B.F.L. Ward,{\it Phys.Rev.} D{\bf 55},1206 (1997); S. Jadach, B.F.L. Ward and Z. Was,{\it Phys. Rev.} D{\bf 63}, 113009 (2001);
S. Jadach, B.F.L. Ward and Z. Was, {\it Comp. Phys. Commun.} {\bf 130}, 260 (2000);
S. Jadach {\it et al.}, {\it ibid.}{\bf 140} 432, 475, (2001).
\bibitem{laut} O. Lauscher and M. Reuter, hep-th/0205062, and references
therein.
\bibitem{reuter2}
A. Bonnanno and M. Reuter, {\it Phys. Rev.} D{\bf 62}, 043008 (2000).
\bibitem{hawk} S. Hawking,{\it Nature (London)} {\bf 248}, 30 (1974);
{\it Commun. Math. Phys.}{\bf 43}, 199 (1975).
%\bibitem{mlg} M.L. Goldberger, private communication, 1972.
%\bibitem{elswh} B.F.L. Ward, to appear.
\bibitem{lewwg}
D. Abbaneo {\it et al.}, hep-ex/0212036; see also, M. Gruenewald, 
hep-ex/0210003, in {\it Proc. ICHEP02}, eds. S. Bentvelsen {\it et al.} ( North-Holland, Amsterdam, 2003) 280.
%\bibitem{sw2}
%S. Weinberg, {\it The Quantum Theory of Fields, vols. 1-3}, 
%( Cambridge University Press, Cambridge, 1995,1996,2000).
\bibitem{thvelt1}
G. 't Hooft and M. Veltman, {\it Ann. Inst. Henri Poincare} 
{\bf XX}, 69 (1974).
\bibitem{schw1} See, for example, J. Schwarz, in {\it Proc. Berkeley Chew Jubilee, 1984}, eds. C. DeTar 
{\it et al.} (World Scientific, Singapore, 1985) p. 106, and references
therein.
\bibitem{superichep02} See for example J. de Boer, in {\it Proc. ICHEP02},
eds. S. Bentvelsen {\it et al.} (North-Holland, Amsterdam, 2003) 353.
\bibitem{dono1} See for example J. Donoghue, 
{\it Phys. Rev. Lett.} {\bf 72}, 2996 (1994);
{\it Phys. Rev.} D{\bf 50}, 3874 (1994); preprint gr-qc/9512024; 
J. Donoghue {\it et al.}, {\it Phys. Lett.} B{\bf 529}, 132 (2002), 
and references therein.
%\bibitem{neut} See for example D. Wark, in {\it Proc. ICHEP02}, eds. S. Bentvelsen {\it et al.} (North-Holland, Amsterdam, 2003) 164; 
%M. C. Gonzalez-Garcia, {\it op. cit.} 186, hep-ph/0211054,
%and references therein.
%\bibitem{pdg2002}K. Hagiwara {\it et al.}, {\it Phys. Rev.} D{\bf 66}, 010001 (2002)1; see also H. Leutwyler and J. Gasser, {\it Phys. Rept.} {\bf 87}, 77 (1982), and references therein.

%\bibitem{Marnelius} R. Marnelius, {\it Acta Phys. Pol.} 
%{\bf B13}, 669 (1982).

%\bibitem{Bjorken} J. D. Bjorken, in {\it Lecture Notes on Current-Induced 
%Reactions}, ed. J. Komer {\it et al.} (Springer, Heidelberg, 1975).

%\bibitem{Bohr} A. Bohr and B. R. Mottelson, {\it Nuclear Structure} 
%(Benjamin, New York, 1969), Vol.~1, pp.~100--102.

%\bibitem{Webb} R. C. Webb, PhD thesis, Princeton University, 1972.

%\bibitem{Toimela} T. Toimela, Helsinki Research Institute for 
%Theoretical Physics, Report No. HU-TFT-82-37, 1982 (unpublished).
\end{thebibliography}

\end{document}



hep-ph0411050.extracted_bib
\begin{thebibliography}{0}
\bibitem{mtw} C. Misner, K.S. Thorne and J.A. Wheeler,
{\it Gravitation},(Freeman, San Francisco, 1973).
\bibitem{sw1} S. Weinberg, {\it Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity},( John Wiley, New York, 1972).
\bibitem{abs} R. Adler, M. Bazin and M. Schiffer, {\it Introduction to General
Relativity },( McGraw-Hill, New York, 1965 ).
\bibitem{sm}S.L. Glashow, {\it Nucl. Phys.} {\bf 22} (1961) 579; 
S. Weinberg, {\it Phys. Rev. Lett.} {\bf 19} (1967) 1264;
A. Salam, in {\it Elementary Particle Theory}, ed. N. Svartholm
(Almqvist and Wiksells, Stockholm, 1968), p. 367;
G.~'t Hooft and M.~Veltman, {\it Nucl. Phys.}B {\bf 44}, 189 (1972)  
and {\bf B50}, 318 (1972); 
G.~'t Hooft, {\it ibid.} {\bf 35}, 167 (1971); M.~Veltman, {\it ibid.} {\bf 7}, 637 (1968);
D. J. Gross and F. Wilczek, 
{\it Phys. Rev. Lett.} {\bf 30} (1973) 1343;
H. David Politzer, {\it ibid.}{\bf 30} (1973) 1346; see also
, for example, F. Wilczek, in {\it Proc. 16th International Symposium on Lepton and Photon Interactions, Ithaca, 1993}, eds. P. Drell and D.L. Rubin 
(AIP, NY, 1994) p. 593, and references therein.
\bibitem{gsw} See, for example, M. Green, J. Schwarz and E. Witten,
{\it Superstring Theory, v. 1 and v.2}, ( Cambridge Univ. 
Press, Cambridge, 1987 ) and references therein.
\bibitem{jp}
See, for example, J. Polchinski, {\it String Theory, v. 1 and v. 2},
(Cambridge Univ. Press, Cambridge, 1998), and references therein.
\bibitem{lqg}
See for example V.N. Melnikov, {\it Gravit. Cosmol.} {\bf 9}, 118 (2003);
L. Smolin, hep-th/0303185, and references therein.
\bibitem{bw1} B.F.L. Ward,{\it Mod. Phys. Lett.}A{\bf 17}, 2371 (2002); 
{\it ibid.}{\bf 19} (2004)143; {\it J. Cos. Astropart. Phys.}{\bf 0402}, 011 (2004).
\bibitem{bw2} B.F.L. Ward, in these {\it Proceedings}, in press.
\bibitem{f1} R. P. Feynman, {\it Acta Phys. Pol.} {\bf 24}, 697 (1963).
\bibitem{f2} R. P. Feynman, {\it Feynman Lectures on Gravitation},
eds. F.B. Moringo and W.G. Wagner (Caltech, Pasadena, 1971).
\bibitem{wein1}
S. Weinberg, in{\it General Relativity}, eds. S.W. Hawking
and W. Israel,( Cambridge Univ. Press, Cambridge, 1979) p.790.
\bibitem{yfs}D.~R.~Yennie, S.~C.~Frautschi, and H.~Suura, Ann. Phys. {\bf 13}, 379 (1961);
see also K.~T.~Mahanthappa, {\it Phys.~Rev.}~{\bf 126}, 329 (1962), for a related analysis.
\bibitem{yfs1} See S. Jadach {\it et al.},{\it Comput. Phys. Commun.} {\bf 102},229 (1997); S. Jadach, M. Skrzypek and B.F.L. Ward,{\it Phys.Rev.} D{\bf 55},1206 (1997); S. Jadach, B.F.L. Ward and Z. Was,{\it Phys. Rev.} D{\bf 63}, 113009 (2001);
S. Jadach, B.F.L. Ward and Z. Was, {\it Comp. Phys. Commun.} {\bf 130}, 260 (2000);
S. Jadach {\it et al.}, {\it ibid.}{\bf 140} 432, 475, (2001).
\bibitem{laut} O. Lauscher and M. Reuter, hep-th/0205062, and references
therein.
\bibitem{reuter2}
A. Bonnanno and M. Reuter, {\it Phys. Rev.} D{\bf 62}, 043008 (2000).
\bibitem{hawk} S. Hawking,{\it Nature (London)} {\bf 248}, 30 (1974);
{\it Commun. Math. Phys.}{\bf 43}, 199 (1975).
%\bibitem{mlg} M.L. Goldberger, private communication, 1972.
%\bibitem{elswh} B.F.L. Ward, to appear.
\bibitem{lewwg}
D. Abbaneo {\it et al.}, hep-ex/0212036; see also, M. Gruenewald, 
hep-ex/0210003, in {\it Proc. ICHEP02}, eds. S. Bentvelsen {\it et al.} ( North-Holland, Amsterdam, 2003) 280.
%\bibitem{sw2}
%S. Weinberg, {\it The Quantum Theory of Fields, vols. 1-3}, 
%( Cambridge University Press, Cambridge, 1995,1996,2000).
\bibitem{thvelt1}
G. 't Hooft and M. Veltman, {\it Ann. Inst. Henri Poincare} 
{\bf XX}, 69 (1974).
\bibitem{schw1} See, for example, J. Schwarz, in {\it Proc. Berkeley Chew Jubilee, 1984}, eds. C. DeTar 
{\it et al.} (World Scientific, Singapore, 1985) p. 106, and references
therein.
\bibitem{superichep02} See for example J. de Boer, in {\it Proc. ICHEP02},
eds. S. Bentvelsen {\it et al.} (North-Holland, Amsterdam, 2003) 353.
\bibitem{dono1} See for example J. Donoghue, 
{\it Phys. Rev. Lett.} {\bf 72}, 2996 (1994);
{\it Phys. Rev.} D{\bf 50}, 3874 (1994); preprint gr-qc/9512024; 
J. Donoghue {\it et al.}, {\it Phys. Lett.} B{\bf 529}, 132 (2002), 
and references therein.
%\bibitem{neut} See for example D. Wark, in {\it Proc. ICHEP02}, eds. S. Bentvelsen {\it et al.} (North-Holland, Amsterdam, 2003) 164; 
%M. C. Gonzalez-Garcia, {\it op. cit.} 186, hep-ph/0211054,
%and references therein.
%\bibitem{pdg2002}K. Hagiwara {\it et al.}, {\it Phys. Rev.} D{\bf 66}, 010001 (2002)1; see also H. Leutwyler and J. Gasser, {\it Phys. Rept.} {\bf 87}, 77 (1982), and references therein.

%\bibitem{Marnelius} R. Marnelius, {\it Acta Phys. Pol.} 
%{\bf B13}, 669 (1982).

%\bibitem{Bjorken} J. D. Bjorken, in {\it Lecture Notes on Current-Induced 
%Reactions}, ed. J. Komer {\it et al.} (Springer, Heidelberg, 1975).

%\bibitem{Bohr} A. Bohr and B. R. Mottelson, {\it Nuclear Structure} 
%(Benjamin, New York, 1969), Vol.~1, pp.~100--102.

%\bibitem{Webb} R. C. Webb, PhD thesis, Princeton University, 1972.

%\bibitem{Toimela} T. Toimela, Helsinki Research Institute for 
%Theoretical Physics, Report No. HU-TFT-82-37, 1982 (unpublished).
\end{thebibliography}