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%% subsection 3.1 \it String Organization [slac-pub-7111-0-0-3-1 in slac-pub-7111-0-0-3: ^slac-pub-7111-0-0-3 >slac-pub-7111-0-0-3-2]
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\sectionLink{slac-pub-7111-0-0-3}{slac-pub-7111-0-0-3}{Above: 3. STRING-INSPIRED METHODS}%
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\subsection{\usemenu{slac-pub-7111::context::slac-pub-7111-0-0-3-1}{\it String Organization}}\label{subsection::slac-pub-7111-0-0-3-1}
The basic motivation for the use of string theory follows from the compact
representation it provides for amplitudes:
at each loop order there is only a
single closed string diagram. As depicted in \fig{StringsFigure},
the string theory diagram contains within it all the Feynman diagrams,
including contributions of the entire tower of superheavy string
excitations. The unwanted superheavy contributions are removed
by taking the ``low-energy limit'' where all external momentum
invariants are much less than the string tension.
This limit picks out different regions of integration in the string
diagram (see \fig{StringsFigure}),
corresponding roughly to particle-like diagrams, but with
different, string-based, rules~\cite{32}.
%FIGURE
%
\begin{figure}
\begin{center}
\epsfig{file=Strings.eps,width=3.0in,clip=}
\end{center}
\vskip -.7 cm \caption[]{
\label{StringsFigure}
A single string diagram implicitly contains all field theory Feynman
diagrams.}
\end{figure}
%
Given knowledge of the string-based rules and organization, one may
also formulate a conventional field-theory framework which mimics them
\cite{38} (at least for one-loop multiparton amplitudes), but
which can be applied more broadly (for example, to amplitudes with
external fermions). At one loop, key ingredients of this
string-inspired framework are: use of a special gauge which is a
hybrid of Gervais-Neveu gauge~\cite{39} and background-field
gauge~\cite{40}; improved color decompositions; systematic
organization of the algebra; and a second-order formalism for fermions
\cite{38,41} which helps make supersymmetry relations
manifest.
Gervais-Neveu gauge, originally derived from the low-energy
limit of tree-level string amplitudes~\cite{39}, has the following
gauge-fixed action (ignoring ghosts):
$$
S^{\rm GN} = \int d^4 x \Bigl( -{1\over 4} \Tr[F^2_{\mu\nu}] -
{1\over 2} \Tr[(\partial \cdot A - i g A^2/\sqrt{2})^2] \Bigr).
\equn\label{GNgauge}
$$
%(The peculiar normalization of the terms in the action is due to the
%unconventional normalization of the group generators $\Tr(T^a T^b) =
%\delta^{ab}$.)
%
The color-ordered Feynman rules derived from this action are depicted
in \fig{GNRulesFigure}; comparing them to the color-ordered vertices
for the standard Lorentz-Feynman gauge (\fig{FeynmanColorFigure}),
we see that the three-point and four-point vertices have, respectively,
half and a third as many terms, showing why the Gervais-Neveu
gauge is simpler for tree-level calculations.
%FIGURE
%
\begin{figure}
\begin{center}
\epsfig{file=GNRules.eps,width=2.in,clip=}
\end{center}
\vskip -.7 cm \caption[]{
\label{GNRulesFigure}
The color-ordered Gervais-Neveu gauge three- and four-point vertices.}
\end{figure}
%
Given this understanding of the string reorganization of tree
amplitudes, one might guess that string theory would best be described
by the Gervais-Neveu gauge at one loop as well. However, the gauge
most closely resembling the string organization of one-loop amplitudes
is a hybrid gauge involving both background-field and Gervais-Neveu
gauges~\cite{38}. To quantize in a background-field
gauge~\cite{40} one splits the gauge field into a classical
background field and a fluctuating quantum field, $A_\mu\ =\ A_\mu^B +
A_\mu^Q$, and imposes the gauge condition $D_\mu^B A_\mu^Q = 0$, where
$D_\mu^B = \del_\mu - {i\over\sqrt{2}}gA_\mu^B$ is the
background-field covariant derivative, with $A_\mu^B$ evaluated in the
adjoint representation. The Feynman-gauge
version of the gauge-fixed action is (again
ignoring ghosts),
$$
S^{\rm Bkgd} = \int d^4 x \Bigl(
-{1\over 4} \Tr[F_{\mu\nu}^2]
-{1\over2}\Tr [(\partial\cdot A^Q
- i g[A_\mu^B,A_\mu^Q]/\sqrt{2})^2] \Bigr).
\equn\label{Backgroundgauge}
$$
The color-ordered background-field gauge vertices which arise
from expanding
\eqn{Backgroundgauge} are depicted in \fig{BackgroundRulesFigure}.
Here we show only the vertices bilinear in the
quantum field $A_\mu^Q$. These suffice for computing the one-loop
effective action $\Gamma[A^B]$, since $A_\mu^Q$ describes
the gluon propagating around the loop while $A_\mu^B$ describes a
gluon emerging from the loop.
%FIGURE
%
\begin{figure}
\begin{center}
\epsfig{file=BackgroundRules.eps,width=2.4in,clip=}
\end{center}
\vskip -.7 cm \caption[]{
\label{BackgroundRulesFigure}
The color-ordered background-field Feynman gauge three- and four-point
vertices. Dashed lines represent either ghosts or scalars.}
\end{figure}
%
Any one-loop diagram can be split into a one-particle-irreducible
(1PI) part, or loop part, along with a set of tree diagrams sewn onto
the loop. Now, $\Gamma[A^B]$ is invariant with respect to $A^B$ gauge
transformations~\cite{40}. Therefore we may use {\it any}
single gauge to compute the trees which are to be sewn onto the 1PI
parts of the diagrams. Indeed, the string-motivated recipe is to use
background-field gauge only for the 1PI or loop vertices, and
Gervais-Neveu gauge for the remaining tree vertices~\cite{38}.
This approach retains the above-noted advantages of Gervais-Neveu
gauge for tree computations, while avoiding the complicated ghost
interactions this nonlinear gauge would entail if it were used inside
the loop. The advantage of the background-field gauge inside the loop
is that the loop momentum appears in only the first term in the
tri-linear gauge vertex in \fig{BackgroundRulesFigure}; the last two
terms contain only the external momentum $k$. (In general, the most
complicated loop integrals to evaluate are those with the most
insertions of the loop momentum in the numerator.) Furthermore, the
first term matches the scalar-scalar-gluon vertex, up to the
$\eta_{\nu\rho}$ factor. Thus in background-field gauge the leading
loop-momentum behavior of one-particle-irreducible graphs with a gluon
in the loop is very similar to that of graphs with a scalar in the
loop. Note also that the interactions of a scalar and of a ghost with
the background field are identical, up to the overall minus sign for a
ghost loop. In the next subsection we elaborate further on these
relations.
%%%%%%%%%%%%%%%%%%%%%%%%