%% slac-pub-7106: page file slac-pub-7106-0-0-4-1.tcx.
%% subsection 4.1 Supersymmetry and background-field gauge [slac-pub-7106-0-0-4-1 in slac-pub-7106-0-0-4: ^slac-pub-7106-0-0-4 >slac-pub-7106-0-0-4-2]
%%%% latex2techexplorer block:
%% latex2techexplorer page setup:
\newmenu{slac-pub-7106::context::slac-pub-7106-0-0-4-1}{
\docLink{slac-pub-7106.tcx}[::Top]{Top}%
\sectionLink{slac-pub-7106-0-0-4}{slac-pub-7106-0-0-4}{Above: 4. Loop-level techniques}%
\subsectionLink{slac-pub-7106-0-0-4}{slac-pub-7106-0-0-4-2}{Next: 4.2. Loop Integral Reduction}%
}
%%%% end of latex2techexplorer block.
\subsection{\usemenu{slac-pub-7106::context::slac-pub-7106-0-0-4-1}{Supersymmetry and background-field gauge}}\label{subsection::slac-pub-7106-0-0-4-1}
At loop level, QCD ``knows'' it is not supersymmetric. However,
one can still rearrange the sum over internal spins propagating
around the loop, in order to take advantage of supersymmetry.
For example, for an amplitude with all external gluons, and a gluon
circulating around the loop, we can use
supersymmetry to trade the internal gluon loop for a scalar loop.
We rewrite the internal gluon loop $g$ (and fermion loop $f$) as a
supersymmetric contribution plus a complex scalar loop $s$,
\bea
g &=& (g+4f+3s)\ -\ 4(f+s)\ +\ s\ =\ A^{N=4}\ -\ 4\,A^{N=1}
\ +\ A^{\rm scalar},
\nonumber \\
\label{eq:SusyDecomp}
f &=& (f+s)\ -\ s\ =\ A^{N=1}\ -\ A^{\rm scalar}.
\eea
Here $A^{N=4}$ represents the contribution of the $N=4$ super Yang-Mills
multiplet, which contains a gluon $g$, four gluinos $f$, and
three complex (six real) scalars $s$; while $A^{N=1}$ gives the
contribution of an $N=1$ chiral matter supermultiplet, one fermion plus
one complex scalar. The advantages of this decomposition are twofold:
\par\noindent
(1) The supersymmetric terms are much simpler than the
nonsupersymmetric ones; not only do they obey SWIs, but we will see
that they have diagram-by-diagram cancellations built into them.
\par\noindent
(2) The scalar loop, while more complicated
than the supersymmetric components, is algebraically simpler
than the gluon loop, because a scalar cannot propagate spin information
around the loop.
\par\noindent
In the context of TQM, this use of supersymmetry could be termed
``internal spin management''.
As an example of how this rearrangement looks, consider the five-gluon
primitive amplitude $A_{5;1}(1^-,2^-,3^+,4^+,5^+)$,
whose components according to \eqn{SusyDecomp} are\,\cite{28}
\bea
A^{N=4} \hskip-2mm &=& \hskip-2mm
\cg \, A^\tree \sum_{j=1}^5 \Biggl[
-{1\over\e^2} \LP {\mu^2\over -s_{j,j+1}}\RP^\e
+ \ln\LP{-s_{j,j+1}\over -s_{j+1,j+2}}\RP\,
\ln\LP{-s_{j+2,j-2}\over -s_{j-2,j-1}}\RP
+ {\pi^2\over6} \Biggr]
\nonumber \\
A^{N=1} \hskip-2mm &=& \hskip-2mm
\cg \, A^\tree \Biggl[ {1\over\e}
+{1\over2}\LB\ln\LP{\mu^2\over -s_{23}}\RP
+\ln\LP{\mu^2\over -s_{51}}\RP\RB + 2 \Biggr]
\nonumber \\
&&\quad + {i\cg\over2}
{{\spa1.2}^2 \LP\spa2.3\spb3.4\spa4.1+\spa2.4\spb4.5\spa5.1\RP\over
\spa2.3\spa3.4\spa4.5\spa5.1}
{\ln\LP {-s_{23}\over -s_{51}}\RP\over s_{51}-s_{23}}
\nonumber \\
A^{\rm scalar} \hskip-2mm &=& \hskip-2mm
{1\over3} A^{N=1} + {2\over9} \cg \, A^\tree
\nonumber \\
&&\hskip-17mm + {i\cg\over3} \Biggl[
- { \spb3.4\spa4.1\spa2.4\spb4.5
\LP\spa2.3\spb3.4\spa4.1+\spa2.4\spb4.5\spa5.1\RP
\over\spa3.4\spa4.5 }
\nonumber \\
&&\hskip30mm
\times { \ln\LP{-s_{23}\over -s_{51}}\RP
-{1\over2}\LP{s_{23}\over s_{51}}-{s_{51}\over s_{23}}\RP
\over (s_{51}-s_{23})^3 }
\nonumber \\
&& \hskip-17mm
- {\spa3.5{\spb3.5}^3\over\spb1.2\spb2.3\spa3.4\spa4.5\spb5.1}
+ {\spa1.2{\spb3.5}^2\over\spb2.3\spa3.4\spa4.5\spb5.1}
+ {1\over2}{\spa1.2\spb3.4\spa4.1\spa2.4\spb4.5\over
s_{23}\spa3.4\spa4.5 s_{51}} \Biggr]
\label{eq:gggggmmppploop}
\eea
where $A^\tree = A_5^\tree(1^-,2^-,3^+,4^+,5^+)$ is given
in \eqn{mhvadjacent}, $\mu$ is the renormalization scale, and
\be \label{eq:cgammadef}
c_\Gamma\ =\ {\Gamma(1+\e)\Gamma^2(1-\e)
\over(4\pi)^{2-\e}\Gamma(1-2\e)}\ .
\ee
These amplitudes contain both infrared and ultraviolet divergences,
which have been regulated dimensionally with $D=4-2\e$,
dropping ${\cal O}(\e)$ corrections.
We see that the three components have quite different analytic
structure, indicating that the rearrangement is a natural one.
As promised, the $N=4$ supersymmetric component
is the simplest, followed by the $N=1$ component.
The non-supersymmetric scalar component is the most complicated,
yet it is still simpler than the direct gluon calculation,
because it does not mix all three components together.
We can understand why the supersymmetric decomposition works by
quantizing QCD in a special gauge, background-field
gauge.\cite{29} The color-ordered rules in \fig{RulesFigure}
were obtained using the Lorentz gauge condition
$\del^\mu A_\mu=0$,
where $A_\mu \equiv A_\mu^a T^a$ with $T^a$ in the fundamental
representation.
After performing the Faddeev-Popov trick to integrate over the
gauge-fixing condition,
one obtains the additional term in the Lagrangian
\be \label{eq:Lorentzgauge}
-{1\over2\xi}\Tr (\del^\mu A_\mu)^2,
\ee
where we chose the integration
weight $\xi=1$ (Lorentz-Feynman gauge) in \fig{RulesFigure}.
To quantize in background-field gauge 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
$D_\mu^B = \del_\mu - \textstyle{i\over\sqrt{2}}gA_\mu^B$
is the background-field covariant derivative, with $A_\mu^B$ evaluated
in the adjoint representation.
Now the Faddeev-Popov integration (for $\xi=1$) leads to the additional
term, replacing \eqn{Lorentzgauge},
\be \label{eq:Backgroundgauge}
-{1\over2}\Tr (D_\mu^B A_\mu^Q)^2
\ =\ -{1\over2}\Tr (\del_\mu A_\mu^Q
- \textstyle{i\over\sqrt{2}} g [A_\mu^B,A_\mu^Q])^2.
\ee
For one-loop calculations we require
only the terms in the Lagrangian that are quadratic in the quantum
field $A_\mu^Q$; $A_\mu^Q$ describes the gluon propagating around the
loop, while $A_\mu^B$ corresponds to the external gluons.
Expanding out the classical Lagrangian $-{1\over4}\Tr(F_{\mu\nu}^2)$
plus \eqn{Backgroundgauge}, one finds that the three-gluon ($QQB$) and
four-gluon ($QQBB$) color-ordered vertices are modified from those
shown in \fig{RulesFigure} to
\bea
V^{QQB}_{\mu\nu\rho} &=& {i\over\sqrt{2}}
\Bigl[ \eta_{\mu\nu}(k-p)_\rho
- 2\eta_{\rho\nu}q_\mu + 2\eta_{\rho\mu}q_\nu \Bigr] \nonumber \\
\label{eq:Backgroundrules}
V^{QQBB}_{\mu\nu\rho\lambda} &=& -{i\over2}
\Bigl[ \eta_{\mu\nu}\eta_{\rho\lambda}
+ 2\eta_{\mu\lambda}\eta_{\nu\rho}
- 2\eta_{\mu\rho}\eta_{\nu\lambda} \Bigr]\ ;
\eea
the remaining rules remain the same.
In background-field gauge the interactions of a scalar and of a ghost
with the background field are identical, and are given by
\bea
V^{ssB}_{\rho} &=& {i\over\sqrt{2}} (k-p)_\rho \nonumber \\
\label{eq:MoreBackgroundrules}
V^{ssBB}_{\rho\lambda} &=& -{i\over2} \eta_{\rho\lambda}\ ;
\eea
of course a ghost loop has an additional overall minus sign.
Now let's use \eqns{Backgroundrules}{MoreBackgroundrules} to compare
the gluon and scalar contributions to an $n$-gluon one-loop amplitude,
focusing on the terms with the most factors of the loop momentum
in the numerator of the Feynman diagrams, because these give
rise to the greatest algebraic complications in explicit computations
(see the next subsection).
The loop momentum only appears in the tri-linear vertices, and
only in the first term in $V^{QQB}_{\mu\nu\rho}$, because $q$ is an
external momentum.
This term matches $V^{ssB}_{\rho}$ up to the $\eta_{\mu\nu}$ factor.
Thus the leading loop-momentum terms for a gluon loop
(including the ghost contribution) are identical to those for a complex
scalar loop: $\eta^\mu_\mu - 2 = 2$ in $D=4$.
In dimensional regularization this result is still true if one uses a
scheme such as dimensional reduction\,\cite{30}
or four-dimensional helicity,\cite{5}
which leaves the number of physical gluon helicities fixed at two.
In fact, as we'll see shortly, the difference between a gluon loop and
a complex scalar loop has {\it two} fewer powers of the loop momentum
in the numerator --- at most $m-2$ powers in a diagram with $m$
propagators in the loop, versus $m$ for the gluon or scalar loop alone.
In summary, a gluon loop is a scalar loop ``plus a little bit more''.
To treat fermion loops in the same way, it is convenient to use a
``second-order formalism'' where the propagator looks more like that
of a boson.\cite{31,32}
It is not necessary to generate the full Feynman rules;
it suffices to inspect the effective action $\Gamma(A)$, which generates
the one-particle irreducible (1PI) graphs. Scattering amplitudes are
obtained by attaching tree diagrams to the external legs of 1PI graphs,
but this process does not involve the loop momentum and is identical
for all internal particle contributions.
The scalar, fermion and gluon contributions to the effective action
(the latter in background-field gauge and including the ghost loop)
are
\bea
\Gamma^{\rm scalar}(A) &=& \ln{\rm det}^{-1}_{[0]}\LP D^2 \RP,
\nonumber \\
\Gamma^{\rm fermion}(A) &=&
{1\over2}\ln{\rm det}^{1/2}_{[1/2]}
\LP D^2-\textstyle{g\over\sqrt{2}}
\hf\sigma^{\mu\nu}F_{\mu\nu} \RP,
\nonumber \\
\label{eq:EffectiveAction}
\Gamma^{\rm gluon}(A) &=&
\ln{\rm det}^{-1/2}_{[1]}\LP D^2-\textstyle{g\over\sqrt{2}}
\Sigma^{\mu\nu}F_{\mu\nu}\RP
+ \ln{\rm det}^{}_{[0]}\LP D^2 \RP,
\eea
where $D$ is the covariant derivative, $F$ is the external field
strength, $\hf\sigma_{\mu\nu}$ ($\Sigma_{\mu\nu}$) is the
spin-${1\over2}$ (spin-1) Lorentz generator, and
${\rm det}{}_{[J]}$ is the one-loop determinant for a particle of
spin $J$ in the loop.
The fermionic contribution has been rewritten in second-order form
using
\be \label{eq:secondorderone}
\ln{\rm det}^{1/2}_{[1/2]}\LP \Dsl \RP
= {1\over2} \ln{\rm det}^{1/2}_{[1/2]}\LP \Dsl^2 \RP
\ee
and
\be \label{eq:secondordertwo}
\Dsl^2
= \hf \{ \Dsl,\Dsl \} + \hf [ \Dsl,\Dsl ]
= D^2 - \textstyle{g\over\sqrt{2}} \hf\sigma^{\mu\nu}F_{\mu\nu} \, .
\ee
We want to compare the leading behavior of each contribution
in \eqn{EffectiveAction} for large loop momentum $\ell$.
The leading behavior possible for an $m$-point 1PI graph is $\ell^m$,
as we saw above in the gluon and scalar cases. The leading term
always comes from the $D^2$ term in \eqn{EffectiveAction}, because
$F_{\mu\nu}$ contains only the external momenta, not the loop momentum.
Using $\Tr_{[0]}(1)=1,\ \Tr_{[1/2]}(1)=\Tr_{[1]}=4$, we see that
the $D^2$ term cancels between the scalar and
fermion loop, and between the fermion and gluon loop;
hence it cancels in any supersymmetric linear combination.
Subleading terms in supersymmetric combinations come from using
one or more factors of $F$ in generating a graph; each $F$ costs one
power of $\ell$.
Terms with a lone $F$ cancel, thanks to
$\Tr\sigma_{\mu\nu} = \Tr\Sigma_{\mu\nu} = 0$,
so the cancellation for an $m$-point 1PI graph is from $\ell^m$
down to $\ell^{m-2}$.
In a gauge other than background-field gauge, the cancellations
involving the gluon loop would no longer happen diagram by
diagram.
\par\noindent
{\bf Exercise:} By comparing the traces of products of two and three
$\sigma_{\mu\nu}$'s ($\Sigma_{\mu\nu}$'s), show that for $A^{N=4}$
the cancellation is all the way down to $\ell^{m-4}$.
\par\noindent
The loop-momentum cancellations are responsible for the much simpler
structure of the supersymmetric contributions to
$A_{5;1}(1^-,2^-,3^+,4^+,5^+)$ in \eqn{gggggmmppploop},
and similarly for generic $n$-gluon loop amplitudes.
As we sketch in the next subsection, loop integrals with fewer powers
of the loop momentum in the numerator can be reduced more simply to
``scalar'' integrals --- integrals with no loop momenta in the
numerator. In the (supersymmetric) case where the $m$-point 1PI
graphs have at most $\ell^{m-2}$ behavior, the set of integrals
obtained is so restricted that such an amplitude
can be reconstructed directly from its absorptive parts\,\cite{33}
(see Section 4.3).
Similar rearrangements can be carried out for one-loop amplitudes with
external fermions.\cite{33,10}
For example, the amplitude with two external quarks
and the rest gluons has many diagrams where a fermion goes part of
the way around the loop, and a gluon the rest of the way around.
It is easy to see that these graphs have an $\ell^{m-1}$ behavior.
If one now subtracts
from each graph the same graph where a scalar replaces the gluon in
the loop, then the background-field gauge rules,
\eqns{Backgroundrules}{MoreBackgroundrules}, show that the difference
obeys the ``supersymmetric'' $\ell^{m-2}$ criterion (even though
in this case it is not supersymmetric). Subtracting and adding back
this scalar contribution is a rearrangement analogous to the $n$-gluon
supersymmetric rearrangement, and does aid practical
calculations.\cite{10}
Finally, these rearrangements can be motivated by the
Neveu-Schwarz-Ramond representation of superstring
theory.\cite{4,5,31,9}
This representation is not manifestly space-time supersymmetric,
but at one loop it corresponds to field theory in
background-field gauge (for 1PI graphs)
and to a second-order formalism for fermions.\cite{31}
At tree-level --- and at loop-level for the trees that have to be
sewn onto 1PI graphs to construct amplitudes ---
string theory corresponds to the
nonlinear Gervais-Neveu gauge,\cite{34,31}
$\del_\mu A_\mu - {i\over\sqrt{2}}g A_\mu A_\mu = 0$.
This gauge choice also simplifies the respective
calculations, though we omit the details here.
String theory may have more to teach us about special gauges
at the multi-loop level.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%