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%% subsection 3.3 Supersymmetry [slac-pub-7106-0-0-3-3 in slac-pub-7106-0-0-3: slac-pub-7106-0-0-3-4]
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\subsection{\usemenu{slac-pub-7106::context::slac-pub-7106-0-0-3-3}{Supersymmetry}}\label{subsection::slac-pub-7106-0-0-3-3}
What does supersymmetry have to do with a non-supersymmetric theory
such as QCD? The answer is that tree-level QCD is ``effectively''
supersymmetric,\cite{14} and the ``non-super\-symmetry'' only leaks
in at the loop level.
To see the supersymmetry of an $n$-gluon tree amplitude is
simple: It has no loops in it, so it has no fermion loops in it.
Therefore the fermions in the theory might as well be gluinos, i.e.
at tree-level the theory might as well be super Yang-Mills theory.
Tree amplitudes with quarks are also supersymmetric, but at the level of
partial amplitudes: after the color information has been stripped off,
there is nothing to distinguish a quark from a gluino.
Supersymmetry leads to extra relations between
amplitudes, supersymmetric Ward identities (SWI),\cite{19}
which can be quite useful in saving computational labor.\cite{14}
To derive supersymmetric Ward identities,\cite{19,3}
we use the fact that the supercharge $Q$ annihilates the vacuum
(we are considering exactly supersymmetric theories,
{\it not} spontaneously or softly broken ones!),
\be \label{eq:startSWI}
0\ =\ \left\langle0\vert [Q, \Phi_1\Phi_2 \cdots \Phi_n] \vert0\right\rangle
\ =\ \sum_{i=1}^n \left\langle0\vert \Phi_1 \cdots [Q,\Phi_i]
\cdots \Phi_n \vert0\right\rangle\ .
\ee
When the fields $\Phi_i$ create helicity eigenstates,
many of the $[Q,\Phi_i]$ terms can be arranged to vanish.
To proceed, we need the precise commutation relations of the
supercharge with the fields $g^\pm(k)$, $\Lambda^\pm(k)$, which
create gluon and gluino states of momentum $k$ ($k^2=0$) and helicity
$\pm$. We multiply $Q$ by a Grassmann spinor parameter $\bar\eta$,
defining $Q(\eta) \equiv \bar\eta^\alpha Q_\alpha$, so that $Q(\eta)$
commutes with the Fermi fields as well as the Bose fields.
The commutators have the form
\bea
\left[ Q(\eta), g^\pm(k) \right] &=&
\mp \Gamma^\pm(k,\eta) \, \Lambda^\pm(k),
\nonumber \\
\label{eq:Qcommutators}
\left[ Q(\eta), \Lambda^\pm(k) \right] &=&
\mp \Gamma^\mp(k,\eta) \, g^\pm(k),
\eea
where $\Gamma(k,\eta)$ is linear in $\eta$, and has its form
constrained by the Jacobi identity for the supersymmetry algebra,
\be \label{eq:QJacobi}
0\ =\ \left[ \left[ Q(\eta), Q(\zeta) \right], \Phi(k) \right]
+ \left[ \left[ Q(\zeta), \Phi(k) \right], Q(\eta) \right]
+ \left[ \left[ \Phi(k), Q(\eta) \right], Q(\zeta) \right]\ ,
\ee
where $\Phi(k)$ is either $g^\pm(k)$ or $\Lambda^\pm(k)$.
Since $[Q(\eta),Q(\zeta)] = -2i\bar\eta \Psl \zeta$,
we need
\be \label{eq:Gammaconstraint}
\Gamma^+(k,\eta) \Gamma^-(k,\zeta)
+ \Gamma^-(k,\eta) \Gamma^+(k,\zeta)
\ =\ -2i \bar\eta\ksl\zeta.
\ee
A solution to~\eqn{Gammaconstraint}, which also has the correct
behavior under rotations around the ${\bf k}$ axis, is ({\it cf.}
\eqn{spinornorm})
\be \label{eq:Gammaexpr}
\Gamma^+(k,\eta)\ =\ \bar\eta u_-(k), \qquad
\Gamma^-(k,\eta)\ =\ \bar\eta u_+(k)\ =\ \bar{u}_-(k)\eta.
\ee
Finally, we choose $\eta$ to be a Grassmann parameter $\theta$,
multiplied by the spinor for an arbitrary massless vector $q$,
and choose $q$ so as to simplify the identities (much
like the choice of reference momentum in $\pol_\pm^\mu(q)$).
Then $\Gamma^\pm(k,\eta)$ become
\be \label{eq:newGammaexpr}
\Gamma^+(k,q)\ =\ \theta \, \langle q^+ | k^- \rangle
\ =\ \theta \, \spb{q}.{k}, \quad
\Gamma^-(k,q)\ =\ \theta \, \langle q^- | k^+ \rangle
\ =\ \theta \, \spa{q}.{k}.
\ee
The simplest case is the like-helicity one. We start with
\bea
0 \hskip-2mm &=& \hskip-2mm
\left\langle0\vert [Q(\eta(q)), \Lambda_1^+ g_2^+ g_3^+ \cdots g_n^+]
\vert0\right\rangle
\nonumber \\
\hskip-2mm &=& \hskip-2mm
- \Gamma^-(k_1,q) \, A_n(g_1^+,g_2^+,\ldots,g_n^+)
+ \Gamma^+(k_2,q) \, A_n(\Lambda_1^+,\Lambda_2^+,g_3^+,\ldots,g_n^+)
\nonumber \\
\label{eq:startSWIallplus}
&&\quad
+\cdots
+\Gamma^+(k_n,q)\,A_n(\Lambda_1^+,g_2^+,\ldots,g_{n-1}^+,\Lambda_n^+).
\eea
Since massless gluinos, like quarks, have only helicity-conserving
interactions in (super) QCD, all of the amplitudes but the first
in~\eqn{startSWIallplus} must vanish. Therefore so must the
like-helicity amplitude $A_n(g_1^+,g_2^+,\ldots,g_n^+)$.
Similarly, with one negative helicity we get
\bea
0 &=& \left\langle0\vert [Q(\eta(q)), \Lambda_1^+ g_2^- g_3^+ \cdots g_n^+]
\vert0\right\rangle
\nonumber \\
&=& - \Gamma^-(k_1,q) \, A_n(g_1^+,g_2^-,g_3^+,\ldots,g_n^+)
- \Gamma^-(k_2,q) \, A_n(\Lambda_1^+,\Lambda_2^-,g_3^+,\ldots,g_n^+),
\nonumber \\
\label{eq:startSWIoneminus}
&&
\eea
where we have omitted the vanishing fermion-helicity-violating
amplitudes. Now we use the freedom to choose $q$, setting $q=k_1$
to show the second amplitude vanishes and setting $q=k_2$
to show the first vanishes. Thus we have recovered
\eqns{treevanishing}{moretreevanishing}.
With two negative helicities, we begin to relate nonzero amplitudes:
\bea
0 &=&
\left\langle0\vert [Q(\eta(q)), g_1^- g_2^- \Lambda_3^+
g_4^+ \cdots g_n^+] \vert0\right\rangle
\nonumber \\
&=&
\Gamma^-(k_1,q) \, A_n(\Lambda_1^-,g_2^-,\Lambda_3^+,\ldots,g_n^+)
+ \Gamma^-(k_2,q) \, A_n(g_1^-,\Lambda_2^-,\Lambda_3^+,\ldots,g_n^+)
\nonumber \\
\label{eq:startSWItwominus}
&&\quad - \Gamma^-(k_3,q) \, A_n(g_1^-,g_2^-,g_3^+,\ldots,g_n^+).
\eea
Choosing $q=k_1$, we get
\be \label{eq:nonzeroSWI}
A_n(g_1^-,g_2^-,g_3^+,g_4^+,\ldots,g_n^+)
\ =\ { \spa1.2\over\spa1.3 } \times
A_n(g_1^-,\Lambda_2^-,\Lambda_3^+,g_4^+,\ldots,g_n^+).
\ee
No perturbative approximations were made in deriving any of the above
SWI; thus they hold order-by-order in the loop expansion.
They apply directly to QCD tree amplitudes,
because of their ``effective'' supersymmetry.
But they can also be used to save some work at the loop level
(see below).
Since supersymmetry commutes with color, the SWI apply to each
color-ordered partial amplitude separately.
Summarizing the above ``MHV'' results (and similar ones including
a pair of external scalar fields), we have
\bea \label{eq:vanishSWI}
A_n^\susy(1^\pm,2^+,3^+,\ldots,n^+) \hskip-2mm &=& \hskip-2mm 0,
\\
A_n^\susy(1^-,2_P^-,3_P^+,4^+,\ldots,n^+)
\hskip-2mm &=& \hskip-2mm
\left({\spa1.2\over\spa1.3}\right)^{2|h_P|}
A_n^\susy(1^-,2_\phi^-,3_\phi^+,4^+,\ldots,n^+).
\nonumber \\
\label{eq:MHVSWI}
&&
\eea
Here no subscript refers to a gluon, while $\phi$ refers to a scalar
particle (for which the ``helicity'' $\pm$ means particle vs. antiparticle),
and $P$ refers to a scalar, fermion or gluon, with respective helicity
$h_P = 0, \hf, 1$.
We can use \eqn{MHVSWI} at the four-point level to obtain
the $\qb qgg$ amplitudes from the four-gluon ones,
\eqns{ggggmmpp}{ggggmpmp}:
\bea \label{eq:qqggmpmp}
A_4^\tree(1_\qb^-,2_q^+,3^-,4^+) &=&
i \, { {\spa1.3}^3\spa2.3 \over \spa1.2\spa2.3\spa3.4\spa4.1 }\ ,
\nonumber \\
\label{eq:qqggmppm}
A_4^\tree(1_\qb^-,2_q^+,3^+,4^-) &=&
i \, { {\spa1.4}^3\spa2.4 \over \spa1.2\spa2.3\spa3.4\spa4.1 }\ .
\eea
{\bf Exercise:} Check the SWI at the five-point level, comparing the
$\qb qggg$ amplitude, \eqn{qqgggmpmpp}, and the $ggggg$ amplitude
from~\eqn{mhvall}.
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