%% slac-pub-7106: page file slac-pub-7106-0-0-2-2.tcx.
%% subsection 2.2 Helicity Nitty Gritty [slac-pub-7106-0-0-2-2 in slac-pub-7106-0-0-2: 0, \quad k_j^0 > 0,
\eea
where $s_{ij}\ =\ (k_i+k_j)^2\ =\ 2 k_i\cdot k_j$, and
\be \label{eq:phiijdef}
\cos\phi_{ij}\ =\ { k_i^1 k_j^+ - k_j^1 k_i^+
\over \sqrt{|s_{ij}| k_i^+ k_j^+} }
\ , \qquad
\sin\phi_{ij}\ =\ { k_i^2 k_j^+ - k_j^2 k_i^+
\over \sqrt{|s_{ij}| k_i^+ k_j^+} }
\ .
\ee
The spinor products are, up to a phase, square roots of Lorentz
products. We'll see that the collinear limits of massless gauge
amplitudes have this kind of square-root singularity,
which explains why spinor products lead to very compact analytic
representations of gauge amplitudes, as well as improved numerical
stability.
We would like the spinor products to have simple properties under
crossing symmetry, i.e. as energies become negative.\cite{13}
We define the spinor product $\spa{i}.{j}$ by analytic continuation
from the positive energy case, using the same
formula, \eqn{posenergyexplicit}, but with $k_i$ replaced by $-k_i$ if
$k_i^0 < 0$, and similarly for $k_j$; and with an extra multiplicative
factor of $i$ for each negative energy particle.
We define $\spb{i}.{j}$ through the identity
\be \label{eq:spbphasedef}
\spa{i}.{j}\spb{j}.{i}
= \langle i^- | j^+ \rangle \langle j^+ | i^- \rangle
= \tr\bigl( \hf(1-\gamma_5) \ksl_i \ksl_j \bigr)
= 2 k_i\cdot k_j = s_{ij}.
\ee
We also have the useful identities:
\par\noindent
Gordon identity and projection operator:
\be \label{eq:spinornorm}
\langle i^\pm | \gamma^\mu | i^\pm \rangle\ =\ 2 k_i^\mu,\qquad\qquad
| i^\pm \rangle \langle i^\pm |\ =\ \hf(1\pm\gamma_5)\ksl_i
\ee
antisymmetry:
\be \label{eq:spinorantisym}
\spa{j}.{i} = -\spa{i}.{j}, \qquad
\spb{j}.{i} = -\spb{i}.{j}, \qquad
\spa{i}.{i} = \spb{i}.{i} = 0 \qquad
\ee
Fierz rearrangement:
\be \label{eq:spinorfierz}
\langle i^+|\gamma^\mu|j^+\rangle \langle k^+|\gamma_\mu|l^+\rangle
\ =\ 2 \, \spb{i}.{k}\spa{l}.{j}
\ee
charge conjugation of current:
\be \label{eq:cccurrent}
\langle i^+|\gamma^\mu|j^+\rangle
\ =\ \langle j^-|\gamma^\mu|i^-\rangle
\ee
Schouten identity:
\be \label{eq:schouten}
\spa{i}.{j} \spa{k}.{l}\ =\
\spa{i}.{k} \spa{j}.{l} + \spa{i}.{l} \spa{k}.{j} .
\ee
In an $n$-point amplitude, momentum conservation,
$\sum_{i=1}^n k_i^\mu = 0$, provides one more identity,
\be \label{eq:momcons}
\sum_{{{i=1} \atop {i\neq j,k}}}^n \spb{j}.{i}\spa{i}.{k}\ =\ 0.
\ee
The next step is to introduce a spinor representation for the
polarization vector for a massless gauge boson of definite helicity
$\pm1$,
\be \label{eq:poldefn}
\pol^\pm_\mu(k,q)\ =\
\pm { \langle q^\mp | \gamma_\mu | k^\mp \rangle
\over \sqrt{2} \langle q^\mp | k^\pm \rangle }\ ,
\ee
where $k$ is the vector boson momentum and $q$ is an auxiliary
massless vector, called the {\it reference momentum}, reflecting the
freedom of on-shell gauge tranformations.
We will not motivate~\eqn{poldefn}, but just show that it has the desired
properties.
Since $\ksl | k^\pm\rangle = 0$, $\pol^\pm(k,q)$ is transverse to $k$,
for any $q$,
\be \label{eq:transverse}
\pol^\pm(k,q) \cdot k\ =\ 0.
\ee
Complex conjugation reverses the helicity,
\be \label{eq:polcc}
(\pol_\mu^+)^*\ =\ \pol_\mu^-\ .
\ee
The denominator gives $\pol_\mu$ the standard normalization
(using~\eqn{spinorfierz}),
\bea \label{eq:polnormnonzero}
\pol^+\cdot(\pol^+)^* &=& \pol^+\cdot\pol^-\ =\
-{1\over2} { \langle q^-|\gamma_\mu|k^-\rangle
\langle q^+|\gamma^\mu|k^+\rangle
\over \spa{q}.{k} \spb{q}.{k} }\ =\ -1,
\nonumber \\
\label{eq:polnormzero}
\pol^+\cdot(\pol^-)^* &=& \pol^+\cdot\pol^+\ =\
{1\over2} { \langle q^-|\gamma_\mu|k^-\rangle
\langle q^-|\gamma^\mu|k^-\rangle
\over {\spa{q}.{k}}^2 }\ =\ 0.
\eea
States with helicity $\pm1$ are produced by $\pol^\pm$. The easiest
way to see this is to consider a rotation around the ${\bf k}$
axis, and notice that the $| k^+ \rangle$ in the denominator
of~\eqn{poldefn} picks up the opposite phase from the state
$| k^- \rangle$ in the numerator; i.e. it doubles the phase from
that appropriate for a spinor (helicity $+\hf$)
to that appropriate for a vector (helicity $+1$).
Finally, changing the reference momentum $q$ does amount to an on-shell
gauge transformation, since $\pol_\mu$ shifts by an amount
proportional to $k_\mu$:
\bea
\pol_\mu^+(\tq) - \pol_\mu^+(q)
\hskip-1.2mm &=& \hskip-1.2mm
{ \langle \tq^- | \gamma_\mu | k^- \rangle
\over \sqrt{2} \spa{\tq}.{k} }
- { \langle q^- | \gamma_\mu | k^- \rangle
\over \sqrt{2} \spa{q}.{k} }
= - { \langle \tq^- | \gamma_\mu \ksl | q^+ \rangle
+ \langle \tq^- | \ksl \gamma_\mu | q^+ \rangle
\over \sqrt{2} \spa{\tq}.{k} \spa{q}.{k} }
\nonumber \\
\label{eq:onshellgauge}
\hskip-1.2mm &=& \hskip-1.2mm
- { \sqrt{2} \spa{\tq}.{q}
\over \spa{\tq}.{k} \spa{q}.{k} } \times k_\mu\ .
\eea
{\bf Exercise:} Show that the completeness relation for these
polarization vectors is that of an light-like axial gauge,
\be \label{eq:polcomplete}
\sum_{\lambda=\pm}
\pol^\lambda_\mu(k,q) \, (\pol^\lambda_\nu(k,q))^*
\ =\ -\eta_{\mu\nu} + { k_\mu q_\nu + k_\nu q_\mu \over k\cdot q }\ .
\ee
A separate reference momentum $q_i$ can be chosen for each gluon
momentum $k_i$ in an amplitude. Because it is a gauge choice,
one should be careful not to change the $q_i$ within the calculation
of a gauge-invariant quantity (such as a partial amplitude).
On the other hand, different choices can be made when calculating
different gauge-invariant quantities.
A judicious choice of the $q_i$ can simplify a
calculation substantially, by making many terms and diagrams vanish,
due primarily to the following identities, where
$\pol_i^\pm(q) \equiv \pol^\pm(k_i,q_i=q)$:
\bea \label{eq:polvanishidone}
\pol_i^\pm(q)\cdot q &=& 0,
\\
\label{eq:polvanishidtwo}
\pol_i^+(q)\cdot \pol_j^+(q) &=&
\pol_i^-(q)\cdot \pol_j^-(q)\ =\ 0,
\\
\label{eq:polvanishidthree}
\pol_i^+(k_j)\cdot \pol_j^-(q) &=&
\pol_i^+(q)\cdot \pol_j^-(k_i)\ =\ 0,
\\
\label{eq:polvanishidfour}
\esl_i^+(k_j) | j^+ \rangle &=&
\esl_i^-(k_j) | j^- \rangle\ =\ 0,
\\
\label{eq:polvanishidfive}
\langle j^+ | \esl_i^-(k_j) &=&
\langle j^- | \esl_i^+(k_j)\ =\ 0.
\eea
In particular, it is useful to choose the reference momenta of
like-helicity gluons to be identical, and to equal the external
momentum of one of the opposite-helicity set of gluons.
We can now express any amplitude with massless external fermions
and vector bosons in terms of spinor products.
Since these products are defined for both positive- and negative-energy
four-momenta, we can use crossing symmetry to extract a number of
scattering amplitudes from the same expression, by exchanging which
momenta are outgoing and which incoming.
However, because the helicity of a positive-energy (negative-energy)
massless spinor has the same (opposite) sign as its chirality,
the helicities assigned to the particles --- bosons as well as fermions
--- depend on whether they are incoming or outgoing.
Our convention is to label particles with their helicity when they
are considered outgoing (positive-energy); if they are incoming the
helicity is reversed.
The spinor-product representation of an amplitude can be related to
a more conventional one in terms of Lorentz-invariant objects,
the momentum invariants $k_i\cdot k_j$ and contractions of the
Levi-Civita tensor $\pol_{\mu\nu\sigma\rho}$ with external momenta.
The spinor products carry around a number of phases. Some of
the phases are unphysical because they are associated with
external-state conventions, such as the definitions of the spinors
$|i^\pm\rangle$. Physical quantities such as cross-sections
(or amplitudes from which an overall phase has been removed),
when constructed out of the spinor products,
will be independent of such choices.
Thus for each external momentum label $i$, if the product
$\spa{i}.{j}$ appears then its phase should be compensated by
some $\spb{i}.{k}$
(or equivalently $1/\spa{i}.{k}=-\spb{i}.{k}/s_{ik}$).
If a spinor string appears in a physical quantity,
then it must terminate, i.e. it has the form
\be \label{eq:longspstring}
\spa{i_1}.{i_2}\spb{i_2}.{i_3}\spa{i_3}.{i_4}\cdots \spb{i_{2m}}.{i_1},
\ee
for some $m$.
Multiplying \eqn{longspstring} by
$1=\spb{i_4}.{i_1}\spa{i_1}.{i_4}/s_{i_1i_4}$, etc.,
we can break up any spinor string into strings of length two and four;
the former are just $s_{ij}$'s (\eqn{spbphasedef}),
while the latter can then be evaluated by performing the Dirac trace:
\bea
\spa{i}.{j}\spb{j}.{l}\spa{l}.{m}\spb{m}.{i} &=&
\tr\Bigl( \hf(1-\gamma_5) \ksl_i \ksl_j \ksl_l \ksl_m \Bigr)
\nonumber \\
\label{eq:fourspstring}
&=& {1\over2} \biggl[ s_{ij}s_{lm} - s_{il}s_{jm} + s_{im}s_{jl}
- 4i \pol(i,j,l,m) \biggr] ,
\eea
where $\pol(i,j,l,m)\ =\
\pol_{\mu\nu\sigma\rho} k_i^\mu k_j^\nu k_l^\rho k_m^\sigma$.
Thus the Levi-Civita contractions are always accompanied by an $i$
and account for the physical phases.
In practice, the spinor products offer the most compact representation
of helicity amplitudes, but it is useful to know the connection
to a more conventional representation.
\par\noindent
{\bf Exercise:} Verify the Schouten identity, \eqn{schouten},
by multiplying both sides by $\spb{j}.{k}\spb{l}.{i}$ and
using \eqn{fourspstring} to simplify.
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