diff --git a/doc/developer_manual/currents.tex b/doc/developer_manual/currents.tex index 6c4a91a..a00596e 100644 --- a/doc/developer_manual/currents.tex +++ b/doc/developer_manual/currents.tex @@ -1,277 +1,358 @@ \section{Currents} \label{sec:currents} The following section contains a list of all the currents implemented in \HEJ. Clean up of the code structure is ongoing. All W+Jet currents are located in \texttt{src/Wjets.cc}. All other currents are defined in \texttt{src/currents.cc}. There is a common header between these files located at \texttt{include/HEJ/currents.hh}. \subsection{Pure Jets} \subsubsection{Quark} \label{sec:current_quark} \begin{align} j_\mu(p_i,p_j)=\bar{u}(p_i)\gamma_\mu u(p_j) \end{align} The exact for depends on the helicity and direction (forward/backwards) for the quarks. Currently all different contractions of incoming and outgoing states are defined in \lstinline!joi!, \lstinline!jio! and \lstinline!joo!. \subsubsection{Gluon} In \HEJ the currents for gluons and quarks are the same, up to a colour factor $K_g/C_F$, where \begin{align} K_g(p_1^-, p_a^-) = \frac{1}{2}\left(\frac{p_1^-}{p_a^-} + \frac{p_a^-}{p_1^-}\right)\left(C_A - \frac{1}{C_A}\right)+\frac{1}{C_A}. \end{align} Thus we can just reuse the results from sec.~\ref{sec:current_quark}. \subsubsection{Single unordered gluon} Configuration $q(p_a) \to g(p_1) q(p_2) g^*(\tilde{q}_2)$~\cite{Andersen:2017kfc} \begin{align} \label{eq:juno} \begin{split} &j^{{\rm uno}\; \mu\ cd}(p_2,p_1,p_a) = i \varepsilon_{1\nu} \left( T_{2i}^{c}T_{ia}^d\ \left(U_1^{\mu\nu}-L^{\mu\nu} \right) + T_{2i}^{d}T_{ia}^c\ \left(U_2^{\mu\nu} + L^{\mu\nu} \right) \right). \\ U_1^{\mu\nu} &= \frac1{s_{21}} \left( j_{21}^\nu j_{1a}^\mu + 2 p_2^\nu j_{2a}^\mu \right) \qquad \qquad U_2^{\mu\nu} = \frac1{t_{a1}} \left( 2 j_{2a}^\mu p_a^\nu - j_{21}^\mu j_{1a}^\nu \right) \\ L^{\mu\nu} &= \frac1{t_{a2}} \left(-2p_1^\mu j_{2a}^\nu+2p_1.j_{2a} g^{\mu\nu} + (\tilde{q}_1+\tilde{q}_2)^\nu j_{2a}^\mu + \frac{t_{b2}}{2} j_{2a}^\mu \left( \frac{p_2^\nu}{p_1.p_2} + \frac{p_b^\nu}{p_1.p_b} \right) \right) , \end{split} \end{align} $j^{{\rm uno}\; \mu}$ is currently not calculated as a separate current, but always as needed for the ME (i.e. in \lstinline!jM2unoXXX!). \subsection{Higgs} Different rapidity orderings \todo{give name of functions} \begin{enumerate} \item $qQ\to HqQ/qHQ/qQH$ (any rapidity order, full LO ME) $\Rightarrow$ see~\ref{sec:V_H} \item $qg\to Hqg$ (Higgs outside quark) $\Rightarrow$ see~\ref{sec:V_H} \item $qg\to qHg$ (central Higgs) $\Rightarrow$ see~\ref{sec:V_H} \item $qg\to qgH$ (Higgs outside gluon) $\Rightarrow$ see~\ref{sec:jH_mt} \item $gg\to gHg$ (central Higgs) $\Rightarrow$ see~\ref{sec:V_H} \item $gg\to ggH$ (Higgs outside gluon) $\Rightarrow$ see~\ref{sec:jH_mt} \end{enumerate} \subsubsection{Higgs gluon vertex} \label{sec:V_H} The coupling of the Higgs boson to gluons via a virtual quark loop can be written as \begin{align} \label{eq:VH} V^{\mu\nu}_H(q_1, q_2) = \mathgraphics{build/figures/V_H.pdf} &= \frac{\alpha_s m^2}{\pi v}\big[ g^{\mu\nu} T_1(q_1, q_2) - q_2^\mu q_1^\nu T_2(q_1, q_2) \big]\, \\ &\xrightarrow{m \to \infty} \frac{\alpha_s}{3\pi v} \left(g^{\mu\nu} q_1\cdot q_2 - q_2^\mu q_1^\nu\right). \end{align} The outgoing momentum of the Higgs boson is $p_H = q_1 - q_2$. As a contraction with two currents this by implemented in \lstinline!cHdot! inside \texttt{src/currents.cc}. The form factors $T_1$ and $T_2$ are then given by~\cite{DelDuca:2003ba} \begin{align} \label{eq:T_1} T_1(q_1, q_2) ={}& -C_0(q_1, q_2)\*\left[2\*m^2+\frac{1}{2}\*\left(q_1^2+q_2^2-p_H^2\right)+\frac{2\*q_1^2\*q_2^2\*p_H^2}{\lambda}\right]\notag\\ &-\left[B_0(q_2)-B_0(p_H)\right]\*\frac{q_2^2}{\lambda}\*\left(q_2^2-q_1^2-p_H^2\right)\notag\\ &-\left[B_0(q_1)-B_0(p_H)\right]\*\frac{q_1^2}{\lambda}\*\left(q_1^2-q_2^2-p_H^2\right)-1\,,\displaybreak[0]\\ \label{eq:T_2} T_2(q_1, q_2) ={}& C_0(q_1, q_2)\*\left[\frac{4\*m^2}{\lambda}\*\left(p_H^2-q_1^2-q_2^2\right)-1-\frac{4\*q_1^2\*q_2^2}{\lambda} - \frac{12\*q_1^2\*q_2^2\*p_H^2}{\lambda^2}\*\left(q_1^2+q_2^2-p_H^2\right)\right]\notag\\ &-\left[B_0(q_2)-B_0(p_H)\right]\*\left[\frac{2\*q_2^2}{\lambda}+\frac{12\*q_1^2\*q_2^2}{\lambda^2}\*\left(q_2^2-q_1^2+p_H^2\right)\right]\notag\\ &-\left[B_0(q_1)-B_0(p_H)\right]\*\left[\frac{2\*q_1^2}{\lambda}+\frac{12\*q_1^2\*q_2^2}{\lambda^2}\*\left(q_1^2-q_2^2+p_H^2\right)\right]\notag\\ &-\frac{2}{\lambda}\*\left(q_1^2+q_2^2-p_H^2\right)\,, \end{align} where we have used the scalar bubble and triangle integrals \begin{align} \label{eq:B0} B_0\left(p\right) ={}& \int \frac{d^dl}{i\pi^{\frac{d}{2}}} \frac{1}{\left(l^2-m^2\right)\left((l+p)^2-m^2\right)}\,,\\ \label{eq:C0} C_0\left(p,q\right) ={}& \int \frac{d^dl}{i\pi^{\frac{d}{2}}} \frac{1}{\left(l^2-m^2\right)\left((l+p)^2-m^2\right)\left((l+p-q)^2-m^2\right)}\,, \end{align} and the K\"{a}ll\'{e}n function \begin{equation} \label{eq:lambda} \lambda = q_1^4 + q_2^4 + p_H^4 - 2\*q_1^2\*q_2^2 - 2\*q_1^2\*p_H^2- 2\*q_2^2\*p_H^2\,. \end{equation} The Integrals as such are provided by \QCDloop{} (see wrapper functions \lstinline!B0DD! and \lstinline!C0DD! in \texttt{src/currents.cc}). In the code we are sticking to the convention of~\cite{DelDuca:2003ba}, thus instead of the $T_{1/2}$ we implement (in the functions \lstinline!A1! and \lstinline!A2!) \begin{align} \label{eq:A_1} A_1(q_1, q_2) ={}& \frac{i}{16\pi^2}\*T_2(-q_1, q_2)\,,\\ \label{eq:A_2} A_2(q_1, q_2) ={}& -\frac{i}{16\pi^2}\*T_1(-q_1, q_2)\,. \end{align} \subsubsection{Peripheral Higgs emission - Finite quark mass} \label{sec:jH_mt} We describe the emission of a peripheral Higgs boson close to a scattering gluon with an effective current. In the following we consider a lightcone decomposition of the gluon momenta, i.e. $p^\pm = E \pm p_z$ and $p_\perp = p_x + i p_y$. The incoming gluon momentum $p_a$ defines the $-$ direction, so that $p_a^+ = p_{a\perp} = 0$. The outgoing momenta are $p_1$ for the gluon and $p_H$ for the Higgs boson. We choose the following polarisation vectors: \begin{equation} \label{eq:pol_vectors} \epsilon_\mu^\pm(p_a) = \frac{j_\mu^\pm(p_1, p_a)}{\sqrt{2} \bar{u}^\pm(p_a)u^\mp(p_1)}\,, \quad \epsilon_\mu^{\pm,*}(p_1) = -\frac{j_\mu^\pm(p_1, p_a)}{\sqrt{2} \bar{u}^\mp(p_1)u^\pm(p_a)}\,. \end{equation} Following~\cite{DelDuca:2001fn}, we introduce effective polarisation vectors to describe the contraction with the Higgs-boson production vertex eq.~\eqref{eq:VH}: \begin{align} \label{eq:eps_H} \epsilon_{H,\mu}(p_a) = \frac{T_2(p_a, p_a-p_H)}{(p_a-p_H)^2}\big[p_a\cdot p_H\epsilon_\mu(p_a) - p_H\cdot\epsilon(p_a) p_{a,\mu}\big]\,,\\ \epsilon_{H,\mu}^*(p_1) = -\frac{T_2(p_1+p_H, p_1)}{(p_1+p_H)^2}\big[p_1\cdot p_H\epsilon_\mu^*(p_1) - p_H\cdot\epsilon^*(p_1) p_{1,\mu}\big]\,, \end{align} We also employ the usual short-hand notation \begin{equation} \label{eq:spinor_helicity} \spa i.j = \bar{u}^-(p_i)u^+(p_j)\,,\qquad \spb i.j = \bar{u}^+(p_i)u^-(p_j)\,, \qquad[ i | H | j\rangle = j_\mu^+(p_i, p_j)p_H^\mu\,. \end{equation} Without loss of generality, we consider only the case where the incoming gluon has positive helicity. The remaining helicity configurations can be obtained through parity transformation. Labelling the effective current by the helicities of the gluons we obtain for the same-helicity case \begin{equation} \label{eq:jH_same_helicity} \begin{split} j_{H,\mu}^{++}{}&(p_1,p_a,p_H) = \frac{m^2}{\pi v}\bigg[\\ &-\sqrt{\frac{2p_1^-}{p_a^-}}\frac{p_{1\perp}^*}{|p_{1\perp}|}\frac{t_2}{\spb a.1}\epsilon^{+,*}_{H,\mu}(p_1) +\sqrt{\frac{2p_a^-}{p_1^-}}\frac{p_{1\perp}^*}{|p_{1\perp}|}\frac{t_2}{\spa 1.a}\epsilon^{+}_{H,\mu}(p_a)\\ &+ [1|H|a\rangle \bigg( \frac{\sqrt{2}}{\spa 1.a}\epsilon^{+}_{H,\mu}(p_a) + \frac{\sqrt{2}}{\spb a.1}\epsilon^{+,*}_{H,\mu}(p_1)-\frac{\spa 1.a T_2(p_a, p_a-p_H)}{\sqrt{2}(p_a-p_H)^2}\epsilon^{+,*}_{\mu}(p_1)\\ & \qquad -\frac{\spb a.1 T_2(p_1+p_H, p_1)}{\sqrt{2}(p_1+p_H)^2}\epsilon^{+}_{\mu}(p_a)-\frac{RH_4}{\sqrt{2}\spb a.1}\epsilon^{+,*}_{\mu}(p_1)+\frac{RH_5}{\sqrt{2}\spa 1.a}\epsilon^{+}_{\mu}(p_a) \bigg)\\ & - \frac{[1|H|a\rangle^2}{2 t_1}(p_{a,\mu} RH_{10} - p_{1,\mu} RH_{12})\bigg] \end{split} \end{equation} with $t_1 = (p_a-p_1)^2$, $t_2 = (p_a-p_1-p_H)^2$ and $R = 8 \pi^2$. Eq.~\eqref{eq:jH_same_helicity} is implemented by \lstinline!g_gH_HC! in \texttt{src/currents.cc} \footnote{\lstinline!g_gH_HC! and \lstinline!g_gH_HNC! includes an additional $1/t_2$ factor, which should be in the Matrix element instead.}. The currents with a helicity flip is given through \begin{equation} \label{eq:jH_helicity_flip} \begin{split} j_{H,\mu}^{+-}{}&(p_1,p_a,p_H) = \frac{m^2}{\pi v}\bigg[\\ &-\sqrt{\frac{2p_1^-}{p_a^-}}\frac{p_{1\perp}^*}{|p_{1\perp}|}\frac{t_2}{\spb a.1}\epsilon^{-,*}_{H,\mu}(p_1) +\sqrt{\frac{2p_a^-}{p_1^-}}\frac{p_{1\perp}}{|p_{1\perp}|}\frac{t_2}{\spb a.1}\epsilon^{+}_{H,\mu}(p_a)\\ &+ [1|H|a\rangle \left( \frac{\sqrt{2}}{\spb a.1} \epsilon^{-,*}_{H,\mu}(p_1) -\frac{\spa 1.a T_2(p_a, p_a-p_H)}{\sqrt{2}(p_a-p_H)^2}\epsilon^{-,*}_{\mu}(p_1) - \frac{RH_4}{\sqrt{2}\spb a.1}\epsilon^{-,*}_{\mu}(p_1)\right) \\ &+ [a|H|1\rangle \left( \frac{\sqrt{2}}{\spb a.1}\epsilon^{+}_{H,\mu}(p_a) -\frac{\spa 1.a T_2(p_1+p_H,p_1)}{\sqrt{2}(p_1+p_H)^2}\epsilon^{+}_{\mu}(p_a) +\frac{RH_5}{\sqrt{2}\spb a.1}\epsilon^{+}_{\mu}(p_a) \right)\\ & - \frac{[1|H|a\rangle [a|H|1\rangle}{2 \spb a.1 ^2}(p_{a,\mu} RH_{10} - p_{1,\mu} RH_{12})\\ &+ \frac{\spa 1.a}{\spb a.1}\bigg(RH_1p_{1,\mu}-RH_2p_{a,\mu}+2 p_1\cdot p_H \frac{T_2(p_1+p_H, p_1)}{(p_1+p_H)^2} p_{a,\mu} \\ & \qquad- 2p_a \cdot p_H \frac{T_2(p_a, p_a-p_H)}{(p_a-p_H)^2} p_{1,\mu}+ T_1(p_a-p_1, p_a-p_1-p_H)\frac{(p_1 + p_a)_\mu}{t_1}\\ &\qquad-\frac{(p_1+p_a)\cdot p_H}{t_1} T_2(p_a-p_1, p_a-p_1-p_H)(p_1 - p_a)_\mu \bigg) \bigg]\,, \end{split} \end{equation} and implemented by \lstinline!g_gH_HNC! again in \texttt{src/currents.cc}. If we instead choose the gluon momentum in the $+$ direction, so that $p_a^- = p_{a\perp} = 0$, the corresponding currents are obtained by replacing $p_1^- \to p_1^+, p_a^- \to p_a^+, \frac{p_{1\perp}}{|p_{1\perp}|} \to -1$ in the second line of eq.~\eqref{eq:jH_same_helicity} and eq.~\eqref{eq:jH_helicity_flip} (see variables \lstinline!ang1a! and \lstinline!sqa1! in the implementation). The form factors $H_1,H_2,H_4,H_5, H_{10}, H_{12}$ are given in~\cite{DelDuca:2003ba}, and are implemented under their name in \texttt{src/currents.cc}. They reduce down to fundamental QCD integrals, which are again provided by \QCDloop. \subsubsection{Peripheral Higgs emission - Infinite top mass} \label{sec:jH_eff} To get the result with infinite top mass we could either take the limit $m_t\to \infty$ in~\eqref{eq:jH_helicity_flip} and~\eqref{eq:jH_same_helicity}, or use the \textit{impact factors} as given in~\cite{DelDuca:2003ba}. Both methods are equivalent, and lead to the same result. For the first one would find \begin{align} \lim_{m_t\to\infty} m_t^2 H_1 &= i \frac{1}{24 \pi^2}\\ \lim_{m_t\to\infty} m_t^2 H_2 &=-i \frac{1}{24 \pi^2}\\ \lim_{m_t\to\infty} m_t^2 H_4 &= i \frac{1}{24 \pi^2}\\ \lim_{m_t\to\infty} m_t^2 H_5 &=-i \frac{1}{24 \pi^2}\\ \lim_{m_t\to\infty} m_t^2 H_{10} &= 0 \\ \lim_{m_t\to\infty} m_t^2 H_{12} &= 0. \end{align} \todo{double check this, see James thesis eq. 4.33} However only the second method is implemented in the code through \lstinline!C2gHgp! and \lstinline!C2gHgm! inside \texttt{src/currents.cc}, each function calculates the square of eq. (4.23) and (4.22) from~\cite{DelDuca:2003ba} respectively. \subsection{W+Jets} \subsubsection{Quark+W} \begin{figure}[!h] \centering \begin{minipage}[b]{0.2\textwidth} \includegraphics[width=\textwidth]{figures/Wbits.pdf} \end{minipage} \begin{minipage}[b]{0.1\textwidth} \centering{=} \vspace{0.7cm} \end{minipage} \begin{minipage}[b]{0.2\textwidth} \includegraphics[width=\textwidth]{figures/Wbits2.pdf} \end{minipage} \begin{minipage}[b]{0.1\textwidth} \centering{+} \vspace{0.7cm} \end{minipage} \begin{minipage}[b]{0.2\textwidth} \includegraphics[width=\textwidth]{figures/Wbits3.pdf} \end{minipage} \caption{The $j_W$ current is constructed from the two diagrams which contribute to the emission of a $W$-boson from a given quark line.} \label{fig:jW} \end{figure} For a W emission we require a fermion. The $j_W$ current is actually a sum of two separate contributions, see Fig: \ref{fig:jW}, one with a W-emission from the initial state, and one with the W-emission from the final state. Mathematically this can be seen as the following two terms: \begin{align} \label{eq:Weffcur1} j_W^\mu(p_a,p_{\ell},p_{\bar{\ell}}, p_1) =&\ \frac{g_W^2}{2}\ \frac1{p_W^2-M_W^2+i\ \Gamma_W M_W}\ \bar{u}^-(p_\ell) \gamma_\alpha v^-(p_{\bar\ell})\nonumber \\ & \cdot \left( \frac{ \bar{u}^-(p_1) \gamma^\alpha (\slashed{p}_W + \slashed{p}_1)\gamma^\mu u^-(p_a)}{(p_W+p_1)^2} + \frac{ \bar{u}^-(p_1)\gamma^\mu (\slashed{p}_a + \slashed{p}_W)\gamma^\alpha u^-(p_a)}{(p_a-p_W)^2} \right). \end{align} - There are a couple of subtleties here. There is a minus sign distinction between the quark-anti-quark cases due to the fermion flow of the propagator in the current. Note that which type of W emission (+ or -) will depend on the quark flavour, and that the handedness of the quark-line will depend on if it is a quark or anti-quark. +\subsubsection{W+uno} +\begin{figure}[t] + \centering + \begin{subfigure}{0.45\textwidth} + \centering + \includegraphics{figures/wuno1} + \caption{} + \label{fig:U1diags} +\end{subfigure} +\begin{subfigure}{0.45\textwidth} + \centering +\includegraphics{figures/wuno2} +\caption{} +\label{fig:U2diags} +\end{subfigure} +\begin{subfigure}{0.45\textwidth} + \centering + \includegraphics{figures/wuno3} +\caption{} +\label{fig:Cdiags} +\end{subfigure} +\begin{subfigure}{0.45\textwidth} +\centering +\includegraphics{figures/wuno4} +\caption{} + \label{fig:Ddiags} + \end{subfigure} + \vspace{0.4cm} + \caption{Examples of each of the four categories of Feynman diagram which + contribute to at leading-order; there are twelve in total. (a) is an example where the gluon and $W$ + boson are emitted from the same quark line and the gluon comes after the + $t$-channel propagator. In (b), the gluon and $W$ boson are emitted from + the same quark line and the gluon comes before the $t$-channel proagator. + In (c) the gluon is emitted from the $t$-channel gluon and in (d) the gluon + is emitted from the $b$--$2$ quark line.} + \label{fig:Wunodiags} +\end{figure} +It is necessary to include subleading processes in W+Jets also. All of +these currents have been built for the Tensor Class detailed in Sec: +\ref{sec:tensor}. Similarly to the pure jet case, the uno currents are +not calculated separately, and only in the ME functions when required +in the \texttt{src/Wjets.cc} file. For unordered emissions a new +current is required, $j_{W,{\rm uno}}$, it is only non-zero for +$h_a=h_1=-$ and hence we have suppressed its helicity indices. It is +derived from the 12 leading-order Feynman diagrams in the QMRK +limit. These are shown in Fig: \ref{fig:Wunodiags}. Using $T^m_{ij}$ +represent fundamental colour matrices between quark state $i$ and $j$ +with adjoint index $m$ we find +\begin{align}\label{eq:wunocurrent} +\begin{split} + j^{d\,\mu}_{\rm W,uno}(p_a,p_1,p_2,p_\ell,p_{\bar{\ell}}) =& \ i \varepsilon_{\nu}(p_1)\ + \bar{u}^-(p_\ell) \gamma_\rho v^-(p_{\bar \ell}) \\ & \quad \times\ + \left(T^1_{2i} T^d_{ia} (\tilde U_1^{\nu\mu\rho}-\tilde L^{\nu\mu\rho}) + + T^d_{2i} T^1_{ia} (\tilde U_2^{\nu\mu\rho}+\tilde L^{\nu\mu\rho}) \right), + \end{split} +\end{align} +where expressions for $\tilde U_{1,2}^{\nu\mu\rho}$ and $\tilde L^{\nu\mu\rho}$ +are given as: +\begin{align} + \label{eq:U1tensor} + \begin{split} + \tilde U_1^{\nu\mu\rho} =&\frac{\langle 2|\nu (\slashed{p}_2+ \slashed{p}_1)\mu (\slashed{p}_a - \slashed{p}_W)\rho P_L |a\rangle }{s_{12}t_{aW}} + \frac{\langle 2|\nu (\slashed{p}_2+ \slashed{p}_1)\rho P_L (\slashed{p}_2+\slashed{p}_1 + \slashed{p}_W)\mu |a\rangle }{s_{12}s_{12W}} \\ + &+ \frac{\langle 2|\rho P_L (\slashed{p}_2+ \slashed{p}_W) \nu + (\slashed{p}_1 + \slashed{p}_2+\slashed{p}_W)\mu |a\rangle}{s_{2W}s_{12W}}. + \end{split} +\end{align} +\begin{align} + \label{eq:U2tensor} + \begin{split} + \tilde U_2^{\nu\mu\rho} =&\frac{\langle 2|\mu (\slashed{p}_a-\slashed{p}_W-\slashed{p}_1)\nu (\slashed{p}_a - \slashed{p}_W)\rho P_L |a\rangle }{t_{aW1}t_{aW}} + \frac{\langle 2|\mu (\slashed{p}_a-\slashed{p}_W- \slashed{p}_1)\rho P_L (\slashed{p}_a-\slashed{p}_1) \nu |a\rangle }{t_{a1W}t_{a1}} \\ + &+ \frac{\langle 2|\rho P_L (\slashed{p}_2+ \slashed{p}_W) \mu + (\slashed{p}_a-\slashed{p}_1)\nu |a\rangle}{s_{2W}t_{a1}}. + \end{split} +\end{align} +\begin{align} +\label{eq:Ltensor} + \tilde L^{\nu\mu\rho} &= \frac{q_2^2}{2t_{aW2}} \left[\frac{\langle 2 |\mu (\slashed{p}_a - \slashed{p}_W) \rho P_L | a\rangle}{t_{aW}} + \frac{\langle 2 |\rho P_L (\slashed{p}_2 + \slashed{p}_W) \mu | a\rangle }{s_{2W}} \right] + \cdot \left( \frac{p_b^{\nu}}{p_b\cdot p_1} + \frac{p_3^{\nu}}{p_3\cdot p_1} \right) \nonumber \\ + &\quad+\frac{1}{t_{aW2}}\left[\frac{\langle 2 |\sigma (\slashed{p}_a - \slashed{p}_W) \rho P_L | a\rangle}{t_{aW}} + \frac{\langle 2 |\rho P_L (\slashed{p}_2 + \slashed{p}_W) \sigma | a\rangle }{s_{2W}} \right] \nonumber \\ + &\vphantom{+\frac{1}{t_{aW2}}}\quad\cdot \left( g^{\sigma \mu} (q_1 +q_2)^\nu + g^{\mu \nu}(-q_2 +p_1)^\sigma+ g^{\nu \sigma}(-p_1 -q_1)^\mu \right). +\end{align} + %%% Local Variables: %%% mode: latex %%% TeX-master: "developer_manual" %%% End: