diff --git a/doc/developer_manual/currents.tex b/doc/developer_manual/currents.tex index 9305029..efe755e 100644 --- a/doc/developer_manual/currents.tex +++ b/doc/developer_manual/currents.tex @@ -1,653 +1,659 @@ \section{Currents} \label{sec:currents_impl} The following section contains a list of all the currents implemented in \HEJ. Clean up of the code structure is ongoing. Each implemented current has its own separate source file (e.g. \texttt{src/.cc}), and associated header file (e.g. \texttt{include/HEJ/.hh}). The processes (and their filename) that are implemented are: Pure jets (\texttt{jets}), $W$+jets (\texttt{Wjets}), $Z/\gamma$+jets (\texttt{Zjets}), $h$+jets (\texttt{Hjets}), $W^\pm W^\pm $+jets (\texttt{WWjets}). The naming convention for the current contraction $\left\|S_{f_1 f_2\to f_1 f_2}\right\|^2$ is \lstinline!ME_[Boson]_[subleading-type]_[incoming]!. For example \lstinline!ME_W_unob_qq! corresponds to the contraction $j_W^\mu j_{\text{uno}, \mu}$ ($qQ\to \bar{q}WQg$). For bosons on the same side as the subleading we drop the connecting underscore, e.g. \lstinline!ME_Wuno_qq! gives $j_{W,\text{uno}}^\mu j_\mu$ ($qQ\to g\bar{q}WQ$). \subsection{Pure Jets} \subsubsection{Quarks and Gluons} \label{sec:current_quark} The basic quark current is \begin{align} \label{eq:j} j^\pm_\mu(p_i,p_j)=\bar{u}^\pm(p_i)\gamma_\mu u^\pm(p_j), \end{align} see also equation~(\ref{eq:j_gen}). The contraction of two currents is \begin{equation} \label{eq:S} \left\|S_{f_1 f_2\to f_1 f_2}\right\|^2 = \sum_{\substack{\lambda_a = +,-\\\lambda_b = +,-}} \left|j^{\lambda_a}_\mu(p_1, p_a)\ j^{\lambda_b\,\mu}(p_n, p_b)\right|^2 = 2\sum_{\lambda = +,-} \left|j^{-}_\mu(p_1, p_a)\ j^{\lambda\,\mu}(p_n, p_b)\right|^2\,. \end{equation} Gluons use the same currents as quarks. The differences in the tree-level matrix elements are absorbed into the colour acceleration multipliers defined in equations (\ref{eq:K_g}) and (\ref{eq:K_q}). \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} using the short-hand notation $j_{ij} = j^{\lambda_1}(p_i, p_j)$. $j^{{\rm uno}\; \mu}$ is currently not calculated as a separate current, but always as needed for the ME (i.e. in \lstinline!ME_unob_XX!). The contraction with a basic quark/gluon current is \begin{equation} \label{eq:S_uno} \left\|S^{\text{uno}}_{q f_2\to g q f_2}\right\|^2 = 2\sum_{\substack{\lambda_a = +,-\\\lambda_g = +,-}} \left[C_F |X + Y|^2 - C_A \Re(XY^*)\right] \end{equation} with \begin{align} \label{eq:X_munu} X^{\mu\nu} ={}& U_1^{\mu\nu} - L^{\mu\nu}, \\ \label{eq:Y_munu} Y^{\mu\nu} ={}& U_2^{\mu\nu} + L^{\mu\nu}, \\ \label{eq:X} X ={}& j_\mu^-(p_b, p_2) \epsilon_\nu^{\lambda_g}(p_g, p_r) X^{\mu\nu},\\ \label{eq:Y} Y ={}& j_\mu^-(p_b, p_2) \epsilon_\nu^{\lambda_g}(p_g, p_r) Y^{\mu\nu}. \end{align} In the current implementation we choose $p_r = p_1$. \subsubsection{Extremal \texorpdfstring{$q\bar{q}$}{qqbar}} In Pure jets we also include the subleading process which arises when an incoming gluon splits into a $q\bar{q}$ pair. This splitting impact factor is related to the unordered current by simple means of a crossing symmetry.\todo[inline]{Current implementation still uses different conventions (James Cockburn's thesis)} \subsubsection{Central \texorpdfstring{$q\bar{q}$}{qqbar}} The final subleading process type in the Pure Jets case is Central $q\bar{q}$. In this process type, we have two currents scattering off of each other, but this time, via an effective vertex, which connects together two FKL chains. Each FKL chain t-channel gluon splits into a $q\bar{q}$ and this results in a quark and anti-quark in between the most forward and backward jets. One can see an example of such a process in Figure \ref{fig:qqbarcen_example}. \begin{figure}[ht] \centering \includegraphics[]{Cenqqbar_jx} \caption{Momentum labeling for a central $q\bar{q}$ process.} \label{fig:qqbarcen_example} \end{figure} As the new central $q\bar{q}$ piece contains the quark propagator, we will treat this as part of the skeleton process. This means that we do not impose strong ordering between the $q\bar{q}$-pair taking \begin{align} \label{eq:cenqqbarraporder} y_1 \ll y_q,y_{\bar{q}} \ll y_n. \end{align} The HEJ Matrix element for this process can be calculated as: \begin{align} \label{eq:Mcentral} i\mathcal{M} &= g_s^4 T^d_{1a} T^e_{nb}\ \frac{j_{\mu}(p_a,p_1)\ X^{ab\, \mu \nu}_{{\rm cen}}(p_q,p_{\bar{q}},q_1,q_3)\ j_{\nu}(p_b,p_n)}{t_{a1}t_{bn}}. \end{align} where $X^{\mu \nu}_{\rm cen}$ is given by: \begin{equation} \label{eq:Xcen} \begin{split} X^{\mu \nu}_{\rm cen} ={}&\frac{f^{ced}T^c_{q\bar{q}}}{s_{q\bar{q}}} \left(\eta^{\mu \nu} X_{sym}^\sigma + V^{\mu \nu \sigma}_{\bar{q}g} \right) \bar{u}(p_q) \gamma^\sigma u(p_{\bar{q}}) \\ & \qquad + \frac{i T^d_{qj}T^e_{j\bar{q}}}{(q_1-p_q)^2} X^{\mu\nu}_{\text{qprop}} - \frac{i T^e_{qj}T^d_{j\bar{q}}}{(q_1-p_{\bar{q}})^2} X^{\mu\nu}_{\text{crossed}}\,, \end{split} \end{equation} with \begin{align} \label{eq:Xsym} X_{sym}^\sigma ={}& q_1^2 \left( \frac{p_a^\sigma}{s_{aq} + s_{a\bar{q}}} + \frac{p_1^\sigma}{s_{1q} + s_{1\bar{q}}} \right) - q_3^2 \left( \frac{p_b^\sigma}{s_{bq} + s_{b\bar{q}}} + \frac{p_n^\sigma}{s_{nq} + s_{n\bar{q}}} \right)\,,\\ \label{eq:V3g} V_{3g}^{\mu\nu\sigma} ={}& (q_1 + p_q + p_{\bar{q}})^\nu \eta^{\mu\sigma} + (q_3 - p_q - p_{\bar{q}})^\mu \eta^{\nu\sigma} - (q_1 + q_3)^\sigma \eta^{\mu\nu}\,,\\ \label{eq:Xqprop} X^{\mu\nu}_{\text{qprop}} ={}& \frac{\langle p_q | \mu (q_1-p_q) \nu | p_{\bar{q}}\rangle}{(q_1-p_q)^2}\,,\\ \label{eq:Xcrossed} X^{\mu\nu}_{\text{crossed}} ={}& \frac{\langle p_q | \nu (q_1-p_{\bar{q}}) \mu | p_{\bar{q}}\rangle}{(q_1-p_{\bar{q}})^2}\,, \end{align} and $q_3 = q_1 - p_q - p_{\bar{q}}$. \todo[inline]{Document formula for current contraction, like equation~(\ref{eq:S_uno})} \subsection{Higgs} % They individual pieces are listed in section~\ref{sec:currents_impl} and the contraction is done in the current generator, section~\ref{sec:cur_gen}. \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{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 $p_H = q_1 - q_2$. This vertex is used in the current contractions for central and extremal Higgs boson emission: \begin{align} \label{eq:S_ff_fHf} \left\|S_{f_1 f_2\to f_1 H f_2}\right\|^2 ={}& \sum_{\substack{\lambda_a = +,-\\\lambda_b = +,-}} \left|j^{\lambda_a}_\mu(p_1, p_a)V_H^{\mu\nu}(q_j, q_{j+1})\ j^{\lambda_b}_\nu(p_n, p_b)\right|^2\,,\\ \label{eq:S_gf_Hf} \left\|S_{g f_2 \to H f_2}\right\|^2 ={}& \sum_{ \substack{ \lambda_{g} = +,-\\ \lambda_{b} = +,- }} \left|\epsilon_\mu^{\lambda_g}(p_g, p_r)\ V_H^{\mu\nu}(p_g, p_g-p_H)\ j_\nu^{\lambda_b}(p_n, p_b)\right|^2\,, \end{align} 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/Hjets.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} \subsection{Vector Boson + Jets} \label{sec:currents_W} \subsubsection{Quark+ Vector Boson} \begin{figure} \centering \begin{minipage}[b]{0.2\textwidth} \includegraphics[width=\textwidth]{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]{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]{Wbits3.pdf} \end{minipage} \caption{The $j_V$ current is constructed from the two diagrams which contribute to the emission of a vector boson from a given quark line.} \label{fig:jV} \end{figure} For a $W, Z$, or photon emission we require a fermion. The current is actually a sum of two separate contributions, see figure~\ref{fig:jV}, one with a vector boson emission from the initial state, and one with the vector boson emission from the final state. This can be seen as the following two terms, given for the example of a $W$ emission~\cite{Andersen:2012gk,Andersen:2020yax} \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 the type of $W$ emission (+ or -) will depend on the quark flavour, and that the handedness of the quark-line is given by whether its a quark or anti-quark. The coupling and propagator factor in eq.~(\ref{eq:Weffcur1}) have to be adapted depending on the emitted boson. The remaining product of currents \begin{equation} \label{eq:J_V} J_{\text{V}}^\mu(p_2,p_l,p_{\bar{l}},p_3)=\left( \frac{ \bar{u}_2 \gamma^\nu (\slashed{p}_2 + \slashed{p}_l + \slashed{p}_{\bar{l}}) \gamma^\mu u_3}{s_{2l\bar{l}}} - \frac{\bar u_2 \gamma^\mu(\slashed{p}_3 + \slashed{p}_l + \slashed{p}_{\bar{l}}) \gamma^\nu u_3}{s_{3l\bar{l}}} \right) [\bar{u}_l \gamma_\nu u_{\bar{l}}] \end{equation} with $s_{il\bar{l}} = (p_i + p_l +p_{\bar{l}})^2$ is universal. The implementation is in \texttt{include/currents.frm} inside the \texttt{current\_generator} (see section~\ref{sec:cur_gen}). To use it inside \FORM use the place-holder \lstinline!JV(h1, hl, mu, pa, p1, plbar, pl)!, where \lstinline!h1! is the helicity of the quark line and \lstinline!hl! the helicity of the lepton line. \subsubsection{Vector boson with unordered emission} \begin{figure} \centering \begin{subfigure}{0.45\textwidth} \centering \includegraphics{vuno1} \caption{} \label{fig:U1diags} \end{subfigure} \begin{subfigure}{0.45\textwidth} \centering \includegraphics{vuno2} \caption{} \label{fig:U2diags} \end{subfigure} \begin{subfigure}{0.45\textwidth} \centering \includegraphics{vuno3} \caption{} \label{fig:Cdiags} \end{subfigure} \begin{subfigure}{0.45\textwidth} \centering \includegraphics{vuno4} \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 vector boson are emitted from the same quark line and the gluon comes after the $t$-channel propagator. In (b), the gluon and vector 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 $b$--$3$ quark line.} \label{fig:Vunodiags} \end{figure} It is necessary to include subleading processes in vector boson + jets also. Similarly to the pure jet case, the unordered currents are not calculated separately, and only in the ME functions when required in the \texttt{src/Wjets.cc} or \texttt{src/Zjets.cc} file. The following shows the derivation of the calculation of this ME within HEJ. We start with a contraction of two currents: \begin{equation} \label{eq:SabsVuno} S_{qQ\to Vgq^\prime Q} = j_{V{\rm uno}\,\mu}^d(p_a,p_1,p_2,p_\ell,p_{\bar\ell})\ g^{\mu \nu}\ T^d_{3b}\ j^{h_b,h_3}_{\nu}(p_b,p_{3}), \end{equation} where $j_{V,{\rm uno}}$ is our new unordered current which is 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 (see figure~\ref{fig:Vunodiags}). 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 V,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}_V)\rho P_L |a\rangle }{s_{12}t_{aV}} + \frac{\langle 2|\nu (\slashed{p}_2+ \slashed{p}_1)\rho P_L (\slashed{p}_2+\slashed{p}_1 + \slashed{p}_V)\mu |a\rangle }{s_{12}s_{12V}} \\ &+ \frac{\langle 2|\rho P_L (\slashed{p}_2+ \slashed{p}_V) \nu (\slashed{p}_1 + \slashed{p}_2+\slashed{p}_V)\mu |a\rangle}{s_{2V}s_{12V}}\,, \end{split}\\ \label{eq:U2tensor} \begin{split} \tilde U_2^{\nu\mu\rho} ={}&\frac{\langle 2|\mu (\slashed{p}_a-\slashed{p}_V-\slashed{p}_1)\nu (\slashed{p}_a - \slashed{p}_V)\rho P_L |a\rangle }{t_{aV1}t_{aV}} + \frac{\langle 2|\mu (\slashed{p}_a-\slashed{p}_V- \slashed{p}_1)\rho P_L (\slashed{p}_a-\slashed{p}_1) \nu |a\rangle }{t_{a1V}t_{a1}} \\ &+ \frac{\langle 2|\rho P_L (\slashed{p}_2+ \slashed{p}_V) \mu (\slashed{p}_a-\slashed{p}_1)\nu |a\rangle}{s_{2V}t_{a1}}\,, \end{split}\\ \label{eq:Ltensor} \begin{split} \tilde L^{\nu\mu\rho} ={}& \frac{1}{t_{aV2}}\left[ \frac{\langle 2 | \sigma (\slashed{p}_a-\slashed{p}_V)\rho|a\rangle}{t_{aV}} +\frac{\langle 2 | \rho (\slashed{p}_2+\slashed{p}_V)\sigma|a\rangle}{s_{2V}} \right]\\ &\times \left\{\left(\frac{p_b^\nu}{s_{1b}} + \frac{p_3^\nu}{s_{13}}\right)(q_1-p_1)^2g^{\mu\sigma}+(2q_1-p_1)^\nu g^{\mu\sigma} - 2p_1^\mu g^{\nu\sigma} + (2p_1-q_1)^\sigma g^{\mu\nu} \right\}\,, \end{split} \end{align} where $s_{ij\dots} = (p_i + p_j + \dots)^2, t_{ij\dots} = (p_i - p_j - \dots)^2$ and $q_1 = p_a-p_2-p_V$. This is actually calculated and used in the code in a much cleaner way as follows: \begin{align}\label{eq:spinorstringVuno} S_{qQ\to Vgq^\prime Q} = i\varepsilon_\nu (p_g) \bar{u}^-(p_2)&\gamma_\rho\nu(p_{\bar{q}})\times T^d_{3b} \bar{u}^{h_3}(p_3)\gamma_\mu u^{h_b}(p_b) \times \nonumber \\ &\left( T^1_{2i}T^d_{ia} \left( \tilde{U}_1^{\nu\mu\rho}-\tilde{L}^{\nu\mu\rho}\right)+T^d_{2i}T^1_{ia}\left(\tilde{U}_2^{\nu\mu\rho}+\tilde{L}^{\nu\mu\rho}\right) \right) \end{align} If we define the objects: \begin{align}\label{eq:VunoX} X &= \varepsilon_\nu(p_g) \left[ \bar{u}^-(p_2)\gamma_\rho\nu(p_{\bar{q}})\right] \left[ \bar{u}^{h_3}(p_3)\gamma_\mu u^{h_b}(p_b)\right] \left( \tilde{U}_1^{\nu\mu\rho}-\tilde{L}^{\nu\mu\rho}\right)\\ Y &= \varepsilon_\nu(p_g) \left[ \bar{u}^-(p_2)\gamma_\rho\nu(p_{\bar{q}})\right] \left[ \bar{u}^{h_3}(p_3)\gamma_\mu u^{h_b}(p_b)\right] \left( \tilde{U}_2^{\nu\mu\rho}+\tilde{L}^{\nu\mu\rho}\right) - \label{eq:WunoY} + \label{eq:VunoY} \end{align} then we can rewrite Equation \eqref{eq:spinorstringVuno} in the much simpler form: \begin{equation} S_{qQ\to Vgq^\prime Q} = iT^d_{3b} \left( T^{1}_{2i}T^d_{ia} X + T^d_{2i}T^1_{ia} Y \right) \end{equation} then, by using: \begin{align} \sum_{\text{all indices}}& T^d_{3b}T^e_{b3}T^1_{2i}T^d_{ia}T^e_{ai}T^1_{i2} = \frac{1}{2}C_F^2C_A \\ \sum_{\text{all indices}}& T^d_{3b}T^e_{b3}T^1_{2i}T^d_{ia}T^1_{ai}T^e_{i2} = \frac{1}{2}C_F^2C_A - \frac{1}{4}C_A^2C_F = -\frac{1}{4}C_F \end{align} giving then, the spin summed and helicity averaged spinor string as: \begin{equation}\label{eq:VunoSumAveS} ||\;\bar{S}_{qQ\to Vgq^\prime Q}\;|| = \frac{1}{4N_C^2} \left( \frac{1}{2}C_F^2C_A\left(|X|^2+|Y|^2\right)-\frac{1}{4}C_F\times2\mathrm{Re}\left(XY^*\right)\right) \end{equation} - +In the case of $Z$ boson emission and if the two incoming particles are (anti-)quarks one has to take into account the possibility that the boson can be emitted from either the backward or the forward leg. The current contractions needed in the second, third and fourth lines in eq.~(\ref{eq:ME_Z}) are: +\begin{align} + ||\;\bar{S}_{\rm top}\;|| &= \frac{1}{4N_C^2} \left( \frac{1}{2}C_F^2C_A\left(|X_{\rm top}|^2+|Y_{\rm top}|^2\right)-\frac{1}{4}C_F\times2\mathrm{Re}\left(X_{\rm top} Y_{\rm top}^*\right)\right) \\ + ||\;\bar{S}_{\rm bot}\;|| &= \frac{1}{4N_C^2} \left( \frac{1}{2}C_F^2C_A\left(|X_{\rm bot}|^2+|Y_{\rm bot}|^2\right)-\frac{1}{4}C_F\times2\mathrm{Re}\left(X_{\rm bot} Y_{\rm bot}^*\right)\right) \\ + ||\;\bar{S}_{\rm int.}\;|| &= \frac{1}{4N_C^2} \left( \frac{1}{2}C_F^2C_A\left(X_{\rm top} X_{\rm bot}^* + Y_{\rm top} Y_{\rm bot}^*\right)-\frac{1}{4}C_F\left(X_{\rm top} Y_{\rm bot}^* + Y_{\rm top} X_{\rm bot}^*\right)\right) +\end{align} +where $X_{\rm top}, Y_{\rm top}$ correspond to the case where the boson is emitted from the unordered current~(\ref{eq:VunoX}, \ref{eq:VunoY}) while $X_{\rm bot}, Y_{\rm bot}$ correspond to boson emission from the quark leg opposite to the unordered current and are obtained by contracting an unordered current~(\ref{eq:juno}) with a boson emission current (see eq.(\ref{eq:Weffcur1}) with $W \to Z$). \subsubsection{\texorpdfstring{$W$}{W}+Extremal \texorpdfstring{$\mathbf{q\bar{q}}$}{qqbar}} \todo{Update when included in $Z$ + jets} The $W$+Jet sub-leading processes containing an extremal $q\bar{q}$ are related by crossing symmetry to the $W$+Jet unordered processes. This means that one can simply perform a crossing symmetry argument on eq.~\ref{eq:wunocurrent} to arrive at the extremal $q\bar{q}$ current required.We show the basic structure of the extremal $q\bar{q}$ current in figure~\ref{fig:qgimp}, neglecting the $W$-emission for simplicity. \begin{figure} \centering \includegraphics[width=0.3\textwidth]{{qqbarex_schem}} \caption{Schematic structure of the $gq \to \bar{Q}Qq$ amplitude in the limit $y_1 \sim y_2 \ll y_3$} \label{fig:qgimp} \end{figure} \begin{figure} \centering \begin{subfigure}{0.45\textwidth} \centering \includegraphics{qqbarex1} \end{subfigure} \begin{subfigure}{0.45\textwidth} \centering \includegraphics{qqbarex2} \end{subfigure} \begin{subfigure}{0.45\textwidth} \centering \includegraphics{qqbarex4} \end{subfigure} \begin{subfigure}{0.45\textwidth} \centering \includegraphics{qqbarex5} \end{subfigure} \begin{subfigure}{0.45\textwidth} \centering \includegraphics{qqbarex3} \end{subfigure} \caption{The five tree-level graphs which contribute to the process $gq \to \bar{Q}Qq$.} \label{fig:qg_qQQ_graphs} \end{figure} We can obtain the current for $g\rightarrow W q \bar{q}$ by evaluating the current for $W$ plus unordered emissions with the normal arguments $p_a \leftrightarrow -p_1 $ interchanged. This is a non-trivial statement: due to the minimality of the approximations made, the crossing symmetry normally present in the full amplitude may be extended to the factorised current. We must again note that swapping $p_a \leftrightarrow -p_1$ will lead to $u$-spinors with momenta with negative energy. These are identical to $v$-spinors with momenta with positive energy, up to an overall phase which is common to all terms, and can therefore be neglected. Mathematically, this is given by: \begin{align}\label{eq:crossedJ} j^\mu_{\rm W,g\to Q\bar{Q}}(p_a,p_1,p_2,p_\ell,p_{\bar{\ell}}) =i \varepsilon_{g\nu} \langle \ell | \rho | \bar \ell \rangle_L \left(T^1_{2i} T^d_{ia} (\tilde U_{1,X}^{\nu\mu\rho}-\tilde L^{\nu\mu\rho}_X) + T^d_{2i} T^1_{ia} (\tilde U_{2,X}^{\nu\mu\rho}+\tilde L_X^{\nu\mu\rho}) \right), \end{align} where the components are now given by \begin{align} \label{eq:U1tensorX} \begin{split} \tilde U_{1,X}^{\nu\mu\rho} =&\frac{\langle 2|\nu (\slashed{p}_a- \slashed{p}_2)\mu (\slashed{p}_1 + \slashed{p}_W)\rho P_L |1\rangle }{t_{a2}s_{1W}} + \frac{\langle 2|\nu (\slashed{p}_a- \slashed{p}_2)\rho P_L (\slashed{p}_a-\slashed{p}_2 - \slashed{p}_W)\mu |1\rangle }{t_{a2}t_{a2W}} \\ &- \frac{\langle 2|\rho P_L (\slashed{p}_2+ \slashed{p}_W) \nu (\slashed{p}_a - \slashed{p}_2-\slashed{p}_W)\mu |1\rangle}{s_{2W}t_{a2W}}\,, \end{split}\\ \label{eq:U2tensorX} \begin{split} \tilde U_{2,X}^{\nu\mu\rho} =&-\frac{\langle 2|\mu (\slashed{p}_a-\slashed{p}_W-\slashed{p}_1)\nu (\slashed{p}_1 + \slashed{p}_W)\rho P_L |1\rangle }{t_{aW1}s_{1W}} + \frac{\langle 2|\mu (\slashed{p}_a-\slashed{p}_W- \slashed{p}_1)\rho P_L (\slashed{p}_a-\slashed{p}_1) \nu |1\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 |1\rangle}{s_{2W}t_{a1}}\,, \end{split}\\ \label{eq:LtensorX} \begin{split} \tilde L^{\nu\mu\rho}_X &= \frac{1}{s_{W12}}\left[-\frac{\langle 2 |\sigma (\slashed{p}_1 + \slashed{p}_W) \rho P_L | 1\rangle}{s_{1W}} + \frac{\langle 2 |\rho P_L (\slashed{p}_2 + \slashed{p}_W) \sigma | 1\rangle }{s_{2W}} \right] \\ &\vphantom{+\frac{1}{t_{aW2}}}\quad\cdot \left( -\left( \frac{p_b^\nu}{s_{ab}} + \frac{p_n^\nu}{s_{an}} \right) (q_1+p_a)^2 g^{\sigma\mu}+ g^{\sigma \mu} (2q_1 +p_a)^\nu - g^{\mu \nu}(2p_a+q_1)^\sigma+ 2g^{\nu \sigma}p_a^\mu \right)\,, \end{split} \end{align} where $q_1=-(p_1+p_2+p_W)$. Notice in particular the similarity to the $W$+uno scenario (from which this has been derived). \subsubsection{Central \texorpdfstring{$\mathbf{q\bar{q}}$}{qqbar} Vertex} The final subleading process in the $W$+Jet case is the Central $q\bar{q}$ vertex. This subleading process does not require an altered current, but an effective vertex which is contracted with two regular \HEJ currents. This complexity is dealt with nicely by the \FORM inside the \texttt{current\_generator/j\_Wqqbar\_j.frm}, which is detailed in section~\ref{sec:contr_calc}. The $W$-emission can be from the central effective vertex (scenario dealt with by the function \lstinline!ME_WCenqqbar_qq! in the file \texttt{src/Wjets.cc}); or from either of the external quark legs (scenario dealt with by \lstinline!ME_W_Cenqqbar_qq! in same file). In the pure jets case, there are 7 separate diagrams which contribute to this, which can be seen in figure~\ref{fig:qq_qQQq_graphs}. In the $W$+Jets case, there are then 45 separate contributions. \begin{figure} \centering \begin{subfigure}{0.45\textwidth} \centering \includegraphics{qqbarcen1} \end{subfigure} \begin{subfigure}{0.45\textwidth} \centering \includegraphics{qqbarcen2} \end{subfigure} \begin{subfigure}{0.45\textwidth} \centering \includegraphics{qqbarcen3} \end{subfigure} \begin{subfigure}{0.45\textwidth} \centering \includegraphics{qqbarcen4} \end{subfigure} \begin{subfigure}{0.45\textwidth} \centering \includegraphics{qqbarcen5} \end{subfigure} \begin{subfigure}{0.45\textwidth} \centering \includegraphics{qqbarcen6} \end{subfigure} \begin{subfigure}{0.45\textwidth} \centering \includegraphics{qqbarcen7} \end{subfigure} \caption{All Feynman diagrams which contribute to $qq' \to qQ\bar{Q}q'$ at leading order.} \label{fig:qq_qQQq_graphs} \end{figure} The end result is of the effective vertex, after derivation, is: \begin{align} \label{eq:EffectiveVertexqqbar} \begin{split} V^{\mu\nu}_{\text{Eff}}=& - i \frac{C_1}{s_{23AB}}\left(X^{\mu\nu\sigma}_{1a}\hat{t_1} + X^{\mu\nu\sigma}_{4b}\hat{t_3} +V^{\mu\nu\sigma}_{3g}\right)J_{\text{V} \sigma}(p_2,p_A,p_B,p_3) \\ &\quad +iC_2X^{\mu\nu}_{Unc}+iC_3X^{\mu\nu}_{Cro}, \end{split} \end{align} where: \begin{align} \begin{split} C_1=&T^e_{1q}T^g_{qa}T^e_{23}T^g_{4b} - T^g_{1q}T^e_{qa}T^e_{23}T^g_{4b} = f^{egc}T^c_{1a}T^e_{23}T^g_{4b}, \\ C_2=&T^g_{1a}T^g_{2q}T^{g'}_{q3}T^{g'}_{4b} \\ C_3=&T^g_{1a}T^{g'}_{2q}T^g_{q3}T^{g'}_{4b} \end{split} \end{align} are the colour factors of different contributions and $J_\text{V}$ is given in equation~(\ref{eq:J_V}). The following tensor structures correspond to groupings of diagrams in figure~\ref{fig:qq_qQQq_graphs}. \begin{eqnarray} \label{eq:X_1a} X_{1a}^{\mu\nu\sigma} &= \frac{-g^{\mu\nu}}{s_{23AB}\hat{t_3}}\left(\frac{p^\sigma_a}{s_{a2} + s_{a3} + s_{aA} + s_{aB}} + \frac{p^\sigma_1}{s_{12} + s_{13} + s_{1A} + s_{1B}}\right) \\ \label{eq:X_4b} X_{4b}^{\mu\nu\sigma} &=\frac{g^{\mu\nu}}{s_{23AB}\hat{t_1}}\left(\frac{p^\sigma_b}{s_{b2} + s_{b3} + s_{bA} + s_{bB}}+ \frac{p^\sigma_4}{s_{42} + s_{43} + s_{4A} + s_{4B}}\right) \end{eqnarray} correspond to the first and second row of diagrams in figure~\ref{fig:qq_qQQq_graphs}. \begin{align} \label{eq:3GluonWEmit} \begin{split} V^{\mu\nu\sigma}_{3g}=\frac{1}{ \hat{t}_1s_{23AB}\,\hat{t}_3} \bigg[&\left(q_1+p_2+p_3+p_A+p_B\right)^\nu g^{\mu\sigma}+ \\ &\quad\left(q_3-p_2-p_3-p_A-p_B\right)^\mu g^{\sigma\nu}- \\ & \qquad\left(q_1+q_3\right)^\sigma g^{\mu\nu}\bigg]J_{\text{V} \sigma}(p_2,p_A,p_B,p_3) \end{split} \end{align} corresponds to the left diagram on the third row in figure~\ref{fig:qq_qQQq_graphs}. One notes that all of these contributions have the same colour factor, and as such we can group them together nicely before summing over helicities etc. As such, the function \lstinline!M_sym_W! returns a contraction of the above tensor containing the information from these 5 groupings of contributions (30 diagrams in total). It is available through the generated header \texttt{j\_Wqqbar\_j.hh} (see section~\ref{sec:cur_gen}). \begin{align} \label{eq:X_Unc} \begin{split} X^{\mu\nu}_{Unc}=\frac{\langle A|\sigma P_L|B\rangle}{\hat{t_1}\hat{t_3}} \bar{u}_2&\left[-\frac{ \gamma^\sigma P_L(\slashed{p}_2+\slashed{p}_A+\slashed{p}_B)\gamma^\mu (\slashed{q}_3+ \slashed{p}_3)\gamma^\nu}{(s_{2AB})(t_{unc_{2}})}\right.+ \\ &\qquad\left. \frac{\gamma^\mu(\slashed{q}_1-\slashed{p}_2)\gamma^\sigma P_L(\slashed{q}_3+\slashed{p}_3)\gamma^\nu}{(t_{unc_{1}})(t_{unc_{2}})}\right. + \\ &\qquad\qquad\left. \frac{\gamma^\mu(\slashed{q}_1-\slashed{p}_2)\gamma^\nu(\slashed{p}_3+\slashed{p}_A+\slashed{p}_B)\gamma^\sigma P_L }{(t_{unc_1})(s_{3AB})}\right]v_3 \end{split} \end{align} corresponds to the diagram on the right of row three in figure~\ref{fig:qq_qQQq_graphs}. This contribution to the current contraction can be obtained in the code with the function \lstinline!M_uncross_W!. \begin{align} \begin{split} \label{eq:X_Cro} X^{\mu\nu}_{Cro}=\frac{\langle A|\sigma P_L|B\rangle}{\hat{t_1}\hat{t_3}} \bar{u}_2&\left[-\frac{ \gamma^\nu(\slashed{q}_3+\slashed{p}_2)\gamma^\mu (\slashed{p}_3+\slashed{p}_A+\slashed{p}_B)\gamma^\sigma P_L}{(t_{cro_1})(s_{3AB})}\right.+ \\ &\qquad\left. \frac{\gamma^\nu(\slashed{q}_3+\slashed{p}_2)\gamma^\sigma P_L(\slashed{q}_1-\slashed{p}_3)\gamma^\mu}{(t_{cro_{1}})(t_{cro_{2}})}\right.+ \\ &\qquad\qquad\left . \frac{\gamma^\sigma P_L(\slashed{p}_2+\slashed{p}_A+\slashed{p}_B)\gamma^\nu(\slashed{q}_1-\slashed{p}_3)\gamma^\mu }{(s_{2AB})(t_{cro_2})}\right]v_3 \end{split} \end{align} corresponds to the last diagram in figure~\ref{fig:qq_qQQq_graphs}. This contribution to the current contraction can be obtained in the code with the function \lstinline!M_cross_W!. \subsubsection{$W^+ W^+$ + jets} The production of same-sign $WW$ + jets is implemented through the contraction of two vector boson currents; see eqn~\ref{eq:J_V} and more generally subsection~\ref{sec:currents_W}. This contraction is available in FORM under the header \texttt{jV\_jV.hh}, while the matrix elements can be found in \texttt{src/WWjets.cc}. There are two distinct contributions to this process which correspond to the possible pairings of the $W$-bosons and incoming legs. Labelling the bosons as \texttt{W1} and \texttt{W2}, the two configurations correspond to \texttt{W1}-forward with \texttt{W2}-backward, and \texttt{W2}-forward with \texttt{W1}-backward. The interference between these contributions is included. For same-flavour decays, \texttt{reconstruct\_intermediate()} will compute the difference between reconstruced mass and reference mass in possible pairings and select the pairing which minimises this quantity. %%% Local Variables: %%% mode: latex %%% TeX-master: "developer_manual" %%% End: