Rational Functions in MultiLeg QCD Amplitudes with Masses Giuseppe De Laurentis University of Edinburgh QCD Meets EW
CERN Find these slides at gdelaurentis.github.io/slides/qcd_meets_ew_feb2024 Introduction Color Ordered Amplitude Coefficients $\circ\,$ To obtain cross sections we have to compute amplitudes $$ \require{color} \require{amsmath} \hat{σ}_{n}=\frac{1}{2\hat{s}}\int d\Pi_{n-2}\;(2π)^4δ^4\big(\sum_{i=1}^n p_i\big)\;|\overline{\mathcal{A}(p_i,h_i,a_i,μ_F, μ_R)}|^2 $$ $\circ\,$ The gauge group dependence is fairly well understood, through color decompositions $$ \mathcal{A}(p_i,h_i,a_i,μ_F, μ_R) = \sum_{\sigma} \mathcal{C}(\sigma \circ a_i) \times A(\sigma \circ \{p_i, h_i\}, \mu_F, \mu_R) $$ $\circ\,$ So we can deal with either color-ordered amplitudes $$ \mathcal{A}(\lambda^\alpha, \tilde \lambda^{\,\dot\alpha}) = \sum_{\substack{\Gamma,\\ i \in M_\Gamma}} \frac{ \sum_{k=0}^{\text{finite}} \, {\color{red}c^{(k)}_{\,\Gamma, i}}(\lambda^\alpha, \tilde \lambda^{\,\dot\alpha}) \, \epsilon^k}{\prod_j (\epsilon - a_{ij})} \, I_{\Gamma,i}\left( (\lambda\tilde\lambda)^{\alpha\dot\alpha}, \epsilon\right) \;, \;\;\text{with} \quad a_{ij} \in \mathbb{Q} $$ $\phantom{\circ}\,$ or finite remainders (arguably better since they retain all the physical info) Catani ('98), Becher, Neubert ('09), Gardi, Magnea ('09) $$ \mathcal{R}^{(\ell-loop)} = \mathcal{A}^{(\ell)}_R - \sum_{i=1}^{\ell} I^{(\ell-i)} \mathcal{A}^{(i-1)} + \mathcal{O}(\epsilon) = \sum_i {\color{red}{r_{i}(\lambda,\tilde\lambda)}} \, h_i(\lambda\tilde\lambda) $$ Number of Indep.