Collinear Limits in QCD from MHV Rules

TL;DR

The paper studies tree-level multi-parton collinear limits in QCD using CSW/MHV rules to express splitting amplitudes. It shows that in the multi-collinear limit only a subset of MHV diagrams contribute, with splitting coefficients that are purely holomorphic or anti-holomorphic, hinting at a twistor-space interpretation. The authors derive general formulas for splitting amplitudes with up to two negative helicities and arbitrary numbers of positive helicity partons, and provide explicit triple-collinear results. These results offer compact building blocks for higher-order QCD computations relevant to jet cross sections and infrared structure.

Abstract

We consider multi-parton collinear limits of QCD amplitudes at tree level. Using the MHV formalism we specify the underlying analytic structure of the resulting multi-collinear splitting functions. We derive general results for these splitting functions that are valid for specific numbers of negative helicity partons and an arbitrary number of positive helicity partons (or vice versa).

Figures (5)

• Figure 1: Factorisation of an $N$-point colour ordered amplitude with gluons $p_1,\ldots,p_n$ collinear into splitting function for $P \to 1, \ldots, n$ multiplied by an $(N-n+1)$-point amplitude.
• Figure 2: MHV topologies contributing to (a) $\mathrm{Split}_+(m_1)$ and (b) $\mathrm{Split}_-(m_1,m_2)$. Negative helicity particles are indicated by solid lines, while arbitrary numbers of positive helicity particles emitted from each vertex are shown as dotted arcs. All particles that are not in the collinear set must be emitted from the left-hand vertex.
• Figure 3: MHV topologies contributing to the two quark collinear limit of the type $\widetilde{\mathrm{split}}(1_q^-,\ldots,m^-,\ldots,n_{\bar{q}}^+\to P_\gamma^-)$. Quarks of type $Q$ ($q$) are shown as green(red)-dotdashed lines and negative helicity gluons as black solid lines. The negative helicity photon is shown as a blue dashed line.
• Figure 4: MHV topologies contributing to the four quark collinear limit of the type $\mathrm{split}(1^+,\ldots,s_{\bar{Q}}^\lambda, (s+1)_Q^{-\lambda}, \ldots t_{\bar{q}}^{\lambda^\prime}, (t+1)_q^{-\lambda^\prime}, \ldots, n^+ \to P^-)$. Quarks of type $Q$ ($q$) are shown as green(red)-dotdashed lines and negative helicity gluons as black solid lines.
• Figure 5: MHV topologies contributing to the four quark collinear limit of the type $\widetilde{\mathrm{split}}(1^+,\ldots,s_{\bar{Q}}^\lambda, (s+1)_q^{-\lambda^\prime}, \ldots t_{\bar{q}}^{\lambda^\prime}, (t+1)_Q^{-\lambda}, \ldots, n^+ \to P^-)$. Quarks of type $Q$ ($q$) are shown as green(red)-dotdashed lines and negative helicity gluons as black solid lines.