$q_T$-slicing with multiple jets

Abstract

Modern collider phenomenology requires unprecedented precision for the theoretical predictions, for which slicing techniques provide an essential tool at next-to-next-to-leading order (NNLO) in the strong coupling. The most popular slicing variable is based on the transverse momentum $q_T$ of a color-singlet final state, but its generalization to final states with jets is known to be very difficult. Here we propose two generalizations of $q_T$ that can be used for jet processes, providing proof of concept with an NLO slicing for $pp \to 2$ jets. We present factorization formulae that enable our approach to NNLO, calculate the NNLO collinear-soft function and demonstrate slicing at this order for $e^+e^- \to 2$ jets. One of these generalizations of $q_T$ only applies to planar Born processes, such as $pp \to 2$ jets, but offers a dramatic simplification of the soft function. We also discuss how our approach can directly be extended to obtain predictions for the fragmentation of hadrons. This presents a promising path for high-precision QCD calculations with multi-jet final states.

Figures (5)

• Figure 1: The $pp \to 2$ jet process. By using the WTA scheme, the transverse momentum perpendicular to the scattering plane $q_x$ (equal to $p_{x,2}$ in the above coordinates), or equivalently the azimuthal decorrelation $\delta \phi$, is a suitable slicing variable. The ingredients in the corresponding factorization are: the hard scattering (yellow), collinear initial- (purple) and final-state (blue) radiation, and soft radiation (pink).
• Figure 2: In the lower panel the NLO correction $\delta\sigma^{\rm NLO}$ (blue dots) obtained using the slicing is plotted as a function of the cut on the azimuthal angle $\delta\phi^{\rm cut}$, showing that this converges for small $\delta\phi^{\rm cut}$ to the correct result (red line) obtained from NLOJET++Nagy:2001xb. In the upper panel the individual terms (green dashed and yellow solid lines) in Eq. \ref{['eq:slicing']} are shown, of which the blue dots are the sum. Jets are defined using the anti-$k_T$ algorithm Cacciari:2008gp with radius $R = 0.5$, and subject to the following cuts on their transverse momentum and rapidity $p_{T,1}>100$ GeV, $p_{T,2}>80$ GeV, $|\eta_{1,2}|<2$. The renormalization and factorization scales are set to $\mu_{R,F} = 2p_{T,1}$. Note the different ranges of the vertical axis for the two panels, and that the numerical uncertainties are smaller than the size of the markers.
• Figure 3: Same as Fig. \ref{['fig:qx']} but using instead a cut on the total transverse momentum $q_T^{\rm cut}$ (with the WTA scheme) for the slicing. This converges faster than $\delta \phi^{\rm cut}$ at the expense of a more complicated soft function, and can also be extended to nonplanar Born processes. Note again the different ranges of the vertical axis for the two panels.
• Figure 4: A comparison of the precision of the slicing with $r_{\rm cut} = \delta\phi^{\rm cut}$ and $q_T^{\rm cut}/p_{T,1}$ for $pp \to 2$ jets with the same kinematics as in Figs. \ref{['fig:qx']} and \ref{['fig:qT']}. Zooming in on the region of small $r_{\rm cut}$, this clearly shows the faster convergence of $q_T$. The curves are obtained from fitting to $a\, r_{\rm cut} \ln r_{\rm cut} + b\, r_{\rm cut}$.
• Figure 5: In the lower panel the NNLO correction $\delta\sigma^{\rm NNLO}$ (blue dots with error bars) obtained using the slicing is plotted as a function of $\theta^{\rm cut}$. This converges for small $\theta^{\rm cut}$ to the correct result $K_2$ (red line), given in Eq. \ref{['eq:K12']}. In the upper panel the individual terms (green dashed and yellow solid curves) in Eq. \ref{['eq:slicing']} are shown, of which the blue dots are the sum.