Heavy quark mass effects in the energy–energy correlation in the back-to-back region

Abstract

We consider heavy quark mass (m) effects in the energy–energy correlation function in \(e^+e^-\mapsto \textrm{hadrons}\) at high energy Q, in the back-to-back (two-jet) region. In the ultra-relativistic limit, \(Q \gg m\), the QCD Sudakov form factor S(b) in impact parameter (b-)space reads:

$$\begin{aligned} \log S(b)= & - \int \limits _{m^2}^{Q^2} \frac{dk^2}{k^2} \left\{ \log \left( \frac{Q^2}{k^2}\right) \, A[\alpha _S(k^2)] \, \right. \\ & \left. + \, B\left[ \alpha _S\left( k^2\right) \right] \right\} \big [ 1 \, - \, J_0\left( b k\right) \big ] \\ & - \,\int \limits _0^{m^2} \frac{dk^2}{k^2} \left\{ \log \left( \frac{Q^2}{m^2}\right) \, A\left[ \alpha _S\left( k^2\right) \right] \, \right. \\ & \left. + \, D\left[ \alpha _S\left( k^2\right) \right] \right\} \big [ 1 \, - \, J_0\left( b k\right) \big ]. \end{aligned}$$

The double-log function \(A(\alpha _S)\) describes the effects of soft gluons, quasi-collinear to the original heavy quark-antiquark pair, while the single-log functions \(B(\alpha _S)\) and \(D(\alpha _S)\) describe hard collinear radiation and soft radiation not log-collinearly enhanced, respectively. The usual logarithmic expansion of the low transverse momenta contribution to the form factor (\(k^2<m^2\)) involves two new function series. An explicit evaluation of the heavy quark form factor at next-to-next-to-leading logarithmic accuracy is presented. The relation of our results with the well-known (classical) dead-cone effect in gauge theories is analyzed. At next-to-leading logarithmic (NLL) accuracy, an improved formula for the dead cone is proposed; the soft term produces an increase of the dead-cone opening angle of a factor \(\sqrt{e} \simeq 1.64872\). However, beyond NLL, a simple physical interpretation of this kind breaks down. The generalization of the above formula to the resummation of different shape variables in \(e^+e^-\) annihilation, or to the resummation of the transverse momentum distributions in hadron collisions, is also briefly discussed.