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 $e^+e^- \to hadrons$ 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{eqnarray} \log S(b) &=& - \int_{m^2}^{Q^2} \frac{dk^2}{k^2} \left\{ \log\left(\frac{Q^2}{k^2}\right) A[\alpha_S(k^2)] + B[\alpha_S(k^2)]\right\} [1-J_0(b k)] \nonumber\\ && - \int_0^{m^2} \frac{dk^2}{k^2} \left\{ \log\left(\frac{Q^2}{m^2}\right) A[\alpha_S(k^2)] + D[\alpha_S(k^2)] \right\} [1-J_0(b k)]. \nonumber \end{eqnarray} The double-log function $A(\alpha_S)$ describes the effects of soft gluons, quasi-collinear to the original 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 physical interpretation of this kind breaks down. The generalization of the above formula to the resummation of shape variables in $e^+e^-$ annihilation, or to the resummation of the transverse momentum distributions in hadron collisions, is also briefly discussed.