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

Ugo Giuseppe Aglietti; Giancarlo Ferrera

doi:10.1140/epjc/s10052-025-13954-z

2023 Impact factor 4.2

Particles and Fields

Open Access

Eur. Phys. J. C (2025) 85: 272
https://doi.org/10.1140/epjc/s10052-025-13954-z

Regular Article - Theoretical Physics

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

Ugo Giuseppe Aglietti¹ and Giancarlo Ferrera²^a

¹ Dipartimento di Fisica, Università di Roma “La Sapienza” and INFN, 00185, Rome, Sezione di Roma, Italy
² Dipartimento di Fisica, Università di Milano and INFN, 20133, Milan, Sezione di Milano, Italy

^a giancarlo.ferrera@mi.infn.it

Received: 13 December 2024
Accepted: 14 February 2025
Published online: 11 March 2025

Abstract

We consider heavy quark mass (m) effects in the energy–energy correlation function in $e^{+} e^{-} \mapsto hadrons$ $e^+e^-\mapsto \textrm{hadrons}$ at high energy Q, in the back-to-back (two-jet) region. In the ultra-relativistic limit, $Q ≫ m$ $Q \gg m$ , the QCD Sudakov form factor S(b) in impact parameter (b-)space reads: $\begin{matrix} log S (b) = & - \int_{m^{2}}^{Q^{2}} \frac{d k^{2}}{k^{2}} {log (\frac{Q^{2}}{k^{2}}) A [α_{S} (k^{2})]) \\ (+ B [α_{S} (k^{2})]} [1 - J_{0} (b k)] \\ - \int_{0}^{m^{2}} \frac{d k^{2}}{k^{2}} {log (\frac{Q^{2}}{m^{2}}) A [α_{S} (k^{2})]) \\ (+ D [α_{S} (k^{2})]} [1 - J_{0} (b k)] . \end{matrix}$ $\begin{aligned} \log S(b)= & - \int \limits _{m^2}^{Q^2} \frac{dk^2}{k^2} \left\{ \log \left( \frac{Q^2}{k^2}\right) \, A[\alpha _S(k^2)] \, \right. \\ & \left. + \, B\left[ \alpha _S\left( k^2\right) \right] \right\} \big [ 1 \, - \, J_0\left( b k\right) \big ] \\ & - \,\int \limits _0^{m^2} \frac{dk^2}{k^2} \left\{ \log \left( \frac{Q^2}{m^2}\right) \, A\left[ \alpha _S\left( k^2\right) \right] \, \right. \\ & \left. + \, D\left[ \alpha _S\left( k^2\right) \right] \right\} \big [ 1 \, - \, J_0\left( b k\right) \big ]. \end{aligned}$ The double-log function $A (α_{S})$ $A(\alpha _S)$ describes the effects of soft gluons, quasi-collinear to the original heavy quark-antiquark pair, while the single-log functions $B (α_{S})$ $B(\alpha _S)$ and $D (α_{S})$ $D(\alpha _S)$ 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 ( $k^{2} < m^{2}$ $$k^2<m^2$$ ) 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 $\sqrt{e} ≃ 1.64872$ $\sqrt{e} \simeq 1.64872$ . 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 $e^{+} e^{-}$ $$e^+e^-$$ annihilation, or to the resummation of the transverse momentum distributions in hadron collisions, is also briefly discussed.

© The Author(s) 2025

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