The two-jet limit is where the EEC becomes most sensitive to soft and collinear gluon radiation. When the correlation angle chi approaches zero, meaning the two particles are nearly back-to-back, large logarithms of the angle proliferate at every order in the strong coupling alpha_S. Resummation techniques, built on a factorization formula involving a hard-virtual factor, a Sudakov form factor and a remainder function, tame these logarithms and yield reliable predictions. But when the quarks are heavy, a genuinely new physical scale, the quark mass, enters the problem, and the clean factorization structure of the massless theory is put under strain. In a previous study, two of the authors had already shown how the logarithmic mass effects, terms of the form alpha_S^n times powers of the logarithm of m squared over Q squared, can be resummed into a massive Sudakov form factor. The new work addresses the non-logarithmic mass effects, which live in the coefficient function and the remainder function instead.

The authors’ first move is conceptually elegant. At tree level, the EEC spectrum contains two delta-function peaks of equal strength, one at chi equals zero and one at chi equals pi, corresponding to the two back-to-back quarks. Beyond this Born approximation, the complete endpoint coefficients of these peaks receive contributions from both virtual corrections and subtle distributional pieces of the real-emission process, in which a gluon is radiated. Separating these endpoint contributions consistently is notoriously delicate. The team sidesteps the difficulty by defining a new observable: a partial EEC cumulant, obtained by normalizing the integrated EEC distribution at a fixed maximal angle chi_M, typically of order pi over two, rather than at pi. At first order in alpha_S, the back-to-back endpoint coefficient cancels exactly in the ratio between numerator and denominator, while the forward endpoint simply never enters. The result can therefore be computed from the real-emission distribution in four space-time dimensions, with no need to disentangle endpoint terms at all.

The authors are careful to stress that this partial cumulant is not a restricted event sample and does not impose a veto on radiation. The EEC is a pairwise correlation, and a single event can contribute through many pairs with angles on either side of chi_M. The new quantity is simply the ratio of two integrals of the same inclusive, infrared- and collinear-safe distribution, and for a fixed normalization angle away from the endpoints, the denominator introduces no new small scale. The logarithmically enhanced behavior as chi goes to zero is therefore still governed by the standard back-to-back Sudakov factor, with no additional non-global evolution to worry about.

With the observable in hand, the team computed the massive EEC spectrum to first order in alpha_S for the process in which an electron and a positron annihilate into a heavy quark–antiquark pair plus a gluon. A non-zero quark mass replaces the simple powers of parton energies familiar from the massless case with square roots, making the calculation substantially harder. The authors reduced the EEC cumulant to a two-dimensional integral over the quark and antiquark energies, performed the first integration analytically, and handled the second numerically, obtaining one-dimensional integral representations that depend parametrically on the correlation angle and on the mass parameter eta, defined as twice the quark mass divided by the hard scale Q. The quark–antiquark correlation proved especially intricate: solving the kinematic constraint equation after squaring produces two candidate solutions, and mapping out exactly where in the angle–energy plane each solution is genuine, spurious or complex required a careful numerical survey. The analysis even revealed a physically intuitive forbidden region: when the quark and antiquark are nearly collinear, the gluon recoils against the pair, and the quark energy can no longer reach its maximal value because the almost-stationary antiquark absorbs part of the available energy.

Validation came from comparison with the literature. The massive EEC function had been computed numerically in the 1980s, and the new results agree well with both earlier calculations, one performed at PETRA kinematics with a center-of-mass energy of 34 GeV and another at 30 GeV with a realistic flavor composition of bottom, charm and light quark pairs. In one comparison, a small overall shift of roughly two percent brings the curves into alignment, a discrepancy the authors attribute to ambiguities in the normalization used in the older work. In the other, agreement is excellent in both normalization and shape across the full angular range.

The deeper lesson of the paper concerns the factorization scheme itself. When the authors extended the standard massless factorization to the massive case and evaluated the remainder functions for mass parameters up to eta of 0.9, they found a strikingly simple pattern: the massive remainder function equals the massless one minus a mass-dependent correction of the form a_eta times one minus the exponential of minus b_eta times chi, valid up to surprisingly large masses of order eta equals 0.5. The coefficient a_eta is of order five, while b_eta scales roughly as one over eta. But this convenient structure hides a pathology. In the massless limit, the correction term does not vanish, so the massive remainder function jumps discontinuously away from the ab initio massless result. The root cause is that the massless limit and the two-jet limit do not commute: taking chi to zero first and then eta to zero gives zero, while the reverse order leaves a finite offset. Physically, a tiny quark mass should barely matter, so this discontinuity is an artifact of the scheme, not of nature.

The remedy is an improved factorization scheme with a smooth massless limit. The key observation is that both the massless remainder function and the mass-dependent correction vanish in the two-jet limit, so the offending term can be moved from the remainder function into the coefficient function without changing the first-order expansion of the resummed distribution. The price is that the new coefficient function acquires an explicit dependence on the correlation angle chi, a feature absent from conventional schemes. In the improved scheme, the remainder function is simply the massless one, all non-logarithmic mass effects are concentrated in the coefficient function, and the massless limit is continuous. Notably, a closely analogous non-commutativity problem and a similar angle-dependent solution had appeared a few years earlier in threshold resummation with massive quarks, suggesting a general pattern in how factorization handles mass scales.

The practical consequences are tangible. Because the improved coefficient function is enhanced in the two-jet region, the mass effects under study are expected to boost the predicted rate there, and the authors note that the new scheme enables phenomenological analyses of EEC data with genuinely improved control over heavy-quark mass effects, complementing their recent global fit that determined the strong coupling and non-perturbative parameters. The massive EEC, being a global, jet-algorithm-independent correlator, also offers a view of mass-regulated radiation, including the dead-cone effect, that differs from and complements heavy-flavor jet observables such as angularities and the Lund b-jet plane. Extending the calculation to second order in alpha_S, likely requiring substantial Monte Carlo work, is the natural next step. For an observable that has anchored precision QCD since the late 1970s, ensuring that heavy quarks are treated with the same rigor as massless ones is a quiet but essential advance in the quest to squeeze every percent of uncertainty out of the strong force.

Subject of Research: Perturbative QCD factorization and resummation of the energy–energy correlation function with heavy-quark mass effects in the two-jet limit

Article Title: Factorization of the energy–energy correlation in the two-jet limit in the massive case

Article References: Factorization of the energy–energy correlation in the two-jet limit in the massive case. (n.d.). https://doi.org/10.1140/epjc/s10052-026-16343-2

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16343-2

Keywords: QCD, energy-energy correlation, heavy quarks, factorization, resummation, Sudakov form factor, two-jet limit, perturbative calculations, electron-positron annihilation, event shapes, strong coupling, theoretical physics