Super-Leading Logarithms in Hadron Colliders

• Super-leading logarithms are logarithmically enhanced contributions that arise when a veto restricts radiation in hadron-collider events, leading to an extra logarithmic power.
• They originate from the interplay of non-global measurement constraints, initial-state collinear enhancement, and Coulomb/Glauber phase effects, breaking the usual real-virtual cancellation.
• Resummation requires careful treatment of color correlations and process-dependent dynamics, significantly impacting precision predictions in gaps-between-jets observables.

Super-leading logarithms are logarithmically enhanced contributions that arise in non-global hadron-collider observables when a veto restricts radiation only in part of phase space. In the standard examples, the hard scale $Q$ is much larger than the veto scale $Q_0$, and the large variable is $L=\ln(Q/Q_0)$. The defining feature is that, beginning with the first genuine double-logarithmic term at four loops, the perturbative expansion contains contributions with one more power of $L$ than expected from ordinary leading-logarithmic counting for non-global observables, for example $\alpha_s^4 L^5$ rather than $\alpha_s^4 L^4$. Their origin is the interplay of non-global measurement constraints, initial-state collinear radiation, and Coulomb/Glauber phases, which spoil naïve color coherence and obstruct the usual real-virtual cancellation (0808.1269, Becher et al., 2021, Becher et al., 2023).

1. Definition and physical setting

The canonical setting is the gaps-between-jets cross section. Two hard jets are produced at scale $Q$, and radiation above a lower scale $Q_0$ is vetoed in the rapidity interval between them. Because the veto is imposed only in part of phase space, the observable is non-global. Ordinary non-global logarithms already arise from the fact that radiation outside the measured region can itself radiate into the vetoed region. Super-leading logarithms are a distinct effect: in hadron collisions, phase factors in the amplitudes produce double-logarithmic corrections that are absent in the corresponding $e^+e^-$ problem (0808.1269, Becher et al., 2021).

The all-order structure derived for generic hadronic $2\to l$ or $Q_0$0 processes can be written schematically as

$Q_0$1

so that the contribution at order $Q_0$2 scales as

$Q_0$3

This is “super-leading” because, from four loops onward, it carries more powers of $Q_0$4 than the usual leading non-global terms. The nomenclature is not entirely uniform. Some analyses describe the whole series, including the three-loop $Q_0$5 precursor, as the SLL contribution because it has the same Glauber-phase origin, while reserving the phrase “first genuine double-logarithmic term” for the four-loop $Q_0$6 contribution (Becher et al., 2021, Becher et al., 2024).

A crucial restriction is that the effect is tied to colored initial states. The basic Glauber/Coulomb phase operator acts on the two incoming partons, so the phenomenon does not arise for analogous lepton-collider event shapes. The effect is also subleading in color and therefore absent in the large-$Q_0$7 approximation commonly used in early non-global-logarithm studies (0902.0477, Becher et al., 2023).

2. Fixed-order discovery in gaps-between-jets observables

The first explicit super-leading term was identified in the gaps-between-jets cross section by analyzing configurations with one gluon outside the gap, dressed by softer virtual corrections. In a color-basis-independent formulation, the leading nonzero contribution for quark-quark scattering was found to be

$Q_0$8

with analogous results for $Q_0$9 and $L=\ln(Q/Q_0)$0 (0808.1269). The structure

$L=\ln(Q/Q_0)$1

is the first super-leading logarithm in the original sense.

The fixed-order diagrammatic analysis was extended to fifth order in $L=\ln(Q/Q_0)$2, where the first two-gluons-outside contribution was obtained. For quark-quark scattering with gluon exchange, the fifth-order result includes

$L=\ln(Q/Q_0)$3

showing explicitly the anticipated $L=\ln(Q/Q_0)$4 pattern (0902.0477). The same analysis verified several cancellation statements that delimit the effect: out-of-gap contributions cancel through $L=\ln(Q/Q_0)$5, there is no two-gluons-outside contribution at $L=\ln(Q/Q_0)$6, and the failure of these cancellations begins only when Coulomb-phase effects and initial-state collinear enhancement are simultaneously present (0902.0477).

These early fixed-order results established two durable points. First, the effect is not an artifact of a special color basis or a specific partonic channel. Second, the logarithmic enhancement is inseparable from exact color correlations: the coefficients involve both leading and subleading powers of $L=\ln(Q/Q_0)$7, and the effect was missed by large-$L=\ln(Q/Q_0)$8 treatments of standard non-global logarithms (0808.1269, 0902.0477).

3. Origin: non-global vetoes, initial-state collinear enhancement, and Coulomb phases

The physical mechanism is a real-virtual mis-cancellation created by a vetoed region together with initial-state color exchange. The decisive kinematic region is one in which a gluon emitted outside the gap becomes collinear to an incoming parton. In that limit the rapidity integration extends to

$L=\ln(Q/Q_0)$9

so an apparently rapidity-type enhancement becomes an extra logarithm in the ordered $L$0 integrals. This is the step that promotes a nominal $L$1 contribution to an $L$2 one at fourth order (0902.0477).

In color space, the origin of the effect is the non-commutativity between the operator governing real Sudakov suppression across the gap and the Coulomb-phase operator. In the basis-independent derivation this appears as

$L$3

If these operators commuted, the real and virtual out-of-gap contributions would cancel. Because they do not, the mismatch survives and produces the first nonzero nested-commutator contribution involving one eikonal exchange and two Coulomb exchanges (0808.1269).

The SCET formulation sharpens the same mechanism. The one-loop anomalous dimension separates into a soft operator $L$4, a soft-collinear operator $L$5, and a Glauber/Coulomb phase operator $L$6. Without Glauber phases, the soft-collinear real and virtual terms cancel into ordinary DGLAP kernels. With Glauber phases, the color structure prevents this cancellation. The leading SLL tower is generated by operator strings of the form

$L$7

or, in the hard-function trace language,

$L$8

Two Glauber insertions are required because a physical cross section is real, and each additional $L$9 insertion contributes another double logarithm (Becher et al., 2023, Becher et al., 2021).

4. All-order structure and resummation

The modern formulation starts from a factorization theorem for non-global hadron-collider observables,

$\alpha_s^4 L^5$0

with hard functions $\alpha_s^4 L^5$1 evolved from a hard scale $\alpha_s^4 L^5$2 to a soft scale $\alpha_s^4 L^5$3 by an RG equation in multiplicity space (Becher et al., 2023). At leading double-logarithmic accuracy, the low-energy matrix elements can be taken at tree level, and the nontrivial structure resides in the hard evolution.

The all-order resummation showed that the SLL contribution is not Sudakov-exponential. For fixed coupling,

$\alpha_s^4 L^5$4

and the resummed partonic result can be written in terms of generalized hypergeometric or Kampé de Fériet functions, with natural expansion variable

$\alpha_s^4 L^5$5

Its asymptotic falloff is much weaker than a standard Sudakov form factor: instead of $\alpha_s^4 L^5$6, the leading terms scale as $\alpha_s^4 L^5$7 or $\alpha_s^4 L^5$8 (Becher et al., 2023). The series is alternating in sign, and individual fixed-order terms can be much larger than the resummed result, which is why resummation is essential (Becher et al., 2023).

A later RG-improved treatment reorganized the evolution operator so that all double-logarithmic corrections exponentiate from the outset into generalized Sudakov factors,

$\alpha_s^4 L^5$9

This makes consistent inclusion of the running coupling possible and yields the first leading-order RG-improved resummation for arbitrary $\alpha_s^4 L^4$0 scattering processes. It also shows that higher-order Glauber exchanges are parametrically suppressed, with the asymptotic scaling

$\alpha_s^4 L^4$1

where $\alpha_s^4 L^4$2 (Böer et al., 2024).

5. Process dependence and phenomenology

The numerical importance of super-leading logarithms is strongly process dependent. In the SCET analysis of arbitrary $\alpha_s^4 L^4$3 channels, $\alpha_s^4 L^4$4 and $\alpha_s^4 L^4$5 processes such as $\alpha_s^4 L^4$6 and $\alpha_s^4 L^4$7jet are structurally suppressed. In these channels, color-algebra sum rules force the first few perturbative orders to cancel: for $\alpha_s^4 L^4$8, the third- and fourth-order contributions vanish and the SLLs start only at five loops, while for $\alpha_s^4 L^4$9 they start at four loops but remain only a few percent at most in the examples studied (Becher et al., 2023). By contrast, $Q$0 dijet channels do not enjoy these cancellations, and the effect can be sizable (Becher et al., 2023).

A full hadronic analysis for

$Q$1

with a gap veto showed that the effect survives PDF convolution and the sum over all partonic channels. For $Q$2, $Q$3, $Q$4, $Q$5, and $Q$6, the total Born cross section was reported as

$Q$7

while the resummed SLL contribution was

$Q$8

corresponding to roughly

$Q$9

The dominant channels were $Q_0$0 and $Q_0$1, which together accounted for about $Q_0$2 of the Born cross section and about $Q_0$3 of the SLL correction (Becher et al., 2024).

The same study found that the relative correction grows as the veto scale decreases, reaching

$Q_0$4

at $Q_0$5, and that the isolated three-loop term alone would overestimate the total resummed SLL effect by about a factor of two (Becher et al., 2024). A further color-space refinement is that, although SLLs are generally suppressed at large $Q_0$6, some interference terms in partonic $Q_0$7 scattering are only linearly suppressed in $Q_0$8, not quadratically suppressed (Becher et al., 2024). This suggests that finite-$Q_0$9 interference can be more consequential than leading-color intuition would indicate.

6. Extensions, variants, and terminological boundaries

The super-leading-logarithm mechanism has been extended beyond the original massless gaps-between-jets problem. For one-jettiness in color-singlet plus jet production, the first coherence-violating contribution was found to scale as

$e^+e^-$0

relative to Born. Because this is one logarithm more dominant than the previously known $e^+e^-$1 pattern, it was termed a “super-super-leading logarithm.” The extra logarithm is not associated with additional poles; rather, it arises from a double-logarithmically large phase-space region created by the geometry of the one-jettiness veto, and the result remains consistent with universal PDFs evaluated at $e^+e^-$2 (Banfi et al., 14 Nov 2025).

Massive final states introduce a different extension. When heavy colored particles appear in the final state, the soft anomalous dimension acquires an additional Coulomb phase,

$e^+e^-$3

which provides a new source of SLLs. In $e^+e^-$4 production this new effect contributes only in the $e^+e^-$5 channel, is numerically comparable to the known Glauber SLLs in part of phase space, and near threshold requires an additional Sommerfeld-type resummation because $e^+e^-$6 as $e^+e^-$7 (Banerjee et al., 28 Oct 2025).

A separate terminological boundary concerns ordinary leading logarithms in effective field theories unrelated to non-global hadron-collider observables. The conference paper on anomalous processes in chiral perturbation theory discusses only ordinary leading chiral logarithms, defined as the highest power of the chiral logarithm at a given loop order, and explicitly does not address super-leading logarithms in the perturbative-QCD sense (Kampf, 2013). This distinction matters because the phrase “leading logarithm” is ubiquitous across field theory, whereas “super-leading logarithm” in modern usage refers to the veto-induced, coherence-violating, Glauber-sensitive structures described above.