Non-Global Logarithms in QCD

• Non-global logarithms are defined as logarithmically enhanced contributions in QCD observables that measure only a subset of the full phase space, contrasting with inclusive Sudakov exponentiation.
• They arise from correlated soft emissions across measurement boundaries, and their resummation is addressed via evolution equations like the Banfi–Marchesini–Smye (BMS) equation in the large-Nc limit.
• Applications of non-global logarithms include jet mass distributions, rapidity-gap observables, and jet substructure techniques, where they significantly alter theoretical predictions and require sophisticated resummation methods.

Non-global logarithms (NGLs) are logarithmically enhanced contributions that arise when a QCD observable is sensitive only to radiation in part of phase space rather than inclusively over the whole event. Their defining mechanism is a correlated soft-emission history across a measurement boundary: radiation in an unconstrained or less-constrained region can radiate softer quanta into a constrained region, producing a real–virtual mismatch that is not captured by ordinary Sudakov exponentiation. They are a generic feature of jet masses, hemisphere observables, rapidity-gap and energy-flow measurements, isolation-cone cross sections, and jet-substructure procedures such as filtering (Banfi et al., 2021, Rubin, 2010, Balsiger et al., 2018).

1. Definition and conceptual scope

A non-global observable constrains radiation only in a restricted region of phase space. In that setting, emissions outside the measured region are not fully inclusive, so a soft gluon emitted in the allowed region can become a new color antenna and radiate an even softer gluon into the vetoed or measured region. This is the canonical origin of NGLs, and it is why they are tied to coherent soft wide-angle radiation rather than to purely collinear dynamics (Banfi et al., 2021, Caron-Huot, 2015).

The contrast with global logarithms is structural. For global observables, the measurement acts on all radiation, and soft/collinear singularities combine into the familiar Sudakov double logarithms, schematically $\alpha_s^n L^{2n}$. For observables sensitive only to soft wide-angle radiation, the leading tower is instead single-logarithmic, $\alpha_s^n L^n$, with next-to-leading terms $\alpha_s^n L^{n-1}$ (Banfi et al., 2021). In exclusive jet observables with several soft scales, the non-global terms appear as logarithms of ratios such as $\ln^2(\Lambda/m_{1,2})$ or $\ln^2(m_1/m_2)$, which are not determined by ordinary single-scale renormalization-group evolution (Hornig et al., 2011).

The term has also been broadened to include correlations induced by the measurement function itself. In recombination algorithms, clustering across a jet boundary can correlate soft phase space even in Abelian contributions. On that basis, clustering logarithms have been interpreted as a class of NGLs: they involve correlated soft emissions, depend on separated phase-space regions, arise from soft singularities, and are not governed by the soft anomalous dimension or resummed by standard RGE (Kelley et al., 2012). This places conventional non-Abelian NGLs and algorithm-induced clustering effects within a common restricted-acceptance framework.

2. Evolution equations and factorization frameworks

At leading logarithmic accuracy and in the planar large-$N_c$ limit, the standard evolution equation is the Banfi–Marchesini–Smye (BMS) equation. In one common form it reads

$$\partial_Y P_{ab}(Y) = -\int_{C_{\rm out}}\frac{d\Omega_c}{4\pi}\, w_{abc}\,P_{ab}(Y) + \int_{C_{\rm in}}\frac{d\Omega_c}{4\pi}\, w_{abc}\,\Big[P_{ac}(Y)P_{bc}(Y)-P_{ab}(Y)\Big],$$

with

$$w_{abc}=\frac{v_a\!\cdot v_b}{(v_a\!\cdot v_c)(v_c\!\cdot v_b)}.$$

The first term is a Sudakov-like source term for direct emission into the veto region, while the nonlinear term describes dipole branching inside the allowed region: a soft gluon splits the parent dipole into two daughter dipoles that then evolve independently at large $N_c$ (Hatta et al., 2017). In transverse-momentum-ordered form, the same evolution appears as a generating-functional equation whose real term creates daughter dipoles and whose virtual term enforces unitarity (Banfi et al., 2021).

Beyond the planar limit, a more general formulation is provided by the color density matrix

$\sigma[U],$

whose renormalization-group evolution is

$\alpha_s^n L^n$0

In this language, NGL resummation becomes evolution under a universal Hamiltonian $\alpha_s^n L^n$1. At one loop the Hamiltonian reproduces the all-$\alpha_s^n L^n$2 generalization of the leading NGL evolution; in the planar limit it reduces to BMS, while in the weak-field limit it becomes the BFKL/BJKP kernel (Caron-Huot, 2015).

Effective-theory factorizations organize the same physics in terms of towers of hard and soft functions. A representative formula is

$\alpha_s^n L^n$3

in which the hard functions describe energetic partons inside the allowed region and the soft functions are Wilson-line matrix elements with the veto imposed on the restricted region (Becher et al., 2023, Balsiger et al., 2018). In SCET$\alpha_s^n L^n$4 observables such as narrow jet broadening, this factorization must additionally accommodate recoil and rapidity separation, but the genuinely non-global content still resides in multi-Wilson-line hard/soft evolution rather than in the rapidity-anomaly sector (Becher et al., 2017).

3. Fixed-order structure, geometry, and jet algorithms

The first non-global contribution appears at $\alpha_s^n L^n$5 from correlated double-soft emission. In filtered Higgs-jet reconstruction, for example, the cumulative distribution is expanded as

$\alpha_s^n L^n$6

with the two-loop term decomposed as

$\alpha_s^n L^n$7

and

$\alpha_s^n L^n$8

The relevant configuration is the standard in–out mismatch: a harder soft gluon is retained by the measurement, while a softer one is emitted outside the retained region, so real and virtual graphs fail to cancel (Rubin, 2010).

In exclusive dijet cross sections with two jet masses and an out-of-jet veto, the leading double logarithms are organized into in–out and in–in sectors. For cone or anti-$\alpha_s^n L^n$9 jets one finds, for equal radii,

$\alpha_s^n L^{n-1}$0

together with an in–in coefficient

$\alpha_s^n L^{n-1}$1

These coefficients depend on jet radius and algorithm, and renormalization-group consistency imposes exact relations among the coefficients of the various region-by-region logarithms (Hornig et al., 2011).

Jet algorithms modify the measurement boundary in qualitatively different ways. In anti-$\alpha_s^n L^{n-1}$2, soft radiation sees an approximately rigid cone boundary, the measurement function factorizes, Abelian exponentiation is preserved, and clustering effects are power suppressed in the relevant limit (Kelley et al., 2012). In contrast, Cambridge/Aachen and $\alpha_s^n L^{n-1}$3 cluster soft gluons across the boundary, which violates Abelian exponentiation starting at two loops and generates clustering logarithms at NLL in the exponent (Kelley et al., 2012, Banfi et al., 2010). For the specific jet-mass distribution in $\alpha_s^n L^{n-1}$4 dijets, a recent four-loop analysis with C/A clustering found that C/A minimizes the impact of both Abelian and non-Abelian NGLs among anti-$\alpha_s^n L^{n-1}$5, $\alpha_s^n L^{n-1}$6, and C/A for that observable (Khelifa-Kerfa, 18 Nov 2025).

A notable substructure-specific feature is that the geometry dependence need not parallel the primary-emission sector. In filtered jets, the primary contribution for $\alpha_s^n L^{n-1}$7 has a collinear enhancement $\alpha_s^n L^{n-1}$8 at small $\alpha_s^n L^{n-1}$9, while the two-loop non-global coefficient

$\ln^2(\Lambda/m_{1,2})$0

has no collinear $\ln^2(\Lambda/m_{1,2})$1 enhancement. That separation of structures is central to the perturbative optimization of filtering (Rubin, 2010).

4. Beyond leading logarithms and beyond large $\ln^2(\Lambda/m_{1,2})$2

The resummation of NGLs beyond LL was a long-standing problem. For observables sensitive only to soft wide-angle radiation, a complete next-to-leading logarithmic framework in the planar large-$\ln^2(\Lambda/m_{1,2})$3 limit is now available in the form of nonlinear integro-differential equations

$\ln^2(\Lambda/m_{1,2})$4

where the NLL kernel contains one-loop real–virtual and two-loop virtual corrections, correlated double-real emission, and a subtraction of the first LL iteration to avoid double counting (Banfi et al., 2021). A four-dimensional finite formulation based on a massless projection

$\ln^2(\Lambda/m_{1,2})$5

renders the kernels point-by-point finite and suitable for numerical implementation (Banfi et al., 2021).

A complementary jet-calculus formulation recasts the same evolution in terms of generating functionals and provides the first complete NLL resummation for a non-global observable in the planar limit. Applied to the transverse-energy distribution in the interjet rapidity region of $\ln^2(\Lambda/m_{1,2})$6 dijets, the calculation found substantial NLL corrections and a significant reduction of perturbative scale uncertainties (Banfi et al., 2021). In a related SCET/RG implementation, the two-loop anomalous dimension is inserted once into a large-$\ln^2(\Lambda/m_{1,2})$7 shower evolution,

$\ln^2(\Lambda/m_{1,2})$8

yielding the first NLL-resummed predictions for gap-between-jets observables at hadron colliders (Becher et al., 2023).

Finite-color effects can also be treated explicitly. For hemisphere mass, NGLs have been computed analytically at finite $\ln^2(\Lambda/m_{1,2})$9 fully through four loops and partially at five loops at LL, with a structured exponentiation pattern and the first genuine finite-$\ln^2(m_1/m_2)$0 correction appearing at four loops (Khelifa-Kerfa et al., 2015). For hadron-collision rapidity gaps, leading NGLs have been resummed directly at the physical color number $\ln^2(m_1/m_2)$1 by stochastic Wilson-line evolution in SU(3), revealing nontrivial higher multipoles—quadrupoles, sextupoles, and eight-line correlators—while showing that the final gap-survival probabilities often remain close to large-$\ln^2(m_1/m_2)$2 predictions (Hatta et al., 2020). For $\ln^2(m_1/m_2)$3jet jet masses at hadron colliders, finite-$\ln^2(m_1/m_2)$4 coefficients are now known through four loops for anti-$\ln^2(m_1/m_2)$5 jets (Khelifa-Kerfa, 2024).

The standard LL BMS equation also becomes unstable in highly boosted or narrow-veto configurations, where anti-collinear emissions violate formation-time ordering and generate double collinear logarithms. In that regime one must use a collinearly improved BMS kernel,

$\ln^2(m_1/m_2)$6

which resums the tower of time-ordering-induced double logarithms (Hatta et al., 2017).

5. Asymptotics, symmetries, and relations to high-energy evolution

The large-logarithm asymptotics of the BMS equation are governed by a buffer-region dynamics. As the control variable grows, real emissions are increasingly suppressed near the jet boundary, leaving only shrinking collinear neighborhoods around the original hard legs; the rest of the active region becomes a “gluonic desert” (Neill, 2016). By matching a collinearly regulated BMS evolution to a BFKL-like collinear limit, the asymptotic form of the NGL distribution is found to be Gaussian: $\ln^2(m_1/m_2)$7 with the slowest decay obtained at

$\ln^2(m_1/m_2)$8

The leading decay width is largely geometry independent, depending only on whether the dipole configuration is in–in or in–out (Neill, 2016).

The BMS equation for hemisphere observables also exhibits a hidden $\ln^2(m_1/m_2)$9 symmetry after stereographic projection of the sphere to the Poincaré disk. This symmetry reduces the functional dependence of the large-$N_c$0 problem and makes analytic higher-loop calculations tractable. Using iterated-integral methods, symbols, coproducts, and classical and Goncharov polylogarithms, the leading hemisphere NGL coefficients were computed analytically through five loops at large $N_c$1, with uniform transcendentality at each order (Schwartz et al., 2014).

A deeper structural relation links NGL evolution to high-energy small-$N_c$2 evolution. The color-density-matrix Hamiltonian is conformally related to the Balitsky–JIMWLK/BFKL evolution equation under a stereographic map from detector angles to transverse coordinates. In conformal theories the Hamiltonians coincide, while in QCD the mismatch is proportional to the $N_c$3-function (Caron-Huot, 2015). The asymptotic Gaussian tail of NGLs then maps directly to the black-disc saturation limit of the Balitsky–Kovchegov equation, with the shrinking angular cutoff playing the role of a saturation scale (Neill, 2016).

6. Phenomenology, substructure, and current directions

NGLs are phenomenologically important whenever a veto or measurement acts only on restricted angular regions. In high-$N_c$4 jet shapes with fixed jet multiplicity, the resummed structure takes the form

$N_c$5

with the non-global factor producing numerically significant modifications to jet-mass distributions; anti-$N_c$6 is preferred in that setting because the rigid-cone behavior keeps clustering corrections under control (Banfi et al., 2010). For gap and isolation observables, naive exponentiation is often inadequate, especially when the vetoed region is small, since non-global terms are enhanced by their dependence on the veto-region size (Balsiger et al., 2018).

In boosted-Higgs substructure, the filtering procedure provides a concrete non-global measurement. After a mass-drop step identifies the two $N_c$7-subjets, the event is reclustered with

$N_c$8

and only the $N_c$9 hardest filtered subjets are retained. The reconstructed mass shift is controlled by soft radiation not retained by filtering, so the observable is explicitly non-global. Semi-analytic optimization balancing perturbative loss against UE/PU contamination leads to a saturation point in $$\partial_Y P_{ab}(Y) = -\int_{C_{\rm out}}\frac{d\Omega_c}{4\pi}\, w_{abc}\,P_{ab}(Y) + \int_{C_{\rm in}}\frac{d\Omega_c}{4\pi}\, w_{abc}\,\Big[P_{ac}(Y)P_{bc}(Y)-P_{ab}(Y)\Big],$$0 and suggests that $$\partial_Y P_{ab}(Y) = -\int_{C_{\rm out}}\frac{d\Omega_c}{4\pi}\, w_{abc}\,P_{ab}(Y) + \int_{C_{\rm in}}\frac{d\Omega_c}{4\pi}\, w_{abc}\,\Big[P_{ac}(Y)P_{bc}(Y)-P_{ab}(Y)\Big],$$1 or $$\partial_Y P_{ab}(Y) = -\int_{C_{\rm out}}\frac{d\Omega_c}{4\pi}\, w_{abc}\,P_{ab}(Y) + \int_{C_{\rm in}}\frac{d\Omega_c}{4\pi}\, w_{abc}\,\Big[P_{ac}(Y)P_{bc}(Y)-P_{ab}(Y)\Big],$$2 is preferred, while $$\partial_Y P_{ab}(Y) = -\int_{C_{\rm out}}\frac{d\Omega_c}{4\pi}\, w_{abc}\,P_{ab}(Y) + \int_{C_{\rm in}}\frac{d\Omega_c}{4\pi}\, w_{abc}\,\Big[P_{ac}(Y)P_{bc}(Y)-P_{ab}(Y)\Big],$$3 loses too much perturbative radiation and $$\partial_Y P_{ab}(Y) = -\int_{C_{\rm out}}\frac{d\Omega_c}{4\pi}\, w_{abc}\,P_{ab}(Y) + \int_{C_{\rm in}}\frac{d\Omega_c}{4\pi}\, w_{abc}\,\Big[P_{ac}(Y)P_{bc}(Y)-P_{ab}(Y)\Big],$$4 aggravates hadronisation and small-$$\partial_Y P_{ab}(Y) = -\int_{C_{\rm out}}\frac{d\Omega_c}{4\pi}\, w_{abc}\,P_{ab}(Y) + \int_{C_{\rm in}}\frac{d\Omega_c}{4\pi}\, w_{abc}\,\Big[P_{ac}(Y)P_{bc}(Y)-P_{ab}(Y)\Big],$$5 effects (Rubin, 2010).

Other observables separate the non-global sector from additional large-logarithm structures rather than eliminating it. Narrow jet broadening is non-global and recoil sensitive, but its rapidity logarithms are tied to the jet function through the collinear anomaly, so the leading non-global factor is the same as for the hemisphere soft function and light jet mass (Becher et al., 2017). For massive colored particles such as top quarks, the dead-cone effect suppresses both collinear radiation and the size of non-global corrections, and LL resummation improves the description of $$\partial_Y P_{ab}(Y) = -\int_{C_{\rm out}}\frac{d\Omega_c}{4\pi}\, w_{abc}\,P_{ab}(Y) + \int_{C_{\rm in}}\frac{d\Omega_c}{4\pi}\, w_{abc}\,\Big[P_{ac}(Y)P_{bc}(Y)-P_{ab}(Y)\Big],$$6 central-jet-veto data (Balsiger et al., 2020).

Current developments extend the domain of NGL phenomenology rather than closing it. Finite-$$\partial_Y P_{ab}(Y) = -\int_{C_{\rm out}}\frac{d\Omega_c}{4\pi}\, w_{abc}\,P_{ab}(Y) + \int_{C_{\rm in}}\frac{d\Omega_c}{4\pi}\, w_{abc}\,\Big[P_{ac}(Y)P_{bc}(Y)-P_{ab}(Y)\Big],$$7 four-loop results are now available for hadron-collider $$\partial_Y P_{ab}(Y) = -\int_{C_{\rm out}}\frac{d\Omega_c}{4\pi}\, w_{abc}\,P_{ab}(Y) + \int_{C_{\rm in}}\frac{d\Omega_c}{4\pi}\, w_{abc}\,\Big[P_{ac}(Y)P_{bc}(Y)-P_{ab}(Y)\Big],$$8jet jet masses (Khelifa-Kerfa, 2024). Flavor-sensitive observables introduce a distinct class of flavor-changing NGLs from soft $$\partial_Y P_{ab}(Y) = -\int_{C_{\rm out}}\frac{d\Omega_c}{4\pi}\, w_{abc}\,P_{ab}(Y) + \int_{C_{\rm in}}\frac{d\Omega_c}{4\pi}\, w_{abc}\,\Big[P_{ac}(Y)P_{bc}(Y)-P_{ab}(Y)\Big],$$9 emission split across a jet boundary; these terms make a naïve net-flavor jet definition infrared unsafe, motivating the subtractive jet flavor prescription in which the problematic soft logarithms are explicitly subtracted (Larkoski, 7 Oct 2025). This suggests that NGLs are not merely a correction to jet shapes or gap fractions, but a structural component of how restricted-phase-space observables are defined in perturbative QCD.