Alaric Parton Shower Framework

• Alaric Parton Shower is a parton-shower framework defined by its matched soft–collinear kernel and global, non-local recoil strategy to achieve NLL accuracy.
• It employs unique kinematic mappings and ordering variables that preserve angular coherence and exact momentum conservation in both e+e- and hadron-collider setups.
• The framework supports recoil-safe NLO matching and MEPS merging, with validations against analytic resummation and collider data confirming its enhanced predictive power.

Alaric is a color-coherent parton-shower framework designed to achieve formal next-to-leading-logarithmic (NLL) accuracy for a class of global, infrared- and collinear-safe observables while exposing recoil effects away from the strict soft and collinear limits as a controlled algorithmic choice. Its defining ingredients are a matched soft–collinear kernel, a recoil strategy in which the recoil momentum is not tied to a local color spectator, and kinematic mappings constructed to be collinear-safe and NLL-safe. The framework was developed first for massless final-state evolution and then extended to hadron colliders, MC@NLO matching, multi-jet merging, and resonance- and width-aware evolution; a related Pythia implementation, Apollo, incorporates Alaric’s global transverse recoil into a partitioned dipole-antenna shower (Höche et al., 2024, Herren, 2024, Preuss, 2024).

1. Development, scope, and nomenclature

Alaric was introduced as a next-to-leading-logarithm accurate parton-shower algorithm implemented within SHERPA, with the explicit goal of providing formally NLL-accurate resummation for global observables while remaining usable in a general-purpose event generator. In the hadron-collider formulation, it is presented as a framework for studying recoil effects away from soft and collinear limits without changing the evolution variable or the splitting functions. The 2024 proceedings contribution describes it as supporting MC@NLO matching, and later work extends it to multi-jet merging and to resonance- and width-aware NLO matching (Höche et al., 2024, Herren, 2024, Höche et al., 30 Jul 2025, Höche et al., 15 Apr 2026).

The literature distinguishes Alaric from Apollo. Alaric is the shower framework implemented in SHERPA, whereas Apollo is a new final-state shower in Pythia that combines “central aspects of the Vincia antenna shower with the global transverse-recoil scheme of the Alaric framework” (Preuss, 2024). The proceedings paper also notes that the acronym “ALARIC” is not expanded in the cited works (Herren, 2024).

Paper	Main scope	Implementation
“The Alaric parton shower for hadron colliders” (Höche et al., 2024)	Hadron-collider algorithm, recoil systematics, MEPS@LO	SHERPA
“ALARIC: A NLL accurate Parton Shower algorithm” (Herren, 2024)	NLL concepts, y23 tests, LEP comparisons	SHERPA
“A partitioned dipole-antenna shower with improved transverse recoil” (Preuss, 2024)	Alaric recoil embedded in a dipole-antenna shower	Pythia
“Recoil-Safe Subtraction, Matching and Merging in e+e- to hadrons” (Höche et al., 30 Jul 2025)	Recoil-safe S-MC@NLO and MEPS@NLO up to five jets	Alaric/SHERPA
“Resonance- and Width-aware Parton Shower Evolution and NLO Matching” (Höche et al., 15 Apr 2026)	Resonance- and width-aware showering near $t\bar t$ threshold	ALARIC/SHERPA

A recurrent theme across these developments is that recoil is not treated as a secondary kinematic detail. Instead, it is part of the formal accuracy statement. This differentiates Alaric from local-recoil dipole or antenna showers whose recoil organization can induce long-range correlations that spoil NLL resummation for global observables (Preuss, 2024).

2. Soft–collinear construction and splitting kernels

The soft limit is organized through color-correlated eikonal radiation. In the proceedings summary, the soft factor for emission of gluon $j$ from legs $i$ and $k$ is written as

$$w_{ik,j}=\frac{p_i\!\cdot p_k}{(p_i\!\cdot p_j)(p_j\!\cdot p_k)}\,,$$

and then decomposed into an angular radiator and the emitted-gluon energy,

$$w_{ik,j}= \frac{W_{ik,j}}{E_j^2}, \qquad W_{ik,j}=\frac{1-\cos\theta_{ik}}{(1-\cos\theta_{ij})(1-\cos\theta_{jk})}\,.$$

Alaric performs a positive partial-fraction partition,

$$W_{ik,j}= \tilde W^i_{ik,j}+\tilde W^k_{ki,j}, \qquad \tilde W^i_{ik,j}= \frac{1-\cos\theta_{ik}}{(1-\cos\theta_{ij})(2-\cos\theta_{ij}-\cos\theta_{jk})},$$

so that the soft term can be associated probabilistically with a specific emitter while preserving angular correlations and populating the full soft phase space (Herren, 2024).

The hadron-collider formulation emphasizes the same logic in a slightly different language. The soft eikonal is partially fractioned to match collinear kernels, and the matched branching kernel replaces the eikonal term by a sum over color spectators plus a purely collinear remainder. This explicit soft–collinear matching, together with splitting kernels evaluated with light-cone fractions determined by the chosen kinematics, is stated to reproduce the exact leading soft and collinear limits. A further distinctive feature is the non-trivial azimuthal dependence of the splitting functions even without explicit spin correlations, allowing the shower to reproduce the complete one-loop soft radiation pattern without angular ordering (Höche et al., 2024).

In the Apollo reformulation, needed to combine Alaric recoil with antenna factorization, the single-soft eikonal is partitioned with an auxiliary reference vector $n_j$. The partitioned kernels are “attributed to a specific emitter,” while the full antenna is recovered by symmetrization. For example, the global reference vector is chosen as

$$n_j^\mu = K^\mu + p_j^\mu, \qquad K^\mu = -\sum_{a\in \mathrm{FS}} p_a^\mu = \sum_{a\in \mathrm{IS}} p_a^\mu ,$$

and the collinear momentum fraction is defined as

$z=\frac{p_i\!\cdot n_j}{(p_i+p_j)\!\cdot n_j}\,.$

The paper states that a “naive” choice $j$0 reproduces Catani–Seymour partitioning but, if combined with local recoil, fails NLL; Apollo instead uses the global $j$1 choice above (Preuss, 2024).

A common misconception is that Alaric is merely a new partition of the eikonal. The papers do not support that reduction. The soft partition, the purely collinear remainder, the azimuthal structure, the evolution variable, and the recoil mapping are treated as a coupled construction. The NLL statements rely on that combination rather than on any single kernel component taken in isolation (Höche et al., 2024, Herren, 2024).

3. Recoil strategy, kinematic mappings, and ordering variables

The recoil strategy is the defining structural feature of Alaric. In the hadron-collider implementation, one picks an emitter $j$2 and a freely chosen recoil momentum $j$3 that defines a hard scale; a color spectator $j$4 sets the azimuthal reference but does not receive recoil by construction. For soft radiation, the shower variables are

$j$5

with mapping

$j$6

$j$7

$j$8

and

$j$9

All momenta constituting $i$0 are then Lorentz transformed by

$i$1

so that momentum conservation is exact and the recoil is distributed globally (Höche et al., 2024).

The proceedings summary characterizes the same idea as “multipole” recoil. Rather than recoiling against the color spectator, Alaric recoils against the total momentum of all other particles, preserves the direction and magnitude of the color spectator, preserves the direction of the emitter, and conserves the invariant mass of the recoiling system. These properties are identified as central to the analytic proof of NLL accuracy because they induce only a mild boost on previously emitted particles and stabilize the angular structure of subsequent emissions (Herren, 2024).

The Pythia/Apollo formulation sharpens the recoil discussion by contrasting Alaric with local-recoil dipole or antenna showers ordered in a $i$2-like variable. In local schemes, the emitted transverse momentum is absorbed by the ends of the emitting dipole or antenna, which induces long-range correlations between emissions on the “back” of the Lund plane and earlier emissions. Alaric instead keeps $i$3 ordering but moves to global transverse recoil: the emitted transverse momentum is balanced by a cumulative recoil of the whole final state or the incoming system, while the emitter recoils only longitudinally. The paper states that this yields formal consistency with NLL for both global and non-global observables (Preuss, 2024).

Alaric has been formulated with more than one ordering variable. The hadron-collider paper discusses an original “$i$4-frame” variant and a new default “$i$5-frame” variant,

$i$6

with the $i$7-frame chosen because it simplifies the setting of upper bounds. Apollo uses

$i$8

described as the collinear-limit counterpart of the Lund $i$9, and states that this choice is global, monotonic along strongly ordered cascades, and preserves soft coherence with the global recoil (Höche et al., 2024, Preuss, 2024).

4. Sudakov evolution, subtraction, fixed-order matching, and merging

Alaric is organized as a standard shower evolution in which matched branching kernels are exponentiated into Sudakov no-emission probabilities. The proceedings formulation writes the Sudakov schematically as

$k$0

while the Apollo formulation gives the dipole-antenna version

$k$1

with $k$2 the partitioned kernels and with the phase space defined by the shower mapping itself (Herren, 2024, Preuss, 2024).

For hadron-collider multi-jet merging, the SHERPA implementation uses MEPS@LO following four steps: a jet veto partitions matrix-element and shower regions, fixed-order events are assigned shower-like histories by clustering, $k$3-reweighting is applied with branchwise scales, and Sudakov reweighting is implemented via pseudo-showers. Because Alaric uses distinct soft and collinear kinematics, multiple histories can exist for the same final state with different underlying Born kinematics, and the merging procedure is constructed to handle this explicitly (Höche et al., 2024).

The 2025 e$k$4e$k$5 study extends Alaric to recoil-safe S-MC@NLO and MEPS@NLO. Its defining statement is that the Monte Carlo counterterm $k$6 is built from the same kernels and, crucially, the same mapping and recoil distribution as the shower, so that $k$7 is locally finite and the integrated counterterm $k$8 matches the Sudakov exponent. The matched cross section is written as

$k$9

and this construction is then embedded in MEPS@NLO merging for $$w_{ik,j}=\frac{p_i\!\cdot p_k}{(p_i\!\cdot p_j)(p_j\!\cdot p_k)}\,,$$0 hadrons with 2-, 3-, 4-, and 5-jet processes at NLO (Höche et al., 30 Jul 2025).

Apollo adopts a parallel but distinct strategy in Pythia. It outlines a unitary matrix-element correction for the first branching and two multiplicative NLO matching schemes, one based on color-ordered projectors and one on Born-local subtraction terms in FKS-like sectors. Both are argued to preserve the logarithmic accuracy of the shower because the first emission is corrected to the exact real matrix element while subsequent emissions use the same ordering variable and Alaric recoil mapping, eliminating a mismatch between hardest-emission generation and the shower (Preuss, 2024).

The practical importance of these constructions is not only fixed-order accuracy. In the Alaric papers, matching and merging are explicitly tied to recoil consistency. This is why the 2025 work describes its subtraction as “recoil-safe”: the local counterterms and the shower are intended to agree not just in singular limits but also in the kinematic realization of recoil (Höche et al., 30 Jul 2025).

5. Validation against analytic resummation, LEP data, and hadron-collider observables

The clearest analytic validation reported for Alaric is the comparison to known NLL resummation in the $$w_{ik,j}=\frac{p_i\!\cdot p_k}{(p_i\!\cdot p_j)(p_j\!\cdot p_k)}\,,$$1 limit. The proceedings contribution states that the leading Lund-plane declustering scale $$w_{ik,j}=\frac{p_i\!\cdot p_k}{(p_i\!\cdot p_j)(p_j\!\cdot p_k)}\,,$$2 in the Cambridge algorithm converges to the analytic NLL result for Alaric, while DIRE does not. The same paper reports good agreement with JADE/OPAL data for $$w_{ik,j}=\frac{p_i\!\cdot p_k}{(p_i\!\cdot p_j)(p_j\!\cdot p_k)}\,,$$3, with the hadron-level difference to DIRE described as small even though only Alaric shows formal NLL convergence (Herren, 2024).

The Apollo study provides a broader NLL validation suite. It reports that the algorithm passes the $$w_{ik,j}=\frac{p_i\!\cdot p_k}{(p_i\!\cdot p_j)(p_j\!\cdot p_k)}\,,$$4 test, meaning a flat distribution in the small-$$w_{ik,j}=\frac{p_i\!\cdot p_k}{(p_i\!\cdot p_j)(p_j\!\cdot p_k)}\,,$$5 limit, and presents extensive comparisons for thrust, heavy-jet mass, jet broadenings, Cambridge $$w_{ik,j}=\frac{p_i\!\cdot p_k}{(p_i\!\cdot p_j)(p_j\!\cdot p_k)}\,,$$6, and fractional energy correlators. In these tests Apollo is reported to tend to the NLL predictions as $$w_{ik,j}=\frac{p_i\!\cdot p_k}{(p_i\!\cdot p_j)(p_j\!\cdot p_k)}\,,$$7, with ratios approaching unity across the resummation region (Preuss, 2024).

At hadron colliders, the SHERPA implementation was assessed in Drell–Yan and QCD jet production. For Drell–Yan lepton-pair production with ME+PS merging up to 3 jets, the paper reports agreement within $$w_{ik,j}=\frac{p_i\!\cdot p_k}{(p_i\!\cdot p_j)(p_j\!\cdot p_k)}\,,$$8–10% for $$w_{ik,j}=\frac{p_i\!\cdot p_k}{(p_i\!\cdot p_j)(p_j\!\cdot p_k)}\,,$$9 and $$w_{ik,j}= \frac{W_{ik,j}}{E_j^2}, \qquad W_{ik,j}=\frac{1-\cos\theta_{ik}}{(1-\cos\theta_{ij})(1-\cos\theta_{jk})}\,.$$0 across rapidities, with mild $$w_{ik,j}= \frac{W_{ik,j}}{E_j^2}, \qquad W_{ik,j}=\frac{1-\cos\theta_{ik}}{(1-\cos\theta_{ij})(1-\cos\theta_{jk})}\,.$$1 sensitivity. Variations of the recoil choice and of the evolution frame were used to quantify systematic effects: choosing $$w_{ik,j}= \frac{W_{ik,j}}{E_j^2}, \qquad W_{ik,j}=\frac{1-\cos\theta_{ik}}{(1-\cos\theta_{ij})(1-\cos\theta_{jk})}\,.$$2 improves the mid-range $$w_{ik,j}= \frac{W_{ik,j}}{E_j^2}, \qquad W_{ik,j}=\frac{1-\cos\theta_{ik}}{(1-\cos\theta_{ij})(1-\cos\theta_{jk})}\,.$$3 description but degrades high-$$w_{ik,j}= \frac{W_{ik,j}}{E_j^2}, \qquad W_{ik,j}=\frac{1-\cos\theta_{ik}}{(1-\cos\theta_{ij})(1-\cos\theta_{jk})}\,.$$4 tails and boosted decorrelations, while replacing the exact mapping-induced momentum fractions $$w_{ik,j}= \frac{W_{ik,j}}{E_j^2}, \qquad W_{ik,j}=\frac{1-\cos\theta_{ik}}{(1-\cos\theta_{ij})(1-\cos\theta_{jk})}\,.$$5 by the shower $$w_{ik,j}= \frac{W_{ik,j}}{E_j^2}, \qquad W_{ik,j}=\frac{1-\cos\theta_{ik}}{(1-\cos\theta_{ij})(1-\cos\theta_{jk})}\,.$$6, or switching between $$w_{ik,j}= \frac{W_{ik,j}}{E_j^2}, \qquad W_{ik,j}=\frac{1-\cos\theta_{ik}}{(1-\cos\theta_{ij})(1-\cos\theta_{jk})}\,.$$7 and $$w_{ik,j}= \frac{W_{ik,j}}{E_j^2}, \qquad W_{ik,j}=\frac{1-\cos\theta_{ik}}{(1-\cos\theta_{ij})(1-\cos\theta_{jk})}\,.$$8, has small impact (Höche et al., 2024).

The same paper reports that inclusive jet $$w_{ik,j}= \frac{W_{ik,j}}{E_j^2}, \qquad W_{ik,j}=\frac{1-\cos\theta_{ik}}{(1-\cos\theta_{ij})(1-\cos\theta_{jk})}\,.$$9 spectra at 13 TeV up to 2 TeV and $$W_{ik,j}= \tilde W^i_{ik,j}+\tilde W^k_{ki,j}, \qquad \tilde W^i_{ik,j}= \frac{1-\cos\theta_{ik}}{(1-\cos\theta_{ij})(2-\cos\theta_{ij}-\cos\theta_{jk})},$$0 show very good agreement, that $$W_{ik,j}= \tilde W^i_{ik,j}+\tilde W^k_{ki,j}, \qquad \tilde W^i_{ik,j}= \frac{1-\cos\theta_{ik}}{(1-\cos\theta_{ij})(2-\cos\theta_{ij}-\cos\theta_{jk})},$$1 versus $$W_{ik,j}= \tilde W^i_{ik,j}+\tilde W^k_{ki,j}, \qquad \tilde W^i_{ik,j}= \frac{1-\cos\theta_{ik}}{(1-\cos\theta_{ij})(2-\cos\theta_{ij}-\cos\theta_{jk})},$$2 or $$W_{ik,j}= \tilde W^i_{ik,j}+\tilde W^k_{ki,j}, \qquad \tilde W^i_{ik,j}= \frac{1-\cos\theta_{ik}}{(1-\cos\theta_{ij})(2-\cos\theta_{ij}-\cos\theta_{jk})},$$3 shows excellent agreement with ATLAS and CMS, and that gap fractions in dijets are excellent overall with a slight excess at the smallest $$W_{ik,j}= \tilde W^i_{ik,j}+\tilde W^k_{ki,j}, \qquad \tilde W^i_{ik,j}= \frac{1-\cos\theta_{ik}}{(1-\cos\theta_{ij})(2-\cos\theta_{ij}-\cos\theta_{jk})},$$4. It also notes that the Les Houches angularity distributions are slightly narrower than data but comparable to earlier SHERPA studies (Höche et al., 2024).

In e$$W_{ik,j}= \tilde W^i_{ik,j}+\tilde W^k_{ki,j}, \qquad \tilde W^i_{ik,j}= \frac{1-\cos\theta_{ik}}{(1-\cos\theta_{ij})(2-\cos\theta_{ij}-\cos\theta_{jk})},$$5e$$W_{ik,j}= \tilde W^i_{ik,j}+\tilde W^k_{ki,j}, \qquad \tilde W^i_{ik,j}= \frac{1-\cos\theta_{ik}}{(1-\cos\theta_{ij})(2-\cos\theta_{ij}-\cos\theta_{jk})},$$6 phenomenology, the 2025 recoil-safe matching paper reports that Alaric+MEPS@NLO at the $$W_{ik,j}= \tilde W^i_{ik,j}+\tilde W^k_{ki,j}, \qquad \tilde W^i_{ik,j}= \frac{1-\cos\theta_{ik}}{(1-\cos\theta_{ij})(2-\cos\theta_{ij}-\cos\theta_{jk})},$$7 pole describes thrust, $$W_{ik,j}= \tilde W^i_{ik,j}+\tilde W^k_{ki,j}, \qquad \tilde W^i_{ik,j}= \frac{1-\cos\theta_{ik}}{(1-\cos\theta_{ij})(2-\cos\theta_{ij}-\cos\theta_{jk})},$$8-parameter, hemisphere broadenings, and sphericity with excellent agreement to LEP data, and that NLO merging halves, or better, the perturbative scale uncertainty in the bulk of distributions relative to LO merging. It also reports less than 10% variations at MEPS@NLO transition points under $$W_{ik,j}= \tilde W^i_{ik,j}+\tilde W^k_{ki,j}, \qquad \tilde W^i_{ik,j}= \frac{1-\cos\theta_{ik}}{(1-\cos\theta_{ij})(2-\cos\theta_{ij}-\cos\theta_{jk})},$$9 variation (Höche et al., 30 Jul 2025).

6. Extensions, misconceptions, and present limitations

Alaric’s formal NLL claims are not universal. The proceedings summary states the guaranteed NLL control for global, IRC-safe event shapes in $n_j$0 annihilation, while also noting that the shower can generate non-global radiation patterns dynamically because it populates the full soft phase space with angular correlations. This does not amount to a claim of systematic beyond-leading control for non-global logarithms, and the paper explicitly says that such control is outside the “guaranteed NLL” set (Herren, 2024).

A second misconception is that Alaric’s accuracy claim is independent of color and spin approximations. The hadron-collider paper says the shower is constructed to fulfill stringent NLL criteria at leading color for recursively infrared safe observables, and the Apollo paper states that to claim full NLL accuracy “across the board,” inclusion of spin correlations and systematic subleading-color effects is needed. Both papers therefore treat the present implementations as formally constrained but still approximated in subleading color and spin structure (Höche et al., 2024, Preuss, 2024).

The original scope was massless final-state evolution, but the framework has already been extended. The 2026 resonance- and width-aware study preserves resonance virtualities by assigning recoil from a decay product to the remainder of the same decay tree and modifies the scalar soft kernel with a width-dependent interpolation factor,

$n_j$1

so that finite-width effects enter both the shower and the subtraction. Near the $n_j$2 threshold in $n_j$3, this width-aware matched shower is reported to avoid the artificial high-mass tails seen with default or purely resonance-aware showers, while the inclusive NLO rate remains scheme independent at the sub-permille level (Höche et al., 15 Apr 2026).

Open limitations are stated plainly across the papers. The hadron-collider implementation has no dedicated Alaric-specific tune yet and exhibits residual recoil-systematic effects in boosted or high-$n_j$4 regions. The Apollo paper lists current scope as massless final-state showers and identifies extensions to initial-state radiation and massive partons as nontrivial because they must preserve NLL safety. The same paper also notes that integrated kernels for all purely collinear pieces in one of its NLO matching schemes remain incomplete, and that global recoil can change intermediate invariant masses, although the common boost to the recoil rest frame preserves total momentum and on-shellness (Höche et al., 2024, Preuss, 2024).

Taken together, these papers define Alaric as a shower framework in which soft coherence, recoil organization, and fixed-order interfaces are designed as a single structure. In SHERPA, that structure has been carried from massless final-state NLL proofs to hadron-collider applications, recoil-safe NLO matching, MEPS@NLO merging up to five jets, and resonance- and width-aware evolution. In Pythia, the same recoil concept underlies Apollo’s partitioned dipole-antenna construction. The resulting picture is not of a single isolated algorithmic trick, but of a coherent shower design in which the treatment of recoil is part of the formal accuracy statement itself (Herren, 2024, Höche et al., 30 Jul 2025, Preuss, 2024).