Dipole Parton Shower: Gluon-Recoil Scheme

• The dipole parton shower gluon-recoil scheme is a method for redistributing momentum between an emitter and its color-connected partner, ensuring exact local four-momentum conservation.
• It employs dipole variables and specific splitting kernels to handle momentum mappings and energy sharing in both final–final and initial–final dipoles.
• The scheme underpins NLL accuracy in Monte Carlo generators by coordinating local and global recoil strategies and matching detailed QCD matrix element structures.

The dipole parton shower gluon-recoil scheme defines the algorithmic treatment of recoil induced by gluon emissions in parton showers that use a dipole or dipole–antenna description of QCD radiation. These schemes govern how momentum and energy are reassigned among partons during a $2\to3$ branching, particularly focusing on how the recoil from gluon emission is distributed between the emitter and its color-connected partner (the recoiler). The gluon-recoil prescription underpins the formal accuracy of parton showers, particularly in reproducing the soft and collinear singularity structure of QCD at the logarithmic accuracy relevant for modern collider simulations.

1. Foundational Principles and Formalism

The dipole gluon-recoil scheme builds on the large-$N_c$ factorization properties of QCD, where radiative emissions naturally associate with color-connected pairs ("dipoles"). For a final–final (FF) dipole with partons $a$ (radiator) and $r$ (recoiler), the emission $a\to b+c$ is handled within the dipole subsystem. The key kinematic step reshuffles momentum as: $$p_{a^*}=p_a+\frac{Q^2}{m_{ar}^2}p_r, \qquad p_{r'}=\left(1-\frac{Q^2}{m_{ar}^2}\right)p_r,$$ where $Q^2$ is the virtuality of the splitting and $m_{ar}^2=(p_a+p_r)^2$ is the dipole invariant mass. This strictly conserves four-momentum locally within the dipole—only the energies, not the directions, of $a$ and $r$ are modified in the dipole rest frame. In the permutation of emissions from both ends (as in the full three-body phase space),

$\frac{d\pT^2}{\pT^2}\,\frac{dz}{1-z} \oplus \frac{d\pT^2}{\pT^2}\,\frac{dz}{z} = \frac{dx_1\,dx_2}{(1-x_1)(1-x_2)},$

ensuring recovery of the full soft and collinear emission measure (Cabouat et al., 2017).

For initial–final (IF) dipoles (initial-state shower), recoil from a resolved emission is strictly absorbed by a unique color partner in the final state, not the entire final-state particle system, facilitating local four-momentum conservation at each branching. This treatment contrasts with the older "global-recoil" approach, where the entire final state absorbs the recoil via a Lorentz transformation (Cabouat et al., 2017).

2. Splitting Kernels and Phase-Space Structure

In the dipole scheme, emission kernels are constructed in terms of dipole variables, e.g.,

$$P_{q\to qg}(z) = C_F\;\frac{1+z^2}{1-z},\qquad P_{g\to gg}(z) = C_A\;\frac{1+z^3}{1-z},$$

with associated differential emission measures

$d\mathcal P_{q\to qg} = C_F\,\frac{\alpha_s}{2\pi}\frac{d\pT^2_{\rm evol}}{\pT^2_{\rm evol}}\frac{1+z^2}{1-z}dz\, d\varphi.$

Global consistency with three-body QCD matrix elements is achieved by summing over permutations of the emitting ends. For gluon-induced splitting (e.g., $g\to gg$), the scheme typically assigns recoil to a unique dipole end, with kernels adapting to the color structure and the need to interpolate between $C_F$ and $C_A/2$ limits via a color-weight correction factor (Cabouat et al., 2017).

3. Momentum Mapping and Kinematic Recoil Assignment

Momentum reconstruction within dipole showers is a critical procedural detail. The momentum of newly produced partons and the recoiler after splitting is determined by energy-sharing variables ($z$) and the azimuthal angle ($\varphi$), such that in the dipole rest frame,

$E_b = z E_{a^*},\quad E_c = (1-z)E_{a^*},$

with vector momenta back-to-back, followed by boosts/rotations to the event frame. In the initial–final case, the mapping is performed in the rest frame of the incoming–final recoil pair, and the explicit mappings guarantee exact four-momentum conservation and on-shellness for all partons,

$$\hat{p}'_c=(1/z-1)\hat{p}_b+\hat{p}_{\rm shift},\quad \hat{p}'_f=\hat{p}_f-\hat{p}_{\rm shift},$$

with the transverse momentum squared of the emission determined by dipole kinematics (Cabouat et al., 2017).

4. Global vs. Local Recoil Strategies and Logarithmic Accuracy

The choice between local recoil (recoil absorbed only by the dipole partner) and global recoil (recoil absorbed by the full final state) impacts the logarithmic accuracy of the parton shower. Local-dipole recoil, as used in the core PYTHIA scheme, ensures local momentum conservation and facilitates straightforward matching to matrix elements for first emissions, but can spoil next-to-leading-log (NLL) accuracy due to incorrect treatment of soft, wide-angle radiation (Preuss, 28 Mar 2024).

Global recoil schemes, such as in Alaric and Apollo (Preuss, 28 Mar 2024) or the "consistent dipole shower" (Forshaw et al., 2020), implement a global Lorentz boost and possibly uniform momentum rescaling after each emission, distributing recoil across the entire ensemble of final-state particles. This is constructed to preserve the correct soft wide-angle radiation pattern and avoid correlations among distant emissions that violate NLL accuracy. Analytically, this ensures the preservation of all single- and double-logarithmic structures, as demonstrated in event-shape observables such as thrust and angular decorrelations (Preuss, 28 Mar 2024, Forshaw et al., 2020).

5. Implementation Algorithms in Monte Carlo Generators

The local-dipole plus gluon-recoil scheme is implemented in modern event generators such as PYTHIA, enabling a modular approach to the interleaved evolution of multiple-parton interactions (MPI), initial-state radiation (ISR), and final-state radiation (FSR):

1. Start evolution at the hard-process scale.
2. Generate trial emissions and MPI at candidate scales.
3. Select the highest scale; identify FSR/ISR/MPI.
4. For an IF dipole emission, perform the backward evolution of the incoming, apply PDF and color-weight corrections, choose an azimuth, reconstruct momenta as per the rest-frame mapping.
5. Repeat until the $p_\perp$ cutoff is reached (Cabouat et al., 2017).

In the DIRE hybrid approach (Höche et al., 2015), a physically motivated evolution variable symmetric between emitter and recoiler is used, interpolating smoothly between dipole and traditional parton shower limits. Handling of negative weights arising from splitting functions away from soft/collinear limits is achieved by a weighted veto algorithm.

6. Theoretical Implications, Advantages, and Limitations

Key theoretical advantages of the dipole gluon-recoil approach include:

• Exact local four-momentum conservation within each dipole, leading to theoretically robust and Lorentz-invariant kinematics (Cabouat et al., 2017).
• Complete soft/collinear singularity coverage via a single IF kernel in deep inelastic scattering (DIS) and similar processes, with direct recovery of the first-order QCD matrix element structure.
• Automatic treatment of azimuthal correlations through the kinematical boost procedure, without recourse to explicit coherence models (Cabouat et al., 2017).
• NLL accuracy in event shapes and multi-jet observables, provided a global recoil map is employed (Preuss, 28 Mar 2024, Forshaw et al., 2020).

Practical limitations include:

• Inadequacies for gluon-splitting ($g\to q\bar q$) channels, which require special handling owing to more complex recoil assignment (Cabouat et al., 2017).
• Impact of gluon polarization and azimuthal asymmetries, which can interfere with color coherence and, in some cases, do not universally improve angular distributions compared to global-recoil schemes.
• Necessity for color-factor correction weights in mixed $qg$ dipoles to interpolate between different Casimir limits (Cabouat et al., 2017).
• No automatic improvements for regions away from soft and collinear limits, where splitting functions may become negative and must be managed with signed-weight algorithms (Höche et al., 2015).

Parameter choices, including the infrared cutoff ($p_{\perp\min}\sim 1$ GeV), $\alpha_s$ scale, and color weights, are set to align with both theoretical requirements and empirical tuning, though additional retuning may be required to absorb multiplicity or other small spectrum shifts (Cabouat et al., 2017).

7. Contemporary Schemes and Numerical Performance

Recent developments such as the Apollo partitioned dipole-antenna recoil scheme (Preuss, 28 Mar 2024) integrate partitioned antenna functions with global Lorentz-boosted recoil, restoring full i↔k symmetry and facilitating matching to fixed-order calculations and matrix-element corrections. The key to achieving NLL accuracy is that the global mapping of recoil prevents hard emissions from distorting the kinematics of subsequent large-angle soft gluons. Numerical studies show strong agreement with analytic NLL predictions and data, outperforming local-dipole algorithms in the resummation region for multi-jet event shapes (Preuss, 28 Mar 2024, Forshaw et al., 2020). The DIRE “midpoint” hybrid shower (Höche et al., 2015) and the rigorous consistency studies of (Forshaw et al., 2020) further establish that the dipole gluon-recoil prescription is central to the resummation properties of modern Monte Carlo parton showers, especially where global recoil and color-coherence preservation are critical.