Logarithmically-accurate showers with massive quarks

This paper presents a formulation of PanScales final-state showers that incorporate massive quarks to achieve next-to-leading logarithmic accuracy while preserving the original accuracy for mass-irrelevant observables, with its validity confirmed through fixed-order tests, all-order resummed comparisons, and phenomenological studies using LEP data.

The Gist

Imagine the universe's fundamental particles as a chaotic, high-speed dance floor. In this dance, heavy particles like bottom and charm quarks are like dancers wearing heavy, clunky boots, while lighter particles are like dancers in ballet slippers.

For decades, physicists have used computer simulations called "parton showers" to predict how these dancers move and interact. However, most of these simulations treated everyone as if they were wearing ballet slippers, ignoring the fact that the heavy dancers move differently. They missed a crucial rule of the dance floor: the "Dead Cone."

The Dead Cone: A Personal Space Bubble

When a heavy dancer (a massive quark) spins, they can't swing their arms (emit energy) as freely as a light dancer. Because they are heavy, they create a personal space bubble around them where no one else can dance too closely. This is the "dead cone." If a simulation ignores this, it predicts too much energy being radiated right next to the heavy dancer, leading to wrong predictions about how the dance floor looks.

The New Solution: PanScales Showers

The authors of this paper have built a new, smarter simulation called PanScales. Think of it as upgrading the dance floor's rulebook to include the heavy dancers' boots.

They didn't just add a simple "no-entry" sign for the dead cone. They rewrote the physics of the dance to ensure that:

1. Heavy dancers keep their energy: Because they can't shed energy as easily in their personal bubble, they retain more of their original speed.
2. Light dancers still dance normally: The new rules don't mess up the predictions for the light dancers; they only change things where the heavy boots matter.
3. The math is perfect: They proved their new rules work by checking them against the most precise mathematical formulas available (like checking a recipe against a master chef's exact measurements).

How They Tested It

To make sure their new simulation wasn't just a guess, they ran three types of tests:

• The "Two-Step" Test (Fixed-Order): They simulated a dance with exactly two extra moves and compared it to the exact mathematical answer. Their simulation matched the math perfectly, even when the heavy dancers were involved.
• The "Crowd Flow" Test (All-Order): They looked at the overall shape of the crowd (the "Lund Tree"). They checked if the simulation correctly predicted how many sub-groups formed in the crowd. Again, the new simulation got it right, capturing the subtle ways heavy mass changes the flow.
• The "Real World" Check (Phenomenology): They compared their simulation to real data from the LEP collider (a famous particle accelerator from the past). They looked at how bottom quarks break apart into other particles.
  • The Result: The old simulations (ignoring mass) predicted the bottom quarks would slow down and break apart too early. The new PanScales simulation, which respects the heavy boots, matched the real-world data much better.

The Bottom Line

This paper introduces a new way to simulate particle collisions that finally treats heavy particles with the respect they deserve. It fixes a long-standing blind spot in physics simulations by correctly modeling the "dead cone" effect.

The authors have made this new code public, allowing other scientists to use it to better understand the heavy particles that make up our universe, ensuring that when we look at the data from massive colliders like the LHC, we aren't looking at a distorted picture caused by ignoring the weight of the dancers.

Technical Summary

Technical Summary: Logarithmically-Accurate Showers with Massive Quarks

1. Problem Statement

Heavy quarks (bottom and charm) are produced abundantly in collider experiments like the LHC and are central to understanding the Higgs boson and top quark physics. However, modeling Quantum Chromodynamics (QCD) radiation in jets containing massive partons presents specific challenges compared to the massless case. The most prominent feature is the "dead-cone" effect, where collinear emissions are suppressed within a cone of size $m_Q/E$ around the heavy quark direction.

From a perturbative QCD perspective, the presence of a mass scale introduces new logarithmically-enhanced terms (e.g., $\ln(m_Q/Q)$) in the expansion. While resummation of these logarithms has been achieved for specific observables (such as fragmentation functions) at Next-to-Leading Logarithmic (NLL) and even Next-to-Next-to-Leading Logarithmic (NNLL) accuracy, a general framework for parton showers that systematically captures these mass effects across diverse classes of observables while preserving the original accuracy of massless showers has been missing. Existing showers often incorporate mass effects only at Leading Logarithmic (LL) accuracy or fail to maintain NLL accuracy for global event shapes when masses are relevant.

2. Methodology

The authors formulate a new class of final-state parton showers, named PanScales, which account for quark masses and achieve NLL accuracy. The framework includes two variants:

• PanLocal: A dipole-based shower with fully local momentum conservation.
• PanGlobal: An antenna-like shower with global transverse-momentum conservation.

2.1 Kinematic Maps and Lightlike Vectors

To handle massive partons while maintaining a Sudakov decomposition structure, the authors introduce lightlike reference vectors ($\bar{n}_i, \bar{n}_j$) derived from the pre-branching massive momenta ($\bar{p}_i, \bar{p}_j$). These vectors are defined such that in the dipole rest frame, their three-momenta match those of the massive legs, differing only in energy components. This choice ensures:

• The coefficients relating the lightlike vectors to the massive ones vanish in the massless limit.
• The direction of the massive particles is preserved, facilitating a meaningful definition of the quasi-collinear momentum fraction $z$.
• Improved numerical stability for logarithmic accuracy tests.

2.2 Emission Probabilities

The differential emission probability is constructed to reproduce the exact QCD matrix elements in two critical limits:

1. Soft Limit: The shower must reproduce the massive eikonal factor, which includes the dead-cone suppression terms proportional to $m^2/(p \cdot k)^2$.
2. Quasi-Collinear Limit: The shower must reproduce the massive DGLAP splitting functions ($P_{Q \to Qg}$ and $P_{g \to Q\bar{Q}}$).

The authors implement these limits by modifying the emission probability factors $D_i$ and $D_j$ (and their PanGlobal counterparts $\bar{D}_i, \bar{D}_j$) to include mass-dependent terms that suppress radiation in the dead-cone region.

2.3 Effective Coupling and Flavor Thresholds

The effective strong coupling $\alpha_s^{\text{eff}}$ is defined in a variable-flavor-number scheme. The authors implement a specific prescription for crossing heavy-quark thresholds:

• Continuity of the effective coupling is enforced at a scale $\mu^{(n_f)}$ slightly shifted from the physical mass $m_Q$ due to the CMW correction ($K_{CMW}$).
• The shift is derived as $\ln \mu^{(n_f)} = \ln m_Q + 5/6$ at $\mathcal{O}(\alpha_s^2)$, ensuring the correct NLL behavior of the running coupling.

2.4 Preservation of Massless Accuracy

A key requirement is that the massive showers must not degrade the accuracy of the original massless formulations.

• PanGlobal retains NNLL accuracy for global observables.
• PanLocal retains Next-to-Next-to-Double Logarithmic (NNDL) accuracy.
  This is achieved by:
• Implementing next-to-leading order multiplicative matching using massless matrix elements, but replacing the shower matrix element with the massive one in the dead-cone region.
• Evaluating NNLL Sudakov corrections using a variable number of flavors ($n_f$).
• Handling double-soft matrix-element corrections by ensuring they reduce to the massless result when masses are negligible and are vetoed below the mass threshold for $g \to Q\bar{Q}$ channels.

2.5 Spin and Color Correlations

• Spin: The Collins-Knowles algorithm is adapted to include mass-dependent helicity channels (e.g., allowing helicity flips in $Q \to Qg$ and equal helicities in $g \to Q\bar{Q}$).
• Color: Subleading color effects are handled using the Nested Ordered Double-Soft (NODS) method, though the authors note that subleading corrections are not fully accounted for within the dead-cone region.

3. Validation and Results

The authors perform a comprehensive suite of tests to validate the logarithmic accuracy of the showers.

3.1 Fixed-Order Tests ($\mathcal{O}(\alpha_s^2)$)

• Phase-Space Contours: The authors verify that a second emission does not alter the kinematics of a prior emission in a way that breaks NLL accuracy. They demonstrate that while the Dire-v1 shower fails this test in the dead-cone region, PanScales showers (both PanLocal and PanGlobal) satisfy the requirement, with deviations in the deep collinear region being power-suppressed ($1/k_t^4$).
• Matrix Element Comparison: Differential comparisons between the shower weights and exact QCD matrix elements (generated via MadGraph) for processes like $e^+e^- \to Q\bar{Q}g$ and $e^+e^- \to Q\bar{Q}g_1g_2$ show excellent agreement in the bulk of phase space and near the dead-cone boundary. Deviations are observed only in regions dominated by power corrections ($k_t^2/Q^2$) or where double-soft corrections are missing (which are $\mathcal{O}(N_c^{-1})$).
• Spin Correlations: Tests of the $a_2/a_0$ spin correlation ratios for sequential splittings confirm perfect agreement with analytic predictions for both massless and massive cases.

3.2 All-Order Logarithmic Tests

• Lund-Tree Shapes (Global Observables): The showers are tested against NLL resummation results for observables like the sum and maximum of transverse momenta in the Lund plane. The results confirm that PanScales showers correctly predict the dead-cone suppression and the running coupling thresholds, achieving NLL accuracy for $\beta_{ps} = 0.5$ (PanLocal) and $\beta_{ps} < 1$ (PanGlobal).
• Non-Global Energy Flow: The accuracy is tested for energy flow into a rapidity slice, an observable sensitive to non-global logarithms (NGLs). The showers reproduce the analytic resummation results (including mass thresholds in the coupling) perfectly, demonstrating that the full structure of the massive eikonal factor is correctly captured.
• Lund Subjet Multiplicity: Tests for the number of subjets above a $k_t$ cutoff show agreement with Next-to-Double Logarithmic (NDL) analytic calculations for both light- and heavy-quark initiated jets.

3.3 Phenomenological Studies

Using LEP data ($Z$-peak events), the authors compare the showers to experimental measurements:

• $b$-quark Fragmentation Function: The massive showers correctly predict the harder fragmentation pattern (higher average $x_B$) observed in data, whereas massless showers fail to reproduce the shape, pushing the average to lower values.
• Total Jet Broadening: The massive showers show improved agreement with data around the scale $B_T \approx m_b/Q$, where mass effects suppress radiation, compared to massless variants.

4. Significance and Claims

The paper claims to present the first final-state parton showers that demonstrably account for NLL logarithmic terms associated with non-zero quark masses.

Key contributions include:

1. Systematic Framework: A unified approach to capture mass effects across different observable classes (global, non-global, and multiplicity) while preserving the high logarithmic accuracy (NNLL/NNDL) of the underlying massless showers.
2. Validation: Rigorous validation through fixed-order tests up to $\mathcal{O}(\alpha_s^2)$ and all-order comparisons to analytic resummations, confirming the correct treatment of dead-cone suppression, flavor thresholds, and spin correlations.
3. Phenomenological Impact: Demonstration that mass corrections are crucial for describing $b$-quark fragmentation functions and that the new showers provide a better description of LEP data than massless alternatives.
4. Public Availability: The showers are implemented in the PanScales code (v0.4.0) and interfaced with Pythia 8 for hadronization.

The authors note that while the current implementation achieves NLL accuracy, it does not capture non-global logarithms in a simplified "dead-cone veto" model (discussed in Appendix C), highlighting the necessity of their full kinematic and probabilistic formulation. Future work is identified as extending these algorithms to initial-state radiation and implementing higher-order ingredients with full mass effects.