IRC-safe jet flavour at leading power

This paper presents a practical method for incorporating leading-power quark mass effects into infrared-collinear-safe jet cross sections at NNLO without altering standard jet definitions, thereby enabling direct comparison with experimental measurements using existing jet algorithms while demonstrating that neglected power corrections can be significant.

The Gist

Imagine you are a chef trying to count the number of specific ingredients (let's say, "flavorful" truffles) hidden inside a giant, chaotic salad bowl. In the world of particle physics, this "salad" is a jet of particles created when protons smash together at the Large Hadron Collider (LHC). The "truffles" are heavy quarks (like bottom quarks), and the "salad" is a mix of many other particles.

For a long time, physicists had a major problem trying to count these truffles accurately using their best mathematical recipes (calculations).

The Problem: The "Ghost" Truffle

The standard way to count truffles is simple: "If you see at least one truffle in a spoonful of salad, call it a 'truffle spoonful'."

However, when physicists tried to do this with extreme precision (a level called NNLO, or "Next-to-Next-to-Leading Order"), their math broke down. Why? Because in the mathematical model they were using, the truffles were treated as having zero weight (massless).

In this zero-weight world, a truffle can split into two tiny, ghost-like truffles that fly off in almost the exact same direction.

• The Glitch: If these two ghosts fly together, they might land in the same spoonful. But if they fly slightly apart, they might land in two different spoonfuls.
• The Result: Because the math treats them as weightless, the probability of them flying apart is infinite. This causes the total count of "truffle spoonfuls" to blow up to infinity. It's like trying to count coins in a storm where the coins can multiply infinitely if the wind blows just right.

The Old Solutions: Changing the Rules

To fix this, previous scientists tried to change the rules of the game:

1. Change the Spoon: They invented new, complicated ways to define what a "spoonful" is, specifically designed to force those two ghosts to stay together.
2. Change the Counting: They changed how they counted the truffles (e.g., "only count if there's an odd number of truffles").

The Catch: These solutions were like changing the rules of basketball mid-game. Experimentalists (the people actually catching the particles) were using standard spoons (standard algorithms). If theorists changed the rules, the experimentalists couldn't compare their real-world data with the new math without doing a massive, error-prone translation job.

The New Solution: Give the Truffles Some Weight

This paper proposes a much simpler fix: Just give the truffles their real weight.

In reality, bottom quarks are heavy. They aren't ghosts. If you give them a little bit of mass in the math, they can't fly apart infinitely easily. The "infinite" problem disappears naturally.

But wait, the author says, "If we give them mass, the math becomes incredibly hard to calculate, and we get new problems with huge numbers (logarithms) that might break the math again."

The Magic Trick: The "Leading Power" Shortcut

The author's breakthrough is a clever shortcut. They realized they don't need to calculate the entire complex, heavy-truffle recipe from scratch. They only need to add a tiny, specific "correction ingredient" to the simple, zero-weight recipe.

Think of it like this:

• The Old Way: Trying to bake a perfect, heavy-cake from scratch every time. It takes forever and is prone to burning.
• The New Way: Bake the simple, zero-weight cake (which is fast and easy). Then, sprinkle a very specific, pre-measured "magic dust" on top. This dust accounts for the weight of the truffles just enough to fix the counting error, without needing to rebuild the whole cake.

Why This is a Big Deal

1. No Rule Changes: The experimentalists can keep using their standard spoons (the anti-kT algorithm). They don't have to learn a new way to count.
2. No Infinite Numbers: By adding this "magic dust" (the mass correction), the math stays finite and stable. The "ghost" truffles are tamed.
3. Speed: It's much faster to calculate than the old "heavy cake" method.
4. Accuracy: The author tested this and found that the "magic dust" works perfectly up to the current level of precision. The only time it might fail is if you try to measure things so precisely that you start noticing tiny, leftover crumbs (called "power corrections") that the dust didn't cover. But for now, the dust is sufficient.

The Surprising Discovery

While testing this, the author found something weird. When they compared their new "standard spoon + magic dust" method against the old "special spoon" methods, the results were different.

• The "special spoons" sometimes counted more truffles than the standard spoon, which seemed backwards.
• The author suspects this is because the "special spoons" accidentally let in some "unreal" truffles (particles with impossible speeds) that the standard spoon, with its mass correction, naturally rejects.

The Bottom Line

This paper provides a practical, easy-to-use tool for physicists to calculate heavy-flavor particle collisions with high precision. It allows theorists and experimentalists to speak the same language without needing to invent new, confusing definitions for what a "jet" is. It's a way to fix a broken math recipe by adding a pinch of salt (mass) rather than rewriting the entire cookbook.

Technical Summary

Technical Summary: IRC-safe jet flavour at leading power

Problem Statement
Standard definitions of jet flavour (specifically the "any-flavour" scheme, where a jet is flavoured if it contains at least one heavy quark) are not infrared-collinear (IRC) safe at next-to-next-to-leading order (NNLO) in QCD. While the "flavour modulo-2" scheme (flavoured if the jet contains an odd number of heavy quarks) is IRC safe at next-to-leading order (NLO), it fails at NNLO. The failure arises from uncancelled singularities associated with the emission of soft, collinear heavy quark-antiquark ($b\bar{b}$) pairs. In standard massless calculations, the double-soft limit of a $b\bar{b}$ pair leads to a non-integrable singularity that does not cancel against virtual corrections when using standard jet algorithms like anti-$k_T$.

Existing solutions proposed in the literature typically require modifying the jet definition (e.g., introducing new clustering parameters or altering distance measures) or modifying the cross-section definition itself. These approaches introduce new free parameters, complicate the comparison between theory and experiment (as experimental collaborations rely on standard algorithms), and often require unfolding procedures that are sensitive to parton-shower modelling. Furthermore, fully massive NNLO calculations are computationally prohibitive due to the complexity of two-loop amplitudes with massive quarks and potential numerical instabilities.

Methodology
The paper proposes a method to recover the IRC-safe massive-quark cross section from the massless one by including leading power (LP) quark mass effects, without modifying the jet definition or the cross-section observable. The core of the approach relies on the factorisation of mass dependence in the cross section.

1. Factorisation of Mass Logs: The dependence on the heavy quark mass ($m_b$) factorises into process-independent "soft" functions ($S$) that describe the emission of soft $b\bar{b}$ pairs. The massive cross section is expressed as a convolution of these $S$ functions with the massless cross section:
   $$F(\sigma(m_b)) = F(S_\emptyset \otimes \sigma(m_b=0)) + F(S_{b\bar{b}} \otimes \sigma(m_b=0)) + \dots + O(m_b/Q)$$
   Here, $S_\emptyset$ represents virtual corrections and $S_{b\bar{b}}$ represents real emissions of soft $b\bar{b}$ pairs.

2. Handling Singularities:

   • Soft Singularities: The double-soft singularity in the massless limit is cancelled by the $S_{b\bar{b}}$ contribution. The authors implement this by numerically integrating the difference between the massive and massless double-soft limits. To handle the UV divergence introduced by omitting the scaleless massless contribution, a subtraction term is constructed using a specific parametrisation of the soft energy ($E_S$) and momentum fractions ($x_S$). This allows the singular integral to be evaluated analytically in $d$ dimensions while the finite part is integrated numerically in four dimensions.
   • Collinear Singularities: The paper argues that collinear logs associated with final-state fragmentation functions cancel due to the Kinoshita-Lee-Nauenberg (KLN) theorem when using the flavour modulo-2 scheme and integrating over momentum fractions. Thus, no additional resummation or modification of PDFs/FFs is required for the final state.
   • Analytical Continuation: The implementation requires defining observables in $d$ dimensions (specifically 5 and 6 dimensions for the two unresolved partons). The authors demonstrate that the physical cross section is independent of the specific analytical continuation chosen, provided the continuation respects the recovery of the 4D limit and depends on higher-dimensional components only through scalar products.

3. Implementation: The method is implemented within the Stripper library, which uses the sector-improved residue subtraction scheme. The implementation extends Stripper to handle processes with no massless partons at the Born level (a necessary step for numerical checks involving $e^+e^- \to t\bar{t}$ with small masses) by using a two-step phase space generation: first generating massless kinematics, then "massifying" them using the RAMBO algorithm. This approach also serves as a powerful optimisation technique for integrating quasi-singularities.

Key Results
The authors present numerical checks and phenomenological applications:

• Numerical Checks:

  • The implementation successfully cancels $\epsilon$-poles between the massless cross section and the $S$-function contributions, rendering the result finite.
  • The result is independent of the arbitrary parameters ($A, B, R$) used in the UV subtraction term.
  • The result is independent of the specific analytical continuation of observables to higher dimensions.
  • A comparison with a fully massive calculation for $e^+e^- \to \text{top-jets}$ (with a small top mass to enhance power corrections) shows that the sum of the massless LP contributions matches the fully massive result up to negligible power corrections.

• Phenomenology ($Z + b$-jet and Inclusive $b$-jet at the LHC):

  • Perturbative Convergence: Through NNLO, the authors do not observe a breakdown of perturbative convergence due to logarithms of the quark mass ($\ln(Q^2/m_b^2)$). The soft mass logs appear to be small enough that resummation is not necessary at this order.
  • Comparison with Flavoured Algorithms: When comparing the LP anti-$k_T$ results (with mass effects included) to results from modified "flavoured" algorithms (like CMP and IFN), significant differences (up to $\sim 10\%$) are observed at low transverse momentum.
  • Power Corrections: The paper finds that power corrections in the quark mass are significant (on the order of 5–10%) and often larger than the logarithmic mass effects. These corrections can have opposite signs for different algorithms (e.g., positive for anti-$k_T$, negative for CMP), explaining why flavoured algorithms sometimes yield higher cross sections than anti-$k_T$ at low $p_T$, contrary to naive expectations.
  • Algorithm Sensitivity: The study suggests that some flavoured jet algorithms (like CMP with certain parameters) may probe unphysical regions of phase space (effectively treating quarks as massless below a scale lower than the physical mass), leading to large, unphysical logarithms.

Significance and Claims
The paper claims that its approach offers a practical solution to the NNLO jet flavour problem without requiring experimental collaborations to adopt new, non-standard jet definitions.

• Preservation of Standard Definitions: The method allows the use of standard jet algorithms (like anti-$k_T$) and standard jet flavour definitions (modulo-2) for theoretical predictions.
• Experimental Compatibility: Experimentalists can continue using standard clustering algorithms. While an unfolding to the flavour modulo-2 scheme is still required to compare with theory, the authors argue this is a standard procedure and is simplified by the fact that the theory prediction is now IRC-safe and matches the massive cross section up to power corrections.
• Computational Efficiency: The method avoids the computational bottleneck of calculating fully massive two-loop amplitudes. The additional contribution (the $S$-function term) is numerically efficient and converges quickly.
• Limitations: The authors modestly note that while logarithmic mass effects are small at NNLO, they might become significant at higher orders (N$^3$LO and beyond). Furthermore, they emphasize that power corrections are currently the dominant source of uncertainty in the low-$p_T$ regime, suggesting that for precision beyond $\sim 5\%$, fully massive calculations may eventually be unavoidable regardless of the jet definition used.

In conclusion, the work provides a robust, leading-power framework for calculating IRC-safe flavoured jet cross sections at NNLO, bridging the gap between theoretical IRC safety and experimental standard practices, while highlighting the critical role of power corrections in heavy-flavour phenomenology.