Reconciling hadronic and partonic analyticity in $b\to s\ell\ell$ transitions

This paper demonstrates that the analytic structure of partonic calculations for $b\to s\ell\ell$ transitions, including anomalous thresholds arising from triangle topologies, fully matches the expectations from hadronic degrees of freedom, thereby validating the use of perturbative operator product expansions to constrain nonlocal charm-loop effects in rare $B$-meson decays.

The Gist

Imagine you are a detective trying to solve a mystery: Is there a new, invisible force of nature hiding inside the decay of a heavy particle called a B-meson?

In the world of particle physics, scientists look at how these B-mesons break apart. Sometimes, they turn into a strange particle and a pair of muons (heavy electrons). The Standard Model (our current best rulebook for physics) predicts exactly how often this should happen. But recent experiments suggest the numbers are slightly off. This "off" feeling could mean we've found New Physics (like a hidden character in the story), or it could just mean we misunderstood the "background noise" of the calculation.

The biggest source of this "background noise" is something called a Charm Loop.

The Problem: The "Ghost" in the Machine

Think of the B-meson decay like a complex Rube Goldberg machine. Inside the machine, a "charm quark" (a type of particle) briefly pops in and out of existence, creating a loop. This loop affects the final result.

Scientists have two ways to calculate the effect of this loop:

1. The Partonic View (The Microscope): They look at the raw ingredients (quarks and gluons) and use math to calculate what happens. This is like looking at the machine's gears and springs.
2. The Hadronic View (The Big Picture): They look at the particles as whole objects (like protons and mesons) and use data from experiments to estimate the effect. This is like looking at the whole machine and guessing how it moves based on how similar machines moved in the past.

The Conflict:
For a long time, there was a worry that the "Microscope" view (Partonic) was missing something. Specifically, the math suggested that in certain tricky situations, the "Microscope" calculation might miss a weird, hidden glitch called an Anomalous Threshold.

An Anomalous Threshold is like a hidden trapdoor in the machine. If you step on it, the whole machine behaves differently. The worry was: If the Microscope calculation ignores this trapdoor, but the Big Picture view sees it, then the two methods won't agree. If they don't agree, we can't trust the Microscope to tell us if we've found New Physics.

The Solution: The Triangle Map

The authors of this paper (Martin Hoferichter, Bastian Kubis, and Simon Mutke) decided to settle the argument. They asked: "Does the Microscope calculation actually see the trapdoor, or is it blind to it?"

To answer this, they used a clever trick. They realized that the complex, messy two-loop diagrams (the Rube Goldberg machine) could be simplified into a much easier shape: a Triangle.

Think of it like this:

• Imagine you are trying to understand the traffic flow in a massive, confusing city (the complex particle interaction).
• Instead of mapping every single street, you realize that all the traffic bottlenecks happen at three specific intersections that form a triangle.
• If you understand the rules of traffic at these three intersections, you understand the whole city.

The authors mapped every complex diagram from the "Microscope" view onto a simple Triangle Diagram.

The Discovery: The Trapdoor is Real!

Once they simplified the math into these triangles, they found something amazing:

1. The Trapdoor Exists: The "Microscope" calculation does have the anomalous thresholds. They are there, just like the "Big Picture" view predicted.
2. They Match Perfectly: The "trapdoor" in the Microscope view (quarks) lines up perfectly with the "trapdoor" in the Big Picture view (mesons). The only difference is a tiny adjustment for the mass of the strange quark (a minor detail in the recipe).
3. The Math Holds Up: They proved that even with these tricky trapdoors, the mathematical rules (called Dispersion Relations) that connect the "before" and "after" of the particle decay still work perfectly.

The Analogy: The Two Maps

Imagine you are trying to navigate a forest.

• Map A (Partonic) is drawn by a satellite looking at individual trees and roots.
• Map B (Hadronic) is drawn by a hiker who has walked the path and knows where the swamps are.

Previously, people worried that the Satellite Map missed the swamps because it was too focused on the trees. If the Satellite Map missed the swamps, you couldn't trust it to guide you to the treasure (New Physics).

This paper is the proof that the Satellite Map does show the swamps. The authors showed that if you look closely at the satellite data (using their Triangle trick), you can see the swamps exactly where the hiker said they were.

Why This Matters

This is a huge victory for particle physicists.

• Trust: It means we can now safely combine the "Microscope" math (which is great for high-energy situations) with the "Big Picture" data (which is great for low-energy situations).
• Confidence: Because we know the math isn't missing any hidden traps, if the experiments still don't match the combined prediction, we can be much more confident that we have actually discovered New Physics beyond the Standard Model.

In short: The authors proved that the two different ways of looking at the universe's building blocks are actually looking at the same thing, including all the weird, hidden quirks. This clears the path for finding the next big discovery in physics.

Technical Summary

1. Problem Statement

Rare $B$-meson decays mediated by $b \to s\ell\ell$ transitions are critical probes for physics beyond the Standard Model (BSM). Recent experimental data (e.g., from LHCb) show deviations from Standard Model (SM) predictions in angular observables and decay rates. However, establishing these deviations as genuine BSM signals requires rigorous control over non-local hadronic matrix elements, specifically those involving charm loops.

The central challenge addressed in this paper is the reconciliation of analyticity between two different theoretical descriptions of these charm-loop effects:

• Hadronic Picture: Describes the process via intermediate hadronic states (e.g., $D\bar{D}$) using dispersion relations and triangle topologies. This approach accounts for "anomalous thresholds" (singularities arising from specific kinematic configurations) which can significantly distort the amplitude.
• Partonic Picture: Uses the Operator Product Expansion (OPE) to calculate the process perturbatively at the quark level (two-loop diagrams).

The Conflict: Previous analyses suggested that the partonic calculation might miss the anomalous effects present in the hadronic picture, particularly for diagrams involving specific topologies (labeled (a) and (c) in the literature). If the partonic calculation is incomplete, its use as a constraint for large spacelike virtualities (where OPE is valid) combined with hadronic dispersion relations would be theoretically inconsistent.

2. Methodology

The authors employ a multi-step approach to demonstrate that the partonic calculation fully captures the analytic structure expected from the hadronic picture:

1. Topological Reduction: They decompose the complex two-loop partonic diagrams for $b \to s\gamma^*$ (featuring charm loops) into effective one-loop triangle topologies. This allows them to utilize established formalisms for analyzing discontinuities and singularities.
2. Parameterization of Discontinuities: Using the general analysis of triangle diagrams (specifically referencing the work of Mutke et al. [26]), they derive explicit parameterizations for the discontinuities (imaginary parts) of the form factors.
   • They introduce a spectral function $\rho(\mu^2)$ to describe the left-hand cut structure.
   • They expand these spectral functions into conformal polynomials to fit numerical data.
3. Numerical Validation: They fit their parameterized discontinuities to the exact numerical results of the two-loop calculations provided in Ref. [13] (implemented in the EOS software).
4. Dispersion Relation Construction: They construct dispersion relations for each gauge-invariant class of diagrams. Crucially, they handle the integration carefully to include contributions from anomalous branch points (singularities on the physical sheet or near the cut) that arise in diagrams (a) and (c).
5. Mapping to Hadronic Degrees of Freedom: They explicitly map the kinematic variables of the partonic anomalous thresholds to those of the hadronic picture (involving $B \to K^{(*)}$ transitions and $D^{(*)}$ mesons) to demonstrate a one-to-one correspondence.

3. Key Contributions

• Resolution of Analyticity Discrepancy: The paper proves that the partonic two-loop calculation does not miss anomalous effects. The anomalous thresholds found in the partonic calculation correspond exactly to those expected in the hadronic description, differing only by hadronization effects and the strange quark mass.
• Explicit Treatment of Anomalous Thresholds: The authors provide a rigorous treatment of diagrams (a) and (c), which were previously suspected to have unaccounted singularities.
  • Diagram (a): Shows an anomalous threshold $s_+$ located on the unitarity cut, generating a real part in the discontinuity.
  • Diagram (c): Features an anomalous threshold coinciding with a pseudothreshold, leading to a pole structure and complex singularity behavior.
• Validation of Dispersion Relations: They demonstrate explicitly that the full partonic form factors satisfy dispersion relations once the anomalous contributions are properly included. This validates the use of OPE constraints in regions of parameter space where perturbative descriptions apply.
• Spectral Function Parameterization: They develop a robust parameterization for the spectral functions $\rho(\mu^2)$ that captures both regular and divergent behaviors (e.g., threshold divergences in crossed-channel gluon exchanges).

4. Results

• Fit Quality: The parameterized discontinuities derived from the triangle topology model reproduce the exact two-loop numerical results from Ref. [13] with high precision across the relevant kinematic range (see Fig. 2 in the paper).
• Dispersion Relation Satisfaction: When the reconstructed form factors via dispersion relations (DR) are compared to the direct two-loop calculations, they match perfectly (see Fig. 3 and Fig. 6). This confirms that the analytic structure is fully captured.
• Trajectory Analysis: The authors analyze the trajectory of the anomalous branch point $s_+$ in the complex plane as a function of kinematic parameters (Fig. 4).
  • In the partonic limit ($b \to s$), the threshold lies on the real axis.
  • In the hadronic limit ($B \to K^{(*)}$), the threshold moves into the complex plane due to the mass differences between $D^{(*)}$ and $D^{(*)}_s$ mesons.
  • Despite these shifts, the underlying mechanism is identical, confirming the "reconciliation."
• Complex Plane Consistency: Comparisons in the complex $s$-plane (Fig. 7) show that the deviations between the DR and 2-loop results are negligible (logarithmic scale), limited only by numerical implementation instabilities at very high energies.

5. Significance

This work is significant for the field of flavor physics and the search for New Physics for several reasons:

1. Justification of Combined Analyses: It provides the theoretical foundation for combining data-driven dispersive approaches (relying on timelike experimental data like $J/\psi$ resonances) with perturbative OPE constraints (valid for large spacelike virtualities). Previously, there was concern that the OPE might miss non-perturbative anomalous effects, rendering such combinations inconsistent.
2. Robustness of BSM Searches: By confirming that the partonic calculation is complete regarding analytic structure, the paper strengthens the reliability of current $b \to s\ell\ell$ anomaly interpretations. It ensures that deviations observed are less likely to be artifacts of missing theoretical inputs (charm loops).
3. Methodological Framework: The paper establishes a general framework for handling anomalous thresholds in perturbative QCD calculations by mapping them to triangle topologies. This methodology can be applied to other processes involving non-local matrix elements and loop-induced transitions.

In conclusion, Hoferichter, Kubis, and Mutke successfully demonstrate that the partonic and hadronic descriptions of $b \to s\ell\ell$ transitions are analytically consistent, resolving a major theoretical hurdle in the precise interpretation of rare B-decay data.