Theoretical framework for lattice QCD computations of $B\to K \ell^+ \ell^-$ and $\bar{B}_s\to \ell^+\ell^- \gamma$ decays rates, including contributions from charming penguin diagrams

This paper proposes a theoretical framework based on spectral-density methods to compute complex, long-distance contributions from "charming penguin" and chromomagnetic operators in $B\to K\ell^+\ell^-$ and $\bar{B}_s\to\gamma\ell^+\ell^-$ decay amplitudes using lattice QCD, addressing key challenges such as on-shell intermediate states and ultraviolet divergences through non-perturbative renormalization.

The Gist

The Big Picture: Catching the Ghosts in the Machine

Imagine you are trying to predict the outcome of a very complex game of billiards. You know the rules (the Standard Model of physics), and you know how the balls usually bounce off each other. But sometimes, the balls hit a hidden, invisible cushion that changes their path in a way you can't easily calculate.

In the world of particle physics, this "hidden cushion" is a specific type of particle interaction called a "Charming Penguin."

This paper is a roadmap for how scientists can finally calculate exactly how these "Charming Penguins" affect the decay of heavy particles called B-mesons. If we can calculate this perfectly, we can see if the universe is playing by the rules we think it is, or if there is "New Physics" (ghosts in the machine) hiding in the shadows.

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1. The Problem: The "Real-Time" vs. "Slow-Motion" Camera

To understand the problem, imagine you are a photographer trying to take a picture of a hummingbird's wings.

• The Reality (Minkowski Space): In the real world, the wings are moving so fast they blur. Sometimes, the bird hovers in mid-air (an "on-shell" state) for a split second before flying away. This creates a complex, blurry image that is hard to analyze.
• The Lattice Simulation (Euclidean Space): To study this, physicists use a supercomputer called Lattice QCD. Think of this as a camera that only takes photos in "slow motion" or "reverse time." It's great for stable things, but when it comes to those fleeting, hovering moments (the Charming Penguins), the camera gets confused. The math breaks down because the "blur" becomes a complex number (with a real and an imaginary part), and standard lattice computers only handle "real" numbers well.

The Analogy: It's like trying to measure the exact speed of a race car by only looking at its shadow on a wall. You can see the shadow, but you can't easily figure out the car's true speed or direction just from the shadow.

2. The Solution: The "Spectral Density" Recipe

The authors of this paper have developed a new recipe to fix the camera. They call it the Spectral Function Reconstruction (SFR) method, combined with a technique called HLT.

Here is how it works, using a cooking analogy:

• The Ingredients: The "Charming Penguin" is a loop of charm quarks (heavy particles) that pops in and out of existence. It's like a ghost that briefly appears, interacts with the B-meson, and vanishes.
• The Problem: Standard math can't handle the ghost because it exists in a "forbidden zone" of energy where the math gets messy (complex numbers).
• The Fix (SFR): Instead of trying to photograph the ghost directly, the authors propose looking at the spectrum of all possible things the ghost could be.
  • Imagine you have a smoothie. You can't see the individual strawberries inside once they are blended. But if you know the recipe (the spectral density), you can mathematically "un-blend" the smoothie to figure out exactly how many strawberries were in there.
  • The SFR method allows physicists to take the "slow-motion" data from the supercomputer and mathematically reconstruct the "real-time" ghost interaction, separating the real parts from the imaginary parts.

3. The "Contact Terms": When Things Get Too Close

There is a second problem. Sometimes, the particles get so close to each other that they touch. In physics, when things touch, the math goes to infinity (a "divergence").

• The Analogy: Imagine two magnets snapping together. If they get too close, the force becomes infinite. In the computer simulation, this looks like a "glitch" where the numbers explode.
• The Fix: The authors show how to "subtract" these infinite glitches. They use a clever trick where they calculate the messy parts separately and then cancel them out, leaving only the clean, physical signal. It's like using noise-canceling headphones: you generate a sound wave that is the exact opposite of the noise, so when they mix, you are left with silence (or in this case, a clean signal).

4. The Test Drive: A "Proof of Concept"

The authors didn't just write the recipe; they tried cooking the dish.

• They ran a simulation on a supercomputer using a specific set of rules (Twisted Mass Fermions).
• They focused on a specific decay: $B \to K \ell^+ \ell^-$ (a B-meson turning into a Kaon and two leptons).
• The Result: They successfully isolated the "Charming Penguin" contribution. The data looked messy at first (because of the "ghosts"), but once they applied their new SFR method, the signal emerged clearly.
• The Catch: The simulation used a "lighter" B-meson than the real one (to save computer time), but the results were promising. It proved that the method works.

5. Why Does This Matter?

Why do we care about these "Charming Penguins"?

• The Mystery: Recently, experiments have seen slight differences between what the Standard Model predicts and what is actually observed in particle decays. Some scientists think this is a sign of New Physics (like a new particle we haven't found yet).
• The Danger: Others think these differences are just because we haven't calculated the "Charming Penguins" correctly yet. If we get the calculation wrong, we might think we found a new particle when we actually just made a math error.
• The Goal: This paper provides the tool to calculate the "Charming Penguins" with high precision.
  • If the calculation matches the experiment, the Standard Model is safe, and the "New Physics" was just a misunderstanding.
  • If the calculation is perfect and the experiment still doesn't match, then we have a genuine discovery of New Physics!

Summary

This paper is a user manual for a new mathematical microscope. It teaches physicists how to look at the blurry, ghostly interactions of heavy particles (Charming Penguins) that were previously impossible to calculate. By using a technique called Spectral Function Reconstruction, they can turn "slow-motion" computer data into a clear picture of reality, helping us decide if the universe is following the rules we know, or if it's hiding a brand new secret.

Technical Summary

1. Problem Statement

The paper addresses a critical bottleneck in the theoretical prediction of Flavor Changing Neutral Current (FCNC) decays, specifically $B \to K\ell^+\ell^-$ and $\bar{B}_s \to \gamma\ell^+\ell^-$.

• The Challenge: While experimental measurements (e.g., by LHCb) have reached high precision, theoretical predictions are limited by hadronic uncertainties. Specifically, the evaluation of matrix elements involving on-shell intermediate states (such as charm-quark loops) leads to complex amplitudes with non-trivial imaginary parts.
• The Limitation of Standard Lattice QCD: Standard Euclidean lattice QCD techniques cannot directly compute these complex amplitudes because the analytic continuation from Minkowski to Euclidean space fails when intermediate states can go on-shell (creating poles in the energy integral).
• The "Charming Penguin" Issue: A major source of uncertainty is the "charming penguin" contribution (diagrams with $c\bar{c}$ loops, operators $O_1^{(c)}$ and $O_2^{(c)}$). These involve long-distance effects near charmonium resonances ($J/\psi, \psi(2S)$) and have traditionally been estimated using phenomenological models (e.g., Vector Meson Dominance), introducing uncontrolled systematic errors.
• Renormalization Issues: The calculation involves "contact terms" where the weak and electromagnetic currents coincide, leading to ultraviolet (UV) divergences. Additionally, the four-quark operators mix with lower-dimensional operators (scalar/pseudoscalar densities) with power divergences (scaling as $1/a$ or $1/a^2$), which must be subtracted non-perturbatively.

2. Methodology

The authors propose a comprehensive framework combining Lattice QCD with Spectral Density Methods, specifically the Spectral Function Reconstruction (SFR) method and the Hansen-Lupo-Tantalo (HLT) technique.

A. Spectral Density Approach (SFR/HLT)

• Core Concept: Instead of attempting a direct analytic continuation, the method reconstructs the physical Minkowski amplitude from Euclidean correlation functions using a spectral representation.
• Time Ordering: The amplitude is split into time-ordered contributions. Contributions where intermediate states are off-shell are real and calculable via standard methods. Contributions where intermediate states can go on-shell (creating imaginary parts) are handled via the spectral density $\rho(E)$.
• The Kernel Smearing: The singular kernel $1/(E - E_{pole} - i\epsilon)$ is approximated by a sum of exponentials (the HLT method):
  $$\frac{1}{E - m - i\epsilon} \approx \sum_{n} g_n(m, \epsilon) e^{-a E_n}$$
  This allows the amplitude to be expressed as a linear combination of Euclidean correlation functions computed at specific time separations.
• Generalization: The paper extends the SFR method (previously used for $n=2$ operators) to $n=3$ operators (required for $\bar{B}_s \to \gamma\ell^+\ell^-$), handling multiple time intervals and simultaneous unitarity cuts.

B. Renormalization and Subtraction

• Contact Terms: The authors demonstrate that while individual time-orderings ($t<0$ and $t>0$) are quadratically divergent, their sum is only logarithmically divergent due to electromagnetic current conservation. They propose a subtraction scheme (Eq. 75) to separate the UV-divergent parts (calculable via standard lattice methods) from the long-distance spectral parts.
• Power Divergence Subtraction: For the operators $O_1^{(c)}$ and $O_2^{(c)}$, mixing with lower-dimensional operators creates power divergences ($1/a^2, 1/a$). The authors detail a non-perturbative subtraction strategy using Twisted-Mass Fermions at maximal twist. By exploiting specific lattice symmetries ($CS$, $S_3$, $R_5^{sp}$) and using "replica" valence quarks with opposite Wilson parameters, they show how to eliminate these power divergences entirely, leaving only finite or logarithmically divergent terms.

C. Exploratory Numerical Study

• Setup: The authors performed a proof-of-principle calculation on a single gauge ensemble ($N_f = 2+1+1$ Wilson-Clover twisted-mass fermions, $a \approx 0.08$ fm).
• Configuration:
  • Physical light, strange, and charm quark masses.
  • A lighter-than-physical $b$-quark mass ($m_b = 2m_c$) to reduce lattice artifacts.
  • Focus on the "charming penguin" diagram where the virtual photon is emitted from the charm loop (Fig 1a).
  • Comparison between $B \to K\ell^+\ell^-$ and a fictitious $B_s \to \eta_{ss'}\ell^+\ell^-$ decay (to test statistical noise reduction).

3. Key Contributions

1. Theoretical Framework: Established a rigorous, first-principles method to compute complex amplitudes with on-shell intermediate states in FCNC decays, removing reliance on phenomenological models for charming penguins.
2. Extension to $n=3$ Operators: Generalized the SFR/HLT formalism to handle trilocal operators required for radiative decays ($\bar{B}_s \to \gamma\ell^+\ell^-$).
3. Renormalization Strategy: Provided a concrete, non-perturbative prescription for subtracting power divergences in four-quark operators using Twisted-Mass fermions, ensuring the physical signal is not obscured by lattice artifacts.
4. Proof-of-Principle Results: Successfully extracted the real and imaginary parts of the "three-times-subtracted" amplitude for the charming penguin contribution.
   • The results showed clear resonance structures corresponding to $J/\psi + K$ and $\psi(2S) + K$ states.
   • The imaginary part exhibited the expected peak behavior, and the real part showed the characteristic sign change across the resonance.
   • Comparison with a Vacuum Saturation Approximation (VSA) model showed qualitative agreement for $O_1^{(c)}$ but revealed significant deviations for $O_2^{(c)}$ due to non-factorizable effects (cancellation between vector and axial components).

4. Results

• Feasibility: The study confirms that the SFR/HLT method can successfully reconstruct the spectral density and the complex amplitude from Euclidean data.
• Resonance Resolution: The method resolved the $J/\psi$ and $\psi(2S)$ resonances. However, the authors note that resolving narrow resonances requires very small smearing parameters ($\epsilon$), which demands extremely high statistical precision.
• Statistical Challenges: The exploratory calculation highlighted that reaching the "asymptotic regime" (where $\epsilon$ is small enough for a smooth polynomial extrapolation to $\epsilon \to 0$) is currently limited by statistical noise, particularly near the resonance peaks.
• Model Comparison: The lattice data for $O_1^{(c)}$ matched the VSA-based model after a normalization factor, but $O_2^{(c)}$ was significantly suppressed compared to the model, suggesting that non-factorizable effects are crucial and cannot be ignored.

5. Significance

• Precision Physics: This framework paves the way for truly quantitative, first-principles predictions of $B \to K\ell^+\ell^-$ and $\bar{B}_s \to \gamma\ell^+\ell^-$ decay rates. This is essential for distinguishing between Standard Model effects and New Physics (NP).
• Resolving Tensions: By accurately calculating the long-distance "charming penguin" contributions, the field can determine if the observed tensions in angular distributions and branching ratios (e.g., the $P_5'$ anomaly) are due to NP or underestimated hadronic effects.
• Methodological Advance: The successful application of SFR/HLT to multi-local operators and the non-perturbative handling of power divergences represent a major step forward in lattice QCD capabilities, applicable to other processes involving on-shell intermediate states (e.g., $K \to \pi\pi$, inclusive decays).

In conclusion, the paper provides the necessary theoretical and numerical tools to compute the most difficult hadronic contributions to rare $B$ decays, moving the field from model-dependent estimates to rigorous lattice QCD calculations. Future work will focus on increasing statistics, performing the physical $b$-quark mass extrapolation, and refining the $\epsilon \to 0$ extrapolation near resonances.