Polarized and un-polarized $\mathcal{R}_{K^*}$ in and beyond the SM

This paper investigates the potential of polarized and unpolarized lepton flavor universality ratios ($R_{K^*}$) involving $\tau$ and $\mu$ leptons in $B \to K^* \ell^+ \ell^-$ decays as sensitive observables to discriminate between various New Physics models, finding that these values are within current experimental reach and could help distinguish between different theoretical extensions of the Standard Model.

The Gist

Imagine the universe is a giant, incredibly complex machine built by a master engineer. For decades, we've been using a blueprint called the Standard Model to understand how this machine works. This blueprint predicts how tiny particles interact with each other. For the most part, the blueprint is perfect. But recently, engineers have noticed a few strange glitches in the machine's behavior that the blueprint can't quite explain.

This paper is like a team of detectives (physicists) trying to figure out if these glitches are just random noise or signs of a hidden, new part of the machine (called New Physics) that we haven't discovered yet.

Here is the story of their investigation, explained simply:

1. The "Universal" Rule and the Glitch

In our particle machine, there are three types of "lepton" particles: electrons, muons, and taus. They are like three siblings who look different (they have different weights) but are supposed to behave exactly the same way when interacting with the rest of the machine. This is called Lepton Flavor Universality.

• The Glitch: Recently, scientists measured how often a heavy particle (a B-meson) decays into a muon versus an electron. The results were slightly off from what the blueprint predicted. It's like if you dropped a heavy ball and a light ball, and they fell at slightly different speeds, even though gravity says they should fall the same.
• The Mystery: The "tau" sibling (the heaviest one) is the hardest to catch. We haven't measured how it behaves in these specific decays yet. The detectives suspect that if we look at the tau, we might find the smoking gun for the "New Physics."

2. The New Detective Tool: Polarization

The authors of this paper realized that just counting how many particles appear isn't enough. They needed a better magnifying glass.

Imagine a spinning top. It can spin in different ways:

• Longitudinal: Spinning like a bullet flying straight.
• Transverse: Spinning like a wheel rolling on the ground.

In the decay of the B-meson, the resulting particle (called a $K^*$) is like that spinning top. The paper suggests we shouldn't just count the tops; we should check how they are spinning (their polarization).

The authors calculated what would happen if we looked at these spinning tops in different scenarios:

1. Standard Model: The blueprint says they should spin a certain way.
2. New Physics: If there are hidden forces, the tops might spin differently, or the ratio of "tau-tops" to "muon-tops" might change depending on their spin.

3. The Investigation: Testing the Scenarios

The team created a list of "suspects" (different theories of New Physics). They ran a simulation to see how each suspect would change the behavior of these spinning tops.

• The Analogy: Imagine you have a bag of marbles. The Standard Model says 50% should be red and 50% blue. But you suspect someone swapped some for green ones.
  • If you just count the marbles, you might miss the swap because the total number looks right.
  • But if you look at how the marbles are spinning (polarization), you might see that the "green" marbles spin differently than the "red" ones.

The paper shows that by looking at the spin (polarization) of the particles, we can tell the difference between the different "suspects" (New Physics theories). Some theories make the "longitudinal" spin change, while others affect the "transverse" spin.

4. The Findings

The detectives did the math and found some exciting things:

• The "Tau" Ratio: They calculated the ratio of Tau particles to Muon particles ($R_{\tau\mu}$). In the Standard Model, this ratio is a specific number.
• The "Spin" Ratios: They broke this down further. What if we only look at the "longitudinal" spin? What if we only look at the "transverse" spin?
• The Result: They found that many of the "New Physics" theories predict values that are very different from the Standard Model, especially when you look at the spin.

It's like finding that while the total number of marbles is the same, the "green" ones only appear when the top is spinning one way, and the "blue" ones only when it spins the other. This gives us a powerful way to separate the different theories.

5. Why This Matters

Currently, we can't measure these "Tau" decays perfectly because the Tau particles are like ghosts—they disappear too quickly and are hard to catch in our detectors (like the LHCb at CERN).

However, the paper says: "Don't give up!"

• As our detectors get better (like upgrading from a blurry camera to a 4K camera), we will be able to measure these spins.
• Once we can measure them, these "spin ratios" will act as a discriminator. They will tell us exactly which version of New Physics is real, rather than just saying "something is wrong."

Summary

Think of this paper as a user manual for a new detective tool.

• The Problem: The universe's blueprint has a few typos.
• The Old Tool: Just counting particles (which is getting harder to use).
• The New Tool: Checking the "spin" (polarization) of the particles.
• The Promise: If we use this new tool, we won't just know that there is new physics; we will know exactly what kind of new physics it is.

The authors are essentially saying: "We've done the homework. When the experimentalists are ready to catch these tricky Tau particles and check their spin, we have the answers ready to tell them which theory is the winner."

Technical Summary

1. Problem Statement

The Standard Model (SM) predicts Lepton Flavor Universality (LFU), implying that interactions of leptons differ only by their masses. However, recent measurements of the LFU ratios $R_{K^{(*)}} = \mathcal{B}(B \to K^{(*)}\mu^+\mu^-)/\mathcal{B}(B \to K^{(*)}e^+e^-)$ have shown deviations from SM predictions, hinting at New Physics (NP). While current data for electron-muon ratios ($R_{K^{(*)}}^{\mu e}$) are becoming consistent with the SM, the ratios involving the third generation, specifically $\tau$-to-$\mu$ ratios ($R_{K^{(*)}}^{\tau\mu}$), remain largely unexplored experimentally due to the difficulty in reconstructing $\tau$ leptons.

The central problem addressed is: Can specific observables involving the polarization of the final state vector meson ($K^*$) in $B \to K^* \ell^+ \ell^-$ decays serve as sensitive probes to distinguish between various New Physics scenarios affecting the $\tau$ and $\mu$ sectors? The authors aim to identify observables that are not only sensitive to NP parameters but also possess discriminatory power to separate different theoretical models.

2. Methodology

The authors employ a theoretical framework combining the Weak Effective Hamiltonian (WEH) with a phenomenological analysis of various NP scenarios.

• Effective Hamiltonian: The study utilizes the standard operator product expansion (OPE) for $b \to s \ell^+ \ell^-$ transitions. The Hamiltonian includes SM operators ($O_7, O_9, O_{10}$) and their chirality-flipped counterparts ($O'_7, O'_9, O'_{10}$).
• Wilson Coefficients (WCs): The analysis considers both SM WCs and NP contributions. The NP is modeled by modifying the WCs $C_9, C_{10}, C'_9, C'_{10}$ for muons and taus.
  • Universal Couplings: Scenarios where NP affects all leptons equally (e.g., $C_9^U$).
  • Lepton Flavor Violating (LFUV) Couplings: Scenarios where NP couples differently to muons and taus (e.g., $C_9^\mu \neq C_9^\tau$).
• Polarization Observables: The authors define a comprehensive set of ratios involving the branching fractions of $B \to K^* \ell^+ \ell^-$ where the $K^*$ meson is either unpolarized, longitudinally polarized ($K^*_L$), or transversely polarized ($K^*_T$).
  • LFU Ratios: $R_{K^*}^{\tau\mu} = \mathcal{B}(B \to K^* \tau^+\tau^-) / \mathcal{B}(B \to K^* \mu^+\mu^-)$.
  • Polarized Ratios: Ratios comparing specific polarizations, such as $R_{K^*}^{\tau\mu, L} = \mathcal{B}(B \to K^*_L \tau^+\tau^-) / \mathcal{B}(B \to K^* \mu^+\mu^-)$, and cross-polarization ratios like $R_{K^*}^{LT}$.
  • Helicity Fractions: Ratios of polarized $\tau$ modes to the total $\tau$ mode ($R_{K^*}^{\tau, L/T}$).
• Scenarios Analyzed:
  • 1D Scenarios (S-I, S-II, S-III): Variations in a single Wilson coefficient (e.g., $C_9^\mu$).
  • Multi-dimensional Scenarios (S-V to S-XIII): Complex scenarios involving combinations of vector and axial-vector couplings, derived from global fits to existing $b \to s \ell \ell$ data (referencing works like Alguero et al.).
• Calculation: The authors compute the differential decay rates, integrating over the dilepton invariant mass squared ($q^2$) in the bin $14 \le q^2 \le s_{max}$ GeV$^2$. They express the NP contributions to the branching ratios as quadratic functions of the Wilson coefficients.

3. Key Contributions

1. Comprehensive Observable Set: The paper introduces and calculates a wide array of polarized LFU ratios ($R_{K^*}^{\tau\mu}$, $R_{K^*}^{\tau\mu, L}$, $R_{K^*}^{\tau\mu, T}$, $R_{K^*}^{LL}$, $R_{K^*}^{TT}$, etc.) which have not been extensively studied in the context of $\tau$-lepton modes.
2. Discriminatory Power Analysis: The authors demonstrate that while some observables are masked by SM uncertainties, specific polarized ratios (particularly those involving transverse polarization or cross-polarization) offer superior sensitivity to distinguish between different NP models.
3. Quantitative Predictions: The study provides precise numerical predictions for these observables in the SM and for 11 distinct NP scenarios (S-I through S-XIII) within the high-$q^2$ region.
4. Correlation Studies: The paper presents correlation plots between the standard ratio $R_{K^*}^{\tau\mu}$ and the polarized variants, showing how these correlations can be used to pinpoint specific NP Wilson coefficients in future experiments.

4. Results

• SM Baseline: The SM prediction for the unpolarized ratio $R_{K^*}^{\tau\mu}$ in the $q^2 \in [14, 19]$ GeV$^2$ bin is approximately 0.41 $\pm$ 0.01.
• 1D Scenarios:
  • Scenario S-II ($C_9^\mu = -C_{10}^\mu$): Predictions for $R_{K^*}^{\tau\mu}$ are close to the SM value but slightly higher, often overlapping with SM uncertainties.
  • Scenarios S-I and S-III: These scenarios predict values significantly lower than the SM (approx. 0.30–0.35), providing a clear distinction, especially in the high-$q^2$ region ($17-19$ GeV$^2$).
• Multi-dimensional Scenarios (D > 1):
  • Scenario S-V: Shows the largest deviation, with $R_{K^*}^{\tau\mu}$ ranging from 0.31 to 0.71, significantly exceeding the SM prediction.
  • Scenario S-XIII: Also shows significant deviation, with values up to 0.61.
  • Polarized Sensitivity: The ratio $R_{K^*}^{\tau\mu, T}$ (transverse) and $R_{K^*}^{\tau\mu, L}$ (longitudinal) exhibit distinct behaviors. For instance, in Scenario S-V, $R_{K^*}^{\tau\mu, L}$ can reach up to 0.31, while $R_{K^*}^{\tau\mu, T}$ reaches 0.40, both deviating from SM expectations.
  • Helicity Fractions: The ratios $R_{K^*}^{\tau, L}$ and $R_{K^*}^{\tau, T}$ (helicity fractions) are found to be less sensitive to NP in some scenarios but provide crucial constraints when combined with other ratios.
• Distinguishing Power: The study finds that observables like $R_{K^*}^{\tau\mu, T}$, $R_{K^*}^{LL}$, and $R_{K^*}^{TT}$ are less masked by SM form factor uncertainties compared to the unpolarized ratio, making them ideal tools to disentangle scenarios like S-V from S-XIII.

5. Significance

• Future Experimental Guidance: With the LHCb and Belle II experiments accumulating more data, the reconstruction of $\tau$ leptons in $B$ decays is becoming increasingly feasible. This paper provides a "roadmap" of specific observables that experimentalists should prioritize to test LFU in the $\tau-\mu$ sector.
• Model Discrimination: The results highlight that measuring polarization-dependent ratios is essential. Unpolarized measurements alone may not be sufficient to distinguish between different NP models (e.g., distinguishing between a pure $C_9$ modification and a $C_9-C_{10}$ correlation). The polarization observables break degeneracies between models.
• New Physics Sensitivity: The analysis confirms that the $\tau-\mu$ sector offers a "window" for NP that is currently open, as the mass suppression effects are different from the $\mu-e$ sector. The proposed observables are sensitive enough to detect NP contributions even if they are subtle in the electron/muon channels.

In conclusion, the paper establishes that polarized lepton flavor universality ratios in $B \to K^* \tau^+ \tau^-$ decays are powerful, theoretically robust tools for probing New Physics. They offer a unique capability to discriminate between competing theoretical models that explain the current flavor anomalies, making them a high-priority target for upcoming high-energy physics experiments.