Impact of Hadronic Resonances on $B\to K^{(*)}\tau^+\tau^-$ decays

This paper proposes a data-driven strategy to predict $B\to K^{(*)}\tau^+\tau^-$ decays by explicitly incorporating hadronic resonance contributions, such as those from $\psi(2S)$, rather than avoiding them, thereby enabling the use of hadron-collider data and enhancing sensitivity to large New Physics effects across the full kinematic spectrum.

The Gist

Imagine you are trying to listen to a specific, quiet conversation happening in a very noisy room. In the world of particle physics, this "conversation" is a rare event where a heavy particle called a B-meson decays into a lighter particle (a Kaon) and a pair of tau leptons (heavy cousins of electrons).

Physicists want to listen to this conversation to see if there are any "ghosts" in the room—evidence of New Physics (particles or forces we don't know about yet) that might be whispering alongside the standard rules of nature.

Here is the problem: The room is full of loud, booming speakers playing music. These speakers are called hadronic resonances (specifically, a particle called $\psi(2S)$). In simpler experiments with lighter particles (like electrons), scientists can just put on noise-canceling headphones or wait for a quiet moment to ignore the music.

But with tau leptons, it's different. When they decay, they leave the room with some "missing energy" (neutrinos), making it impossible to tell exactly when the conversation happened or to filter out the music. If you try to listen at a hadron collider (like the LHC), you hear the conversation and the music mixed together.

The Paper's Solution: "The Data-Driven Mix"

Instead of trying to silence the music (which is impossible here), the authors of this paper decided to learn the music so well they can predict exactly how it sounds.

1. The Problem: Previous predictions for these tau decays tried to ignore the "music" (the resonances) by only looking at specific quiet time slots. But at the LHC, you can't pick and choose time slots; you hear everything from start to finish. If you ignore the music in your prediction, your math will be wildly wrong—off by a factor of 10!
2. The Strategy: The authors used a "data-driven" approach. They looked at a similar, easier-to-hear conversation: the decay of B-mesons into muons (lighter cousins of taus). In this muon conversation, the "music" (resonances) is clearly visible and has been measured perfectly by the LHCb experiment.
3. The Transfer: They realized the "music" (the resonance effects) depends on the B-meson and the Kaon, not on whether the final particles are muons or taus. So, they took the "sheet music" measured from the muon decays and applied it to the tau decays.

The Key Findings

• The Music is Loud: When they included this "music" (the $\psi(2S)$ resonance) in their predictions for the Standard Model (the known rules of physics), the predicted rate of these decays jumped by ten times. It's like realizing the quiet conversation was actually happening at a volume 10x louder than you thought because of the background noise.
• When New Physics is Strong: If there is a massive amount of "New Physics" (a very loud ghost whispering), it eventually drowns out the music. In that case, the music matters less. However, for small or moderate amounts of New Physics, the music is still the dominant factor.
• The "Cut" Mistake: The paper warns that if scientists try to "cut out" the noisy part of the data (by ignoring the resonance region), they will get the wrong answer. Even if New Physics is huge, ignoring the resonance region makes the predicted signal look half as big as it actually is. To compare with real experiments, you must include the whole noisy spectrum.

The Big Picture

The authors created a new "map" for these decays. They showed that:

1. You cannot ignore the background noise (resonances) when studying tau decays at the LHC.
2. By using data from muon decays to model the noise, they can make accurate predictions for tau decays.
3. This allows experiments like LHCb and CMS to correctly interpret their data. If they see a signal, they can now tell if it's just the "music" (Standard Model) or if there's a real "ghost" (New Physics) hiding in the mix.

In short, the paper teaches us that to hear the faint whispers of new physics, we first have to learn to sing along with the loud, known background noise.

Technical Summary

Technical Summary: Impact of Hadronic Resonances on $B \to K^{(*)}\tau^+\tau^-$ Decays

Problem Statement
Neutral-current semileptonic $B$ decays ($b \to s\ell^+\ell^-$) are significantly affected by hadronic resonances across the dilepton invariant-mass squared spectrum ($q^2$). For light leptons ($\ell = e, \mu$), these resonant regions can be excluded via kinematic cuts to isolate short-distance physics. However, for $\tau$ modes, the presence of missing energy from neutrinos in $\tau$ decays prevents the direct reconstruction of $q^2$. Consequently, experiments at hadron colliders like LHCb and CMS cannot apply $q^2$ cuts to exclude resonances; they must measure rates over the full kinematically accessible $q^2$ range, which includes the resonant region. Currently, theoretical predictions for these inclusive measurements in the Standard Model (SM) and beyond are lacking, creating a barrier to interpreting potential New Physics (NP) signals in $b \to s\tau^+\tau^-$ transitions, which are theoretically linked to the $R(D^{(*)})$ and $B \to K^{(*)}\nu\nu$ anomalies.

Methodology
The authors adopt a data-driven approach to incorporate hadronic resonance effects into predictions for $B \to K^{(*)}\tau^+\tau^-$ decays, rather than attempting to avoid them. The core strategy involves transferring nonlocal hadronic amplitudes extracted from recent LHCb measurements of $B \to K^{(*)}\mu^+\mu^-$ decays to the $\tau$ mode. This transfer is justified because the nonlocal amplitudes are properties of the meson transition ($B \to K$ or $B \to K^*$) and are largely independent of the final-state lepton flavor (up to negligible QED effects).

The methodology proceeds as follows:

1. Local Contributions: Calculated using standard techniques with form factors from the HPQCD collaboration.
2. Nonlocal Contributions: Extracted from LHCb fits to $B^0 \to K^{*0}\mu^+\mu^-$ and $B^+ \to K^+\mu^+\mu^-$ data. These include charmonium resonances (specifically $\psi(2S)$, $J/\psi$, and heavier $c\bar{c}$ states) and nonlocal effects from $D\bar{D}^{(*)}$ loops.
3. Implementation:
   • For $B \to K\tau^+\tau^-$, the nonlocal effects are modeled as a shift in the effective Wilson coefficient $C_9^\tau(q^2)$, utilizing a dispersion relation approach with relativistic Breit-Wigner functions for resonances.
   • For $B \to K^*\tau^+\tau^-$, the effects are incorporated as helicity-dependent shifts in the transversity amplitudes, as resonances depend on the $K^*$ polarization.
4. Approximations: The authors also evaluate a "simplified" data-driven approach using the Narrow Width Approximation (NWA) for the dominant $\psi(2S)$ resonance and compare it with the full data-driven fit and the NWA without interference.

Key Results

• Standard Model Predictions: Including the full $q^2$ range and the $\psi(2S)$ resonance enhances the SM branching ratios by approximately one order of magnitude compared to predictions that exclude the resonance region (e.g., $q^2 > 15 \text{ GeV}^2$). The predicted SM values are:
  • $\mathcal{B}(B^+ \to K^+\tau^+\tau^-)_{\text{SM}} = 1.86^{+0.17}_{-0.16} \times 10^{-6}$
  • $\mathcal{B}(B^0 \to K^{*0}\tau^+\tau^-)_{\text{SM}} = 1.78^{+0.25}_{-0.27} \times 10^{-6}$
  • (Values for $q^2_{\text{min}} = 14.18 \text{ GeV}^2$ and $15 \text{ GeV}^2$ are also provided in Table I).
• Dominance of $\psi(2S)$: The $\psi(2S)$ resonance is the dominant nonlocal contribution. Contributions from other resonances ($J/\psi$ tail, $\psi(3770)$, etc.) and $D\bar{D}^{(*)}$ loops are numerically negligible (orders of magnitude smaller).
• Interference Effects: In the SM, the interference between the short-distance amplitude and the $\psi(2S)$ resonance is negligible because the resonance contribution changes sign across the peak while the short-distance term is nearly constant, leading to cancellation upon integration. However, for large NP contributions, interference can become relevant, though the net effect often remains small due to similar cancellation mechanisms.
• Impact of Neglecting Resonances: The paper quantifies the error ($\delta$) introduced by neglecting resonances. In the SM, this error is $\sim 90\%$. Even with moderate NP (5$\times$ SM local contribution), the error remains $\sim 20\%$. Only for very large NP (where the short-distance contribution dominates the resonance) does the error become negligible.
• SMEFT Correlations: Within a Simplified SMEFT framework linking $b \to s\tau^+\tau^-$ to $R(D^{(*)})$ and $B \to K^{(*)}\nu\nu$, the authors show that for currently preferred anomaly values, the branching ratio is predicted to be of order $10^{-5}$. In this regime, the resonance contribution becomes sub-dominant to the NP-enhanced short-distance term, but the full phase space integration still yields a branching ratio roughly twice as large as a cut at $q^2 > 15 \text{ GeV}^2$.

Significance and Claims
The paper claims that its primary contribution is providing the necessary theoretical framework to interpret $B \to K^{(*)}\tau^+\tau^-$ measurements from hadron colliders (LHCb, CMS), which cannot resolve the $q^2$ spectrum to exclude resonances. By including resonant effects via a data-driven approach, the authors:

1. Enable direct comparison between theory and experimental data over the full kinematic range.
2. Demonstrate that exploiting the resonant phase space allows for the probing of large NP contributions that would otherwise be obscured or misinterpreted if resonances were ignored.
3. Show that the narrow-width approximation and simplified models can recover the full data-driven results for integrated branching ratios, offering practical tools for cross-checks.
4. Highlight that neglecting resonances leads to substantial underestimation of branching ratios in the SM and moderate NP scenarios, potentially masking or distorting the interpretation of experimental limits.

The authors conclude that their approach is essential for future analyses of $b \to s\tau^+\tau^-$ processes and can be extended to other modes like $B_s \to \phi\tau^+\tau^-$ and $\Lambda_b \to \Lambda\tau^+\tau^-$, which are inaccessible to $B$-factories like Belle II and thus rely entirely on hadron-collider data.