Rare and very rare decays at the LHCb experiment

This paper presents recent LHCb results on rare and very rare decays of third-generation particles, including first searches and the most stringent limits to date for processes such as $b \to s \tau^+\tau^-$, $b \to s \tau^\pm e^\mp$, $b \to s \mu^\pm e^\mp$, and $\tau^- \to \mu^-\mu^+\mu^-$, which serve as sensitive probes for physics beyond the Standard Model.

The Gist

Imagine the universe as a giant, incredibly complex machine built by a master engineer (the Standard Model of physics). For decades, we've been testing this machine, and it works perfectly. But scientists suspect there might be hidden gears or secret levers we haven't found yet—parts of the machine that belong to a "New Physics" blueprint we haven't discovered.

The LHCb experiment at CERN is like a team of ultra-precise detectives. Instead of smashing the machine apart to see what's inside (which is what other experiments do), these detectives are looking for glitches. They are watching for the machine to do something it's supposed to do, but do it in a way that is so rare, so strange, or so forbidden that it proves a hidden lever is being pulled.

This paper is a report card on how well these detectives are doing at finding those glitches. Here is what they found, explained simply:

1. The Strategy: Looking for the "Impossible"

In our everyday world, if you flip a coin, it lands heads or tails. In the quantum world of subatomic particles, there are rules about what can happen.

• The "Forbidden" Moves: Some things are strictly banned by the rules of the Standard Model, like a particle changing its "flavor" (identity) in a way that shouldn't happen, or breaking the rule that "lepton number" (a type of particle count) must be conserved.
• The "Rare" Moves: Other things are allowed but so unlikely they happen maybe once in a trillion tries.
• The Detective Work: The LHCb team is watching billions of collisions. They are looking for those one-in-a-trillion events. If they see even a handful of these "impossible" or "super-rare" events, it means the "New Physics" blueprint is real.

2. The Specific Cases They Investigated

The paper details several specific "crime scenes" they investigated:

A. The "Ghostly" Disappearing Act ($b \to s\tau^+\tau^-$)

• The Scenario: A heavy particle (containing a 'bottom' quark) is supposed to decay into a lighter one and two heavy 'tau' particles.
• The Challenge: The tau particles are like ghosts; they decay instantly into other things and leave behind invisible "neutrinos" (like smoke that vanishes). You can't see the whole picture.
• The Result: The detectives looked for this specific ghostly pattern. They didn't find it. They set a new record for how rare this event must be, effectively saying, "If this happens, it's even rarer than we thought." This helps rule out some theories about why other particles behave strangely.

B. The "Identity Theft" Cases (Lepton Flavour Violation)

• The Scenario: Imagine a particle that is supposed to be a "muon" suddenly turning into an "electron" or a "tau" without any reason. In the Standard Model, this is like a cat turning into a dog. It's strictly forbidden.
• The Detective Work: They looked for cases where a heavy particle decayed into a mix of different "families" of particles (like a tau and an electron together).
• The Result: No identity theft was found. They tightened the rules, making it even harder for these "impossible" transformations to exist. This is a huge win because finding even one would be a smoking gun for new physics.

C. The "Double Trouble" Violation (Lepton Number Violation)

• The Scenario: Imagine a rule that says, "For every particle created, an anti-particle must also be created." Lepton Number Violation is like breaking that rule—creating two particles without their matching anti-partners.
• The Detective Work: They looked for a specific decay where a particle turned into two muons of the same charge (like two positive muons).
• The Result: Nothing. The universe still seems to be keeping its balance sheet perfectly. They improved the limits on how often this could happen, pushing the boundaries of our knowledge.

D. The "Silent Annihilation" (Loop-Suppressed Decays)

• The Scenario: Some particles are supposed to annihilate each other in a very specific, complicated way that involves "virtual" particles popping in and out of existence (like a quantum loop).
• The Result: They looked for a specific, very quiet decay ($B^0 \to \phi\phi$) and didn't see it. They set a new, stricter limit on how often this can happen.

3. How They Did It (The Toolkit)

Finding these needles in a haystack is hard because the "haystack" (background noise from normal collisions) is huge.

• The Filter: They used advanced computer algorithms (called "Boosted Decision Trees") that act like a super-smart bouncer. The bouncer checks every single particle collision and says, "No, that's just a random mess," or "Yes, that looks like the rare event we want."
• The Reconstruction: Since they can't see everything (like the ghostly neutrinos), they use math to reconstruct the missing pieces, much like a detective figuring out a crime by looking at the footprints left behind.

4. The Bottom Line

Did they find New Physics?
Not yet. They didn't find any "smoking guns" or "forbidden" events.

Is that bad?
Actually, it's very good science. By not finding these rare events, they have drawn a tighter map of where New Physics cannot be. They have pushed the boundaries of the known universe further out.

What's Next?
The LHCb detector is getting an upgrade (like giving the detectives better glasses and a faster computer). With more data coming in the future (Run 3 and beyond), they will be able to see even rarer events. If the "New Physics" is hiding in the shadows, the LHCb team is getting closer and closer to shining a light on it.

In short: The universe is still playing by the rules, but the LHCb team is checking the rules with such extreme precision that if the universe does break a rule, they will be the first to know.

Technical Summary

1. Problem Statement and Motivation

The paper addresses the search for physics beyond the Standard Model (SM) through the study of rare and very rare decays of third-generation particles, specifically $b$-hadrons and $\tau$ leptons.

• Theoretical Context: In the SM, Flavor-Changing Neutral Currents (FCNC) are forbidden at tree level, Lepton-Flavor Violation (LFV) is highly suppressed (even with neutrino masses), and Lepton-Number Violation (LNV) is strictly forbidden.
• Sensitivity: These decays proceed via higher-order loop diagrams involving virtual particles, making them sensitive to energy scales far beyond the direct reach of current collider energies.
• The Challenge: Many of these processes have predicted SM branching fractions ($\mathcal{B}$) well below current experimental sensitivity (e.g., $\mathcal{O}(10^{-55})$ for $\tau^- \to \mu^- \mu^+ \mu^-$ in the SM). Consequently, any observation of these decays or significant enhancements over SM predictions would constitute unambiguous evidence of New Physics (NP).
• Specific Goals: The paper reports on searches for:
  • SM-suppressed FCNC: $b \to s \tau^+ \tau^-$.
  • LFV transitions: $b \to s \tau^\pm e^\mp$, $b \to s \mu^\pm e^\mp$, and $\tau^- \to \mu^- \mu^+ \mu^-$.
  • LNV processes: $B^- \to D^{(*)+} \mu^- \mu^-$.
  • Loop-suppressed annihilation: $B^0 \to \phi \phi$.

2. Methodology and Analysis Techniques

The analysis utilizes data from LHCb Run 1 ($\sqrt{s} = 7, 8$ TeV) and Run 2 ($\sqrt{s} = 13$ TeV), corresponding to integrated luminosities of $9 \text{ fb}^{-1}$ (combined) and $5.4 \text{ fb}^{-1}$ (Run 2 only).

Detector Capabilities:
The LHCb detector is a single-arm forward spectrometer ($2 < \eta < 5$) optimized for heavy-flavor physics. Key features include:

• Excellent vertex resolution for reconstructing decay vertices.
• Superior particle identification (PID), particularly for muons, essential for background suppression.

Background Suppression Strategies:
Three primary background types are addressed:

1. Combinatorial: Random track combinations mimicking signal topology.
2. Partially Reconstructed: Decays where final-state particles are missed.
3. Misidentification (MisID): Particles (e.g., pions) reconstructed as other species (e.g., muons).

Analysis Tools:

• Selection: Multivariate classifiers, specifically Boosted Decision Trees (BDTs), are employed to separate signal from background.
• Signal Extraction: Fits are performed on the reconstructed invariant mass or the BDT output. Templates are derived from simulation and control data (using kernel density estimations).
• Statistical Limits: In the absence of a significant signal, upper limits on branching fractions are set using the $CL_s$ method.

3. Key Contributions and Results

A. Searches for $b \to s \tau^+ \tau^-$ Transitions

• Channels: $B^0 \to K^+ \pi^- \tau^+ \tau^-$ and $B^0_s \to K^+ K^- \tau^+ \tau^-$ (with $\tau^+ \to \mu^+ \bar{\nu}_\tau \nu_\mu$).
• Challenge: The presence of multiple neutrinos prevents full invariant mass reconstruction.
• Method: The $B$-meson vertex is reconstructed from hadrons; muons are selected via PID. A multiclass BDT separates signal from semileptonic backgrounds in bins of dihadron mass.
• Results: No significant signal observed.
  • Set world's most stringent limits on $b \to s \tau^+ \tau^-$ branching fractions.
  • Specific limits: $\mathcal{B}(B^0 \to K^{*0} \tau^+ \tau^-) < 2.8 \times 10^{-4}$ (95% CL) and $\mathcal{B}(B^0_s \to \phi \tau^+ \tau^-) < 4.7 \times 10^{-4}$ (95% CL).
  • Implication: Results are recast as constraints on Wilson coefficients ($C_9, C_{10}$), providing limits on NP scenarios linked to $R(D^{(*)})$ anomalies.

B. Lepton Flavor Violation (LFV) Searches

1. $B^0 \to K^{*0} \tau^\pm e^\mp$:
   • Uses three-prong $\tau$ decays. A constrained mass ($m_{fit}$) is reconstructed using the known $\tau$ mass.
   • Result: $\mathcal{B} < 5.9 \times 10^{-6}$ (90% CL). This is the most stringent constraint to date.
2. $B^+ \to \pi^+ \mu^\pm e^\mp$:
   • Fully reconstructible final state. Uses two BDTs to suppress combinatorial and partially reconstructed backgrounds.
   • Result: $\mathcal{B} < 1.8 \times 10^{-9}$ (90% CL). This improves previous bounds by two orders of magnitude.
3. $\tau^- \to \mu^- \mu^+ \mu^-$:
   • SM prediction is negligible ($\sim 10^{-55}$).
   • Result: $\mathcal{B} < 1.9 \times 10^{-8}$ (90% CL). This result is comparable to recent Belle II limits.

C. Lepton Number Violation (LNV) Searches

• Channel: $B^- \to D^{(*)+} \mu^- \mu^-$.
• Context: Probes Majorana neutrinos. SM prediction is $\mathcal{O}(10^{-23})$.
• Method: BDT separates signal from combinatorial background; specific handling for pion misidentification and in-flight decays.
• Result:
  • $\mathcal{B}(B^- \to D^+ \mu^- \mu^-) < 3.8 \times 10^{-8}$ (90% CL).
  • $\mathcal{B}(B^- \to D^{*+} \mu^- \mu^-) < 4.5 \times 10^{-8}$ (90% CL).
  • These limits improve upon previous searches by an order of magnitude.

D. Loop-Suppressed Annihilation Search

• Channel: $B^0 \to \phi \phi$.
• Context: OZI- and Cabibbo-suppressed in the SM ($\mathcal{B} \sim 10^{-8}$).
• Result: $\mathcal{B} < 1.3 \times 10^{-8}$ (90% CL). This improves the previous limit by a factor of two.

4. Significance and Future Outlook

• Scientific Impact: This work establishes the most stringent limits to date for a wide range of rare and forbidden decays. While no evidence of New Physics was found, the exclusion regions significantly constrain NP models, particularly those attempting to explain anomalies in $R(D^{(*)})$ and $b \to s \ell^+ \ell^-$ transitions.
• Systematic Limitations: The dominant uncertainties in many analyses stem from the limited size of data-derived background templates and external inputs (e.g., $\tau$ production fractions).
• Future Prospects: The paper highlights that Run 3 and beyond, utilizing the upgraded LHCb detector with a fully software-based trigger and significantly larger datasets, will further enhance sensitivity. This will allow for more stringent tests of the SM and potentially reach the sensitivity required to observe these rare processes if they exist at the predicted NP scales.