Prospects for Measuring $CP$-Violation in $B_s^0 \rightarrow ϕμ^+μ^-$ via Time-Dependent Angular Analysis

This paper investigates the prospects for measuring $CP$-violation in $B_s^0 \rightarrow \phi\mu^+\mu^-$ decays at the LHC by introducing new time-dependent angular observables, demonstrating that future Run 3–5 datasets will enable their extraction with high precision and significantly enhance sensitivity to $CP$-violating short-distance effects and Wilson coefficients.

The Gist

Imagine the universe as a giant, high-speed racetrack where tiny particles called B-mesons are the race cars. Specifically, this paper focuses on a rare type of car, the $B_s^0$, which is made of a heavy "bottom" quark and a strange "strange" quark.

The scientists at MIT are asking a big question: Can we catch these cars in the act of breaking the rules of the Standard Model (the rulebook of physics)?

Here is a breakdown of their work using simple analogies:

1. The Mystery of the "Ghost" Switch

In the Standard Model, certain things are forbidden. It's like a rule that says, "You cannot turn a bottom quark into a strange quark unless you take a very long, complicated detour." Because this process is so rare and difficult, it's the perfect place to look for "New Physics"—secret rules or invisible forces that the current rulebook doesn't know about.

The specific race they are watching is the $B_s^0 \to \phi \mu^+ \mu^-$ decay.

• The Car: The $B_s^0$ meson.
• The Crash: It decays (crashes) into a $\phi$ particle (which quickly turns into two kaons) and two muons (heavy electrons).
• The Twist: The $B_s^0$ is a "ghost" car. It has a magical ability to switch identities. It can turn into its anti-car ($\bar{B}_s^0$) and back again while it is flying down the track. This is called mixing.

2. The Time-Dependent Camera

Usually, physicists take a snapshot of the crash and measure the angles of the debris. But because these cars switch identities so fast, a single snapshot isn't enough. You need a slow-motion video.

The authors propose a new way to analyze the crash by looking at time.

• The Analogy: Imagine watching a spinning top. If you only look at it once, you see a blur. If you watch it spin over time, you can see exactly how it wobbles.
• The Innovation: They have written a new mathematical "script" (a Probability Density Function) that describes exactly how the angles of the debris change as the $B_s^0$ oscillates between its two identities over time. This allows them to see patterns that were previously invisible.

3. The "Tagging" Problem

To understand the wobble, you need to know which way the car was spinning when it started.

• Untagged (Blind): Sometimes, you don't know if the car started as a $B_s^0$ or an anti-$B_s^0$. You just see the crash.
• Tagged (Labeled): Sometimes, you can look at the other debris from the collision to figure out what the car was at the start. This is called flavour tagging.

The paper shows that even if you can't "tag" every single car (which is hard to do), you can still get useful data. However, if you can tag them, you unlock a whole new set of secrets.

4. The New "Optimized" Rulers

The scientists realized that the standard way of measuring these angles is like trying to measure the length of a shadow when the sun is moving; the shadow gets distorted by "hadronic form-factors" (messy background noise from the strong nuclear force).

To fix this, they invented new, optimized rulers (observables).

• The Metaphor: Instead of measuring the raw shadow, they created a special lens that cancels out the sun's movement.
• The Result: These new rulers (called $M_i$ and $Q_i$) are much cleaner. They are less affected by the messy background noise, making it easier to spot if a "New Physics" force is pushing the car off course.

5. The Future Race (LHC Runs 3, 4, and 5)

The authors ran thousands of computer simulations (pseudoexperiments) to predict what will happen when the Large Hadron Collider (LHC) collects more data in the future (Runs 3, 4, and 5).

• The Prediction: By the end of the LHC's current era (Run 5), they expect to have enough data to measure these new angles with incredible precision.
• The Payoff:
  • They can measure the "mixing" effects (the $H_i$ and $Z_i$ observables) for the first time.
  • They can measure the "tagged" observables (like the famous $P'_5$ equivalent) with a precision that rivals current measurements of other particles.
  • Most importantly, these new measurements will tighten the constraints on the "Wilson Coefficients." Think of these coefficients as the dials on the engine of the universe. If the dials are set to the Standard Model values, the car runs smoothly. If the dials are slightly off, it means New Physics is at work.

The Bottom Line

This paper is a blueprint for a future experiment. It says:

"If we use a slow-motion camera to watch these rare particle crashes, and we use our new, noise-canceling rulers, we will be able to detect tiny cracks in the Standard Model that we couldn't see before. By the time the LHC finishes its current run, we will have enough data to either confirm the current rulebook or find the first clear evidence of a new, hidden law of physics."

They found that even without perfect "tagging" (knowing the car's starting identity), the time-dependent analysis is powerful enough to reveal these secrets, but having the tags makes the picture crystal clear.

Technical Summary

Technical Summary: Prospects for Measuring CP-Violation in $B^0_s \to \phi\mu^+\mu^-$ via Time-Dependent Angular Analysis

Problem and Motivation
Transitions of $b$-quarks to $s$-quarks ($b \to s\ell\ell$) are forbidden at tree-level in the Standard Model (SM) and proceed only via higher-order diagrams, making them sensitive probes for New Physics (NP). While recent tensions in branching fractions, angular distributions, and lepton universality tests have been observed in $B^0 \to K^{*0}\mu^+\mu^-$ decays, the $B^0_s \to \phi\mu^+\mu^-$ decay offers a complementary channel. The $B^0_s$ system is theoretically advantageous because the narrow-width approximation for the $\phi$ meson holds to a higher degree than for the $K^{*0}$, potentially allowing for more precise SM calculations. However, the $B^0_s \to \phi\mu^+\mu^-$ decay involves a CP-conjugate final state and neutral meson mixing, introducing time-dependent CP-violation effects that complicate the analysis. Furthermore, many angular observables (such as the analogue of $P'_5$) are only accessible if the initial flavour of the $B^0_s$ meson is determined (flavour-tagging). This work investigates the feasibility of performing a full time-dependent angular analysis for this decay at hadron colliders, addressing the challenge of extracting a complete set of angular observables, including those requiring flavour-tagging, and introducing new observables with reduced hadronic uncertainties.

Methodology
The authors develop a comprehensive framework for the time-dependent differential decay rate of $B^0_s \to \phi\mu^+\mu^-$, incorporating $B^0_s$-$\bar{B}^0_s$ mixing.

1. Theoretical Formalism: The decay rate is parameterized using four kinematic variables: the dilepton invariant mass squared ($q^2$) and three helicity angles ($\vartheta_K, \vartheta_\ell, \phi$). The authors derive the full probability density function (PDF) for both tagged and untagged scenarios, expressing the rate in terms of angular coefficients ($J_i, h_i, s_i$) and their CP-conjugates. These are normalized to define time-dependent observables: $S^t_i$ (CP-symmetric), $A^t_i$ (CP-asymmetric), $H_i$ (mixing-induced CP-symmetric), and $Z_i$ (mixing-induced CP-asymmetric).
2. Optimised Observables: To mitigate uncertainties from hadronic form factors, the authors construct a set of "optimised" observables ($M_i, Q_i, P^t_i, AP^t_i$). These are ratios of angular coefficients designed such that form-factor dependencies cancel at leading order, similar to the $P'_i$ observables in $B^0 \to K^{*0}\mu^+\mu^-$. This includes new observables specifically for the time-dependent mixing terms ($H_i, Z_i$).
3. Pseudoexperiment Simulation: Sensitivities are estimated using large ensembles of pseudoexperiments generated with jax. The simulation incorporates:
   • Signal and Background: Signal yields are extrapolated from LHCb Run 2 data to Run 3, Run 4, and Run 5 integrated luminosities (up to 300 fb$^{-1}$). Backgrounds are modeled as combinatorial (flat in angles, exponential in mass) with a 10% level.
   • Detector Effects: The simulation includes a time-dependent acceptance function, a decay-time resolution model (Gaussian smearing with width dependent on decay time), and flavour-tagging algorithms (Same-Side and Opposite-Side) with realistic efficiencies and mistag probabilities.
   • Fitting: An unbinned maximum-likelihood fit is performed to the pseudodata to extract the angular observables and subsequently constrain the Wilson coefficients ($C_9, C_{10}$) within an Effective Field Theory (EFT) framework.

Key Contributions

• First Full Time-Dependent Analysis: This work presents the first complete parameterization and analysis of the time-dependent angular structure of $B^0_s \to \phi\mu^+\mu^-$, explicitly deriving the PDF altered by mixing effects for both tagged and untagged cases.
• New Optimised Observables: The authors introduce a suite of optimised observables ($M_i, Q_i$) for the time-dependent coefficients $h_i$ and $s_i$, as well as adapted versions of the $P'_i$ observables ($P^t_i$) for the $B^0_s$ system. These are designed to reduce dependence on hadronic form factors.
• Feasibility Demonstration: Contrary to previous literature suggesting otherwise, the authors demonstrate that a flavour-tagged, time-dependent angular analysis is feasible at hadron colliders with realistic background levels and resolution.
• Global Fits: The paper performs global fits to extract Wilson coefficients, comparing the constraining power of different analysis setups (tagged vs. untagged, time-dependent vs. time-integrated).

Results

• Sensitivity to Observables: Using pseudoexperiments scaled to LHCb Run 3, Run 4, and Run 5 statistics, the authors find that all angular observables can be extracted with Run 3 statistics.
  • Untagged vs. Tagged: Observables accessible via the untagged sum ($S^t_i, H_i$) can be measured with high precision. Observables requiring flavour-tagging ($A^t_i, Z_i, P^t_i$) are also accessible; for instance, $P^{t'}_5$ (the $B^0_s$ analogue of $P'_5$) is projected to reach a precision comparable to current $B^0 \to K^{*0}\mu^+\mu^-$ measurements by the end of Run 5.
  • Mixing Terms: The $H_i$ and $Z_i$ observables exhibit similar sensitivities, despite $H_i$ being suppressed by mixing terms ($\sinh(y\Gamma t)$).
  • Optimised Observables: The new optimised observables ($M_i, Q_i$) show a marked increase in sensitivity to CP-violating short-distance effects compared to their unoptimised counterparts. Specifically, $Q'_4$ and $M'_8$ demonstrate significant gains in sensitivity to imaginary and real parts of $C_9$, respectively.
• Wilson Coefficient Constraints: Global fits to the Wilson coefficients ($C_9, C_{10}$) show a significant increase in precision when time-dependent observables or flavour-tagging are included.
  • The inclusion of the full set of observables (including those requiring tagging) nearly doubles the constraint on the imaginary part of the Wilson coefficients compared to untagged-only fits.
  • The inclusion of $H_i$ observables provides improved per-$q^2$ sensitivity, allowing for a more precise study of the variation of Wilson coefficients across different $q^2$ regions.

Significance and Claims
The paper claims that a dedicated heavy-flavour experiment (using LHCb Upgrade I performance as a benchmark) can perform a complete time-dependent angular analysis of $B^0_s \to \phi\mu^+\mu^-$ with $\mathcal{O}(10^3)$ signal events. The authors assert that this analysis will:

1. Provide the first measurements of the full set of angular observables for this decay mode.
2. Significantly improve constraints on short-distance New Physics parameters (Wilson coefficients), particularly the imaginary parts which are currently less constrained.
3. Offer a crucial cross-check to $B^0 \to K^{*0}\mu^+\mu^-$ anomalies, potentially clarifying the origin of deviations in $C_9$ by studying their $q^2$ dependence with reduced hadronic uncertainties.
4. Demonstrate that even untagged time-dependent measurements yield substantial improvements over existing time-integrated analyses, though tagged analyses provide the maximal sensitivity.

The work concludes that the prospects for measuring CP-violation and constraining New Physics in the $B^0_s \to \phi\mu^+\mu^-$ channel are highly promising for the upcoming LHC runs.