How to tame penguins: Advancing to high-precision measurements of $\phi_d$ and $\phi_s$

This paper utilizes $SU(3)$ flavour symmetry and recent CP asymmetry measurements from LHCb and Belle-II to constrain penguin decay contributions, thereby enabling high-precision extractions of the Standard Model phases $\phi_d$ and $\phi_s$ and outlining future prospects for the HL-LHC and Belle-II experiments.

The Gist

Here is an explanation of the paper "How to tame penguins," translated into simple, everyday language using analogies.

The Big Picture: The Universe's "Mixing" Mystery

Imagine the universe has a set of rules called the Standard Model. It's like the rulebook for how tiny particles interact. One of the most important rules involves B-mesons (tiny particles made of a bottom quark).

These particles have a weird habit: they can spontaneously turn into their own anti-particles and then turn back. It's like a coin that flips itself from "Heads" to "Tails" and back again while sitting on the table. Physicists call this mixing.

The paper is about measuring the speed and angle of this flipping. These angles are called $\phi_d$ and $\phi_s$. If we measure them perfectly, we can check if the Standard Model's rulebook is correct. If the numbers don't match the prediction, it means there is a hidden player in the game—New Physics (like dark matter or extra dimensions) that we haven't found yet.

The Problem: The "Penguin" Noise

Here is the catch: When we try to measure these angles, the particles don't just flip; they also decay (break apart) in a specific way.

Most of the time, the decay happens the "easy" way (like a tree falling naturally). But sometimes, a sneaky, complex process happens that physicists jokingly call a "Penguin" (because the Feynman diagram for this process looks a bit like a penguin).

• The Golden Path: The "tree" process gives us the clean angle we want to measure.
• The Penguin Path: This is a messy, background noise. It adds a tiny, unwanted twist to the angle.

The Analogy: Imagine you are trying to listen to a specific singer (the "Golden Mode") in a concert hall to hear their pitch perfectly. But, there is a tiny, annoying echo (the "Penguin") bouncing off the walls. If you don't account for that echo, you might think the singer is singing a slightly different note than they actually are.

Right now, our microphones (the LHC and Belle-II experiments) are getting so sensitive that we can hear the singer very clearly. But the echo is now loud enough that it might trick us into thinking the singer is off-key, when really, the echo is just distorting the sound.

The Solution: Using "Control Modes" as a Mirror

The authors of this paper propose a clever way to cancel out the echo. They use SU(3) Flavour Symmetry.

The Analogy: Imagine you want to know how much the echo distorts the singer's voice. You can't just listen to the singer alone. Instead, you find a different singer who sings the exact same song but in a room where the echo is louder.

1. The Main Stage: We have our "Golden" decays ($B^0_d \to J/\psi K^0$, etc.). The echo is quiet here, but we need to know exactly how quiet it is to get a perfect measurement.
2. The Control Room: We look at "Control Modes" (like $B^0_s \to J/\psi K^0_S$ or $B^0_d \to D^+D^-$). In these decays, the "Penguin" echo is much louder and easier to hear.

By measuring the loud echo in the Control Room, we can use math (specifically the symmetry of the universe) to calculate exactly how much that same echo is whispering in the Main Stage. Once we know the size of the echo, we can subtract it from our Main Stage measurement to get the true angle.

What They Did in This Paper

The authors gathered all the latest data from the LHCb (at CERN) and Belle-II (in Japan) experiments. They looked at:

• The "Golden" decays (the main singers).
• The "Control" decays (the loud echo rooms).

They built a massive mathematical model (a "fit") that solves for everything at once:

1. What is the true angle ($\phi_d$ and $\phi_s$)?
2. How big is the Penguin echo?

The Result:
They found that, currently, the Penguin echo is actually quite small. The "noise" isn't ruining the measurement yet.

• $\phi_d$ is measured to be about 45.7 degrees.
• $\phi_s$ is measured to be about -3.7 degrees.

However, they warn that this is a temporary victory. As the experiments get even more powerful in the next 10 years (the HL-LHC and future Belle-II runs), the "Golden" measurements will become so precise that even a tiny, unmeasured echo will ruin the result.

The Future: Why We Need to Measure the "Control Modes"

The paper concludes with a strong message: Don't ignore the control modes.

The Analogy: Imagine you are building a skyscraper. You have a perfect blueprint (the Standard Model). You are building the top floor (the Golden measurements) with incredible precision. But if you don't measure the foundation (the Control modes) with the same precision, the whole building might tilt.

The authors argue that if we only focus on the "Golden" decays and ignore the "Control" decays, the uncertainty from the Penguin echo will eventually become the biggest error in our measurements. We will be so precise that the "echo" will look like a new discovery, when it's just a calculation error.

Summary

• Goal: Measure the universe's mixing angles to find New Physics.
• Obstacle: A sneaky "Penguin" effect adds a tiny error to the measurement.
• Method: Use "Control Modes" (where the error is loud and obvious) to calculate and remove the error from the "Golden Modes."
• Conclusion: We are doing well right now, but to reach the extreme precision needed for the next 10 years, we must measure the "Control Modes" just as carefully as the "Golden Modes." If we don't, the "Penguin" will trick us.

Technical Summary

Here is a detailed technical summary of the paper "How to tame penguins: Advancing to high-precision measurements of $\phi_d$ and $\phi_s$" (Nikhef-2025-007).

1. Problem Statement

The complex phases $\phi_d$ and $\phi_s$, associated with the mixing of neutral $B^0_q$ and $\bar{B}^0_q$ mesons ($q \in \{d, s\}$), are critical observables for testing the Standard Model (SM) and searching for New Physics (NP).

• The Challenge: Experimentally, these phases are extracted from mixing-induced CP asymmetries in "golden modes" ($B^0_d \to J/\psi K^0$, $B^0_s \to J/\psi \phi$, and $B^0_s \to D^+_s D^-_s$).
• The Limitation: While the leading-order contribution comes from tree-level topologies, subleading penguin topologies introduce hadronic phase shifts ($\Delta\phi_q$).
• The Consequence: The measured effective phase is $\phi^{\text{eff}}_{q,f} = \phi_q + \Delta\phi_q$. If $\Delta\phi_q$ is not precisely constrained, it can mimic or hide New Physics contributions. As experimental precision improves (e.g., at HL-LHC and Belle-II), these hadronic uncertainties will become the dominant systematic error, potentially limiting the sensitivity to BSM physics.

2. Methodology

The authors employ a strategy based on SU(3) flavor symmetry of QCD to constrain the unknown penguin contributions.

• Control Modes: The core idea is to relate the "golden modes" to "control modes" where penguin topologies are enhanced relative to the tree topology.
  • For $B^0_d \to J/\psi K^0$: The ideal control mode is $B^0_s \to J/\psi K^0_S$ (U-spin partner). Secondary control modes include $B^0_d \to J/\psi \pi^0$ and charged modes ($B^+ \to J/\psi \pi^+, B^+ \to J/\psi K^+$).
  • For $B^0_s \to J/\psi \phi$: The control mode is $B^0_d \to J/\psi \rho^0$.
  • For $B^0_s \to D^+_s D^-_s$: The control mode is $B^0_d \to D^+ D^-$ (U-spin partner).
• Theoretical Framework:
  • Decay amplitudes are parametrized by a tree normalization factor ($N_f$) and a penguin contribution characterized by a magnitude ($b_f$) and a strong phase ($\rho_f$).
  • Under exact SU(3) symmetry, the penguin parameters of the control modes are related to those of the golden modes (e.g., $a' e^{i\theta'} = a e^{i\theta}$).
  • The analysis uses time-dependent CP asymmetries ($A_{CP}^{\text{dir}}$ and $A_{CP}^{\text{mix}}$) to extract these parameters.
• Fit Strategies: The paper defines and compares six fit strategies:
  1. Pseudoscalar Fit: Uses $B^0_s \to J/\psi K^0_S$ and $B^0_d \to J/\psi \pi^0$ (requires external $\phi_s$).
  2. Vector Fit: Uses $B^0_d \to J/\psi \rho^0$ (requires external $\phi_d$).
  3. $B \to J/\psi X$ Fit: Simultaneous fit of $B \to J/\psi P$ and $B \to J/\psi V$ channels.
  4. $B \to DD$ Fit: Uses $B^0_d \to D^+ D^-$ (requires external $\phi_d$).
  5. Nominal Fit: A simultaneous global fit of all three golden modes and their primary control modes ($B \to J/\psi X$ and $B \to DD$ combined).
  6. Extended Fit: Includes additional SU(3) partners ($B^+ \to J/\psi \pi^+, B^+ \to J/\psi K^+, B^0_s \to J/\psi \bar{K}^{*0}$).

3. Key Contributions

• First Simultaneous Analysis: This is the first study to perform a simultaneous global fit of all three golden channels ($B^0_d \to J/\psi K^0$, $B^0_s \to J/\psi \phi$, $B^0_s \to D^+_s D^-_s$) and their respective control modes, fully accounting for correlations between $\phi_d$ and $\phi_s$.
• Integration of New Data: The analysis incorporates the latest experimental results, including:
  • LHCb measurements of $\phi^{\text{eff}}_s$ in $B^0_s \to D^+_s D^-_s$ and CP asymmetries in $B^0_d \to D^+ D^-$.
  • Belle-II measurements of CP asymmetries in $B^0_d \to J/\psi \pi^0$.
  • LHCb measurements of direct CP asymmetry differences in $B^+ \to J/\psi \pi^+$ vs $B^+ \to J/\psi K^+$.
• Robustness Check: The authors explicitly quantify the impact of non-factorizable SU(3) symmetry breaking (the main theoretical uncertainty). They demonstrate that while factorizable breaking affects branching ratios, the CP asymmetry-based method is primarily sensitive to non-factorizable effects, which are found to be small and manageable.

4. Results

Using the Nominal Fit strategy with current data:

• Mixing Phases:
  • \phi_d = 45.7^{+1.1}_{-1.0}^\circ
  • \phi_s = -0.065^{+0.018}_{-0.017} \approx -3.72^{+1.03}_{-0.97}^\circ
  • These values are consistent with the SM but include corrections for penguin effects.
• Penguin Parameters:
  • The penguin magnitudes ($a$) are found to be small ($a_{J/\psi P} \approx 0.14$, $a_{J/\psi V} \approx 0.05$, $a_{DD} \approx 0.008$).
  • Consequently, the hadronic shifts $\Delta\phi_q$ are small, confirming that the "golden modes" are indeed robust, but the uncertainty on these shifts is currently significant.
• Correlations: The fit reveals a strong correlation between $\phi_d$ and the penguin parameter $a_{J/\psi P}$. This indicates that the precision on $\phi_d$ is currently limited by the precision of the control mode measurements, not the golden mode itself.
• Future Projections:
  • Excluding Penguin Control Modes: If control modes are not measured with higher precision, the uncertainty on $\phi_d$ will only improve by ~15-25% by the end of HL-LHC.
  • Including Penguin Control Modes: If control modes receive equal attention, the precision on $\phi_d$ improves by 50% (HL-LHC) and $\phi_s$ by 30%.
  • Without controlling penguins, the uncertainty on $\phi_s$ will be dominated by the hadronic shift uncertainty, negating the gains from increased luminosity.

5. Significance

• BSM Sensitivity: The paper establishes that to reach the projected precision goals of the HL-LHC and Belle-II (e.g., $\sigma(\phi_s) < 10$ mrad), the community must prioritize the measurement of control modes. Failure to do so will result in hadronic uncertainties becoming the bottleneck for New Physics searches.
• Strategic Guidance: It provides a concrete roadmap for experimentalists, highlighting that $B^0_s \to J/\psi K^0_S$ is the ideal U-spin partner for $B^0_d \to J/\psi K^0$ and that polarisation-dependent measurements for $B^0_s \to J/\psi \phi$ are crucial.
• Theoretical Validation: The work validates the SU(3) symmetry approach as a viable tool for high-precision flavor physics, showing that current data is consistent with small penguin effects and that the method is robust against expected symmetry breaking.

In summary, the paper argues that "taming the penguins" (controlling subleading hadronic effects via SU(3) symmetry and control modes) is not just a theoretical exercise but a prerequisite for the next generation of flavor physics experiments to successfully test the Standard Model.