Viability of $A_4$, $S_4$ and $A_5$ Flavour Symmetries in Light of the First JUNO Result

This paper evaluates the viability of $A_4$, $S_4$, and $A_5$ discrete flavour symmetries by incorporating the first JUNO measurement of $\sin^2\theta_{12}$ into global neutrino oscillation data, revealing that the number of compatible mixing pattern cases is reduced from five to three for normal ordering and from four to two for inverted ordering at the $3\sigma$ confidence level.

The Gist

Imagine the universe as a giant, complex orchestra. For decades, physicists have been trying to figure out the sheet music for the "neutrino section" of this orchestra. Neutrinos are ghostly, tiny particles that zip through everything, and they have a strange habit called flavor oscillation: they can change their identity (from electron to muon to tau) as they travel.

To understand how they change, scientists look at three specific "mixing angles" (like the volume knobs for different instruments). One of these, called $\theta_{12}$ (the solar mixing angle), has been a bit of a mystery.

For years, theorists have tried to explain these angles using Flavor Symmetries. Think of these symmetries as the "architectural blueprints" of the universe. Just as a building might be designed with perfect symmetry (like a square or a triangle), the universe might have hidden mathematical rules (groups named A4, S4, and A5) that dictate exactly how these neutrino knobs should be set.

The Old Map vs. The New GPS

Until recently, scientists had a rough map of where these knobs should be. They had a "best guess" based on data from many experiments (like a global survey). Based on this rough map, they found five possible blueprints (for normal mass ordering) and four (for inverted ordering) that seemed to fit the data well enough to keep in the running.

Then, the JUNO experiment (a massive detector in China) turned on. Imagine that after years of guessing the location of a hidden treasure with a blurry map, JUNO suddenly handed us a high-definition GPS signal with pinpoint accuracy.

In just 59 days, JUNO measured the solar mixing angle ($\sin^2 \theta_{12}$) with incredible precision—much better than anyone expected.

The Great Filter

The authors of this paper took those old "blueprints" (the A4, S4, and A5 theories) and ran them through the new, high-precision JUNO data. It was like taking five different keys that might have opened a door and testing them against a lock that is now known to be extremely picky.

Here is what happened:

1. The "Almost" Keys Failed: Two of the five blueprints (specifically those related to patterns called B1 and B1A5) were close, but the new JUNO data showed they were just a tiny bit off. They were rejected with high confidence.
2. The "Maybe" Key Got Weaker: Another blueprint (B2A5) was still technically possible for one type of neutrino mass, but it became much less likely.
3. The Survivors:
   • For the "Normal Ordering" scenario (where the heaviest neutrino is the heaviest), 3 out of 5 blueprints survived.
   • For the "Inverted Ordering" scenario, 2 out of 4 survived.

The Winner Takes All?

Among the survivors, one blueprint stands out as the "most comfortable fit." It's called Case B2S4 (which corresponds to a pattern known as TM1 mixing).

• Analogy: Imagine you are trying to fit a square peg into a round hole. Most theories were like trying to force a square peg into a slightly smaller square hole. The JUNO data made the hole even more precise. The B2S4 theory is like a peg that fits the hole perfectly, with just a tiny bit of wiggle room left.

What Does This Mean for the Future?

The paper concludes that the JUNO experiment has acted as a powerful filter. It has eliminated many of the "pretty" mathematical theories that looked good on paper but don't match the reality of the universe as precisely as we now know it.

• The Good News: We are narrowing down the list of suspects. We are closer to finding the true "blueprint" of the universe.
• The Next Step: The remaining theories (especially the B2S4 one) will be put under even more pressure by future JUNO data. If the B2S4 theory survives the next round of ultra-precise measurements, it will be a massive victory for physicists, suggesting that the universe really is built on this specific type of mathematical symmetry.

In short: The JUNO experiment took a blurry photo of the universe's neutrino rules and turned it into a 4K image. This new clarity has knocked out several theories, leaving us with a much shorter, more promising list of candidates for how the universe is constructed.

Technical Summary

1. Problem Statement

The paper addresses the viability of specific lepton mixing patterns predicted by non-Abelian discrete flavour symmetries ($A_4$, $S_4$, and $A_5$). These symmetries, effective at high energy scales and broken down to residual symmetries at low energies, often yield "sharp" predictions for the solar neutrino mixing angle, $\theta_{12}$ (specifically $\sin^2 \theta_{12}$), and the Dirac CP-violating phase, $\delta$.

The central problem is that the JUNO experiment has recently released its first physics result with unprecedented precision on $\sin^2 \theta_{12}$ and the solar mass-squared difference $\Delta m^2_{21}$. The authors investigate whether the mixing patterns derived from $A_4$, $S_4$, and $A_5$ symmetries remain compatible with global neutrino oscillation data when constrained by this new, high-precision JUNO measurement.

2. Methodology

The authors employ a statistical analysis framework to test the compatibility of theoretical mixing patterns with experimental data.

• Theoretical Framework:

  • The study focuses on Pattern B (where $G_e = Z_k$ or $Z_m \times Z_n$ and $G_\nu = Z_2$) and Pattern C (where $G_e = Z_2$ and $G_\nu = Z_2$).
  • These patterns lead to "sum rules" relating $\sin^2 \theta_{12}$ and $\cos \delta$ to the mixing angles $\theta_{13}$ and $\theta_{23}$, and fixed parameters ($\theta^\circ_{ij}$) determined by the specific symmetry group.
  • Specific cases analyzed include B1, B1A5, B2S4, B2A5, B2A5II, and C9A5.
  • Case B1 corresponds to Tri-maximal mixing 2 (TM2), and Case B2S4 corresponds to Tri-maximal mixing 1 (TM1).

• Data Inputs:

  1. Global Fit Data: The latest global analysis of neutrino oscillation data by the NuFIT collaboration (September 2024), including Super-Kamiokande atmospheric data. This provides best-fit values and $1\sigma$/$3\sigma$ ranges for $\sin^2 \theta_{12}$, $\sin^2 \theta_{13}$, $\sin^2 \theta_{23}$, and $\delta$ for both Normal Ordering (NO) and Inverted Ordering (IO).
  2. JUNO Result: The first JUNO measurement: $\sin^2 \theta_{12} = 0.3092 \pm 0.0087$. This reduces the relative uncertainty to 2.81%, a factor of 1.6 improvement over previous measurements.

• Statistical Procedure:

  • The authors construct an approximate likelihood function $L(\alpha) = \exp(-\chi^2(\alpha)/2)$.
  • The total $\chi^2$ is approximated by summing one-dimensional projections for $\sin^2 \theta_{12}$ and $\sin^2 \theta_{13}$, and a two-dimensional projection for the correlated pair $(\sin^2 \theta_{23}, \delta)$, based on NuFIT's public data.
  • Step 1: Calculate compatibility of each mixing pattern with the global NuFIT data alone.
  • Step 2: Re-evaluate compatibility by replacing the NuFIT $\sin^2 \theta_{12}$ term with the specific JUNO $\chi^2$ function (Eq. 18).
  • Renormalization Group (RG) Effects: The authors briefly discuss RG running effects, concluding they are negligible for NO (if $m_{lightest} < 0.01$ eV) and potentially suppressed for IO under specific Majorana phase conditions, thus validating the low-energy analysis.

3. Key Contributions

• Updated Viability Analysis: The paper updates the previous analysis (from 2018) using the most recent NuFIT 6.0 global fit data.
• Impact of JUNO: It provides the first rigorous statistical assessment of how the high-precision JUNO measurement of $\sin^2 \theta_{12}$ impacts the survival of specific discrete flavour symmetry models.
• Prediction Refinement: It derives updated likelihood profiles for the atmospheric mixing angle ($\sin^2 \theta_{23}$) and the CP phase ($\delta$) for the surviving models, both before and after the JUNO constraint.

4. Results

A. Compatibility with Global Data (Pre-JUNO)

Using only the NuFIT 6.0 global fit (3$\sigma$ confidence level):

• Normal Ordering (NO): 5 cases were compatible: B1, B1A5, B2S4, B2A5, C9A5.
• Inverted Ordering (IO): 4 cases were compatible: B1, B1A5, B2S4, C9A5.
  • Note: Case B2A5 was compatible in the previous study but is now disfavored at 3.2$\sigma$ for IO. Case C5 (common to $S_4$ and $A_5$) was excluded at >4.9$\sigma$.

B. Compatibility with JUNO Result (Post-JUNO)

After incorporating the JUNO measurement of $\sin^2 \theta_{12}$:

• Normal Ordering (NO): The number of viable cases drops from 5 to 3.
  • Disfavored: B1 (TM2) and B1A5 are disfavored at 3.8$\sigma$. B2A5II is disfavored at 3$\sigma$.
  • Surviving: B2S4 (TM1), B2A5, and C9A5.
  • Most Favored: Case B2S4 is the most compatible, with a significance of 1.8$\sigma$.
• Inverted Ordering (IO): The number of viable cases drops from 4 to 2.
  • Disfavored: B1 (TM2) and B1A5 are disfavored at 3.3$\sigma$. B2A5 is disfavored at 3.5$\sigma$.
  • Surviving: B2S4 (TM1) and C9A5.
  • Most Favored: Case B2S4 is the most compatible, with a significance of 1.1$\sigma$.

C. Predictions for Other Parameters

• CP Phase ($\delta$): The JUNO measurement does not significantly narrow the width of the likelihood profiles for $\cos \delta$ (which is driven by the uncertainty in $\theta_{23}$), but it reduces the overall compatibility of the models. Only B2S4 and B2A5 (for NO) / B2S4 (for IO) survive the 3$\sigma$ test.
• Atmospheric Angle ($\sin^2 \theta_{23}$):
  • For the surviving B2S4 case (IO), the allowed range covers almost the entire experimental 3$\sigma$ range.
  • For B2S4 (NO), the allowed range is restricted to $\sin^2 \theta_{23} \in [0.438, 0.513]$.
  • For B2A5 (NO), the range is slightly reduced to $[0.523, 0.559]$ after JUNO.

5. Significance and Conclusion

• Strong Constraints: The first JUNO result acts as a powerful filter for flavour symmetry models. It eliminates the popular TM2 (B1) and B1A5 patterns at high confidence levels ($>3\sigma$) for both mass orderings.
• Survival of TM1: The B2S4 pattern (corresponding to TM1 mixing) emerges as the most robust candidate, remaining compatible with data at $<2\sigma$ for both NO and IO.
• Future Prospects: The authors conclude that while JUNO has already ruled out several scenarios, future JUNO measurements will critically test the remaining viable cases.
• Decisive Test: If the B2S4 (TM1) pattern survives future tests, a sufficiently precise measurement of the CP-violating phase $\delta$ will be required to definitively confirm or falsify this specific symmetry realization in nature.

In summary, the paper demonstrates that the high-precision measurement of $\sin^2 \theta_{12}$ by JUNO significantly reduces the landscape of viable discrete flavour symmetries, leaving TM1 (B2S4) as the leading candidate among the $A_4$, $S_4$, and $A_5$ models.