Time-of-Flight Constraints on Neutrino Millicharge from Supernova Neutrinos in Galactic Magnetic Fields

This paper proposes a framework to reinterpret supernova time-of-flight limits on neutrino mass as constraints on neutrino millicharge by leveraging their shared $E_\nu^{-2}$ energy dependence, deriving bounds ranging from $\sim 10^{-17}\, e$ for SN1987A to $\sim 10^{-20}\, e$ for future Galactic supernovae using a line-of-sight-dependent magnetic delay kernel.

The Gist

The Big Idea: Neutrinos with a Tiny Spark

Imagine a neutrino as a ghost. It's a tiny particle that zips through the universe, passing through planets, stars, and even your body without ever bumping into anything. In our current understanding of physics (the Standard Model), these ghosts are perfectly neutral—they have no electric charge at all.

But what if they aren't perfectly neutral? What if they have a tiny, almost invisible "spark" of electricity? Physicists call this a "millicharge." It's not enough to make the neutrino stick to a magnet or get zapped by a lightning bolt, but it's just enough to make it react very slightly to magnetic fields.

This paper asks: If neutrinos have this tiny spark, how would we know?

The Race: A Cosmic Time-Travel Experiment

The authors propose a clever way to catch these "sparking" neutrinos by looking at supernovas (exploding stars).

1. The Setup: When a star explodes, it sends out a massive burst of neutrinos all at once. Think of it like a starting gun firing a thousand runners at the exact same moment.
2. The Journey: These runners (neutrinos) have to travel a huge distance to reach Earth. Along the way, they pass through the Galactic Magnetic Field—imagine this as a giant, invisible, swirling ocean of magnetic currents that fills our entire galaxy.
3. The Twist:
   • Normal Neutrinos (No Spark): If a neutrino has no charge, the magnetic ocean doesn't care about it. It swims in a perfectly straight line.
   • Millicharged Neutrinos (Tiny Spark): If a neutrino has even a tiny spark, the magnetic ocean pushes it slightly. It doesn't stop the neutrino, but it forces it to take a slightly curved, zig-zag path instead of a straight line.

The Delay: Why the Curved Path Matters

Here is the key insight: A curved path is longer than a straight path.

Even though the neutrinos are traveling near the speed of light, taking a slightly longer route means they arrive at Earth a tiny bit later than they would have if they went straight.

• The Analogy: Imagine two runners on a track. One runs in a straight line. The other is forced to run in a slight, winding curve because of a gentle wind. Even if they run at the same speed, the one on the curve arrives later.
• The Energy Factor: The paper notes that this delay depends heavily on the neutrino's energy. High-energy neutrinos are "sturdier" and get pushed less, while lower-energy ones get pushed more. This creates a specific pattern: lower-energy neutrinos arrive later than high-energy ones.

The Detective Work: Reusing Old Clues

The authors realized that scientists have been looking for a different kind of delay for decades: neutrino mass delay.

• The Old Theory: We know neutrinos have mass. Just like a heavy runner might be slightly slower than a light one, a massive neutrino takes a tiny bit longer to travel than a massless one. Scientists have used the arrival times of neutrinos from the famous SN1987A supernova (an explosion seen in 1987) to set limits on how heavy neutrinos can be.
• The New Connection: The authors noticed that the delay caused by a tiny electric charge (millicharge) looks mathematically identical to the delay caused by mass. Both create a delay that gets bigger for lower-energy neutrinos.

So, they didn't need new data. They just needed to re-interpret the old data. They said: "If we assume the delay we saw in 1987 wasn't caused by mass, but by a tiny electric charge instead, how big could that charge be?"

The Results: How Small is the Spark?

By running their new "translation" tool on the data from SN1987A and projecting what future, more sensitive detectors (like DUNE, Hyper-Kamiokande, and JUNO) might see, they found:

1. SN1987A Limits: Based on the 1987 explosion, the neutrino's electric charge must be incredibly small—less than about $10^{-17}$ times the charge of an electron. (That's a decimal point followed by 16 zeros and then a 1).
2. Future Limits: If a supernova happens in our own galaxy (a "Galactic Core-Collapse Supernova") and we catch it with next-generation detectors, we could push that limit down to $10^{-20}$.

Why the Direction Matters

The paper also highlights that the "magnetic ocean" isn't the same everywhere.

• The Map: The authors used a detailed map of our galaxy's magnetic field (the JF12 model).
• The Result: If a supernova happens in a part of the sky where the magnetic field is strong and the path is long, the delay is bigger, and we can set stricter limits on the charge. If it happens in a "quiet" part of the galaxy, the limits are weaker. It's like trying to hear a whisper: if the wind is howling (strong magnetic field), you can tell if someone is whispering; if it's dead silent, a whisper is harder to distinguish from background noise.

Summary

This paper is a "translation" project. It takes existing rules about how long it takes neutrinos to travel (Time-of-Flight) and rewrites them. Instead of asking, "How heavy are neutrinos?" it asks, "How much electric charge do they have?"

By using the known magnetic fields of our galaxy as a giant filter, the authors show that if neutrinos have even a microscopic electric charge, the "zig-zag" they take through space would delay their arrival. By checking the arrival times of neutrinos from exploding stars, we can prove that if they do have a charge, it is so small it is almost impossible to imagine.

Technical Summary

1. Problem Statement

The paper addresses the search for a non-zero electric charge (millicharge, $q_\nu$) on neutrinos. While the Standard Model predicts neutrinos are electrically neutral, various Beyond the Standard Model (BSM) theories allow for tiny charges. Existing constraints come from laboratory scattering, stellar cooling, and cosmological arguments, but these probe different physical mechanisms.

The authors focus on a specific, complementary probe: the propagation of supernova neutrinos through the Galactic Magnetic Field (GMF). A millicharged neutrino experiences a Lorentz force, causing a slight deflection. In the ultra-relativistic limit, this deflection increases the path length, inducing a geometric time delay ($\Delta t$) relative to a neutral particle. Crucially, this delay scales as $q_\nu^2 E_\nu^{-2}$, which is identical to the energy dependence of the time-of-flight (ToF) delay induced by neutrino mass ($m_\nu^2 E_\nu^{-2}$).

The Core Challenge: Previous estimates (e.g., Barbiellini & Cocconi, 1987) assumed a uniform magnetic field and simplified statistical treatments. The paper aims to modernize this approach by:

1. Reinterpreting existing and projected supernova ToF limits on neutrino mass as limits on neutrino millicharge.
2. Replacing the uniform field assumption with a realistic, line-of-sight-dependent model of the Galactic Magnetic Field (GMF).

2. Methodology

A. Theoretical Framework: Dispersion Coefficient Translation

The authors establish a phenomenological framework where the total propagation delay is expressed as a dispersion coefficient $A_{ToF}$ scaling with $E^{-2}$:
$$\Delta t_{prop}(E) \simeq \frac{A_{ToF}}{E^2}$$
In standard analyses, $A_{ToF}$ is attributed solely to mass ($A_m \propto m_\nu^2$). The authors generalize this to include millicharge contributions ($A_q \propto q_\nu^2$) and other effects:
$$A_{ToF} = A_m(m_\nu^2) + A_q(q_\nu^2) + A_{other}$$
By taking published upper limits on $A_{ToF}$ (derived from neutrino mass constraints) and assuming $A_m = A_{other} = 0$, they derive an upper bound on $q_\nu$.

B. Magnetic Delay Kernel Calculation

The millicharge-induced delay coefficient $A_q$ is calculated using the Jansson-Farrar (JF12) model for the Galactic Magnetic Field, which includes three components:

1. Regular Field ($B_{reg}$): Large-scale coherent field.
2. Turbulent Field ($B_{turb}$): Isotropic random component.
3. Striated Field ($B_{str}$): Anisotropic component aligned with the regular field but fluctuating in sign.

The time delay for a neutrino traveling distance $L$ is derived in the small-angle approximation. The total delay coefficient is the sum of contributions from these components:
$$A_q = A_q^{reg} + A_q^{turb} + A_q^{str}$$

• Regular Component: Calculated using a geometric kernel $K(x, x')$ (related to a Brownian bridge covariance) integrated over the transverse magnetic field components along the line of sight.
• Stochastic Components (Turbulent/Striated): Treated as a random walk. The delay depends on the field variance and the correlation length ($\lambda_B$) of the magnetic domains.

The final bound on the millicharge fraction $\epsilon_\nu \equiv q_\nu/e$ is given by:
$$|\epsilon_\nu| \leq \left[ \frac{A_{lim}}{F(L, \ell, b)} \right]^{1/2}$$
where $F(L, \ell, b)$ is the magnetic delay kernel, a function of the source distance ($L$) and Galactic coordinates (longitude $\ell$, latitude $b$).

C. Data Inputs

The authors apply this translation to:

• SN 1987A: Both model-independent limits (based on burst duration) and model-dependent Bayesian analyses.
• Future Galactic Core-Collapse Supernovae (CCSNe): Projected sensitivities for detectors including Super-Kamiokande, IceCube, DUNE, and JUNO.

3. Key Contributions

1. Unified Dispersion Formalism: The paper provides a rigorous mathematical mapping between neutrino mass limits and millicharge limits, demonstrating that any ToF analysis constraining an $E^{-2}$ dispersion term can be reinterpreted for millicharge.
2. Realistic GMF Modeling: Unlike previous works using uniform fields, this study utilizes the full JF12 model. It explicitly calculates the delay kernel for specific lines of sight, accounting for the regular, turbulent, and striated components.
3. Line-of-Sight Dependence: The authors demonstrate that the sensitivity to millicharge is not a universal constant but varies significantly with the direction of the supernova. Directions near the Galactic plane (where the regular field is strong) yield much tighter constraints than off-plane directions (like SN 1987A).
4. Uncertainty Quantification: The study propagates uncertainties from the JF12 magnetic field parameters and the stochastic nature of the turbulent field to provide 1$\sigma$ confidence bands on the derived limits.

4. Results

The translated bounds on the neutrino millicharge ($\epsilon_\nu$) are summarized as follows:

• SN 1987A:

  • Due to its location off the Galactic plane ($b \approx -32^\circ$), the magnetic kernel is suppressed.
  • Limits range from $\sim 10^{-17} e$ (model-independent) to $\sim 10^{-18} e$ (model-dependent).
  • These are weaker than future forecasts but serve as a historical benchmark.

• Future Galactic CCSNe (at $L \approx 10$ kpc):

  • For a typical in-plane direction ($\ell=45^\circ, b=0^\circ$), the limits improve significantly.
  • Super-Kamiokande: $\sim 4.5 \times 10^{-19} e$.
  • IceCube: $\sim 7.8 \times 10^{-20} e$.
  • JUNO: $\sim 2.2 \times 10^{-20} e$.
  • DUNE (Optimistic): $\sim 5.0 \times 10^{-20} e$.
  • DUNE (Conservative): $\sim 1.2 \times 10^{-19} e$.

• Directional Spread:

  • The sensitivity varies by a factor of $\sim 3$ depending on the Galactic longitude and latitude.
  • The most favorable directions (near the Galactic center or dense disk regions) can approach the $10^{-20} e$ regime, and in optimistic scenarios, potentially reach $\sim 10^{-21} e$ (though the paper notes the $10^{-20}$ level is the robust "low" regime for next-gen bursts).

• Comparison with Other Bounds:

  • The derived limits are competitive with stellar cooling bounds ($\sim 10^{-14} e$) and reactor/accelerator bounds ($\sim 10^{-12} - 10^{-13} e$).
  • They do not yet reach the "neutrality of matter" indirect bound ($\sim 10^{-21} e$), which relies on charge conservation assumptions rather than direct propagation effects.

5. Significance and Implications

• Complementarity: This method probes neutrino millicharge through a distinct physical mechanism (Lorentz deflection over astrophysical distances) compared to laboratory scattering or stellar cooling. It is particularly sensitive to the propagation properties of neutrinos.
• Model Independence: The translation relies only on the $E^{-2}$ scaling of the delay, making it applicable to any ToF analysis that constrains a single effective dispersion coefficient, regardless of the specific supernova emission model used.
• Future Prospects: The paper highlights that the next Galactic supernova, observed by next-generation detectors (DUNE, JUNO, Hyper-K), has the potential to probe millicharges down to the $10^{-20} e$ level. This would significantly tighten constraints on BSM scenarios involving kinetic mixing or charge dequantization.
• Caveats: The authors note that if the neutrino mass is non-zero and the analysis becomes sensitive to individual mass eigenstates (rather than a single effective coefficient), the simple translation to a single millicharge bound becomes model-dependent. Additionally, the limits are highly sensitive to the specific line of sight and the accuracy of the Galactic Magnetic Field model.

In conclusion, the paper successfully modernizes the concept of using supernova time-of-flight delays to constrain neutrino millicharge, transforming a rough order-of-magnitude estimate into a precise, line-of-sight-dependent sensitivity forecast for the next generation of neutrino astronomy.