Stodolsky effect in the framework of Generalised Neutrino Interactions

This paper investigates the Stodolsky effect within the framework of Generalised Neutrino Interactions for both Dirac and Majorana neutrinos, finding that non-standard and tensor interaction terms yield non-zero contributions beyond the Standard Model, with specific implications for detecting the cosmic neutrino background and analyzing C$\nu$B asymmetry.

The Gist

Here is an explanation of the paper "Stodolsky effect in the framework of Generalised Neutrino Interactions," translated into simple, everyday language with creative analogies.

The Big Picture: Catching the Ghosts

Imagine the universe is filled with a "wind" made of invisible ghosts. These aren't scary ghosts, but neutrinos—tiny, nearly massless particles that zip through everything, including you, the Earth, and the Sun, without ever stopping.

Scientists have been trying to catch these "ghosts" for decades. One specific type of ghost is the Cosmic Neutrino Background (CνB). These are the leftovers from the Big Bang, the very first moments of the universe. They are everywhere, but they are so shy and weak that detecting them is considered the "Holy Grail" of neutrino physics.

This paper asks a very specific question: If we can't catch a single ghost, can we feel the wind they create?

The Core Idea: The Stodolsky Effect

The authors are studying a phenomenon called the Stodolsky effect.

The Analogy: The Magnetic Compass
Imagine you have a tiny, super-sensitive compass (an electron with a specific spin). Usually, this compass just sits there. But now, imagine a river of invisible ghosts (neutrinos) flowing past it.

According to this paper, as these ghosts flow past, they don't just pass through; they give the compass a tiny, tiny nudge. This nudge changes the energy of the compass depending on which way it's pointing.

• If the compass points "North," it gets a tiny push.
• If it points "South," it gets a slightly different push.

This difference in energy creates a torque (a twisting force). If you had a giant, super-sensitive spinning top (a ferromagnet) made of billions of these compasses, the collective nudge from the ghost wind might make the top twist ever so slightly.

The Twist: New Rules of Physics

The Standard Model of physics (the rulebook we currently use) predicts a very specific, tiny amount of twisting. But the authors of this paper say, "What if the rulebook is incomplete?"

They introduce Generalised Neutrino Interactions (GNIs). Think of this as expanding the rulebook to include "exotic" ways neutrinos might interact with electrons. They looked at every possible mathematical shape these interactions could take (like scalar, vector, and tensor shapes).

The Discovery:

1. The "Tensor" Connection: They found that if the universe follows these new, expanded rules, there is a specific type of interaction called a "tensor interaction" that acts like a secret handshake between the neutrinos and the electrons. This handshake creates a much stronger "nudge" than the standard rules predict.
2. Dirac vs. Majorana: Neutrinos might be their own antiparticles (Majorana) or distinct from them (Dirac).
   • If they are Majorana, the "nudge" is very weak and depends only on standard interactions.
   • If they are Dirac, the "nudge" can be significantly stronger if those "tensor interactions" exist.

The Challenge: How Small is "Small"?

The authors did the math and found that even with these new interactions, the effect is incredibly small.

• The Scale: The energy shift is about $10^{-36}$ electron-volts.
• The Analogy: This is like trying to measure the weight of a single grain of sand by weighing a mountain. It is so small that current technology is barely, if at all, able to detect it.

However, they calculated that if we use a super-strong magnet (like Neodymium, used in headphones and motors) and a super-sensitive torsion balance (a scale that twists on a fiber), we might be able to feel this twist in the future.

The "Asymmetric" Scenario

The paper also explores a "what if" scenario. Currently, we assume there are equal numbers of neutrinos and anti-neutrinos in the background wind. But what if the wind is "lopsided"? What if there are more neutrinos than anti-neutrinos?

If the wind is lopsided, the "nudge" becomes much more noticeable, even with the standard rules of physics. This suggests that if we ever detect this effect, it could tell us two things at once:

1. That the Stodolsky effect is real.
2. That the universe has a hidden imbalance between matter and antimatter in its neutrino soup.

Summary: What Does This Mean for Us?

Think of this paper as a blueprint for a new kind of detector.

1. The Goal: Detect the "ghost wind" of the Big Bang (CνB).
2. The Method: Don't try to catch a single ghost. Instead, build a giant, sensitive spinning top and wait for the wind to make it twist.
3. The Innovation: The authors updated the physics rules to include "exotic" interactions. They found that if these exotic rules exist, the wind pushes harder, making it easier (theoretically) to detect.
4. The Reality Check: The push is still incredibly weak. We need better technology (like super-sensitive magnetic gyroscopes) to feel it.

In a nutshell: This paper is a theoretical roadmap. It tells experimentalists, "If you build a detector sensitive enough to feel a cosmic whisper, and if the universe plays by these new, expanded rules, you might finally hear the wind of the Big Bang." It turns the search for the universe's oldest ghosts into a game of "feel the breeze."

Technical Summary

Here is a detailed technical summary of the paper "Stodolsky effect in the framework of Generalised Neutrino Interactions" by Siddhartha Bandyopadhyay and Ujjal Kumar Dey.

1. Problem Statement

The detection of the Cosmic Neutrino Background (CνB) is considered the "holy grail" of neutrino physics but remains experimentally elusive. While direct detection via inverse beta decay (e.g., the PTOLEMY experiment) faces challenges related to the uncertainty principle, alternative methods involving the elastic scattering of relic neutrinos on macroscopic targets have been proposed. Specifically, the Stodolsky effect describes the interaction of relic neutrinos with atomic electrons, causing an energy shift between spin states ($\pm 1/2$) and inducing a microscopic torque on ferromagnetic materials.

Previous calculations of this effect were largely confined to the Standard Model (SM) or specific Non-Standard Neutrino Interaction (NSI) frameworks. The authors identify a gap in the literature: a comprehensive analysis of the Stodolsky effect using the most general form of neutrino interactions (Generalised Neutrino Interactions or GNIs) involving all possible Lorentz-invariant operators (scalar, pseudoscalar, vector, axial-vector, and tensor) up to dimension six. Furthermore, they aim to resolve discrepancies found in previous literature regarding the energy shift in the SM limit for Dirac neutrinos.

2. Methodology

The authors employ a model-independent Effective Field Theory (EFT) approach to derive the energy shift of an electron in the presence of a neutrino background.

• Theoretical Framework:

  • They utilize the most general dimension-6 effective Lagrangian respecting $SU(3)_c \otimes U(1)_{em}$ symmetry. This Lagrangian includes 10 distinct Lorentz structures (operators) coupling neutrinos and electrons, parameterized by dimensionless coupling constants ($\epsilon_k$ and $\tilde{\epsilon}_k$).
  • The analysis is performed for both Dirac and Majorana neutrinos, imposing specific constraints on the coupling parameters for the Majorana case (e.g., $\epsilon_{L,\alpha\beta} = -\tilde{\epsilon}_{L,\beta\alpha}$).
  • The interaction Hamiltonian is derived by switching from the neutrino flavor basis to the mass eigenbasis using the PMNS matrix.

• Calculation of Energy Shift:

  • The energy shift $\Delta E_e$ is calculated as the expectation value of the interaction Hamiltonian between electron and neutrino states.
  • The authors treat external states as wave-packets and perform spatial averaging over a volume $V$ much larger than the wave-packet distribution.
  • They utilize trace techniques with Dirac spinors ($u$ and $v$) to evaluate the matrix elements for various operator structures (Vector, Axial, Scalar, Pseudoscalar, Tensor).
  • Crucially, they isolate terms dependent on the electron spin ($S_e$) to determine the energy splitting between spin-up and spin-down states, which drives the torque.

• Scenarios Analyzed:

  1. Standard CνB Scenario: Assuming standard cosmological decoupling where neutrino and antineutrino number densities are symmetric (or helicity-asymmetric in a specific way).
  2. Asymmetric CνB Scenario: Considering deviations from standard abundances, such as a lepton asymmetry ($\eta_\nu \sim O(10^{-2})$) or local overdensities.
  3. Flavor Eigenstates: Extending the formalism to terrestrial/astrophysical neutrino beams (e.g., from reactors or accelerators) where neutrinos are treated as flavor eigenstates.

3. Key Contributions and Results

A. Generalized Energy Shift Expressions

The authors derive analytical expressions for the energy shift $\Delta E_e$ for both Dirac and Majorana neutrinos.

• Vanishing Terms: They demonstrate that Scalar and Pseudoscalar interactions do not contribute to the energy splitting (spin-dependent part) because these terms are independent of the electron spin orientation.
• Dirac Neutrinos: The energy shift depends on:
  • NSI parameters (Vector/Axial couplings).
  • Tensor interaction parameters ($\epsilon_T, \tilde{\epsilon}_T$).
  • The difference in number densities between neutrinos and antineutrinos of specific helicities.
• Majorana Neutrinos: The energy shift depends only on NSI (Vector/Axial) parameters. Tensor contributions cancel out due to the properties of the charge conjugation matrix in the Majorana formalism.

B. Resolution of SM Discrepancy

The authors identify a sign error in a previous calculation (Ref. [27]).

• In the Standard Model (SM) limit with standard relic abundances, they find that the energy shift for Dirac neutrinos vanishes completely.
• This contradicts the previous claim that a non-zero helicity asymmetry contribution exists in the SM. The authors prove that the SM contribution is zero for Dirac neutrinos in the standard scenario, aligning with Refs. [40, 68].
• For Majorana neutrinos, the SM contribution also vanishes in the standard scenario.

C. Impact of Tensor Interactions

A significant finding is that while the SM contribution vanishes, Tensor interactions provide a non-zero contribution for Dirac neutrinos even in the standard CνB scenario.

• The energy splitting scales with the tensor coupling parameters ($\epsilon_T$).
• Using current experimental constraints on these parameters, the estimated energy splitting is in the range of $10^{-37}$to$10^{-36}$ eV.

D. Asymmetric Backgrounds

The paper explores the "non-standard" scenario where there is an asymmetry between neutrino and antineutrino number densities (or helicity states for Majorana).

• In this scenario, the Standard Model contribution becomes non-zero (order of $10^{-38}$ eV).
• The total energy splitting can reach up to $10^{-36}$ eV depending on the magnitude of the NSI and tensor parameters.
• This suggests that a lepton asymmetry in the early universe could make the Stodolsky effect more detectable.

E. Experimental Feasibility

The authors estimate the macroscopic torque on a ferromagnetic target (specifically Neodymium alloys, $Nd_2Fe_{14}B$).

• For a macroscopic object with aligned spins, the induced acceleration is estimated to be $10^{-25}$to$10^{-26}$ cm/s².
• They compare this to the sensitivity of modern torsion balances and levitated ferromagnetic oscillators, noting that while current technology is at the edge of detectability ($\sim 10^{-36}$ eV sensitivity), future advancements could probe these dynamical effects.

F. Flavor Eigenstates

For terrestrial neutrino beams (e.g., DUNE), the authors provide formulas to calculate the energy shift using flux distributions. However, they conclude that due to the significantly lower flux of terrestrial neutrinos compared to the CνB, the resulting energy splittings are negligible, making terrestrial sources currently unsuitable for probing the Stodolsky effect.

4. Significance

• Theoretical Completeness: This work provides the most general theoretical framework for the Stodolsky effect, incorporating all possible Lorentz-invariant operators and distinguishing clearly between Dirac and Majorana scenarios.
• Correction of Literature: It corrects a specific error in previous SM calculations regarding the energy shift of Dirac neutrinos, clarifying that the SM effect is zero in the standard scenario.
• New Physics Probe: It highlights Tensor interactions as a unique probe for Dirac neutrinos in the CνB, a channel previously unexplored in this context.
• Cosmological Implications: The study links the detectability of the CνB via the Stodolsky effect to cosmological parameters like lepton asymmetry, suggesting that non-standard cosmological histories could enhance the signal.
• Experimental Guidance: By quantifying the expected energy shifts and torques, the paper sets benchmarks for future ultra-sensitive experiments (torsion balances, ferromagnetic gyroscopes) aiming to detect the Cosmic Neutrino Background.

In conclusion, the paper argues that while the Stodolsky effect is extremely small, Generalised Neutrino Interactions (particularly Tensor terms for Dirac neutrinos) and potential CνB asymmetries could enhance the signal to a level where it might be accessible to next-generation precision measurement technologies.