Big Bang Nucleosynthesis and the Neutrino-Extended Standard Model Effective Field Theory

This paper demonstrates that Big Bang nucleosynthesis constraints on GeV-scale heavy neutral leptons within the neutrino-extended Standard Model Effective Field Theory provide a crucial upper bound on the theory's cutoff scale for masses above 100 MeV, thereby defining specific target regions for future laboratory searches.

The Gist

Imagine the universe as a giant, bustling kitchen. In the very first moments after the Big Bang, this kitchen was incredibly hot and chaotic, cooking up the basic ingredients of everything we see today: hydrogen, helium, and a tiny bit of lithium. This cooking process is called Big Bang Nucleosynthesis (BBN).

For decades, scientists have had a perfect recipe for this cosmic cooking. When they calculate how much helium and hydrogen should exist based on the Standard Model (our current best theory of physics), the numbers match what we actually observe in the universe. It's a perfect match.

But, there's a mystery. We know neutrinos (tiny, ghostly particles) have mass, but our standard recipe doesn't explain how. To fix this, physicists propose the existence of a new, heavier cousin to the neutrino called a Heavy Neutral Lepton (HNL). Think of the HNL as a "secret ingredient" that might have been added to the cosmic soup.

The Problem: The "Too Heavy" Ingredient

If this secret ingredient (the HNL) is too heavy or stays in the soup too long, it ruins the recipe.

• Scenario A: If the HNL is stable and hangs around, it acts like extra weight in the pot, changing how fast the universe expands. This messes up the cooking time, resulting in the wrong amount of helium.
• Scenario B: If the HNL decays (breaks apart) too late, it dumps extra energy into the soup, again ruining the flavor balance.

So, cosmologists can say: "If this secret ingredient exists, it must disappear (decay) before the cooking is done, or it must be so light it doesn't weigh anything down."

The New Approach: The "Effective Field Theory" Cookbook

Usually, when physicists look for these particles, they try to build them in giant particle accelerators (like the Large Hadron Collider) or look for them in neutrino detectors. They are looking for the particle directly.

However, this paper introduces a clever new way to look for them using a concept called Effective Field Theory (EFT).

Think of EFT as a simplified recipe card. Instead of trying to understand the entire, complex machinery of the universe (the "UV completion"), we just look at the interactions we can see at low energies. We treat the heavy new physics as if it were just a set of "rules" or "operators" that tell the particles how to behave.

The authors ask: "If we assume these simplified rules exist, what does the cosmic cooking (BBN) tell us about the limits of these rules?"

The Detective Work: Two Sides of the Same Coin

The paper finds that we can trap the "secret ingredient" between two walls:

1. The Lower Wall (Laboratory Searches):
   Experiments on Earth (like the LHC or future beam-dump experiments) look for these particles. If the particle interacts too strongly with normal matter, we would have seen it by now. This sets a minimum limit on how "weakly" the particle can interact. If it interacts too strongly, it's already been caught.

2. The Upper Wall (Cosmic Cooking/BBN):
   This is the paper's main contribution. If the particle interacts too weakly, it might not have been produced enough in the early universe to matter. But if it interacts just right to be produced, but then hangs around too long, it ruins the helium recipe.

   The authors show that BBN acts as a ceiling. It says: "You cannot make the interaction so weak that the particle survives too long, or the universe's helium levels would be wrong."

The "Goldilocks" Zone

By combining these two limits, the paper identifies a "Goldilocks Zone" for these particles.

• They can't be too strong (or we'd have seen them in labs).
• They can't be too weak (or they'd ruin the Big Bang cooking).

This creates a specific, narrow target region where future experiments should look. It's like telling a treasure hunter: "Don't dig in the deep ocean (too weak to exist), and don't dig in the shallow sand (we already checked there). Dig right here, in this specific patch of mud."

The Metaphor: The Invisible Guest

Imagine a party (the early universe).

• The Hosts (Standard Model): They are cooking a meal.
• The Guest (HNL): A mysterious person who might show up.
• The Lab Searches: We are checking the guest list and the door cameras. If the guest is too loud or flashy (strong interaction), we would have seen them.
• The BBN Check: We are checking the leftovers. If the guest stayed too long and ate too much, the food would be gone. If they stayed too long and didn't eat, they might have knocked over the table.

This paper says: "We know the guest isn't too loud (Lab limits). But we also know that if they stayed too long, the party would have been a disaster (BBN limits). So, the guest must have arrived and left within a very specific, narrow window of time."

Why This Matters

This is a powerful new tool. Instead of just guessing where to look, physicists now have a map.

• For Masses above 100 MeV: The "BBN Ceiling" is very effective. It tells us that if these particles exist, they must decay within a fraction of a second after the Big Bang.
• Future Experiments: This gives upcoming experiments (like DUNE, SHiP, and ANUBIS) a clear target. They know exactly what kind of "decay speed" and "interaction strength" they need to look for to either find the particle or prove it doesn't exist in that range.

In short, the paper uses the history of the universe's kitchen to set strict rules on where new physics can hide, turning a vague search into a precise hunt.

Technical Summary

1. Problem Statement

The existence of neutrino masses implies physics beyond the Standard Model (SM), often modeled by introducing Heavy Neutral Leptons (HNLs), also known as sterile neutrinos. While minimal scenarios (where HNLs interact only via active-sterile mixing) are well-constrained, more general scenarios involve HNLs interacting through higher-dimensional operators in the Neutrino-extended Standard Model Effective Field Theory ($\nu$SMEFT).

In these general $\nu$SMEFT scenarios, laboratory searches (colliders, beam dumps) typically provide lower bounds on the New Physics (NP) scale $\Lambda$ (i.e., $\Lambda > \Lambda_{\text{min}}$). However, these bounds can always be evaded by increasing $\Lambda$, leaving the parameter space open. The paper addresses the need for complementary upper bounds on $\Lambda$ to define closed "target regions" for future experiments. Specifically, the authors investigate whether Big Bang Nucleosynthesis (BBN) constraints can provide these upper bounds for HNLs with masses in the MeV–GeV range.

2. Methodology

The authors employ a multi-faceted approach combining cosmological thermal history, effective field theory (EFT) matching, and experimental sensitivity projections:

• Theoretical Framework ($\nu$SMEFT and $\nu$LEFT):

  • They extend the SM with right-handed (RH) neutrino fields.
  • They analyze dimension-6 operators in $\nu$SMEFT that involve one HNL field.
  • They perform Renormalization Group Equation (RGE) evolution from the UV scale ($\Lambda$) down to the electroweak scale ($v$) and further to the chiral symmetry-breaking scale ($\sim 1$ GeV) to match onto the low-energy $\nu$LEFT framework.
  • They categorize operators into Right-Handed (RH) currents (inspired by Left-Right Symmetric Models) and Leptoquark (LQ) scenarios.

• Cosmological Analysis (BBN Constraints):

  • Thermal History: They solve the Boltzmann equation to determine the HNL yield ($Y_N$) and freeze-out temperature ($T_f$). They establish conditions for relativistic freeze-out ($T_f \gtrsim M_4$), which allows for a simplified analytic estimate of the HNL abundance without specifying the UV completion.
  • BBN Impact: They analyze how HNLs affect primordial element abundances:
    • Stable HNLs: If HNLs are stable during BBN, they contribute to the energy density, altering the expansion rate ($H$) and spoiling light element predictions. This sets an upper bound on the HNL mass/yield.
    • Decaying HNLs: If HNLs decay during BBN ($\tau \sim 0.1$ s), they can inject energy or alter the neutron-to-proton ratio ($n/p$). Specifically, decays into charged pions enhance the $n \leftrightarrow p$ conversion rate, increasing Helium-4 abundance. This imposes a strict upper bound on the HNL lifetime ($\tau_N \lesssim 0.023$ s for pion decays).
  • Deriving the Upper Bound: By linking the lifetime bound to the production rate (which scales as $\Lambda^{-4}$), they derive an upper bound on the NP scale $\Lambda$.

• Laboratory Constraints:

  • Neutrinoless Double Beta Decay ($0\nu\beta\beta$): They calculate contributions to $0\nu\beta\beta$ mediated by HNLs, particularly for Majorana HNLs, providing lower bounds on $\Lambda$.
  • Displaced Vertex (DV) Searches: They project sensitivities for future experiments (ANUBIS, CODEX-b, FASER, MATHUSLA, SHiP, DUNE) and recast existing limits (BEBC, NA62, etc.) into the $\nu$SMEFT parameter space.

• Benchmark Scenarios:

  • They define three Flavor-Benchmark (FB) scenarios based on dominant production modes: D-meson decay (FB1), Kaon decay (FB2), and B-meson decay (FB3).
  • They compare simplified EFT scenarios with a full UV completion: the minimal Left-Right Symmetric Model (mLRSM).

3. Key Contributions

• Establishment of Upper Bounds: The paper demonstrates that BBN constraints provide robust upper bounds on the EFT cutoff scale $\Lambda$ for HNL masses $M_4 \gtrsim 100$ MeV. This complements the lower bounds from laboratory experiments, effectively "closing" the allowed parameter space in specific regions.
• Validation of Relativistic Freeze-out: They rigorously verify the condition for relativistic freeze-out ($T_f \gtrsim M_4$) for various operators. They show that for masses $M_4 \gtrsim 0.2$ GeV, the simplified BBN lifetime bounds are valid, justifying the use of analytic estimates over full numerical Boltzmann solutions for a wide range of masses.
• Complementarity Analysis: They map out the "target regions" where future experiments can probe the $\nu$SMEFT parameter space. They show that BBN, $0\nu\beta\beta$, and DV searches cover different but overlapping mass ranges, with BBN being crucial for excluding high-$\Lambda$ (weakly coupled) scenarios that would otherwise be invisible to colliders.
• UV Completion Comparison: They validate that simplified EFT operator analyses (turning on one or two Wilson coefficients) serve as excellent approximations for more complex UV-complete models like the mLRSM, particularly regarding the shape of exclusion contours.

4. Key Results

• Mass Range $0.1 < M_4 < 60$ GeV:
  • Low Mass ($M_4 \lesssim 0.7$ GeV): For scenarios involving couplings to first-generation fermions (electrons/up/down quarks), the combination of BBN and current $0\nu\beta\beta$ limits excludes HNLs with masses up to $\sim 0.7$ GeV. Future $0\nu\beta\beta$ experiments could extend this exclusion to $\sim 0.9$ GeV.
  • Intermediate Mass ($0.7 < M_4 < 5$ GeV): In this region, BBN constraints disfavor NP scales $\Lambda > \mathcal{O}(250)$ TeV. Future DV experiments (specifically ANUBIS and SHiP) are projected to probe $\Lambda$ up to $\mathcal{O}(60-70)$ TeV, closing the window between the BBN upper bound and the $0\nu\beta\beta$ lower bound.
  • Muon Coupling: For scenarios where HNLs couple primarily to muons (making $0\nu\beta\beta$ inapplicable), the combination of BBN and DV searches closes the parameter space for $0.1 < M_4 < 0.5$ GeV.
• Leptoquark (LQ) Scenarios: These scenarios exhibit richer phenomenology due to neutral current contributions. However, the exclusion limits are comparable to RH-current scenarios, with BBN and $0\nu\beta\beta$ dominating the low-mass region and DV searches dominating the intermediate mass region.
• mLRSM Results:
  • Without $W_L-W_R$ mixing ($\xi=0$): HNLs are excluded for $0.1 < M_4 < 0.77$ GeV. Future $0\nu\beta\beta$ extends this to $1.0$ GeV.
  • With mixing ($\xi=0.3$): The excluded range extends to $0.82$ GeV (current) and $1.16$ GeV (future).
  • The simplified EFT analysis accurately reproduces the constraints found in the full mLRSM calculation.

5. Significance

• Defining Target Regions: This work is significant because it moves beyond setting isolated lower or upper bounds. By combining cosmological and terrestrial constraints, it identifies finite, well-defined target regions in the $\nu$SMEFT parameter space. This gives experimentalists clear goals: if HNLs exist in these specific mass/coupling windows, they must be discovered by upcoming experiments (DUNE, SHiP, ANUBIS) or ruled out.
• Cosmology as a Precision Tool: It reinforces the role of BBN not just as a probe of early universe physics, but as a critical, model-independent constraint on particle physics models involving light, feebly interacting particles.
• EFT Validity: The study validates the use of EFT techniques for HNLs in the early universe, showing that one can derive robust cosmological bounds without needing a full UV completion, provided the reheating temperature is sufficiently high.
• Guidance for Future Searches: The paper highlights that for HNL masses below the B-meson mass, DV searches are the most powerful terrestrial probes, but they are only effective if the NP scale is not too high. BBN provides the necessary "ceiling" to ensure these searches are not chasing an impossible parameter space.

In summary, the paper establishes that BBN constraints are essential for closing the parameter space of GeV-scale HNLs in general EFT scenarios, transforming the search from an open-ended exploration into a targeted hunt with clear discovery or exclusion prospects.