Does thermal leptogenesis in a canonical seesaw rely on initial memory?

This paper demonstrates that thermal leptogenesis in the canonical type-I seesaw framework retains a "memory" of asymmetries generated by heavier right-handed neutrinos through flavor-projection effects, which partially survive the washout of the lightest neutrino and significantly modify the final $B-L$ asymmetry beyond the predictions of classical Boltzmann equations.

The Gist

The Big Mystery: Why Are We Here?

Imagine the Big Bang as a giant party where equal amounts of "matter" (us) and "antimatter" (the anti-us) were created. In a perfect world, they should have met, canceled each other out, and left nothing but empty space. But they didn't. For some reason, a tiny bit of matter survived, and that's why you, me, and the stars exist.

Scientists call this the Baryon Asymmetry. The leading theory to explain this is called Leptogenesis. It suggests that heavy, invisible particles (Right-Handed Neutrinos) decayed in the early universe, creating a slight imbalance that eventually turned into the matter we see today.

The Old Story: The "Clean Slate" Theory

For a long time, scientists believed a very simple story about how this happened. They imagined three heavy particles, let's call them N1 (the lightest), N2 (medium), and N3 (the heaviest).

The old theory went like this:

1. N3 and N2 decayed first, creating some imbalance.
2. Then, N1 woke up and started decaying.
3. Because N1 was so active, it acted like a giant eraser. It wiped out any imbalance N2 or N3 had created.
4. Conclusion: Only N1 matters. The universe has no "memory" of what N2 or N3 did. The final result depends entirely on N1.

The New Discovery: The Universe Has a Memory

This paper argues that the "Clean Slate" theory is wrong. The authors used a more advanced mathematical tool (called Density Matrix Equations) to look closer at the process. They found that the universe does have a memory.

Here is the analogy they use:

The "Flavor Vectors" Analogy

Imagine the heavy particles (N1, N2, N3) are artists painting on a canvas.

• N1 paints a red line.
• N2 paints a blue line.
• N3 paints a green line.

In the old theory, everyone thought N1's red paint would completely cover up the blue and green lines, leaving only red.

But the authors discovered that the "paint" isn't just a single color; it has a specific direction or angle (called "flavor").

• Sometimes, N2 paints a blue line that is perfectly parallel to N1's red line. In this case, N1 does erase it.
• However, often N2 paints a blue line that is perpendicular (at a 90-degree angle) to N1's red line.

If N2 paints a line that is perpendicular to N1, N1's "eraser" (which only works along its own red line) cannot reach it. The blue line survives!

This is the "Memory Effect." Even though N1 is active and trying to wipe the slate clean, it misses the parts of the imbalance created by N2 and N3 because they are pointing in a different direction.

The Four Scenarios

The authors checked this idea under four different conditions (based on how "strong" the erasing power of each particle is):

1. All Strong: Everyone is a strong eraser. Even here, if the angles are right, N2 and N3 leave a mark.
2. N1 is Weak: N1 is a weak eraser. N2 and N3 leave a huge mark.
3. N2 is Weak: N2 is a weak eraser. Its mark survives easily.
4. N3 is Weak: N3 is a weak eraser. Its mark survives easily.

In almost every case, they found that the "perpendicular" marks survived, changing the final amount of matter in the universe.

Why This Matters for Experiments

The paper also connects this to a real-world experiment called Neutrinoless Double Beta Decay. This is an experiment trying to prove that neutrinos are their own antiparticles.

• The Old View: If you use the simple "N1 only" theory, the experiment needs to look for very heavy particles to explain the universe's matter.
• The New View: Because of the "Memory Effect" (the perpendicular angles), the universe can create the right amount of matter with lighter particles than we thought.

This means the "Memory Effect" opens up a new range of possibilities. It suggests that experiments like nEXO and LEGEND (future detectors) might actually be able to find the evidence for this theory, whereas the old theory said they wouldn't be sensitive enough.

Summary

• Old Idea: The lightest particle (N1) wipes out all history. Only N1 matters.
• New Idea: N1 is like a broom that only sweeps in one direction. If the other particles (N2, N3) leave their "dirt" in a different direction, the broom misses it.
• Result: The universe keeps a "memory" of the heavier particles. This changes the math, allows for lighter particles to explain our existence, and brings the theory within reach of upcoming experiments.

Technical Summary

Technical Summary: Thermal Leptogenesis and Initial Memory in the Canonical Seesaw

Problem Statement
In the standard treatment of thermal leptogenesis within the type-I seesaw framework, a prevailing assumption holds that the final $B-L$ (baryon minus lepton) asymmetry is determined solely by the CP-violating, out-of-equilibrium decay of the lightest right-handed neutrino (RHN), denoted as $N_1$. Under this paradigm, any asymmetry generated by the decays of heavier RHNs ($N_2$ and $N_3$) is presumed to be completely erased by subsequent washout processes mediated by $N_1$. This assumption effectively decouples the dynamics of the heavier states from the final asymmetry, allowing for a simplified analysis using classical Boltzmann equations that track only total lepton number densities. However, this "flavor-blind" approximation relies on the premise that lepton doublets produced by different RHNs are indistinguishable and perfectly aligned in flavor space, a condition that may not hold given the specific Yukawa coupling textures required by low-energy neutrino data.

Methodology
The authors revisit the validity of the single-RHN dominance assumption by employing the density-matrix formalism rather than classical Boltzmann equations. This approach is necessary to consistently track the evolution of coherence among different heavy-neutrino flavor states and the corresponding asymmetries. The methodology involves:

1. Formalism: Solving the density-matrix equations (DMEs) that include decay, inverse decay, and relevant scattering processes ($\Delta L = 1$ and $\Delta L = 2$). The formalism accounts for "flavor-projection effects," where the lepton states produced by distinct RHNs are treated as vectors in flavor space. If these vectors are not parallel, components of the asymmetry generated by $N_2$ and $N_3$ can be orthogonal to the washout direction of $N_1$, thereby surviving the erasure process.
2. Parameter Space Constraints: The study imposes consistency with low-energy neutrino mass and mixing data (oscillation parameters within the $3\sigma$ range). Using the Casas-Ibarra parameterization, the authors analyze the washout regimes defined by the parameter $K_i = \tilde{m}_i / m^*$. They demonstrate analytically and numerically that, given current neutrino data, at most one of the three RHNs can reside in the weak washout regime ($K_i < 1$).
3. Dynamical Regimes: Based on the constraint that only one RHN can be in the weak washout regime, the authors divide the parameter space into four distinct dynamical cases:
   • Case I: All RHNs in the strong washout regime.
   • Case II: $N_1$ in weak washout; $N_2, N_3$ in strong.
   • Case III: $N_2$ in weak washout; $N_1, N_3$ in strong.
   • Case IV: $N_3$ in weak washout; $N_1, N_2$ in strong.
4. Quantification: The "memory effect" is quantified by the parameter $\delta$, representing the percentage deviation of the final $B-L$ asymmetry when all RHNs are included versus when only $N_1$ is considered.

Key Contributions and Results

• Existence of the Memory Effect: The paper demonstrates that asymmetries generated by heavier RHNs ($N_2, N_3$) generally possess components misaligned in flavor space relative to $N_1$. Consequently, these components are partially protected from $N_1$-mediated washout. This "memory effect" persists even when $N_1$ remains dynamically relevant and cannot be captured by classical Boltzmann frameworks.
• Regime-Dependent Behavior:
  • In the All Strong Washout regime, the memory effect varies significantly (ranging from 70% to 130% in specific benchmark points) depending on the Yukawa coupling texture and mass hierarchies ($M_2/M_1, M_3/M_1$). In cases where the projection matrices indicate near-orthogonality between $N_2/N_3$ and $N_1$ decay directions, the contribution of heavier states is substantial.
  • In regimes where $N_1$ is in weak washout, the pre-existing asymmetry from $N_2$ and $N_3$ persists largely because $N_1$ lacks the efficiency to wash it out, leading to large memory effects.
  • In regimes where $N_2$ or $N_3$ is in weak washout, the asymmetry from these states is protected, and their contribution significantly modifies the final $B-L$ asymmetry, often preventing underproduction or overproduction scenarios predicted by single-flavor approximations.
• Analytical vs. Numerical Discrepancy: The authors compare their full numerical DME solutions with previous analytical approximations. They find that for moderate mass hierarchies (specifically $M_2/M_1 \lesssim 15-20$), analytical approximations systematically underestimate the baryon asymmetry by 10–50%. This deviation arises because off-diagonal entries in the density matrix contribute appreciably when $N_2$ remains thermally populated during the $N_1$ dynamics epoch.
• Connection to Neutrinoless Double Beta Decay ($0\nu\beta\beta$): The study investigates the correlation between the lightest RHN mass ($M_1$) and the effective neutrino mass ($m_{\beta\beta}$) relevant for $0\nu\beta\beta$ experiments. The inclusion of projection effects allows for successful leptogenesis at significantly lower values of $M_1$ for a fixed $m_{\beta\beta}$ compared to the classical Boltzmann treatment. This expands the viable parameter space into the sensitivity range of current and future $0\nu\beta\beta$ experiments (e.g., KamLAND-Zen, nEXO, LEGEND).

Significance and Claims
The paper claims to establish that the standard reduction of hierarchical leptogenesis to an effective single-RHN description is not universally valid. The "memory effect" is a general phenomenon arising from flavor projection effects, distinct from the specific "N2-dominated" scenarios previously discussed in literature which often relied on specific washout hierarchies or classical equations.

The authors assert that:

1. Flavor alignment is not guaranteed: The assumption that lepton vectors from different RHNs are parallel is generally false and depends critically on the Yukawa coupling texture.
2. Density matrix effects are crucial: Accurate determination of the final baryon asymmetry requires solving density-matrix equations, particularly when mass hierarchies are not extreme.
3. Experimental Implications: By incorporating projection effects, the viable parameter space for thermal leptogenesis is significantly enlarged. This enlargement brings a substantial portion of the parameter space into the reach of neutrinoless double beta decay experiments, providing a potential pathway to test the seesaw mechanism and leptogenesis dynamics through low-energy observables.

The work concludes that the interplay between the dynamics of leptogenesis and low-energy lepton-number-violating observables is non-trivial, and ignoring flavor projection effects can lead to incorrect conclusions regarding the viability of the canonical seesaw framework for explaining the baryon asymmetry of the universe.