Leptogenesis from the Dirac CP-violating phase in the minimal left-right symmetric model

This paper demonstrates that in the Minimal Left-Right Symmetric Model with generalized parity, the low-energy Dirac CP-violating phase alone can successfully generate the observed baryon asymmetry of the universe, offering a testable link between neutrino physics and cosmology that depends critically on the neutrino mass ordering and scale.

The Gist

The Cosmic Mystery: Why is There Something Rather Than Nothing?

Imagine the universe as a giant, cosmic kitchen. According to the laws of physics, when you cook up energy, you should create equal amounts of "matter" (the ingredients) and "antimatter" (the anti-ingredients). If you mix them perfectly, they should annihilate each other instantly, leaving behind nothing but a cold, empty soup of light.

But here we are. The universe is full of stars, planets, and you. This means something went wrong in the recipe. At some point, the universe produced a tiny bit more matter than antimatter. This leftover "extra" is what makes up everything we see today. Physicists call this the Baryon Asymmetry.

The big question is: Where did that extra matter come from?

The Recipe Book: The Minimal Left-Right Symmetric Model

For decades, scientists have tried to write a "recipe book" to explain this. One of the most promising recipes is called the Minimal Left-Right Symmetric Model (MLRSM).

Think of this model as a mirror. In our everyday world, left and right are just directions. But in this model, the universe has a hidden "mirror dimension."

• Left-handed particles are like our familiar world.
• Right-handed particles are their mirror twins, which we haven't seen yet because they live at incredibly high energy levels (like a secret ingredient hidden in a locked vault).

This model suggests that heavy, mirror-twin particles (Right-Handed Neutrinos) existed in the early universe. When they decayed (died out), they might have spilled a little extra matter into the mix, saving the universe from total annihilation.

The Missing Link: The "Secret Sauce" of Neutrinos

The problem with this recipe is that we don't know the exact measurements. We know the ingredients (neutrinos) exist, but we don't know the "mixing ratios" (how they interact).

Usually, to calculate how much matter was created, you need to know two things:

1. Low-energy settings: Things we can measure in labs today (like the "flavor" of neutrinos).
2. High-energy settings: Things that happened billions of years ago at energies we can't reach with our current particle colliders.

It's like trying to bake a cake today based on a recipe from 10,000 years ago, but the part of the recipe that says "how much sugar to add" is written in a language we don't speak. Usually, scientists have to guess the "sugar" (high-energy parameters), which makes the prediction unreliable.

The Breakthrough: Finding the "Real" Ingredient

This paper, by Chen and Zhang, found a clever way to solve the "missing sugar" problem.

They looked at the rules of the MLRSM model again. They realized that if the universe follows a specific symmetry rule (called Parity, or P-symmetry), the "mixing matrix" (the recipe for how particles interact) has a special property: it must be "Hermitian."

The Analogy:
Imagine you are trying to balance a scale.

• On one side, you have the Light Neutrinos (the ones we can see).
• On the other side, you have the Heavy Neutrinos (the hidden ones).
• The "Hermitian" rule is like a strict law of physics that says: "The weight on the left side must perfectly match the weight on the right side, but flipped."

Because of this strict rule, the authors realized they didn't need to guess the high-energy parameters anymore. The math forced the "Right-Handed" mixing matrix to be real numbers (no imaginary, complex twists).

The Magic Ingredient: The Dirac Phase

Once they forced the "Right-Handed" part to be simple and real, they looked at the "Left-Handed" part (the neutrinos we know). They found something amazing:

The only thing left that could create the imbalance between matter and antimatter was a single number called the Dirac CP-violating phase (let's call it $\delta$).

The Metaphor:
Think of the universe as a giant orchestra.

• The Majorana phases (other complex numbers in the recipe) are like the musicians tuning their instruments to be perfectly in sync (Conserving CP).
• The Dirac phase ($\delta$) is the conductor waving the baton slightly off-beat.

The authors showed that if the "Right-Handed" orchestra is perfectly tuned (real), the only way to get the "extra matter" is if the conductor ($\delta$) waves the baton in a specific, slightly "off" way. If the conductor waves perfectly straight, the universe would have been empty. If the conductor waves just right, we get a universe full of stars.

What They Did and What They Found

The authors ran massive computer simulations (like running the universe through a video game engine) to test this idea. They tested four different scenarios of how the heavy particles might have decayed.

The Results:

1. It Works: They found that the single "off-beat" wave of the conductor ($\delta$) is enough to create exactly the amount of matter we see in the universe today.
2. It's Picky: The recipe is very sensitive. The result depends heavily on:
   • The Mass Ordering: Are the lightest neutrinos heavy or light? (Like choosing between a small or large cake pan).
   • The Lightest Mass: How heavy is the smallest neutrino?
   • The Angle ($\delta$): The exact angle of the conductor's wave.
3. A Testable Prediction: This is the best part. Because the result is so sensitive to the angle $\delta$, future experiments (like DUNE or Hyper-K) that measure this angle precisely will be able to say: "Yes, this model is correct," or "No, this model is wrong."

The Big Picture

In simple terms, this paper says:

"We don't need to guess the secret high-energy ingredients of the universe. If the universe follows the 'Left-Right Mirror' rules, the only thing that matters is a specific angle in the neutrino's behavior. If we measure that angle correctly, we can prove exactly how the universe got its matter."

It turns a mystery about the beginning of the universe into a testable experiment for the near future. It's like finding the missing page of the universe's recipe book, and realizing that the only thing we need to check is the temperature of the oven.

Technical Summary

Here is a detailed technical summary of the paper "Leptogenesis from the Dirac CP-violating phase in the minimal left-right symmetric model."

1. Problem Statement

The origin of the Baryon Asymmetry of the Universe (BAU) is a fundamental open question in cosmology. Leptogenesis, the generation of a lepton asymmetry that is subsequently converted into a baryon asymmetry via sphaleron processes, is a leading theoretical framework. However, a major obstacle in connecting low-energy neutrino physics (observed in oscillation experiments) to high-energy leptogenesis is the presence of unknown high-energy parameters.

Specifically, in standard seesaw models, the neutrino Dirac mass matrix ($M_D$) depends on an arbitrary complex orthogonal matrix ($R$) in the Casas-Ibarra parametrization. This $R$-matrix introduces unknown high-energy CP-violating phases that obscure the link between the low-energy Dirac CP-violating phase ($\delta$) and the final baryon asymmetry. The paper addresses the question: Can the observed BAU be generated solely by the low-energy Dirac CP-violating phase ($\delta$) without relying on unknown high-energy parameters?

2. Methodology and Framework

The authors investigate this problem within the Minimal Left-Right Symmetric Model (MLRSM), which extends the Standard Model with a gauge group $SU(2)_L \times SU(2)_R \times U(1)_{B-L}$.

• Symmetry Constraints: The model assumes Generalized Parity ($P$) as the discrete left-right symmetry. Under $P$-transformation, the Yukawa coupling matrices are Hermitian ($Y = Y^\dagger$). This leads to a Hermitian Dirac neutrino mass matrix ($M_D = M_D^\dagger$).
• Determining $M_D$: Unlike standard seesaw models where $M_D$ is unknown, the Hermiticity of $M_D$ in the MLRSM allows it to be uniquely determined by the light neutrino mass matrix ($M_\nu$), the heavy neutrino mass matrix ($M_N$), and the vacuum expectation values (VEVs) of the scalar triplets. This eliminates the arbitrary $R$-matrix uncertainty.
• Constraint on Mixing Matrices: The Hermiticity of $M_D$ imposes strict constraints on the right-handed mixing matrix ($V_R$). Specifically, the trace of powers of the matrix $H$ (where $M_D = \sqrt{M_N} H \sqrt{M_N^*}$) must be real. This yields three conditions:
  $$\text{Im}\left[\text{Tr}(H H^\dagger)^n\right] = 0, \quad n=1, 2, 3$$
• Parameter Reduction: To solve these constraints, the authors make minimal assumptions:
  1. They set all phases in the $V_R$ matrix to zero (making $V_R$ real).
  2. They find that for a real $V_R$, the Majorana phases in the light neutrino mixing matrix ($V_L$) must be CP-conserving (i.e., $0$or$\pi$) to satisfy the trace conditions.
  3. Consequently, the Dirac CP-violating phase ($\delta$) remains the sole source of CP violation in the system.
• Leptogenesis Scenarios: The study numerically explores four distinct thermal leptogenesis scenarios in the MLRSM, categorized by the mass hierarchy between the Right-Handed Neutrinos ($N$) and the Scalar Triplet ($\Delta$), and the dominance of Type-I vs. Type-II seesaw contributions:
  • Scenarios 1 & 2: $m_N < m_\Delta$ (Decay of RHNs).
  • Scenarios 3 & 4: $m_N > m_\Delta$ (Decay of Triplet).
  • Each scenario is further divided by whether Type-I or Type-II seesaw dominates the light neutrino mass.

3. Key Contributions

• Elimination of High-Energy Ambiguity: The paper demonstrates that in the MLRSM with Parity symmetry, the hermiticity of $M_D$ naturally constrains the high-energy sector, removing the need for the arbitrary Casas-Ibarra $R$-matrix.
• Isolation of the Dirac Phase: By enforcing the real-trace conditions derived from $P$-symmetry, the authors show that the Majorana phases are forced to be CP-conserving. This isolates the Dirac phase $\delta$ as the unique generator of the baryon asymmetry.
• Comprehensive Numerical Analysis: The authors perform a full numerical scan of the Boltzmann equations for all four leptogenesis scenarios, considering both Normal Ordering (NO) and Inverted Ordering (IO) of neutrino masses, and three benchmark values for the lightest neutrino mass ($m_{min}$).

4. Results

• Feasibility of Dirac-Phase Leptogenesis: The numerical results confirm that viable leptogenesis is achievable in all four scenarios using only the Dirac CP-violating phase $\delta$. The models successfully reproduce the observed baryon asymmetry ($Y_B \approx 8.7 \times 10^{-11}$) with the correct sign.
• Sensitivity to Parameters:
  • Mass Ordering: The predicted asymmetry shows distinct behaviors for Normal vs. Inverted ordering.
  • Lightest Neutrino Mass ($m_{min}$): The results are highly sensitive to the absolute mass scale.
  • Dirac Phase ($\delta$): The dependence on $\delta$ is non-trivial and sharp. For a fixed $m_{min}$, varying $\delta$ can shift the predicted $Y_B$ by one to two orders of magnitude.
• Intersection with Experimental Data: The study identifies specific regions in the $(\delta, m_{min})$ parameter space where the predicted $Y_B$ matches the experimental value within the $3\sigma$ confidence level of the Dirac phase (as measured by current oscillation data like NuFit).
• No Universal Behavior: There is no single functional form linking $\delta$ to $Y_B$ across all scenarios; the relationship is highly scenario-dependent, reflecting the complex interplay of the MLRSM parameters.

5. Significance and Implications

• Testable Framework: This work provides a rigorous, testable framework connecting low-energy neutrino oscillation data to the early universe. Future precision measurements of $\delta$ (by experiments like DUNE, JUNO, and Hyper-K) and the neutrino mass ordering will directly constrain the viability of thermal leptogenesis in the MLRSM.
• Probing High-Energy Scales: By inverting the relationship, the observed baryon asymmetry can be used to probe the mass scale of the heavy particles ($m_H$) in the MLRSM. Since these scales ($10^{12} - 10^{14}$ GeV) are far beyond the reach of current colliders, this offers a rare indirect window into physics at the Grand Unification scale.
• Theoretical Economy: The study achieves a "minimal" explanation for the BAU by relying on the symmetries of the model rather than ad-hoc assumptions about Yukawa couplings, strengthening the case for Left-Right Symmetry as a viable extension of the Standard Model.

In conclusion, the paper establishes that the Dirac CP-violating phase alone can account for the matter-antimatter asymmetry of the universe within the Minimal Left-Right Symmetric Model, provided the model respects Parity symmetry and the Majorana phases are CP-conserving. This creates a direct, high-precision link between neutrino oscillation experiments and cosmological observations.