Baryogenesis in $SU(2)_{L}$ multiplet models

This paper demonstrates that baryogenesis via sphalerogenesis can be successfully realized in Standard Model extensions featuring new $SU(2)_L$ multiplet fields (such as fermionic quintuplets and septuplets) with TeV-scale masses and CP-violating Yukawa interactions, yielding a phenomenologically viable and testable scenario consistent with current electron electric dipole moment limits and future collider searches.

The Gist

The Big Mystery: Why Are We Here?

Imagine the universe right after the Big Bang as a perfectly balanced scale. On one side, there was matter (stuff that makes up stars, planets, and you). On the other side, there was antimatter (the "evil twin" of matter). In a perfect universe, these two should have annihilated each other instantly, leaving behind nothing but empty light.

But here we are. The universe is full of matter. The antimatter is almost entirely gone. This is the Baryon Asymmetry of the Universe (BAU). The big question is: How did the universe cheat the scale and keep the matter?

To do this, the universe needed three specific ingredients (known as the Sakharov conditions):

1. Breaking the rules: A way to destroy or create matter without creating equal antimatter.
2. Bias: A preference for matter over antimatter (CP violation).
3. Chaos: A moment where things weren't calm and balanced (departure from thermal equilibrium).

The Problem with the Standard Model

For a long time, physicists thought the "Electroweak Phase Transition" (a moment when the universe cooled down and particles got mass) was the perfect chaotic event to create this imbalance. However, computer simulations showed that in our current Standard Model, this transition is too smooth—like water slowly turning into ice. It's not chaotic enough to generate the massive amount of matter we see today.

The New Idea: "Sphalerogenesis"

The authors of this paper propose a different mechanism called Sphalerogenesis.

Think of the early universe as a giant, bumpy landscape with deep valleys (stable states) and high hills (unstable states).

• The Sphaleron: Imagine a ball sitting right on the very top of a hill. It's unstable. If it rolls one way, it creates matter; if it rolls the other, it creates antimatter. In a perfect, balanced world, it rolls left and right equally.
• The "Leak": The authors suggest that as the universe cooled, these "hills" (sphalerons) didn't just disappear all at once. Instead, they slowly "froze out" one by one.
• The Bias: If you can make the ball slightly heavier on one side, it will roll that way more often. The paper argues that new, heavy particles (which we haven't found yet) act like a hidden weight, tilting the hill so the ball rolls toward matter more often than antimatter.

The New Players: The "Multiplet" Family

To create this tilt, the authors introduce new particles called SU(2)L multiplets.

• The Analogy: Imagine the Standard Model particles are a standard deck of cards. The new particles are like a special, oversized deck of cards (quintuplets and septuplets) that interact with the old cards in a very specific, secret way.
• The Interaction: These new cards have a "CP-violating" handshake. When they interact with the universe's energy fields, they leave a tiny, invisible "fingerprint" (a mathematical operator called the EW-Weinberg operator) that biases the sphaleron hills.

The Sweet Spot: Heavy but Not Too Heavy

The paper does some heavy math to figure out how heavy these new particles need to be.

• The Result: They need to be about 1 TeV (Tera-electronvolt) heavy. To put that in perspective, that's about 1,000 times heavier than a proton, but light enough that we might be able to create them in the next generation of particle colliders (like the High-Luminosity LHC).
• The Tension: There is a catch. If these particles are too heavy, they can't explain the matter we see. If they are too light, they break other rules. The authors found a "Goldilocks zone" where the mass is just right.

The Detective Work: Checking the Evidence

How do we know if this is true? The authors check two main "crime scenes":

1. The Electron's Spin (Electric Dipole Moment):

   • Imagine an electron as a tiny spinning top. If the universe is perfectly symmetric, the top spins perfectly straight. If there is a bias (CP violation), the top wobbles slightly.
   • Experiments like ACME are looking for this wobble. The authors show that their new particles would cause a wobble, but it's just small enough to have escaped detection so far. However, the next generation of experiments (ACME III) will be sensitive enough to catch this wobble. If they don't find it, this theory is dead.

2. The Particle Collider (LHC):

   • The Large Hadron Collider (LHC) smashes protons together to create new particles.
   • The authors predict that if we smash hard enough at the High-Luminosity LHC (HL-LHC), we should see these new "multiplet" particles popping out, specifically looking for events where a single lepton (like an electron or muon) appears with nothing else (a "mono-lepton" signature).

The Dark Matter Twist

There is one final plot twist. Usually, physicists think these heavy particles could also be Dark Matter (the invisible stuff holding galaxies together).

• The Conflict: To be Dark Matter, these particles usually need to be very heavy (10+ TeV). But to explain the matter/antimatter imbalance, they need to be lighter (around 1 TeV).
• The Conclusion: The authors suggest that while these particles explain why we have matter, they probably aren't the Dark Matter. We will need a different explanation for the invisible stuff in the universe.

Summary

This paper proposes a clever solution to the mystery of why the universe exists:

1. New, heavy particles tilted the playing field in the early universe.
2. This tilt caused a slow, gradual process (sphalerogenesis) to favor matter over antimatter.
3. The math works out perfectly if these particles are around 1 TeV.
4. The best part: We don't have to wait forever to know if this is true. The next generation of electron experiments and the upgraded Large Hadron Collider will either find these particles (and the wobble in the electron) or rule this theory out completely.

It's a concrete, testable story about how the universe cheated the odds to let us exist.

Technical Summary

1. Problem Statement

The origin of the Baryon Asymmetry of the Universe (BAU) remains a central unsolved problem in particle cosmology. The observed baryon-to-entropy ratio is constrained to $n_B/s \approx 8.6 \times 10^{-11}$. Standard Model (SM) extensions often rely on a strong first-order electroweak (EW) phase transition to satisfy the Sakharov condition of departure from thermal equilibrium. However, lattice simulations confirm that the SM EW phase transition is a smooth crossover, rendering standard EW baryogenesis ineffective.

A proposed alternative mechanism, sphalerogenesis, generates BAU during a smooth EW symmetry breaking via the gradual decoupling of CP-violating sphaleron-like processes. While this mechanism works in principle, calculations within the SM alone yield a BAU three orders of magnitude too small. The paper addresses the question: What ultraviolet (UV) complete theory can provide sufficient CP violation to make sphalerogenesis viable while remaining consistent with current experimental bounds?

2. Methodology

The authors employ a multi-step theoretical framework combining Effective Field Theory (EFT), lattice-inspired rates, and specific UV completions:

• Effective Field Theory (SMEFT): They introduce the EW-Weinberg operator (a dimension-six CP-violating operator involving $SU(2)_L$ gauge fields) as the source of CP asymmetry in sphaleron transitions.
  $$\mathcal{L}_{CP} \supset -\frac{g}{3\Lambda^2} f_{ijk} \tilde{W}^i_{\mu\nu} W^{j\nu\rho} W^{k\mu}_{\rho}$$
• Reduced Model for Sphalerons: They utilize a reduced 1+1 dimensional dynamical system description of the sphaleron configuration. The CP asymmetry ($A_{CP}$) is derived from the difference in transition rates for positive and negative Chern-Simons numbers, induced by the cubic term in the effective Hamiltonian arising from the Weinberg operator.
• Boltzmann Equation: They formulate and numerically solve a Boltzmann equation for the baryon number density ($n_B$). The source term ($P(T)$) arises from the decoupling of sphaleron-like configurations, while the washout term ($\Gamma_B(T)$) accounts for active configurations.
• UV Completion: They apply this framework to $SU(2)_L$ multiplet models, specifically introducing new fermionic quintuplets ($r=5$) and septuplets ($r=7$) and complex scalars. These fields possess CP-violating Yukawa interactions that generate the EW-Weinberg operator at the two-loop level.
• Precision Constraints: The analysis incorporates:
  • Electron Electric Dipole Moment (EDM): They use the full three-loop calculation for the electron EDM in these models, which is significantly stronger than the EFT estimate due to threshold corrections.
  • Collider Searches: They compare viable parameter spaces against current LHC mono-lepton searches and projected High-Luminosity LHC (HL-LHC) sensitivities.

3. Key Contributions

• Refined CP Asymmetry Calculation: The authors generalize previous work by introducing two independent deformation parameters ($\alpha, \beta$) for sphaleron-like field configurations (rather than a single parameter). They find this leads to a slight enhancement in the produced baryon asymmetry but confirms that the single-parameter approximation captures the essential physics.
• Full Three-Loop EDM Constraint: A critical contribution is the application of the full three-loop electron EDM calculation (Ref. [13]) to the sphalerogenesis scenario. They demonstrate that the EDM constraint in $SU(2)_L$ multiplet models is approximately three times stronger than the naive EFT estimate ($d_e^{Full} \simeq 3 d_e^{EFT}$). This drastically compresses the viable parameter window.
• UV-Complete Realization: They provide a concrete, testable UV completion for sphalerogenesis, moving beyond the effective field theory description to specific particle content (fermionic quintuplets/septuplets).

4. Key Results

• Viable Mass Scale: To reproduce the observed BAU, the new $SU(2)_L$ multiplet fields must have masses of $O(1)$ TeV.
• Parameter Space:
  • The cutoff scale $\Lambda$ required to explain the BAU lies roughly between 33 TeV and 66 TeV in the EFT approximation.
  • However, when applying the full three-loop EDM constraint, the lower bound on the mass scale shifts significantly, compressing the viable window.
  • Successful baryogenesis favors higher representations (e.g., $r \geq 5$) with fermion masses around 1 TeV.
• Experimental Testability:
  • Electron EDM: The viable parameter regions are within the reach of future EDM experiments like ACME III (sensitivity $\sim 10^{-31}$ e cm).
  • Collider Physics: The HL-LHC (High-Luminosity LHC) via mono-lepton searches is expected to thoroughly probe the parameter space where the BAU is reproduced.
• Dark Matter Tension: The paper highlights a tension between baryogenesis and the standard thermal freeze-out mechanism for Dark Matter (DM). Successful sphalerogenesis prefers light multiplets ($\sim 1$ TeV), whereas the conventional freeze-out mechanism for $SU(2)_L$ multiplets with $r \geq 5$ typically requires masses $> 10$ TeV to avoid overproduction. This suggests that if these models are correct, DM must be generated via a non-freeze-out mechanism.

5. Significance

This work provides the first phenomenologically testable ultraviolet completion of the sphalerogenesis mechanism. It bridges the gap between the abstract effective field theory of sphaleron decoupling and concrete particle physics models.

The study demonstrates that:

1. Sphalerogenesis is a viable mechanism for baryogenesis if new $SU(2)_L$ multiplets exist at the TeV scale.
2. The mechanism is tightly constrained by precision EDM measurements, necessitating a full multi-loop analysis rather than simple EFT estimates.
3. The predicted parameter space is not only consistent with current null results but is fully accessible to upcoming experiments (ACME III and HL-LHC).
4. It offers a specific target for future collider searches (mono-lepton signatures) and suggests a new direction for Dark Matter model building (non-thermal production) to reconcile BAU and DM relic abundance.