Revisiting constraints on magnetogenesis from baryon asymmetry

This paper revisits the viability of primordial U(1)$_Y$ magnetic fields as a unified origin for both intergalactic magnetic fields and the baryon asymmetry of the universe, finding that maximally helical fields can successfully explain both phenomena while identifying a narrow parameter window for non-helical fields contingent on precise Higgs dynamics during the electroweak crossover.

The Gist

The Big Mystery: Two Cosmic Puzzles

Imagine the universe as a giant, expanding balloon. Scientists have been trying to solve two major mysteries about this balloon:

1. The Invisible Web: We know there are magnetic fields floating in the empty space between galaxies (the "cosmic voids"). They are incredibly weak, but they are there. We don't know where they came from. Did they appear right after the Big Bang, or were they created later by stars and galaxies?
2. The Matter-Antimatter Imbalance: When the universe began, it should have created equal amounts of matter (like us) and antimatter (the evil twin that annihilates matter on contact). If that happened, everything would have cancelled out, leaving only light. But we are here! There is way more matter than antimatter. We need to know why the universe decided to keep the matter.

For a long time, scientists thought these two mysteries were unrelated. However, this paper suggests they might be two sides of the same coin.

The Old Theory: A Broken Bridge

Previously, scientists proposed a scenario involving primordial magnetic fields (magnetic fields created in the very first fraction of a second of the universe).

Think of the early universe as a hot, chaotic soup. In this soup, there were magnetic fields swirling around. As the universe cooled down, a major event happened called the Electroweak Crossover. Imagine this as the universe freezing from a liquid into a solid.

• The Old Idea: When the universe "froze," these magnetic fields were supposed to twist and turn. This twisting was thought to act like a machine that could separate matter from antimatter, creating the imbalance we see today.
• The Problem: When scientists did the math, they found a fatal flaw. To create enough magnetic fields to explain what we see in the voids today, the "machine" would have to produce too much matter. It would be like trying to fill a teacup with a firehose; the universe would have been flooded with matter, making it impossible for us to exist as we do.

Because of this "flood," scientists concluded that this theory was dead. The magnetic fields couldn't be the origin of our matter.

The New Twist: The "Higgs" Valve

This paper, written by Hamada, Mukaida, and Uchida, says: "Wait a minute! We missed a crucial part of the machine."

They revisited the physics of the Higgs field (the field that gives particles mass). They realized that during the "freezing" of the universe, the Higgs field doesn't just sit there; it actively participates.

The Analogy: The Leaky Bucket
Imagine the process of creating matter is like filling a bucket with water (matter) using a hose (the magnetic fields).

• The Old View: The hose was wide open. It filled the bucket so fast it overflowed (too much matter).
• The New View: The Higgs field acts like a smart valve or a leak in the bucket. Depending on how the Higgs field behaves, it can drain away most of the excess water.

The authors found that if this "Higgs valve" works efficiently, it can drain away the extra matter that the magnetic fields were trying to create. This changes the rules completely.

The Two Scenarios: Helical vs. Non-Helical

The paper explores two types of magnetic fields, using the analogy of twisted ropes vs. straight ropes.

1. The Twisted Ropes (Helical Fields)

Imagine a magnetic field that is twisted like a corkscrew or a DNA strand. This is called a "helical" field.

• The Result: If the "Higgs valve" is very efficient (draining away 99.9999999% of the excess matter), these twisted ropes can perfectly explain both the magnetic fields in space today and why we have more matter than antimatter.
• The Catch: The valve has to be incredibly precise. If it's even slightly off, we get too much matter again. But if it works, it's a beautiful solution where one cause creates two effects.

2. The Straight Ropes (Non-Helical Fields)

Imagine a magnetic field that isn't twisted, just straight lines.

• The Problem: These fields create "ripples" in the distribution of matter. Imagine dropping a stone in a pond; the water isn't flat anymore. In the early universe, these ripples would have messed up the formation of the first elements (like Deuterium) during the Big Bang.
• The Result: The paper finds a tiny "window" where these straight ropes could exist without causing a disaster, but only if the Higgs valve is extremely good at its job (even more precise than the twisted ropes).

The Conclusion: A New Hope

The main takeaway is that a theory previously thought to be impossible is actually alive and kicking, provided we account for the subtle behavior of the Higgs field.

• Before: "Magnetic fields can't explain our existence because they'd create too much matter."
• Now: "If the Higgs field acts as a precise regulator, magnetic fields created in the first second of the universe could be the reason we have magnetic fields today AND the reason we exist at all."

Why Does This Matter?

This paper is like finding a missing piece of a jigsaw puzzle. It suggests that the universe's magnetic fields and the existence of human beings might share a common birthright. It opens up a new "window" for scientists to look for evidence. If we can measure the magnetic fields in the voids with enough precision, we might be able to test if this "Higgs valve" theory is true, giving us a deeper understanding of the very first moments of our existence.

In short: The universe might have used a cosmic "twist" in its magnetic fields to create matter, and the Higgs field acted as a fine-tuner to make sure we didn't get too much of a good thing.

Technical Summary

1. Problem Statement

The origin of intergalactic magnetic fields (IGMFs) and the baryon asymmetry of the universe (BAU) are two major open questions in cosmology.

• IGMFs: Observations of TeV blazars suggest the existence of IGMFs in cosmic voids with a lower bound of $B_{\text{void}} > 10^{-17}$ G. A leading hypothesis is that these are relics of primordial hypermagnetic fields ($U(1)_Y$) generated before the Electroweak Symmetry Breaking (EWSB).
• The Conflict: Previous literature (e.g., Ref. [30]) argued that primordial $U(1)_Y$ magnetic fields strong enough to explain the observed IGMFs would inevitably generate a baryon asymmetry far exceeding the observed value ($\eta_B \sim 9 \times 10^{-11}$) or create unacceptable baryon isocurvature perturbations that violate Big Bang Nucleosynthesis (BBN) constraints. Consequently, the scenario where primordial fields explain both IGMFs and BAU was considered non-viable.

The Core Issue: The previous constraints relied on theoretical assumptions regarding how magnetic helicity decays during the electroweak crossover and how this decay couples to the generation of baryon number via the Adler-Bell-Jackiw (ABJ) anomaly. The authors argue that recent advances in understanding the hot electroweak theory, specifically the role of the Higgs field dynamics, introduce a crucial uncertainty that was previously overlooked.

2. Methodology

The authors revisit the magnetogenesis-to-baryogenesis conversion process using a refined theoretical framework based on recent lattice studies and effective field theory (Ref. [44]).

• Theoretical Framework:

  • They analyze the transition of $U(1)_Y$ magnetic fields to $U(1)_{\text{em}}$ fields during the electroweak crossover.
  • They introduce a gauge-independent effective mixing angle ($\theta_{\text{eff}}$) that interpolates between the hypermagnetic field and the electromagnetic field, accounting for the Higgs condensate dynamics.
  • They distinguish between two types of magnetic flux: unconfined (massless, $U(1)_{\text{em}}$-like) and confined (massive, $Z$-boson like).

• Mechanism of Baryogenesis:

  • Baryon number ($B+L$) is generated via the ABJ anomaly when the Chern-Simons number ($N_{\text{CS}}$) or magnetic helicity ($H_Y$) changes.
  • The authors identify two competing relaxation processes for the decay of confined helicity:
    1. Sphaleron-driven ($N_{\text{CS}}$ change): The confined flux decays by generating $SU(2)_L$ gauge fields, leading to significant baryon production.
    2. Higgs-driven ($N_H$ change): The confined flux decays via Higgs dynamics (e.g., Nambu monopole pair production) without generating $SU(2)_L$ gauge fields. In this case, the confined helicity decay does not generate baryon number.

• Parameterization:

  • They introduce an efficiency parameter $\alpha$ ($0 \le \alpha \le 1$) to quantify the fraction of confined helicity decay that proceeds via the sphaleron channel (generating baryons) versus the Higgs channel (suppressing baryons).
    • $\alpha = 1$: Full sphaleron contribution (standard assumption).
    • $\alpha = 0$: Full Higgs compensation (no baryon generation from confined decay).
    • Intermediate values: Mixed dynamics.

• Calculations:

  • They derive formulas for the net baryon asymmetry ($\eta_B$) arising from both unconfined helicity dissipation (always present) and confined helicity decay (dependent on $\alpha$).
  • They calculate the resulting baryon isocurvature perturbations for non-helical fields, which are constrained by primordial deuterium abundance.
  • They evolve the magnetic field parameters ($B_i, \xi_{M,i}$) from the electroweak scale ($T \sim 130$ GeV) to the present day using Magnetohydrodynamic (MHD) scaling laws (conservation of magnetic helicity for helical fields and Hosking integral for non-helical fields).

3. Key Contributions

1. Refined Theoretical Uncertainty: The paper identifies that the previous exclusion of primordial magnetogenesis scenarios was based on the assumption that confined helicity decay always generates baryons. The authors demonstrate that Higgs dynamics can suppress this generation, introducing the parameter $\alpha$.
2. Re-evaluation of Constraints: By varying $\alpha$, they map out the allowed parameter space for primordial magnetic fields that satisfy both the observed BAU and IGMF lower bounds.
3. Dual Origin Scenario: They establish a viable window where a single primordial mechanism (helical $U(1)_Y$ fields) can simultaneously explain the origin of the IGMFs and the BAU.

4. Results

The authors present constraints on the initial magnetic field strength ($B_i$) and coherence length ($\xi_{M,i}$) at the electroweak scale ($T \approx 130$ GeV).

• Maximally Helical Fields ($\epsilon_i = 1$):

  • Standard Case ($\alpha = 1$): The constraints are tight. Fields strong enough to explain IGMFs ($B \sim 10^{-16}$ G today) would overproduce baryons.
  • Higgs-Compensated Case ($\alpha \lesssim 10^{-9}$): If the Higgs dynamics suppress confined helicity decay by a factor of $10^{-9}$ or more, the baryon overproduction constraint is relaxed.
  • Conclusion: In this specific window, maximally helical primordial fields can explain both the observed IGMFs and the BAU. The required initial conditions are roughly $B_i \sim 10^{-16}$ G and $\xi_{M,i} \sim 10^{-1}$ Mpc (inflationary generation).

• Non-Helical Fields ($|\epsilon_i| \ll 1$):

  • These fields are constrained by baryon isocurvature perturbations (spatial fluctuations in baryon density).
  • Standard Case ($\alpha = 1$): Strong constraints exclude fields capable of explaining IGMFs.
  • Higgs-Compensated Case ($\alpha \lesssim 10^{-10}$): If the Higgs dynamics are efficient enough ($\alpha \sim 10^{-10}$), the isocurvature constraints are relaxed.
  • Conclusion: A window opens for non-helical fields to explain IGMFs without violating BBN constraints, provided $\alpha$ is sufficiently small.

• MHD Evolution: The authors show that fields generated at the electroweak scale evolve via MHD processes (inverse cascade for helical, Hosking integral conservation for non-helical) to reach the present-day values observed by blazars.

5. Significance

• Revival of Primordial Magnetogenesis: This work overturns the previous consensus that primordial $U(1)_Y$ magnetic fields are ruled out as the source of IGMFs due to baryon constraints. It demonstrates that the scenario is viable if the specific dynamics of the Higgs field during the electroweak crossover are taken into account.
• Unified Origin: It proposes a compelling unified origin for two distinct cosmological phenomena: the baryon asymmetry and the intergalactic magnetic fields.
• Theoretical Implications: The results highlight the critical importance of non-perturbative Higgs dynamics (specifically the interplay between confined flux decay and monopole production) in the early universe. The parameter $\alpha$ represents a new target for future lattice QCD/Electroweak simulations to determine its precise value.
• Observational Outlook: The paper provides specific predictions for the initial conditions of primordial magnetic fields that future observations of CMB distortions, BBN abundances, and IGMF measurements can test.

In summary, the paper argues that maximally helical primordial magnetic fields can be the origin of both the intergalactic magnetic fields and the baryon asymmetry, provided that the Higgs dynamics during the electroweak crossover suppress the baryon-generating channel of confined helicity decay with a precision of $\alpha \lesssim 10^{-9}$. Similarly, non-helical fields are viable if $\alpha \lesssim 10^{-10}$.