Non-anomalous axions: lessons from the Majoron

1. Problem Statement

The paper addresses a fundamental ambiguity in the theoretical construction of Majoron models (Nambu–Goldstone bosons associated with spontaneous lepton number violation) and the broader class of Axion-Like Particles (ALPs).

• The Conventional View: In standard Majoron models, the global symmetry spontaneously broken is typically identified as Lepton number ($L$). Since $L$ is anomalous under the Standard Model (SM) electroweak gauge groups ($SU(2)_L \times U(1)_Y$), the resulting Majoron is often treated as a pseudo-Nambu–Goldstone boson (pNGB) or an axion-like particle with a mass generated by these anomalies.
• The Inconsistency: This view leads to two major issues:
  1. Domain Wall Problem: If the broken symmetry is effectively discrete (due to the anomaly breaking the continuous $U(1)_L$ down to a $Z_n$ subgroup), the spontaneous symmetry breaking (SSB) can generate topological defects known as domain walls, which are cosmologically problematic.
  2. Misidentification of Origin: There is a confusion regarding whether the observation of an effective coupling of the form $a F \tilde{F}$ (where $a$ is the boson and $F$ is a gauge field strength) uniquely implies an anomalous origin.
• The Core Question: When a complex scalar field spontaneously breaks both an anomalous symmetry (like $L$) and a non-anomalous symmetry (like $B-L$) simultaneously, which symmetry does the resulting Nambu–Goldstone boson (NGB) actually correspond to?

2. Methodology

The author employs a rigorous field-theoretic analysis using the following steps:

• Framework Definition: The paper constructs a general Lagrangian containing a symmetry-invariant part ($\mathcal{L}_{inv}$) and explicit symmetry-breaking terms ($\mathcal{O}_{B-L}, \mathcal{O}_{B+L}, \mathcal{O}_B, \mathcal{O}_L$).
• Symmetry Transformation Analysis:
  • The author analyzes the transformation of the scalar field $\phi$ (which acquires a Vacuum Expectation Value, VEV) under different global symmetries ($L$, $B$, and linear combinations like $B-L$).
  • Using the polar parametrization $\phi = e^{i q_\phi \pi(x)} \rho$, the author tracks how the NGB field $\pi(x)$ (the Majoron) shifts under these transformations.
• Anomaly vs. Explicit Breaking:
  • The author distinguishes between symmetry breaking caused by quantum anomalies (which generate a $\theta F \tilde{F}$ term) and explicit breaking by operators in the Lagrangian.
  • The key argument involves showing that if one attempts to define the Majoron via the anomalous $L$ symmetry, the anomaly term appears "spurious" because it can be reabsorbed by a $B$ transformation, given that $B-L$ is anomaly-free.
• Perturbative vs. Non-Perturbative Analysis: The paper examines the origin of the Majoron mass. It demonstrates that if $B-L$ is conserved, the Majoron remains massless even if $L$ is broken by anomalies, provided no explicit $B-L$ breaking operators exist.
• Toy Model Construction: To illustrate the phenomenological consequences, the author constructs a specific extension of the SM involving:
  • A global $B-L$ symmetry.
  • An SM-singlet scalar $\phi$ with $B-L = -2$.
  • New heavy Vector-Like (VL) charged leptons that mix with SM leptons.
  • Calculation of the one-loop diagram generating the $J \gamma \gamma$ coupling.

3. Key Contributions and Results

A. The Majoron is Associated with $B-L$, not $L$

The paper proves that when a scalar VEV breaks both an anomalous symmetry ($L$) and a non-anomalous symmetry ($B-L$) simultaneously, the resulting physical NGB is uniquely associated with the non-anomalous symmetry ($B-L$).

• Reasoning: The shift symmetry protecting the masslessness of the NGB is determined by the symmetry that remains exact (or is only broken by explicit terms, not anomalies). Since $B-L$ is free of gauge anomalies, the Majoron is the NGB of $B-L$.
• Implication: The "anomalous" coupling to gauge bosons often attributed to the Majoron is actually a consequence of the specific basis chosen for the transformation, not an intrinsic property of the boson's origin.

B. Resolution of the Domain Wall Problem

• Result: Majoron models are safe from the domain wall problem as long as there is no explicit breaking of $B-L$.
• Mechanism: Even if $L$ is broken by the electroweak anomaly (which would normally imply a discrete residual symmetry and domain walls), the Majoron remains a true NGB of the continuous $B-L$ symmetry. Since $B-L$ is conserved by non-perturbative effects (unlike $L$), no discrete symmetry is broken, and no domain walls form. This resolves recent confusion in the literature regarding singlet Majoron models.

C. Non-Uniqueness of $a F \tilde{F}$ Couplings

• Result: The observation of an effective coupling $a F \tilde{F}$ (e.g., Majoron-photon or Axion-gluon) does not uniquely indicate that the boson originates from an anomalous symmetry.
• Mechanism: The paper demonstrates that non-anomalous NGBs can generate effective $a F \tilde{F}$ interactions through non-anomalous processes (e.g., loops involving new heavy fermions).
  • In the toy model, the Majoron couples to photons via a one-loop diagram with vector-like leptons.
  • The resulting amplitude satisfies the Adler zero condition (consistent with shift symmetry) but mimics the topological structure of an anomalous coupling.
• Significance: Experiments searching for axion-like couplings (such as $J Z \tilde{Z}$ or $J \gamma \gamma$) cannot automatically assume the detected particle is an ALP arising from an anomaly. Conversely, detecting such a coupling could serve as evidence for the existence of new charged fermions (like the vector-like leptons in the toy model) rather than just an anomalous symmetry.

D. Parallelism with the QCD Axion

The author applies this logic to the QCD axion. While the QCD axion is traditionally linked to the anomalous Peccei-Quinn (PQ) symmetry, the paper argues that physically, the axion mass arises from the explicit breaking of the anomaly-free combination of axial and PQ symmetries by quark masses. The "anomalous" description is a convenient duality but not the fundamental origin of the mass generation mechanism.

4. Significance and Implications

1. Theoretical Clarity: The paper provides a definitive framework for identifying the true symmetry origin of NGBs in models with simultaneous anomalous and non-anomalous breaking. It corrects the common misconception that the Majoron is the NGB of $L$.
2. Cosmological Safety: It establishes that Majoron models do not suffer from the domain wall problem provided $B-L$ is not explicitly broken, removing a major cosmological constraint on these models.
3. Experimental Interpretation:
   • Search Strategies: Searches for axion-like particles (ALPs) via $a F \tilde{F}$ couplings must be re-evaluated. A positive signal does not confirm an anomalous origin.
   • New Physics: The presence of such couplings in a non-anomalous context (like the Majoron) serves as a potential "smoking gun" for new heavy charged fermions that mix with SM fermions.
4. General Applicability: While focused on the Majoron, the result is general: any NGB arising from the simultaneous breaking of an anomalous and a non-anomalous symmetry is associated with the non-anomalous one.

Conclusion

Antonio Herrero-Brocal demonstrates that the Majoron is fundamentally the Nambu–Goldstone boson of the $B-L$ symmetry, not $L$. This re-identification resolves the domain wall problem and reveals that effective anomalous-looking couplings ($a F \tilde{F}$) can arise from non-anomalous mechanisms involving new heavy fermions. This insight is crucial for correctly interpreting experimental data in the search for axions and Majorons and for understanding the phenomenological signatures of physics beyond the Standard Model.