An Axial $U_A(1)_{L_\mu-L_\tau}$: UV Completion and Experimental Searches

Technical Summary: An Axial $U(1)_{L_\mu-L_\tau}$: UV Completion and Experimental Searches

Problem Statement
While the conventional vector-like $U(1)_{L_\mu-L_\tau}$ extension of the Standard Model (SM) is well-studied and anomaly-free, a purely axial realization of this symmetry presents distinct theoretical and phenomenological challenges. In an axial model, left- and right-handed charged leptons must carry opposite charges to generate an axial current. This assignment forbids the standard diagonal SM Yukawa operators ($\bar{L}_\ell H \ell_R$), necessitating a mechanism for charged-lepton mass generation that is consistent with gauge anomaly cancellation. Furthermore, existing experimental constraints on $L_\mu-L_\tau$ bosons are predominantly derived assuming a vector current. Replacing the vector current with an axial one fundamentally alters the phenomenology: it shifts different Wilson coefficients in neutrino trident production, reverses the sign of the contribution to the muon anomalous magnetic moment ($g-2)_\mu$, and modifies production and decay rates due to longitudinal polarization enhancements. There is a lack of a unified treatment that develops the necessary ultraviolet (UV) completion for an axial model alongside a re-interpretation of experimental limits.

Methodology and Model Setup
The authors propose a renormalizable, anomaly-free UV completion for an axial $U(1)_{L_\mu-L_\tau}$ gauge symmetry. The model extends the SM gauge group by introducing:

1. Vector-like charged leptons ($E_\mu, E_\tau$) with specific charge assignments opposite to their SM counterparts.
2. A complex SM-singlet scalar ($S$) responsible for breaking the new gauge symmetry.

Mass Generation Mechanism:
The opposite charges of left- and right-handed leptons forbid direct Yukawa couplings. Instead, muon and tau masses are generated via a "universal-seesaw" mechanism involving the mixing of SM leptons with the heavy vector-like leptons and the singlet scalar. After spontaneous symmetry breaking, the light lepton masses are suppressed by the heavy vector-like mass scale ($M_E$), while the heavy states decouple.

Benchmark Assumptions:
To isolate the axial-vector phenomenology, the authors adopt a specific benchmark:

• Flavor Structure: Off-diagonal Yukawa couplings ($y_{\mu\tau}$) are set to zero to satisfy Charged Lepton Flavor Violation (CLFV) constraints.
• Decoupling Limit: Vector-like lepton masses are set to $M_E > 10$ TeV, ensuring light-heavy mixing angles ($\theta_{L,R}$) are small ($\lesssim 10^{-2}$). This suppresses corrections to electroweak precision observables.
• Scalar Decoupling: The singlet scalar mass is set to $m_s \gtrsim 10$ GeV. This avoids stringent constraints on light scalars and suppresses scalar-mediated contributions to observables, allowing the study of the axial-vector sector independently.
• Kinetic Mixing: The vector-like leptons have opposite $U(1)_{L_\mu-L_\tau}$ charges and identical hypercharges, causing their one-loop contributions to kinetic mixing to cancel, effectively setting the mixing parameter to zero at the matching scale.

Under these assumptions, the phenomenology is described by two primary parameters: the $A'$ mass ($m_{A'}$) and the gauge coupling ($g_X$).

Key Contributions and Results

1. Neutrino Trident Production ($\nu_\mu N \to \nu_\mu N \mu^+ \mu^-$):
   The authors perform a full calculation including finite muon mass effects. Unlike the vector model, which shifts the vector weak coefficient ($C_V$), the axial mediator shifts the axial weak coefficient ($C_A$). This distinction is critical for correctly interpreting constraints from CHARM II and CCFR experiments.

2. Muon Beam-Dump Searches (NA64$\mu$):
   The study reinterprets beam-dump limits for invisible $A'$ decays. Due to axial-current non-conservation, the longitudinal mode of the $A'$ is enhanced, particularly for $m_{A'} \ll m_\mu$. This leads to a production rate scaling as $m_\mu^2/m_{A'}^2$ relative to the vector benchmark, significantly tightening constraints in the low-mass region.

3. Muon Anomalous Magnetic Moment ($(g-2)_\mu$):
   The axial contribution to $(g-2)_\mu$ is calculated to be negative and enhanced by the longitudinal polarization, contrasting with the positive contribution of a vector boson. In the benchmark where the scalar is decoupled ($m_s \gtrsim 10$ GeV), the scalar's positive contribution is negligible, making the axial contribution the dominant effect. This results in a stringent consistency bound, particularly for light mediators, where the exclusion contour is tighter than in the vector case.

4. Invisible Meson Decays:
   The paper analyzes $\pi^+ \to \mu^+ \nu_\mu A'$ and $K^+ \to \mu^+ \nu_\mu A'$. For the pion channel, the $A'$ is treated as invisible since $m_{A'} < m_\pi - m_\mu < 2m_\mu$. For kaons, the branching fraction into neutrinos vs. muons is explicitly calculated to reinterpret NA62 limits.

5. Four-Muon Resonance Searches (BaBar, Belle II, CMS):
   For $m_{A'} \gg m_\mu$, the difference between vector and axial final-state radiation (FSR) rates becomes negligible (differing only by terms of order $m_\mu^2/m_{A'}^2$). The authors demonstrate that existing limits on vector models can be rescaled by the ratio of branching fractions to constrain the axial model.

6. Future Muon Collider Probes:
   The paper investigates the discrimination power of a future muon collider. While total event rates for $\mu^+\mu^- \to \tau^+\tau^-$ are similar for vector and axial mediators, the angular distributions differ significantly. Specifically, the forward-backward asymmetry ($A_{FB}$) retains direct sensitivity to the chirality of the interaction, offering a robust method to distinguish the axial model from the vector benchmark at high energies.

Significance and Claims
The paper claims to provide the first unified treatment of an anomaly-free axial $L_\mu-L_\tau$ model that connects the UV structure (mass generation via vector-like leptons and a singlet scalar) with a comprehensive re-evaluation of experimental constraints.

The authors emphasize that the axial model cannot be treated as a simple "chiral" version of the vector benchmark. Key physical distinctions include:

• The shift of $C_A$ rather than $C_V$ in trident production.
• The negative, longitudinally enhanced contribution to $(g-2)_\mu$.
• The necessity of a specific UV completion to generate fermion masses and cancel anomalies.

The work concludes that while rate-based observables at high energies ($m_{A'} \gg m_\mu$) may allow for simple rescaling of vector limits, low-energy observables and angular distributions at colliders are essential for correctly identifying and constraining the axial nature of the force. The results are presented as conditional constraints on a specific two-dimensional slice of the parameter space, with the understanding that relaxing the scalar decoupling or heavy-lepton assumptions would reintroduce mass-dependent constraints.