UV-Complete Models for a Light Axial Gauge Boson

This paper introduces three anomaly-free, left-right symmetric universal seesaw models featuring a light axial gauge boson ($U(1)_a$) that accommodates small neutrino masses, offers a dark matter candidate via a new Dirac fermion, and exhibits distinct phenomenological constraints on the gauge coupling and boson mass depending on the specific model variant.

The Gist

Imagine the Standard Model of physics as a giant, incredibly complex Lego castle. For decades, scientists have been trying to find the missing bricks that explain things like why the universe has more matter than antimatter, what dark matter is, and why neutrinos (tiny ghost-like particles) have such tiny masses.

This paper, written by physicists Bhaskar Dutta, Aparajitha Karthikeyan, and Rabindra N. Mohapatra, proposes a new set of "blueprints" to add a specific, missing piece to this castle: a Light Axial Gauge Boson.

Here is the breakdown of their idea in simple terms, using some everyday analogies.

1. The New Particle: The "Spin-Switching" Messenger

In the Standard Model, forces are carried by messengers (like photons for light). The authors propose a new messenger particle, let's call it the "Axial Boson" (A).

• The Analogy: Imagine a normal messenger (like a photon) who just delivers a package. This new Axial Boson is like a gymnast. It doesn't just deliver a message; it specifically interacts with particles based on which way they are "spinning" (a quantum property called chirality).
• The Twist: It only talks to particles spinning one way (left-handed) and ignores the other, or vice versa. This "axial" behavior is rare and makes it very special. The authors suggest this particle is very light (sub-GeV), meaning it's much lighter than the Higgs boson, perhaps even lighter than a proton.

2. The Three Blueprints (Models A, B, and C)

To make this new particle fit into the laws of physics without breaking them (specifically, without causing mathematical "glitches" called anomalies), the authors built three different versions of the theory. Think of these as three different architectural styles for the same house.

• Model A (The "Shared Room" House):

  • In this version, the new Axial Boson shares a room with the Standard Model's Higgs boson. Because they are roommates, they influence each other heavily.
  • The Catch: This creates a strict rule. The strength of the new force (the coupling) cannot get too strong, or it would mess up the properties of the Z-boson (a known particle). It's like a shared apartment where if one roommate turns up the music too loud, the whole building shakes. This sets a "speed limit" on how strong this new force can be.
  • Bonus: This model naturally solves the "Strong CP Problem" (a mystery about why the universe doesn't behave differently if you swap left and right in the strong nuclear force).

• Model B (The "Independent" House):

  • Here, the new Axial Boson has its own separate space. It doesn't share a room with the Higgs.
  • The Benefit: Because it's independent, there is no "speed limit" on its strength. It can be stronger or weaker without breaking the Z-boson.
  • The Dark Matter Connection: This model easily includes a "Dark Matter" candidate. Imagine a new, invisible particle that only talks to the Axial Boson. It's like a ghost that can only be seen by the Axial Boson, making it a perfect candidate for the invisible stuff holding galaxies together.

• Model C (The "Family-Selective" House):

  • This version is picky. It only talks to the third generation of particles (the heavy ones: Tau leptons and Top quarks) and ignores the lighter ones (electrons and up/down quarks).
  • Why do this? This is designed to explain a specific weird signal seen in the MiniBooNE experiment, where scientists saw too many low-energy events. By being selective, this model can explain that signal without contradicting other experiments that look at electrons.

3. Solving the "Neutrino Mass" Mystery

Neutrinos are weird because they have mass, but it's incredibly tiny. Why?

• The Analogy: Imagine a heavy boulder (a heavy particle) connected to a tiny pebble (a neutrino) by a very stiff spring. If you push the boulder, the pebble barely moves.
• The Paper's Solution: These models use a mechanism called the "Seesaw." They introduce heavy, invisible particles. The interaction between the light neutrinos and these heavy particles "dilutes" the mass, making the neutrinos we see incredibly light. The paper shows how their specific blueprints make this math work out naturally.

4. The Dark Matter Candidate

Every good theory needs a dark matter candidate.

• In these models, they add a new, heavy, invisible fermion (a type of matter particle).
• How it works: This particle is like a "dark citizen" who only speaks the language of the new Axial Boson. It can't talk to normal matter directly. However, it can annihilate with other dark matter particles to create Axial Bosons, which then decay into normal particles. This process determines how much dark matter is left over in the universe today.

5. Why Should We Care? (The Real-World Test)

The authors didn't just draw these models on paper; they checked them against real data.

• The "Speed Limit" Check: For Model A, they calculated exactly how strong the new force can be before it breaks the known rules of the Z-boson.
• The "MiniBooNE" Connection: They suggest that Model C could explain the strange "low energy excess" seen in the MiniBooNE experiment. It's like finding a key that fits a lock that has been puzzling scientists for years.
• Future Hunting: They outline where scientists should look next. If this Axial Boson exists, experiments like beam-dumps (shooting particles into blocks of metal) or colliders might see it decaying into pairs of electrons or muons.

Summary

This paper is a proposal for a new, lightweight particle that acts like a "spin-switching" messenger. The authors built three different versions of a theory to include this particle:

1. Model A: Strict rules, shares space with the Higgs, sets a limit on the force's strength.
2. Model B: Flexible rules, great for explaining Dark Matter.
3. Model C: Picky rules, only talks to heavy particles, potentially solving the MiniBooNE mystery.

All three versions successfully explain why neutrinos are so light, solve a major puzzle about the strong nuclear force, and provide a candidate for Dark Matter, all while staying consistent with current experimental data. It's a "UV-complete" model, meaning it's a solid, mathematically consistent foundation that doesn't fall apart when you look at it up close.

Technical Summary

1. Problem Statement

The Standard Model (SM) has faced increasing pressure from experimental anomalies that suggest the existence of light, sub-GeV particles. Notable examples include:

• The ATOMKI anomaly (excited Beryllium decay).
• The MiniBooNE low-energy excess.
• The Muon $(g-2)$ discrepancy.

While many effective field theories (EFTs) have been proposed to explain these via a light axial vector boson ($A$), they often lack a consistent UV-complete framework. Specifically, there is a need for models that:

1. Are anomaly-free.
2. Explain the smallness of neutrino masses.
3. Provide a Dark Matter (DM) candidate.
4. Address the Strong CP problem.
5. Remain consistent with current low-energy constraints and LHC data.

2. Methodology

The authors construct three distinct classes of UV-complete models based on the Left-Right Symmetric Universal Seesaw framework. The gauge group for all models is:
$$SU(3)_c \times SU(2)_L \times SU(2)_R \times U(1)_X \times U(1)_a$$
where $U(1)_a$ is the new axial gauge symmetry.

Key Methodological Steps:

• Anomaly Cancellation: The authors assign specific $U(1)_a$ charges to SM fermions and new vector-like fermions (BSM) to ensure all gauge anomalies (including gravitational) cancel out.
• Symmetry Breaking: The models utilize a multi-stage symmetry breaking mechanism involving Higgs doublets ($\chi_{L,R}$) and singlet scalars ($\eta_i$). The breaking scale of the singlets ($\langle \eta \rangle$) determines the mass of the axial boson ($m_A$).
• Fermion Mass Generation: They employ the Universal Seesaw mechanism for charged fermions and various seesaw mechanisms (Type I, Type II, or mixed) for neutrinos.
• Strong CP Solution: The models incorporate Parity (P) symmetry ($L \leftrightarrow R$), ensuring that the vacuum expectation values (VEVs) of the scalar fields are real, thereby naturally solving the Strong CP problem without an axion.
• Dark Matter: A vector-like Dirac fermion ($\zeta$) is introduced as a DM candidate, coupled to the SM via the $U(1)_a$ portal.

3. Key Contributions and Model Variants

The paper presents three primary models (A, B, and C) and a variation (A'), distinguished by their $U(1)_a$ charge assignments and Higgs content:

Model A (and A')

• Symmetry: Uses the axial counterpart of the $U(1)_X$ symmetry.
• Charge Assignment: The SM Higgs doublets ($\chi_{L,R}$) share the $U(1)_a$ quantum number.
• Neutrino Mass:
  • Model A: Uses Type I seesaw with singlet fermions ($N$) that are $U(1)_a$ neutral.
  • Model A': Uses Type II seesaw with triplet Higgs fields ($\Delta_{L,R}$), removing the need for singlet fermions.
• Dark Matter: A vector-like fermion $\zeta$ coupled via $U(1)_a$.
• Novel Feature: Because the SM Higgs carries $U(1)_a$ charge, the axial boson mixes with the SM $Z$ boson. This leads to a "level crossing" phenomenon where the $Z$ mass eigenvalue changes drastically as the coupling $g_a$ increases.

Model B

• Symmetry: Uses the gauged axial $B-L$ symmetry.
• Charge Assignment: The SM Higgs doublets are neutral under $U(1)_a$, but the heavy singlet fermions ($N_{L,R}$) carry non-zero $U(1)_a$ charges.
• Neutrino Mass: Type I seesaw where Majorana masses for $N$ are proportional to the singlet VEV ($v_\eta$).
• Phenomenology: The axial boson decouples from the $Z$ boson at the tree level (no mixing), allowing $g_a$ to be larger without violating $Z$-pole constraints.

Model C

• Symmetry: Uses the family-non-universal $B - 3L_\tau$ symmetry.
• Charge Assignment: Only the third generation (tau sector) and specific BSM fields carry $U(1)_a$ charges.
• Phenomenology: This model is specifically designed to explain the MiniBooNE low-energy excess via dark matter scattering or neutrino interactions, as it avoids constraints from electron/muon experiments.

4. Key Results

A. Neutrino Masses

All models successfully generate small neutrino masses:

• Model A: Achieves masses via Type I seesaw with arbitrary Majorana masses for singlets.
• Model A': Uses Type II seesaw suppression via small triplet VEVs.
• Model B: Requires small Yukawa couplings ($h_\nu \lesssim 10^{-6}$) because Majorana masses are tied to the TeV-scale symmetry breaking.
• Model C: Generates masses for one neutrino pair at tree level and a second pair at one-loop level, leaving one pair massless (or light via higher-dimensional operators).

B. Dark Matter

• A vector-like fermion $\zeta$ serves as the DM candidate.
• Thermal Freeze-out: Requires $g_\zeta \sim 0.016 \sqrt{m_\zeta/\text{GeV}}$.
• Freeze-in: Possible for very small couplings ($g_a \sim 10^{-6}$).
• Model C Specifics: A large DM charge is required to explain the MiniBooNE excess via scattering.

C. The "Level Crossing" Limit (Model A)

A major theoretical result is the derivation of an upper limit on the gauge coupling $g_a$ for Model A.

• Because the Higgs doublet shares the $U(1)_a$ charge, the $Z$ boson mixes with the new axial boson $A$.
• As $g_a$ increases, the mass eigenvalues of the neutral gauge bosons undergo a "level crossing," drastically altering the properties of the physical $Z$ boson.
• To remain consistent with the observed $Z$ mass and couplings, $g_a$ is constrained (e.g., $g_a \lesssim 7 \times 10^{-3}$ for $v_R, v_\eta \sim 5$ TeV).
• Note: Models B and C do not suffer from this constraint because the Higgs is neutral under $U(1)_a$.

D. Phenomenological Constraints

The authors perform a comprehensive scan of experimental constraints:

• Flavor Changing Neutral Currents (FCNCs): Constraints from $D-\bar{D}$ and $K-\bar{K}$ mixing imply $v_\eta \gtrsim 2$ TeV.
• Low Energy Experiments:
  • Solar/Reactor Neutrinos: Coherent Elastic Neutrino-Nucleus Scattering (CE$\nu$NS) and Elastic Neutrino-Electron Scattering (E$\nu$ES) from XENONnT, LZ, and COHERENT.
  • Meson Decays: Constraints from $\pi^0 \to e^+e^-$ and $\eta \to \mu^+\mu^-$ are critical. The axial nature suppresses some contributions, but tree-level effects in Model A (via mixing) are tightly constrained.
  • Beam Dump Experiments: NA64, E137, and others constrain invisible decays.
  • Colliders: BaBar, LEP, and LHC (monophoton/monojet) searches.
• Model C Specifics: Constraints from DONuT (tau neutrino appearance) and $\Upsilon$ decays to $\tau^+\tau^-$ are dominant.

5. Significance and Conclusion

• UV Completeness: The paper provides the first set of fully UV-complete, anomaly-free models for a light axial vector boson that simultaneously address neutrino masses, dark matter, and the Strong CP problem.
• Theoretical Constraint: The discovery of the "level crossing" limit in Model A is a novel theoretical insight, demonstrating that if the SM Higgs carries a new axial charge, the gauge coupling is strictly bounded by $Z$-pole precision data.
• Phenomenological Viability: Models B and C are shown to be viable in the sub-GeV to GeV mass range, with parameter spaces that can explain current anomalies (like MiniBooNE) while satisfying all current experimental bounds.
• Future Probes: The models predict signatures accessible to future high-intensity beam experiments (electron and proton beam dumps) and dark matter direct detection experiments, particularly for the invisible decay channels.

In summary, this work bridges the gap between effective field theory explanations of low-energy anomalies and fundamental high-energy physics, offering testable predictions for the next generation of experiments.