Neutron Dark Decay Mechanism

• The neutron dark decay mechanism is defined by baryon-number violating transitions leading to decays into a dark sector fermion, often accompanied by mesons, photons, or dark gauge bosons.
• It employs effective operators and chiral perturbation theory to predict non-standard kinematics, such as softer meson momenta and photon spectra compared to canonical beta decay.
• The mechanism opens experimental avenues to probe dark sector states and dark forces via nucleon decay, offering a potential complementary approach to traditional dark matter searches.

The neutron dark decay mechanism refers to hypothesized channels in which the neutron decays into non-Standard-Model particles – typically into a dark sector fermion, possibly accompanied by mesons, photons, or light dark sector gauge bosons. Unlike the canonical β-decay (n → p + e⁻ + ν̄ₑ), these processes are characterized by baryon number violation at high scales and result in missing-energy final states with non-standard kinematics. This paradigm provides new decay modes that have potential implications for particle physics, cosmology, and nucleon decay searches. The mechanism arises from high-scale baryon-number violating operators and may be realized in the presence or absence of light dark sector forces.

1. Baryon-Number Violating Couplings and Neutron Portal Operators

The foundation of the neutron dark decay mechanism is the introduction of a baryon-number violating operator that couples a sub-GeV dark sector fermion ($X$) to Standard Model quarks. Specifically, the relevant right-handed operator is:

$O_{BV} = \frac{(X\,u\,d\,d)_R}{M^2}$

where $u$ and $d$ denote up and down quark Weyl fields (right-chirality), $X$ is the dark sector fermion, and $M$ is a new physics scale, $M \gg 1$ TeV, which suppresses baryon number violation (Davoudiasl, 2014). This operator mediates nucleon–dark sector fermion mixing, leading to neutron (or proton) decay channels that produce $X$ in the final state. The high suppression scale is required to avoid rapid baryon number violation incompatible with experimental bounds.

The hadronic matrix element of this operator is typically encapsulated in a parameter $\beta$, which is extracted from lattice QCD calculations. The operator leads to effective interactions which, after electroweak and QCD renormalization group running and hadronization, generate both mass mixing between neutron and $X$ and direct $N\to X$ transition operators relevant for decay calculations.

2. Exotic Nucleon Decay Channels: Meson and Photon Emission

The baryon-number violating operator induces two principal classes of nucleon decay channels:

• Mesonic Channels: Nucleon decays into $X$ and an on-shell meson. Representative processes include
  • $p \to X + \pi^+$
  • $n \to X + \pi^0$
  • $n \to X + \eta$

These modes are described within chiral perturbation theory. The effective Lagrangian comprises:

• $\mathcal{L}_0$: $\Delta B=0$ nucleon–meson interactions (chiral couplings $D$, $F$; pion decay constant $f_\pi$)
• $\mathcal{L}_1$: $\Delta B=1$ operators responsible for direct $n\to X$ and $n\to X + \text{meson}$ transitions

The total decay amplitude receives two contributions: (i) from $n{-}X$ mixing followed by standard meson emission, and (ii) from a direct transition via $\mathcal{L}_1$. For $p\to X\pi^+$, the decay width is

$$\Gamma(p \to X \pi^+) = \frac{\beta^2 c_1^2 |\vec{p}_{\pi^+}|}{32\pi f_\pi^2 m_p^2} \left[ (A_p^2 + B_p^2) f(m_p, m_{\pi^+}) + (B_p^2 - A_p^2) m_p m_X \right]$$

with kinematic structures depending on chiral parameters, nucleon and $X$ masses, and the new physics scale.

• Radiative Channels: Channels such as $n \to X + \gamma$ exploit the neutron’s magnetic dipole operator:

$$\mathcal{L}_\text{mag} = \frac{ie}{2m_p} \bar{n} \sigma^{\mu\nu} q_\nu F_2(0) n\, A_\mu$$

with $F_2(0) \simeq -1.91$. The corresponding decay width,

$$\Gamma(n \to X + \gamma) = \frac{\alpha \beta^2 c_1^2 F_2(0)^2}{16 m_p^2 m_n^3}(m_n^4 - m_X^4)$$

is sensitive to the overlap of the neutron and $X$ masses and the available phase space.

In both cases, the kinematics of the visible decay products differ from standard channels. For $m_X$ closer to $m_N$ the recoil meson is significantly softer, and in the radiative mode the photon carries less energy than in $n \to \nu \gamma$.

3. Dark Sector Gauge Interactions and the Role of $Z_d$

A further layer arises if $X$ participates in a dark $U(1)_d$ interaction mediated by a light vector boson $Z_d$ ($m_{Z_d} < m_X$) (Davoudiasl, 2014). If (for instance) only $X_L$ carries a $U(1)_d$ charge $Q_d$, the decay channel $n\to X + Z_d$ becomes possible through $n{-}X$ mixing and the dark gauge interaction. The corresponding spin-averaged amplitude includes both standard and Goldstone boson longitudinal terms (with explicit $1/m_{Z_d}^2$ enhancement if $m_{Z_d} \to 0$). The decay width (to leading order) is:

$$\Gamma(n \to X Z_d) \simeq \frac{Q_d^2 \alpha_d}{8} \left( \beta c_1\, \frac{m_X}{m_n^{3/2}} \right)^2 \left( \frac{\mu_+^2}{\mu_-^2} + \frac{\mu_-^2}{m_{Z_d}^2} \right)$$

with $\alpha_d = g_d^2/(4\pi)$, $\mu_\pm^2 = m_n^2 \pm m_X^2$.

The signature is distinctive: for $Z_d$ with minimal SM couplings (via kinetic mixing parameter $\epsilon$), its dominant decay is $Z_d \to \ell^+\ell^-$ ($\ell=e$ or $\mu$), with a branching ratio near unity if no lighter $U(1)_d$-charged states exist. The result is a neutron decay to $X$ plus a narrow dilepton resonance at $m_{Z_d}$, yielding a visible and clean experimental signal.

4. Experimental Consequences and Comparison to Standard Decay Searches

The kinematic and topological features of nucleon dark decays diverge sharply from those in Standard Model or canonical GUT-motivated nucleon decay:

• Meson momenta are reduced due to missing mass carried by $X$; events may fail standard signal windows.
• Radiative channels result in photons with softer spectra than neutrino final states.
• $Z_d$ final states produce dilepton resonances absent in the Standard Model.

Standard searches (e.g., in Super-Kamiokande and other large underground experiments) optimize cuts for well-modeled $n\to \nu + \pi^0$ or $p\to \nu + \pi^+$ events with specific kinematics, potentially leaving nucleon dark decay events undetected due to nonstandard kinematic distributions.

Consequently, the consideration of these channels motivates new or reinterpreted searches for:

• Events with lower-momentum mesons and missing energy (not just invisible neutrino signatures),
• Neutron decays with softer photons,
• Direct reconstruction of narrow dilepton resonances coinciding with known nucleon masses.

A tailored analysis strategy involving full kinematic reconstruction and minimal bias on missing energy channels is required.

5. Implications for Dark Sector and Dark Matter Physics

A key implication is the opening of an experimental portal from baryon number violation into otherwise inaccessible dark sector states. Since $X$ can have negligible direct coupling to ordinary matter or negligible cosmological abundance, its detection via nucleon decay represents a unique method for probing dark sector physics (Davoudiasl, 2014).

The mechanism described allows several possibilities:

• If $X$ is or is related to dark matter, this operator connects the dark and visible sectors through a rare but characteristic baryon number violating process.
• If a new vector boson $Z_d$ exists, the process $n \to X + Z_d \to X + \ell^+\ell^-$ provides a "smoking gun" signature.
• Since many low-mass dark sector models predict feeble direct interaction rates, nucleon decay processes represent a critical complementary probe, especially for light or sequestered dark sectors.

6. Model Parameters, Rates, and Constraints

The relevant parameters include:

• $M$: the scale suppressing baryon number violation; requirements that conventional nucleon decay remain rare set $M \gg 1$ TeV.
• $m_X$, $m_{Z_d}$: the dark fermion and dark vector boson masses, determining available phase space and decay rates.
• $g_d$, $Q_d$, $\epsilon$: the strength and structure of the dark $U(1)_d$ interaction and its mixing with hypercharge, which determine $Z_d$ decay patterns and lifetimes.
• $\beta$: hadronic matrix element from lattice QCD.
• Chiral couplings ($D$, $F$), $f_\pi$: parameters controlling meson coupling strengths in nucleon decays.

The partial decay widths given above (for both mesonic and radiative/dark sector channels) must satisfy experimental bounds from conventional nucleon decay and radiative decay searches. Strong limits on $n\to \gamma +$ invisible require that $m_X$ is close to $m_n$ to suppress visible energy releases, or that branching ratios are sufficiently small.

7. Experimental Strategies and Future Prospects

Nucleon decay searches in underground experiments, previously focused on classic signatures (e.g., $p\to e^+\pi^0$, $n\to \nu\pi^0$), can be extended to target dark decay signatures:

• Dedicated analyses for events with missing energy and softer visible spectra.
• Reconstruction of pointed dilepton resonances at variable invariant mass.
• Kinematic studies targeting the low-momentum regime for $\pi$ and $K$ in proton or neutron decay.

In addition, experiments can reinterpret existing data in light of alternative kinematics. This may already be sensitive to these modes if appropriately binned and selection criteria are revised. Future searches may take advantage of new reconstruction techniques, improved detection of low-energy leptons and photons, or separate identification of final-state missing energy. The unique features of $n \to X + Z_d$ with $Z_d \to \ell^+\ell^-$ are particularly promising for high-background-rejection signatures.

Further theoretical developments may enhance sensitivity to these processes by refining hadronic matrix elements, chiral effective theory predictions, and incorporating nuclear effects for nucleons bound in matter.

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In summary, the neutron dark decay mechanism arises via high-scale baryon-number violating interactions coupling a sub-GeV dark sector fermion to SM quarks, leading to nucleon decays with novel missing-energy and visible signatures. These processes feature distinctive kinematics, can evade conventional nucleon decay searches, and provide an avenue for probing otherwise inaccessible dark sector states, especially in scenarios involving new dark forces or light gauge bosons (Davoudiasl, 2014). The unique experimental signatures motivate dedicated searches and analysis strategies, offering potential new access to baryon number violation and the dark sector.