[Paper Review] Forbidden and invisible Z boson decays in covariant theta-exact noncommutative standard model

R. Horvat, Amon Ilakovac+3 more|arXiv (Cornell University)|Apr 27, 2012

Noncommutative and Quantum Gravity Theories7 citations

TL;DR

This paper presents a $θ$-exact, covariant formulation of the noncommutative Standard Model using Seiberg-Witten maps to derive full expressions for triple neutral gauge boson and $\rm{U(1)_{Y}}$-neutrino interactions, enabling reliable predictions across all energy scales. It finds that $Z\to\gamma\gamma$ and $Z\to\bar{\nu}\nu$ decays receive nontrivial contributions from noncommutativity, with the invisible decay mode providing a strong bound of $\Lambda_{\rm{NC}} \gtrsim 120$ GeV when $\vec{E}_{\theta} \neq 0$, surpassing previous perturbative approximations.

ABSTRACT

The triple neutral gauge boson and direct U(1)_Y-neutrino interactions, being absent in ordinary field theory, can arise quite naturally in noncommutative gauge field theories. Using non-perturbative methods and a Seiberg-Witten map based covariant approach to noncommutative gauge theory, we have found theta-exact expressions for both interactions, thereby eliminating previous restrictions to low-energy phenomena. In particular we obtain for the first time the covariant, theta-exact, triple neutral gauge boson interactions within the noncommutative Standard Model gauge sector including an additional gauge-field deformation freedom. Finally we discuss implications for Z->2gamma and Z->neutrino-pair decays, and show that our results behave quite reasonably throughout all interaction energy scales.

Motivation & Objective

• To overcome the limitations of low-energy, θ-expanded approximations in noncommutative field theories for high-energy processes.
• To derive covariant, θ-exact expressions for triple neutral gauge boson and U(1)_Y-neutrino interactions using Seiberg-Witten maps.
• To provide reliable predictions for forbidden and invisible Z boson decays across all energy scales, avoiding divergences and unphysical behavior.
• To constrain the noncommutativity scale $\Lambda_{\rm{NC}}$ using experimental data on $Z\to\gamma\gamma$ and $Z\to\bar{\nu}\nu$ decays.
• To demonstrate that the θ-exact approach yields finite, well-behaved results at all energy scales, unlike perturbative θ-expansion.

Proposed method

• Employing a Seiberg-Witten map-based covariant approach to noncommutative gauge theory to derive θ-exact expressions for gauge and fermion interactions.
• Using non-perturbative methods to construct exact forms of the triple neutral gauge boson interaction and the $\rm{U(1)_{Y}}$-neutrino coupling.
• Introducing a gauge deformation freedom parameter $\kappa_g$ to control divergences and improve UV/IR behavior.
• Expanding the interaction Lagrangian in powers of gauge fields while retaining full θ-dependence to maintain covariance.
• Computing decay widths $\Gamma(Z\to\gamma\gamma)$ and $\Gamma(Z\to\bar{\nu}\nu)$ using the θ-exact interaction vertices.
• Comparing theoretical predictions with experimental bounds from PDG 2011 to derive constraints on $\Lambda_{\rm{NC}}$.

Experimental results

Research questions

• RQ1 Can θ-exact, covariant expressions for triple neutral gauge boson and $\rm{U(1)_{Y}}$-neutrino interactions be derived in the noncommutative Standard Model?
• RQ2 How do $Z\to\gamma\gamma$ and $Z\to\bar{\nu}\nu$ decay rates behave across all energy scales in the θ-exact framework?
• RQ3 What are the phenomenological bounds on the noncommutativity scale $\Lambda_{\rm{NC}}$ from these decays, especially given current experimental limits?
• RQ4 How does the inclusion of gauge deformation freedom affect the finiteness and energy-scale behavior of the decay amplitudes?
• RQ5 Can the θ-exact approach resolve the unphysical divergences and scale dependence seen in θ-expanded models at high energies?

Key findings

• The θ-exact formulation yields finite, well-behaved decay rates for $Z\to\gamma\gamma$ and $Z\to\bar{\nu}\nu$ across all energy scales, unlike perturbative θ-expansion methods.
• The $Z\to\gamma\gamma$ decay rate remains finite and convergent, with a theoretical upper bound of $\Gamma(Z\to\gamma\gamma)/\Gamma_{\rm{tot,SM}} < 5.2 \times 10^{-5}$, consistent with current experimental limits.
• For $\Lambda_{\rm{NC}} \gtrsim 100$ TeV, the $Z\to\gamma\gamma$ rate becomes unobservable at the LHC, indicating weak sensitivity to current bounds.
• The invisible $Z\to\bar{\nu}\nu$ decay mode provides a stronger constraint: $\Lambda_{\rm{NC}} \gtrsim 120$ GeV when $\vec{E}_{\theta} \neq 0$, derived from $\Delta\Gamma \lesssim 1$ MeV.
• The bound is independent of the deformation parameter $\kappa_g$ and holds for both $\kappa=0$ and $\kappa=1$, with the effect vanishing only when $\vec{E}_{\theta} = 0$.
• The results show closed-form, convergent behavior in all figures, confirming the robustness of the θ-exact approach for high-energy phenomenology.