Curvature-Projection as an Effective Field Operator in Quantum Chromodynamics

! 1! Curvature-Projection as an Effective Field Operator in Quantum Chromodynamics Borros Arneth, Philipps University Marburg, Justus Liebig University Giessen, Germany, [email protected] Abstract We show that the mass–charge binding and curvature-projection mechanism proposed for explaining hadronic and leptonic mass hierarchies can be expressed as a legitimate effective operator within Quantum Chromodynamics (QCD) and Quantum Field Theory (QFT). By introducing a scalar curvature mediator that couples to local mass and absolute-charge densities, integrating it out yields a non-local gauge-invariant interaction −𝑔!" ∫ 𝜇(𝑥) 𝐷(𝑥−𝑦) 𝜎(𝑦) 𝑑#𝑥 𝑑#𝑦. This term represents an entropic correction to the QCD vacuum energy associated with gluonic curvature fluctuations and is numerically consistent with lattice determinations of the topological susceptibility 𝜒$ . The resulting operator preserves Lorentz invariance, locality, and renormalizability up to the confinement scale, providing a rigorous QFT basis for curvature-induced mass–charge coupling and charge-orientation projection. 1 Introduction Non-perturbative QCD is dominated by confinement, chiral symmetry breaking, and topological structure encoded in the field strength 𝐹%& ' and its dual 𝐹 .'%& [1–3]. Instantons, monopole condensation, and dual superconductivity of the vacuum provide geometric insight into color confinement [4–6]. Lattice calculations of the topological susceptibility 𝜒$ demonstrate that curvature fluctuations contribute directly to hadron masses [7–9]. To describe these effects analytically, we introduce a curvature-mediated effective field theory that reproduces the empirically observed mass–charge correlations in baryons [10– 12]. The construction respects all axioms of QFT and embeds seamlessly into the QCD Lagrangian. 2 Effective-Field Derivation Starting from the QCD Lagrangian ! 2! ℒ()* =−1 4𝐹%& '𝐹'%& +4𝜓 ¯+(𝑖 /𝐷−𝑚+)𝜓+ + we supplement a real scalar curvature field 𝜙(𝑥) representing coarse-grained gluonic curvature modes: ℒ,=1 2(∂%𝜙)-−1 2𝑚, -𝜙-+𝜆!𝜙 𝜇(𝑥)+𝜆"𝜙 𝜎(𝑥) where 𝜇(𝑥)=> 𝑚.𝛿(𝑥−𝑥.)@@ .and 𝜎(𝑥)=4 ∣𝑞/∣ 𝛿(𝑥−𝑥/)@ / are gauge-invariant composite operators. Functional integration over 𝜙 gives the non-local term ℒ011 =−𝑔!" ∫ 𝜇(𝑥) 𝐷(𝑥−𝑦) 𝜎(𝑦) 𝑑#𝑦,@@@@@@@@@𝑔!" = 𝜆!𝜆" with 𝐷(𝑥−𝑦) the propagator of 𝜙. The resulting Hamiltonian 𝐻 E!" =−𝑔!"∑𝑚.∣ 𝑞/∣𝐷./ is the curvature-projection operator previously introduced phenomenologically. Gauge invariance is preserved because both 𝜇 and 𝜎 are color singlets, and renormalizability follows for energies 𝐸 <𝑚,. 3 Physical Interpretation The effective vertex represents exchange of curvature quanta between regions of mass density and charge amplitude, generating an entropic correction to the bound-state energy: 𝐸!" ≃ −𝐺!" 𝑅 𝑀2𝑄345 Its sign corresponds to curvature orientation: outward (positive) or inward (negative) flux through the color manifold determines electric-charge sign. Minimization of the total free-energy functional ! 3! 𝐹 =𝐸()* +𝐸!" +𝛽𝑆6789 selects the physically realized projection, reproducing empirical charge-mass orderings within isomultiplets. 4 Consistency with Lattice QCD and QFT Identifying 𝐺!"/𝑅 with a curvature-susceptibility scale yields 𝐺!" ∼ 𝜒$ :/#/Λ()* giving effective binding energies ≈ tens of MeV for radii 0.5–0.8 fm—consistent with lattice data [7–9]. 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