Curvature Projection and Internal Charge Amplitude in Hadron Mass Splitting

! 1! Curvature Projection and Internal Charge Amplitude in Hadron Mass Splitting Borros Arneth, Philipps University Marburg, Justus Liebig University Giessen, Germany, [email protected] Abstract Electric charge and rest mass, though distinct in the Standard Model, display systematic correlations in hadron multiplets. We show that both emerge from a unified projection principle acting on a curvature-mediated Hilbert manifold representing QCD color flux. Integrating out a mediator field coupled to mass and absolute charge densities yields a natural mass–charge binding operator. Charge corresponds to curvature orientation: outward flow produces positive charge, inward flow negative, and symmetric flow neutrality. A unified variational model including electromagnetic self-energy and a β selfconsistency loop reproduces strange-baryon mass splitting (Σ, Ξ) to ≲1 MeV with constituent quark inputs. The effective coupling agrees in scale with lattice QCD measurements of topological susceptibility. Internal absolute charge amplitude, rather than net charge, governs the mass–charge binding, explaining the observed ordering of Σ⁺, Σ⁰, and Σ⁻ masses. 1 Introduction Nonperturbative Quantum Chromodynamics (QCD) explains most of hadronic mass through gluon confinement and chiral dynamics, yet it does not predict how electric charge sign and magnitude affect hadron masses within flavor multiplets. Empirical data show that charged baryons often differ by a few MeV from their neutral counterparts, a deviation beyond electromagnetic self-energy alone. Lattice studies demonstrate that the QCD vacuum possesses rich topological structure— characterized by nonzero topological susceptibility and curvature fluctuations—which can influence mass generation and vacuum alignment [1–3]. The confinement mechanism itself appears linked to monopole condensation and dual-superconductor dynamics [4, 5]. High-precision lattice calculations including QED confirm that charge-dependent effects persist after electromagnetic corrections [6–8]. We propose that these residual correlations originate from a mass–charge binding operator emerging from projection on a curvature manifold of the QCD color flux. This projection enforces minimal curvature entropy, selecting curvature orientation and thus the sign of electric charge. The resulting framework integrates the mass–charge operator, ! 2! electromagnetic corrections, and a β self-consistency loop into a unified variational model consistent with lattice QCD observables. 2 Diagram–Hilbert Projection and Effective Operator The internal Hilbert space of a hadron is expressed as ℋ =⨂ !ℋ!, with local projectors 𝑃!. Define the constituent mass and absolute charge densities as 𝜇((𝐱)=,𝑚!𝑃!𝛿(𝐱−𝐱!) ! ,000000000000𝜎2(𝐱)= ,∣𝑞!∣𝑃!𝛿(𝐱−𝐱!) ! A mediator field 𝜙(𝐱) couples linearly to these densities: 𝑆"#$ =∫𝑑%𝑥 𝜙(𝐱)(𝜆&𝜇( +𝜆'𝜎2) Integrating out 𝜙 yields the effective operator 𝐻 =&' = −𝑔&' ,𝑚!∣𝑞(∣ 𝐷(𝐱!,𝐱() !)( where 𝐷 is the mediator kernel and 𝑔&' =𝜆&𝜆'. For localized bound states with characteristic radius 𝑅, 𝐸&' ≃−𝐺&' 𝑅𝑀*𝑄+,- where 𝑀*=∑𝑚!! and 𝑄+,- =∑∣ 𝑞!∣ ! is the internal absolute charge amplitude—the sum of absolute constituent charges. ! 3! 3 Curvature Orientation and Entropic Projection The diagram–Hilbert manifold carries curvature two-form Ω. Its local orientation corresponds to charge sign: • outward curvature → positive charge, • inward curvature → negative charge, • symmetric curvature → neutral. Minimizing the total functional ℱ$.$ =⟨𝐻 =/01 +𝐻 =&'⟩+𝛽𝑆2345[Ω] selects the curvature orientation (and hence charge) that minimizes curvature entropy subject to color confinement. The β self-consistency loop stabilizes the radius 𝑅 and accounts for self-field feedback. 4 Molecular Charge Amplitude and Internal Binding The mass–charge operator acts not on individual quarks but on the composite hadron, coupling its total rest mass to the molecular internal charge amplitude 𝑄+,- =∑∣𝑞!∣ ! For baryons: Particle Quark content Net charge 𝑄+,- Proton (uud) +1 1.67 5/3 Neutron (udd) 0 1.33 4/3 Σ⁺ (uus) +1 1.67 5/3 Σ⁰ (uds) 0 1.33 4/3 Σ⁻ (dds) −1 1.00 3/3 Even neutral baryons (e.g. Σ⁰, n) have substantial internal charge amplitude, hence finite curvature binding. Mass differences follow directly from variations in 𝑄+,-: larger internal charge amplitude enhances curvature binding (lower mass). 5 Variational Energy Model Each hadron satisfies ! 4! 𝐸(𝑅)=𝑀*−𝑎 𝑅+𝑏𝑅−𝐾 𝑅𝑀*𝑄+,- +𝐶67 𝑅+𝐵8 𝑅9. Fitting to light baryons using constituent quark masses (𝑚:≈330, 𝑚;≈335, 𝑚<≈ 500 MeV) yields 𝐾 ≈0.06 MeV=>,𝐵8≈ 4×10? MeV⋅fm,𝑝 = 1, with radius solutions 𝑅@≈ 0.5–0.8 fm and residuals ≲ 1 MeV for Σ, Ξ splitting. 6 Add/Subtract Rule and Symbolic Examples General Projection Rule 𝑀A=𝑀B!"#$ +(𝐸.$CD4 A−𝐸.$CD4 B!"#$)+(𝐸&' A−𝐸&' B!"#$) Whether to add or subtract depends on which particle is the curvature-balanced projection baseline. (a) Proton–Neutron Baseline = proton (stronger binding): 𝑀E−𝑀9=𝐸&' (E) −𝐸&' (9) >0⇒neutron heavier. (b) Σ Multiplet Baseline = Σ⁰ (balanced curvature): 𝐸&' (H%)<𝐸&' (H&)⇒𝑀H%=𝑀H&−Δ𝐸&' so charged Σ baryons are lighter. State 𝑀* (MeV) 𝑄+,- Δ𝐸&'(MeV) Mass rel. to Σ⁰ Σ⁰ (uds) 1165 1.33 0 baseline Σ⁺ (uus) 1160 1.67 −6 Σ⁰ − 6 MeV Σ⁻ (dds) 1170 1.00 −4 Σ⁰ − 4 MeV ! 5! 7 Lattice and Topological Consistency The effective coupling corresponds to curvature energy densities of order 10 MeV at R ≈ 0.5 fm—comparable to fluctuations implied by lattice-determined topological susceptibility 𝜒I∼(200MeV)J [1, 2, 9]. A testable prediction is that the lattice correlator 𝐶KL(𝑟) = ⟨𝜇(0)𝜎(𝑟)⟩ and its integral 𝐼KL(𝑅) scale as 1/𝑅. Lattice QCD + QED studies of isospin mass splittings already achieve sub-MeV precision [6–8], enabling quantitative tests. 8 Microscopic Mechanisms Two QCD mechanisms naturally produce the required ∣𝑞 ∣ coupling: 1. Instanton-induced ’t Hooft interactions [10, 11]: linearization via a Hubbard– Stratonovich field introduces scalar couplings linking quark mass and topological charge densities. 2. Monopole condensation in the dual-superconductor model [4, 12]: curvature fluctuations of color flux tubes couple to condensate amplitude, generating an effective interaction proportional to ∣𝑞 ∣. Both mechanisms are consistent with confinement and nonzero 𝜒I. 9 Discussion and Outlook This work unifies hadronic mass splitting, curvature orientation, and charge sign in a single entropic–topological framework. 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