Macroscopic Emergence of the NFW from Infrared Loop Interactions

We investigate whether the Navarro--Frenk--White (NFW) halo profile can arise from the infrared, static limit of loop--corrected gravity. The NFW profile accurately describes dark matter halos in cosmological simulations, yet its theoretical origin remains unsettled. Starting from a spectral (Källén--Lehmann) representation of the dressed graviton two--point function, we analyze the infrared structure of the polarization operator and separate monopole and curvature--dependent contributions. After renormalization of the monopole term (implemented by imposing a vanishing monopole contribution in the infrared), the remaining curvature sector governs the static response. Under the assumption of strong infrared spectral concentration around a single characteristic scale rH⁻¹, the spectral integral reduces to an effective Yukawa kernel of a single scale. Acting with the curvature operator yields, in the static limit, the cumulative form{equation}g̃(s)=1-(1+s)e⁻ˢ, s={rH}{r},{equation}which coincides in functional form with the NFW mass profile. The relative deviation from the exact NFW form induced by residual spectral width is parametrically suppressed in the galactic weak--field regime. A direct consequence is the scaling relation{equation}vₘₐₓ≈ 0.547\,√{{G Mₜₒₜ}{rH}},{equation}which reproduces characteristic Milky Way rotation velocities for observed halo parameters. If the curvature sector is removed, the model reduces to a pure cosmological constant background without halo structure. These results indicate that an NFW--type profile can emerge macroscopically from infrared loop effects under controlled renormalization conditions. \\\: NFW, infrared gravity, loop corrections, spectral methods, \\ renormalization, galaxy dynamics