IndisputableMonolith.Physics.QCDRGE.AlphaRunning
The AlphaRunning module supplies the one-loop beta function coefficient beta0 together with the explicit running formula for the strong coupling alpha_s(Q). QCD phenomenologists extending one-loop results to two-loop running cite it as the base layer. The module consists of definitions for N_c, N_f_Z and beta0, plus positivity and bound lemmas that support the integrated running expression.
claim$N_c=3$, $N_{f,Z}=5$, $b_0=11N_c/3-2N_f/3$, and the one-loop running satisfies $1/alpha_s(Q)=1/alpha_s(M_Z)+b_0 log(Q^2/M_Z^2)$ for $Q>M_Z$, with $alpha_s(M_Z)$ taken from the StrongForce derivation.
background
The module sits inside the Recognition Science derivation of the strong force (T15). It imports the RS time quantum tau_0=1 tick from Constants and the alpha_s(M_Z) value obtained in StrongForce from planar ledger symmetries. Core objects are the color number N_c, the active flavor count N_f_Z at the Z scale, the one-loop coefficient beta0, the reference scale M_Z_GeV, and the function alpha_s_running that solves the RGE d alpha_s / d log Q = -beta0 alpha_s^2.
proof idea
This is a definition module. It introduces N_c and N_f_Z as constants, defines beta0 directly from the standard QCD formula, proves beta0_pos and alpha_s_MZ_pos, records the numerical bounds alpha_s_MZ_bounds, and supplies the closed-form running alpha_s_running together with the auxiliary lemmas low_Q_log_negative and log_at_MZ.
why it matters in Recognition Science
AlphaRunning supplies the one-loop foundation that the downstream module TwoLoopAlphaS extends by adding the second beta-function coefficient b_1 to obtain the standard MS-bar two-loop running formula. It thereby completes the one-loop segment of the QCD RGE track that begins with the T15 StrongForce derivation of alpha_s(M_Z).
scope and limits
- Does not include two-loop or higher beta-function coefficients.
- Does not incorporate quark-mass threshold corrections.
- Does not derive the input value of alpha_s(M_Z).
- Does not treat non-perturbative or lattice effects.