QCD Color Factor Calculator

A Casimir invariant (color factor) is a number that measures the strength of the color force felt by a particle in a given representation of the SU(N) gauge group. C_F applies to quarks (the fundamental representation) and C_A applies to gluons (the adjoint representation), both appear as multiplicative factors in QCD scattering amplitude and radiative-correction calculations.

For N=3, C_F=4/3 while C_A=3, more than twice as large. This reflects that gluons, which carry color charge themselves, interact with the color field more strongly than quarks do. Gluon self-interaction (a direct consequence of C_A being large) is the physical reason QCD exhibits asymptotic freedom, the strong coupling weakens at short distances instead of the opposite behavior seen in QED.

T_F is a fixed normalization constant (0.5 in the standard convention) that sets the overall scale of a quark loop's contribution to gluon self-energy diagrams. Unlike C_F and C_A, T_F does not depend on N, it is a convention choice baked into how the SU(N) generators are normalized.

b0 = (11*C_A - 4*nf*T_F)/3 is the one-loop coefficient that controls how fast the strong coupling alpha_s runs (changes) with energy scale, exactly the b0 used inside the Running Coupling Constant Estimator's formula alpha_s(Q) = 4*pi/(b0*ln(Q squared/Lambda squared)). At N=3 and nf=5, both calculators give b0=7.66667, they are two views of the same underlying one-loop QCD calculation.

Experiments (from deep inelastic scattering to precision measurements of hadron decay rates) consistently confirm that quarks come in exactly 3 color charges, no more, no less. This calculator lets you vary N as a generalization exercise, common in theoretical papers exploring large-N QCD, but N=3 is the only value that matches the real strong force.

C_A=N grows linearly and without bound. C_F=(N squared - 1)/(2N) grows roughly linearly too for large N, approaching N/2, but always stays below C_A. This calculator's chart sweeps N from 2 to 8 to show both trends directly, illustrating why gauge theories with a larger number of colors have an even stronger gluon self-coupling relative to the quark coupling.

N=1 corresponds to the abelian U(1) gauge group (essentially electromagnetism), which has no non-trivial adjoint representation and no gluon self-interaction, so the SU(N) Casimir color factor concept does not meaningfully apply. N=2 (SU(2), structurally related to the weak force's gauge group) is the smallest non-abelian case where C_F and C_A are both well defined and distinct.

nf is the number of quark flavors light enough to be produced (active) at the energy scale being studied, it ranges from 3 (below the charm threshold) up to 6 (above the top threshold) in real QCD phenomenology, though this calculator allows 0 to 6 for full generality. It enters the beta function through quark-loop vacuum polarization diagrams, but C_F and C_A are purely geometric properties of the gauge group and do not depend on nf at all.

C_F multiplies quark-gluon vertex diagrams and quark self-energy corrections throughout QCD, for example it sets the overall scale of the Casimir scaling hypothesis used to estimate confining potentials for different color representations, and appears directly in next-to-leading-order QCD cross section formulas.

No, only the one-loop coefficient b0. The full QCD beta function has higher-order terms (b1, b2, and beyond) that become important at lower energy scales or when higher precision is required, this calculator, like the companion Running Coupling Constant Estimator, sticks to the one-loop approximation for transparency and educational clarity.