Cabibbo Angle and CKM Matrix Element Calculator

The Cabibbo angle theta_c is the mixing angle Nicola Cabibbo introduced in 1963 to explain why the weak force couples quarks in combinations that are rotated relative to their mass eigenstates, defined by sin(theta_c) = V_us, roughly 13 degrees.

In Cabibbo's original two-generation picture, the down and strange quarks mix into weak eigenstates d' = cos(theta_c) d + sin(theta_c) s, so the coupling strength between an up quark and a strange quark, V_us, directly equals sin(theta_c) by definition.

Yes. Even though the full Standard Model has three quark generations described by the larger CKM matrix, the (1,2) element of that matrix is still called V_us and its arcsine is still called the Cabibbo angle, it remains a standard shorthand for the dominant quark mixing strength.

The CKM matrix is required to be unitary if there are exactly three quark generations and no new physics, meaning each row's squared magnitudes must sum to 1. For the first row, V_ud squared + V_us squared + V_ub squared should equal 1 if the Standard Model with three generations is complete.

Using current PDG central values, the first-row sum comes out to about 0.9985, measurably below 1 by more than the combined experimental uncertainty. This 2 to 3 sigma level tension is sometimes called the Cabibbo angle anomaly or Vud/Vus tension, and is an active topic in precision electroweak physics, it is not necessarily evidence of new physics, since it may also reflect underestimated theoretical uncertainties in the V_ud and V_us extractions.

arcsin(V_us) uses the definition directly. arctan(V_us/V_ud) instead assumes cos(theta_c) equals V_ud exactly, which is only an approximation once a third generation and V_ub are present. The two values differ by a few thousandths of a degree, small but real, and the gap is itself a diagnostic of how close the two-generation approximation is to the full three-generation result.

V_ud is approximately 0.97373, V_us is approximately 0.2243, and V_ub is approximately 0.00382, the current Particle Data Group central values for the first row of the CKM matrix. All three are fully editable so you can test other measurements or textbook values.

V_ud is extracted mainly from superallowed nuclear beta decays and neutron decay to about 4 to 5 significant figures. V_us is extracted mainly from kaon decays and hyperon decays to a similar precision, small shifts in either value directly move both the Cabibbo angle and the unitarity sum.

Barely. V_ub is about 60 times smaller than V_us, so its square contributes only around 0.0015% to the first-row unitarity sum, it is included here for completeness and to compute the full three-term unitarity check, but it has almost no effect on the numerical result.

The CKM matrix elements V_ud, V_us, and V_ub are dimensionless coupling strengths between 0 and 1. This calculator reports the resulting Cabibbo angle in degrees, since that is the conventional unit used in most textbooks and papers, though the underlying calculation works entirely in radians internally.

Nicola Cabibbo introduced the mixing angle in 1963, before the charm quark was even discovered, to fix an inconsistency between the observed rates of strangeness-conserving and strangeness-changing weak decays. It was later generalized to three generations by Kobayashi and Maskawa in 1973, whose extension explained CP violation and earned them the 2008 Nobel Prize in Physics.