Unitarity triangles and CP-violating invariants

Take the inner product of the down and bottom columns of V. Unitarity gives $V_{ud}V_{ub}^*+V_{cd}V_{cb}^*+V_{td}V_{tb}^*=0$. Each term is a complex number, hence a directed vector in a plane. Placing these three vectors head to tail closes a triangle. The construction requires their relative complex phases as well as magnitudes. Three positive lengths satisfying a geometric inequality are not by themselves proof that independently measured weak amplitudes satisfy the complex unitarity relation.

For plotting, divide by the nonzero middle product and choose an overall orientation so the vertices are zero, one, and $z=\bar\rho+i\bar\eta=-V_{ud}V_{ub}^*/(V_{cd}V_{cb}^*)$. The closed vectors can then be written as z, one minus z, and minus one. This normalized triangle has base one and height the absolute imaginary part of z. Its area is convenient, but it is not the area of the original CKM-product triangle until the removed length scale is restored.

A quark rephasing multiplies every entry of one row or column by a common phase. The quartet $V_{ud}V_{cs}V_{us}^*V_{cd}^*$ contains each relevant row and column phase once and its inverse once, so they cancel. Define J as the imaginary part of this quartet. Other distinct-row, distinct-column quartets give plus or minus J according to their index ordering. A sign therefore has meaning only after fixing the quartet convention. The unsigned triangle area is always half the absolute value of J.

Use the standard parameterization $V=R_{23}U_{13}(\delta)R_{12}$, where the real rotations have positive sine above and negative sine below the diagonal, and the 13 entry of the complex rotation is $s_{13}e^{-i\delta}$. Direct multiplication gives J as a product of three sines, appropriate cosines, and the sine of delta. A phase of ninety degrees does not guarantee a large CP invariant: a small mixing angle can suppress the whole product. Vanishing any necessary angle collapses every unitarity triangle.

Within the three-family Standard Model quark sector, a nonzero J with nondegenerate same-charge quark masses signals a physical CP phase. Exact degeneracy would permit additional rotations and can remove observable CP violation even if a chosen matrix representation looks complex. This qualification distinguishes a convenient matrix invariant from the full statement about the mass and interaction theory. The strong CP parameter is a separate issue; this discussion concerns the charged-current flavor sector and does not claim to classify every possible source of CP violation.

Experiments do not photograph this triangle. They infer side lengths and angles from decay rates, mixing frequencies, and interference asymmetries, with process-specific strong-interaction inputs. Comparing these constraints overdetermines a common geometry. A disagreement can test the assumed three-family unitary description or reveal underestimated inputs, but requires a complete uncertainty analysis. Conversely, a neatly drawn triangle built from one global fit is not an independent experimental verification of each side. Its diagnostic power comes from comparing genuinely distinct measurements under transparent dynamical assumptions.