Unitary triangle

Graphic source:
CKMfitter collaboration

The top of the triangle
CP violation was first observed in 1964 by studying two particles containing the strange quark: the short-lived K_S and the longer-lived K_L. In a CP-conserving world, the K_S would decay to two particles called pions, and the K_L would decay to three. While this is very nearly true, the K_L was also found to decay to two pions about 0.1% of the time. This manifestation of CP violation is described by the parameter ε_K, and measurements of it constrain the peak of the triangle to lie somewhere on the light green, boomerang-shaped region.

The angle β
By studying the interference between decays of neutral B⁰ mesons, which consist of an anti-bottom quark and a down quark, and their antiparticles ⁰, physicists determine the angle β of the unitarity triangle. It is the “golden measurement” for the two B factories, BaBar at SLAC in California, and Belle at KEK in Japan. Because what they measure is sin(2β), there are actually four possible solutions for the angle, shown as the four blue jets radiating from the bottom-right corner of the triangle.

The other angles: α and γ
These two angles are more difficult to measure than β. Quantum effects, so-called “penguin” processes, interfere with direct measurements of α, while measurements of γ require studies of rare decay processes. The B factories have been able to measure these angles, adding two constraints to the triangle. Like β, the angle α produces four allowed regions for the upper vertex, shown as blue arc-shaped areas. The angle γ constrains the location of the peak of the triangle to the two wedges radiating from the bottom-left corner of the triangle.

The triangle’s left side
Measuring the rate at which bottom quarks decay into up and charm quarks, physicists determine two elements of the CKM matrix called V_ub and V_cb. The ratio of these two elements gives the length of the left side of the triangle, the end of which must lie in the dark green circle. Many experiments have measured V_ub and V_cb, and the current best measurements come from the B factories at SLAC and KEK.

The triangle’s right side
The B⁰ meson can spontaneously turn into a ⁰ meson, its antiparticle. The rate at which this transformation occurs has been measured by a number of experiments and constrains the length of the right side of the triangle, placing its end in the yellow ring (largely covered by the orange ring). Studying the B_s meson, which contains an anti-bottom quark and a strange quark, and finding no evidence for its transformation into the antiparticle _s, physicists know that the end point of the right side must lie within the orange ring only. Experiments at Fermilab are currently improving this measurement, and may soon observe B_s to _s mixing for the first time.