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unitaritytriangle

Unitarity triangle is a triangle in the complex plane that encodes the unitarity of the Cabibbo–Kobayashi–Maskawa (CKM) matrix, the quark mixing matrix in the Standard Model. Because the CKM matrix is unitary, certain sums of products of its elements must vanish. The most studied case is the bd unitarity triangle, derived from the relation Vud Vub* + Vcd Vcb* + Vtd Vtb* = 0. Interpreted as vectors in the complex plane, these three terms form the sides of a closed triangle. In the Wolfenstein parametrization the apex of this triangle—the point where the second and third sides meet—is given by the complex ratio -Vud Vub*/Vcd Vcb*, often denoted by the coordinates (rho_bar, eta_bar).

The internal angles of the triangle are labeled alpha, beta, and gamma, and they correspond to CP-violating

Experimentally, the unitarity triangle provides a framework to test the Standard Model: independent measurements of the

phases
in
B-meson
decays.
They
can
be
written
in
terms
of
CKM
elements
as
alpha
≡
arg(-Vtd
Vtb*/Vud
Vub*),
beta
≡
arg(-Vcd
Vcb*/Vtd
Vtb*),
and
gamma
≡
arg(-Vud
Vub*/Vcd
Vcb*).
The
side
lengths
are
proportional
to
the
magnitudes
|Vud
Vub*|,
|Vcd
Vcb*|,
and
|Vtd
Vtb*|,
and
the
area
of
the
triangle
is
proportional
to
the
Jarlskog
invariant
J,
a
measure
of
CP
violation
in
the
quark
sector.
sides
and
the
angles,
obtained
from
CKM
elements
and
CP-violating
observables
in
B
decays,
are
combined
in
global
fits.
Consistency
supports
the
CKM
mechanism
of
CP
violation;
inconsistencies
could
indicate
new
physics.
Multiple
unitarity
triangles
exist,
but
the
bd
triangle
is
the
most
extensively
studied.