Hypercharge

Hypercharge is one of two quantum numbers of the SU(3) model of hadrons, alongside isospin I₃. The isospin alone is sufficient for two quark flavours—namely,
u
and
d
—whereas presently six flavours of quarks are known.

SU(3) weight diagrams (see below) are 2-dimensional with the coordinates referring to two quantum numbers, I₃ (also known as I_z), which is the z-component of isospin and Y, which is the hypercharge (the sum of strangeness S, charm C, bottomness B′, topness T, and baryon number B). Mathematically, hypercharge is

${\displaystyle Y=S+C+B^{\prime }+T+B.}$

Strong interactions conserve hypercharge, but weak interactions do not.

The Gell-Mann–Nishijima formula relates isospin and electric charge

${\displaystyle Q=I_{3}+{\tfrac {1}{2}}Y,}$

where I₃ is the third component of isospin and Q is the particle's charge.

Isospin creates multiplets of particles whose average charge is related to the hypercharge by:

${\displaystyle Y=2{\bar {Q}}.}$

since the hypercharge is the same for all members of a multiplet, and the average of the I₃ values is 0.

The SU(2) model has multiplets characterized by a quantum number J, which is the total angular momentum. Each multiplet consists of 2J + 1 substates with equally-spaced values of J_z, forming a symmetric arrangement seen in atomic spectra and isospin. This formalizes the observation that certain strong baryon decays were not observed, leading to the prediction of the mass, strangeness and charge of the
Ω⁻
baryon.

The SU(3) has supermultiplets containing SU(2) multiplets. SU(3) now needs two numbers to specify all its sub-states which are denoted by λ₁ and λ₂.

(λ₁ + 1) specifies the number of points in the topmost side of the hexagon while (λ₂ + 1) specifies the number of points on the bottom side.

• The nucleon group (protons with Q = +1 and neutrons with Q = 0) have an average charge of +1/2, so they both have hypercharge Y = 1 (baryon number B = +1, S = C = B′ = T = 0). From the Gell-Mann–Nishijima formula we know that proton has isospin I₃ = +1/2, while neutron has I₃ = −1/2.
• This also works for quarks: for the up quark, with a charge of +2/3, and an I₃ of +1/2, we deduce a hypercharge of 1/3, due to its baryon number (since three quarks make a baryon, a quark has a baryon number of 1/3).
• For a strange quark, with charge −1/3, a baryon number of 1/3 and strangeness of −1 we get a hypercharge Y = −2/3, so we deduce an I₃ = 0. That means that a strange quark makes an isospin singlet of its own (same happens with charm, bottom and top quarks), while up and down constitute an isospin doublet.

Hypercharge was a concept developed in the 1960s, to organize groups of particles in the "particle zoo" and to develop ad hoc conservation laws based on their observed transformations. With the advent of the quark model, it is now obvious that hypercharge Y is the following combination of the numbers of up (n_u), down (n_d), strange (n_s), charm (n_c), top (n_t) and bottom (n_b):

${\displaystyle Y={\tfrac {1}{3}}\left(n_{u}+n_{d}-2n_{s}+4n_{c}-2n_{b}+4n_{t}\right).}$

In modern descriptions of hadron interaction, it has become more obvious to draw Feynman diagrams that trace through individual quarks composing the interacting baryons and mesons, rather than counting hypercharge quantum numbers. Weak hypercharge, however, remains of practical use in various theories of the electroweak interaction.

• Semat, Henry; Albright, John R. (1984). Introduction to Atomic and Nuclear Physics. Chapman and Hall. ISBN 978-0-412-15670-0.