Hypercharge

Not to be confused with Weak hypercharge.

In particle physics, the hypercharge (from hyperonic + charge) Y of a particle is related to the strong interaction, and is distinct from the similarly named weak hypercharge, which has an analogous role in the electroweak interaction. The concept of hypercharge combines and unifies isospin and flavour into a single charge operator.

Definition

Hypercharge in particle physics is a quantum number relating the strong interactions of the SU(3) model. Isospin is defined in the SU(2) model while the SU(3) model defines hypercharge.

SU(3) weight diagrams (see below) are 2-dimensional with the coordinates referring to two quantum numbers, 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

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

and conservation of hypercharge implies a conservation of flavour. Strong interactions conserve hypercharge, but weak interactions do not.

Relation with electric charge and isospin

Main article: Gell-Mann–Nishijima formula

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

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

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:

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

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

SU(3) model in relation to hypercharge

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 2 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.

SU(3) singlet weight diagram, where Y is Hypercharge and I3 is the third component of isospin	SU(3) triplet weight diagram
SU(3) septet / octet / nonet weight diagram Note similarity with both charts on the right. The number used to describe the weight diagram depends on whether the particle(s) occupying the center of the diagram have one, two, or three distinct names.	Mesons of spin 0 form a nonet, K = kaon, π = pion, η = eta meson	The octet of light spin-1/2 baryons described in SU(3). n = neutron, p = proton, Λ = Lambda baryon, Σ = Sigma baryon, Ξ = Xi baryon
SU(3) decuplet weight diagram Note similarity with chart on the right.	A combination of three u, d or s-quarks with a total spin of 3/2 form the so-called baryon decuplet. The lower six are hyperons. S = strangeness, Q = electric charge

Examples

• 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 you need 3 quarks to make a baryon, a quark has 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.

Practical obsolescence

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 (if one only includes the up, down and strange quarks out of the total 6 quarks in the Standard Model), 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):

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

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.

References

• Henry Semat, John R. Albright (1984). Introduction to atomic and nuclear physics. Chapman and Hall. ISBN 978-0-412-15670-0.