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In particle physics, the **hypercharge** (a portmanteau of hyperonic and charge) *Y* of a particle is a quantum number conserved under the strong interaction. The concept of hypercharge provides a single charge operator that accounts for properties of isospin, electric charge, and flavour. The hypercharge is useful to classify hadrons; the similarly named weak hypercharge has an analogous role in the electroweak interaction.

## Definition

Hypercharge is one of two quantum numbers of the SU(3) model of hadrons, alongside isospin *I*_{3}. 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, *I*_{3} (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

Strong interactions conserve hypercharge, but weak interactions do not.

## Relation with electric charge and isospin

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

where *I*_{3} 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:

since the hypercharge is the same for all members of a multiplet, and the average of the *I*_{3} 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 2*J* + 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 *λ*_{1} and *λ*_{2}.

(*λ*_{1} + 1) specifies the number of points in the topmost side of the hexagon while (*λ*_{2} + 1) specifies the number of points on the bottom side.

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.A combination of three up, down or strange 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*I*_{3}= +1/2, while neutron has*I*_{3}= −1/2. - This also works for quarks: for the
*up*quark, with a charge of +2/3, and an*I*_{3}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*I*_{3}= 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 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}):

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

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