Hypercharge

Template:Short description Template:About Template:Multiple issues Template:Flavour quantum numbers 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 Template:Mvar₃. The isospin alone was sufficient for two quark flavours — namely Template:Subatomic particle and Template:Subatomic particle — whereas presently 6 flavours of quarks are known.

SU(3) weight diagrams (see below) are 2 dimensional, with the coordinates referring to two quantum numbers: Template:Mvar₃ (also known as Template:Mvar_z), which is the Template:Math component of isospin, and Template:Mvar, which is the hypercharge (defined by strangeness Template:Mvar, charm Template:Mvar, bottomness Template:Mvar, topness Template:Mvar, and baryon number Template:Mvar). Mathematically, hypercharge is <ref>Template:Citation</ref>

<math>Y = B + C - S + T' - B' ~. </math>

Strong interactions conserve hypercharge (and weak hypercharge), but weak interactions do not.

Relation with electric charge and isospin

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} The Gell-Mann–Nishijima formula relates isospin and electric charge

<math> Q = I_3 + \tfrac{1}{2}Y,</math>

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:

<math> Y = 2 \bar Q.</math>

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

These definitions in their original form hold only for the three lightest quarks.

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 Template:Nowrap 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 [[Omega baryon|Template:SubatomicParticle 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.

Template:Nowrap specifies the number of points in the topmost side of the hexagon while Template:Nowrap specifies the number of points on the bottom side.

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Examples

• The nucleon group (protons with Template:Nowrap and neutrons with Template:Nowrap) have an average charge of Template:Sfrac, so they both have hypercharge Template:Nowrap (since baryon number Template:Nowrap and Template:Nowrap). From the Gell-Mann–Nishijima formula we know that proton has isospin Template:Nowrap while neutron has Template:Nowrap
• This also works for quarks: For the up quark, with a charge of Template:Sfrac, and an Template:Mvar₃ of Template:Sfrac, we deduce a hypercharge of Template:Sfrac, due to its baryon number (since three quarks make a baryon, each quark has a baryon number of Template:Sfrac).
• For a strange quark, with electric charge Template:Sfrac, a baryon number of Template:Sfrac, and strangeness −1, we get a hypercharge Template:Nowrap so we deduce that Template:Nowrap That means that a strange quark makes an isospin singlet of its own (the same happens with charm, bottom and top quarks), while up and down constitute an isospin doublet.
• All other quarks have hypercharge Template:Nowrap.

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 strong hypercharge, Template:Mvar, is the following combination of the numbers of up (Template:Mvar_u), down (Template:Mvar_d), strange (Template:Mvar_s), charm (Template:Mvar_c), top (Template:Mvar_t) and bottom (Template:Mvar_b):

<math> Y = \tfrac{1}{3} n_\textrm{u} + \tfrac{1}{3} n_\textrm{d} + \tfrac{4}{3} n_\textrm{c} - \tfrac{2}{3} n_\textrm{s} + \tfrac{4}{3} n_\textrm{t} - \tfrac{2}{3} n_\textrm{b} ~.</math>

In modern descriptions of hadron interaction, it has become more obvious to draw Feynman diagrams that trace through the individual constituent quarks (which are conserved) composing the interacting baryons and mesons, rather than bothering to count strong hypercharge quantum numbers. Weak hypercharge, however, remains an essential part of understanding the electroweak interaction.

References

• Template:Cite book