Hypercharge

In particle physics, the hypercharge (represented by Y) of a particle is related to the strong interaction, and it should not be confused with similarly named weak hypercharge, which has an analogous role in the electroweak interaction.

Hypercharge

Hypercharge in particle physics is a quantum number relating the strong interactions of the Special Unitary of a 3*3 matrix algebraic structure or SU(3) model. Note that isospin is defined in the SU(2) model while the SU(3) model defines hypercharge.

SU(3) weight diagrams are 2 dimensional with the coordinates referring to two quantum numbers, T₃, which is the third component of isospin and Y the hypercharge which is the sum of the Strangeness (S) and Baryon number (B).

Y = S + B

Relation with Electric charge and Isospin

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

$(1)\; qquad\; Q\; =\; I\_z\; +\; \{1\; over\; 2\}\; Y$

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

Hypercharge is also a term used to refer to the Conservation of strangeness and is a combination of the conservation of charge, isospin, and baryon number, which is expressed below:

$(2)\; qquad\; Y\; =\; S\; +\; B\; =\; 2(Q\; -\; I\_3)$

Note that hypercharge may not be conserved in weak nuclear interactions.

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

$(3)\; qquad\; Y\; =\; 2\; bar\; Q$.

which is easily derived from (2), 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 formalises the observation that certain strong baryon decay were not observed leading to the prediction of the mass, strangeness and charge of the Ω− hyperon.

The SU(3) has supermultiplets containing SU(2) multiplets. SU(3) now needs 2 numbers to specify all its substates 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.

Examples

• The nucleon group (proton plus neutron) have an average charge of (1 + 0)/2 = +1/2, so they both have hypercharge Y = 1 (baryon number B = +1, flavor charges set to 0). From the Gell-Mann–Nishijima formula we know that proton has isospin +1 - 1/2 = +1/2, while neutron is the 0 − 1/2 = −1/2.
• This also works for quarks: for the up quark, with a charge of +2/3, and an I_z 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 I_z = 0. That means that a strange quark makes a 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 "subatomic 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, down and strange quarks ($n\_u$) , ($n\_d$), ($n\_s$):

$(4)\; qquad\; Y\; =\; \{1\; over\; 3\}\; (n\_u\; +\; n\_d\; -\; 2\; n\_s)$

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.

See also

• Flavour (particle physics), baryon number and quark flavor charges: isospin, strangeness, charm, bottomness, topness
• Weak isospin, Weak hypercharge, standard model
• Murray Gell-Mann
• Richard Feynman

References

• Henry Semat, John R. Albright (1984). Introduction to atomic and nuclear physics. Chapman and Hall. ISBN 0-412-15670-9.