Standard Model and Beyond · Problem 3.21

Gauge Theories

Physics — Particle Physics · u-spin-hyperon-magnetic-moments

Problem

From (3.89), the electromagnetic current can be written as

J^\mu_Q = J^{3\mu} + \frac{1}{\sqrt{3}} J^{8\mu}

, where

J^3_\mu

and

J^8_\mu

are elements of an

SU(3)

octet of currents.

J^\mu_Q

is a singlet under the

U

-spin subgroup of

SU(3)

, which has generators

U^3 = -\frac{1}{2}T^3 + \sqrt{\frac{3}{4}}T^8

U^\pm = T^6 \pm iT^7

. (a) Use

U

-spin to show that in the

SU(3)

limit the hyperon magnetic moments are related by

\mu_{\Sigma^+} = \mu_p, \quad \mu_n = \mu_{\Xi^0} = -\frac{1}{2}\mu_{\Sigma^0} + \frac{3}{2}\mu_\Lambda

\mu_{\Sigma^-} = \mu_{\Xi^-}, \quad \mu_{\Sigma\Lambda} = \frac{\sqrt{3}}{2}(\mu_{\Sigma^0} - \mu_\Lambda),

where

\mu_{\Sigma\Lambda}

is the transition moment observed in

\Sigma^0 \to \Lambda\gamma

. (b) Use isospin to show that

\mu_{\Sigma^0} = \frac{1}{2}(\mu_{\Sigma^+} + \mu_{\Sigma^-})

. (c) Use

SU(3)

to show that

\mu_{\Sigma^0} = -\mu_\Lambda

, allowing all of the magnetic moments to be expressed in terms of

\mu_p

and

\mu_n

. Hint: use the analog of (3.102).

Intuitive explanation

The key to this problem is recognizing that $U$ -spin, isospin, and $SU(3)$ symmetries constrain how the electromagnetic current couples to different baryon states. Since $J^\mu_Q$ is a $U$ -spin singlet, it cannot connect states that differ in $U$ -spin quantum number—this selection rule forces relationships between magnetic moments of particles in the same $U$ -spin multiplet. Similarly, isospin and $SU(3)$ symmetries impose their own constraints by requiring the current to respect these group structures. The art is identifying which symmetry applies to which pair of particles and using the corresponding selection rules or multiplet structure to derive the moment relations.

Intuitive explanation

Related theory

Magnetic Force And Magnetic Field

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