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 Incoming links: Eta mixing

Past exam of the particle and nuclear physics course of the University of Cambridge / 2016 / Paper 1 / 1 / b / Solution 2026-10-20

Use natural units and write the common light constituent quark mass as

m

, and the strange one as

M

. For two spin one-half constituents,

S_{1} \cdot S_{2} = \frac{1}{2} [S (S + 1) - \frac{3}{2}] = {- 3/4, + 1/4, S = 0, S = 1.

The relative hyperfine contributions are

- 3

and

+ 1

. Absorb their normalization and spatial overlap into pair-dependent parameters

h_{ℓℓ}, h_{ℓ s}, h_{ss}

. The meson hyperfine mass splitting model gives

m_{π} m_{K} m_{ϕ} = 2 m - 3 h_{ℓℓ}, = m + M - 3 h_{ℓ s}, = 2 M + h_{ss} . m_{ρ} m_{K^{*}} = 2 m + h_{ℓℓ}, = m + M + h_{ℓ s},

A common color-magnetic coefficient gives

h_{ij} \propto 1/ (m_{i} m_{j})

, but its numerical value is not needed.

In the specified Eta octet state used for the eta meson the light-flavor probability is

1/3

and the strange-flavor probability

2/3

. Assuming a flavor-diagonal mass operator,

m_{η} = \frac{1}{3} (2 m - 3 h_{ℓℓ}) + \frac{2}{3} (2 M - 3 h_{ss}) = \frac{2}{3} m + \frac{4}{3} M - h_{ℓℓ} - 2 h_{ss} .

Inserting the five mass expressions into the proposed combination makes the coefficient of

h_{ℓ s}

equal to

4 - 4 = 0

. Those of

m, M, h_{ℓℓ}, h_{ss}

are

2/3, 4/3, - 1, - 2

. Consequently

m_{η} = - \frac{1}{3} m_{π^{0}} - 2 m_{ρ^{0}} + 4 m_{K^{* 0}} + \frac{4}{3} m_{K^{0}} - 2 m_{ϕ} .

This is an identity within the assumed mass operator and octet flavor assignment. It is not an exact relation among measured masses: eta mixing and additional spin-independent binding effects are outside those assumptions.

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