 Solution

ID: past-exam-of-the-groups-and-representations-course-of-the-university-of-oxford/2021/3/b/solution

w = (w_{1}, w_{2})^{T}

q (w) = \frac{2 w _{1} + w _{2}}{3}, y (w) = \frac{w _{1} + 2 w _{2}}{3} .

Apply these to the meson octet weights in solution. The results, with electric charges measured in units of the positive elementary electric charge, are

w (2, - 1) (- 2, 1) (1, 1) (- 1, 2) (- 1, - 1) (1, - 2) (0, 0) twice q 1 - 1 10 - 1 00 y 0011 - 1 - 1 0 pseudoscalar octet member π^{+} π^{-} K^{+} K^{0} K^{-} \overset{ˉ}{K}^{0} π^{0}, η_{8}

The table illustrates the meson octet of pseudoscalar mesons; the same weight calculation applies to another meson species arranged in the same representation. Here flavour hypercharge is

Y = B + S

, with

B

S

3 \otimes \overline{3}

is retained, its additional trivial representation is a ninth meson with

(q, y) = (0, 0)

. This singlet is separate from the octet and need not share its mass.

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