 Solution

ID: past-exam-of-the-quantum-field-theory-course-of-the-university-of-oxford/2017/3/b/solution

The two real fields have canonical normalization because

ϕ^{*} ϕ = \frac{ϕ _{1}^{2} + ϕ _{2}^{2}}{2}, \partial_{μ} ϕ^{*} \partial^{μ} ϕ = \frac{1}{2} \sum_{a = 1}^{2} \partial_{μ} ϕ_{a} \partial^{μ} ϕ_{a} .

Therefore

L = \frac{1}{2} a = 1 \sum 2 [(\partial ϕ_{a})^{2} - m_{0}^{2} ϕ_{a}^{2}] - \frac{λ _{0}}{16} (ϕ_{1}^{4} + 2 ϕ_{1}^{2} ϕ_{2}^{2} + ϕ_{2}^{4}) .

The interaction is invariant under

O (2)

rotations and reflections of

(ϕ_{1}, ϕ_{2})

. The continuous rotations are the original complex-field

U (1)

symmetry. The factor

1/16

follows from both the original

1/4

and the square of the

1/2

ϕ^{*} ϕ

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