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ID: past-exam-of-the-advanced-quantum-field-theory-for-particle-physics-course-of-the-university-of-oxford/2019/3/b/iv/solution

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Past exam of the advanced quantum field theory for particle physics course of the University of Oxford / 2019 / 3 / b / iv / Solution by

Codex  0 2026-10-25

Write

H = ϕ_{1}

and

P = ϕ_{2}

, both Hermitian matrices. Through quadratic order around the chosen vacuum expectation value,

X = Φ^{†} Φ = C^{2} 1 + 2 C H + \frac{1}{2} (H^{2} + P^{2} + i [H, P]) .

The trace of a commutator vanishes. The needed matrix traces are therefore

tr X tr X^{2} (tr X)^{2} = N C^{2} + 2 C tr H + \frac{1}{2} tr (H^{2} + P^{2}), = N C^{4} + 22 C^{3} tr H + C^{2} tr (3 H^{2} + P^{2}) + O ((H, P)^{3}), = N^{2} C^{4} + 22 N C^{3} tr H + N C^{2} tr (H^{2} + P^{2}) + 2 C^{2} (tr H)^{2} + O ((H, P)^{3}) .

Using

m^{2} = - (α + Nβ) C^{2}

, all linear terms cancel. The quadratic scalar potential reduces to

V^{(2)} = α C^{2} tr H^{2} + β C^{2} (tr H)^{2},

with no

P^{2}

term. The kinetic term is

L_{kin}^{(2)} = \frac{1}{2} tr [(\partial H)^{2} + (\partial P)^{2}],

since its mixed term is the trace of a commutator. A Hermitian matrix has

N^{2}

independent real components, so

P

describes exactly

N^{2}

massless scalar modes.

To determine the other masses, take

T_{0} = 1 / N

and traceless Hermitian

T_{A}

with

tr (T_{A} T_{B}) = δ_{A B}

, and expand

H = h_{0} T_{0} + \sum_{A = 1}^{N^{2} - 1} h_{A} T_{A}

. These are canonically normalized real scalar fields. The scalar mass matrix is diagonal in this basis:

m_{h_{0}}^{2} = 2 C^{2} (α + Nβ) = - 2 m^{2}, m_{h_{A}}^{2} = 2 α C^{2} (A = 1, \dots, N^{2} - 1), m_{P_{A}}^{2} = 0 (A = 0, \dots, N^{2} - 1) .

Thus

H

contains one massive singlet and a degenerate multiplet of

N^{2} - 1

massive fields in the Adjoint representation of the unbroken diagonal

S U (N)

. Both squared masses are strictly positive for the specified parameters. The

P

fields are the

N^{2}

Goldstone bosons associated with the broken generators:

N^{2} - 1

traceless relative left-right rotations and one overall phase rotation.

Goldstone's theorem guarantees those massless directions. It does not alone forbid additional accidental massless modes; the positive scalar mass matrix computed here supplies the further conclusion that every

H

component is massive. The fields describe a global symmetry, so these Goldstone bosons are physical rather than being absorbed into massive gauge bosons.

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