116 The occupation number n k is given by the Bose-Einstein distribution where we are neglecting the chemical potential. Substituting this into Eq. (A.10) gives The 1 2 term that appears in Eq. (A.10) is not present in Eq. (A.12) as it can be removed by one of the following: Eq. (A.12) can be expressed as The proof is as follows. The definition of the J term is where for our derivation we define n$_{ω}$$_{k}$ = 1 e ω k T − 1 (A.11) 〈 φ 2 〉 T = 1 2 π 2 ∫ ∞ 0 k 2 ω$_{k}$ ( e ω k T − 1 ) d k (A.12) − Having n$_{ω}$$_{k}$ ≫ 1 2 − Subtracting the vacuum − Performing normal ordering 〈 φ 2 〉 T = T 2 4 π $_{2}$J (1) − ( | m$_{φ,}$$_{eff}$ | T , 0 ) (A.13) J ( ν ) ∓ ( κ, τ ) ≡ ∫ ∞ κ ( x 2 − κ $^{2}$) ν 2 e x τ ∓ 1 d x + ∫ ∞ κ ( x 2 − κ $^{2}$) ν 2 e x + τ ∓ 1 d x (A.14) κ ≡ | m$_{φ,}$$_{eff}$ | T τ ≡ µ T (A.15)