\frac{1}{vg} = \int_{∣ k ∣ < Λ} \frac{d ^{2} k}{( 2 π ) ^{2}} \int_{- \infty}^{\infty} \frac{d ω}{2 π} \frac{1}{ω ^{2} + v ^{2} k ^{2} + m ^{2}} .

1/ (2 v^{2} k^{2} + m^{2})

, and radial momentum integration gives

\frac{1}{vg} = \frac{v ^{2} Λ ^{2} + m ^{2} - m}{4 π v ^{2}} .

\frac{1}{v g _{c}} = \frac{Λ}{4 π v} .

Taking the cutoff to infinity in the difference gives

\frac{1}{g} - \frac{1}{g _{c}} = - \frac{m}{4 π v},

and hence

m = \frac{4 π v ( g - g _{c} )}{g g _{c}} .

A positive mass solution exists only for

g > g_{c}

. For

g < g_{c}

, the symmetric saddle cannot enforce the constraint with

m^{2} > 0

; instead the

O (N)

O (N - 1)

, the field acquires Néel order, and the ordered phase contains massless Goldstone bosons.

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