It is instructive to see how the scale invariance at the upper critical dimension becomes a scale invariance below this dimension. For small external wave vectors the vertex functions Γ Γ acquire additional exponents, for example Γ Γ 2 ( k ) ∼ ∼ k 2 − − η η ( d ) . If these exponents are inserted into a matrix A ( d ) (which only has values in the first column) the condition for scale invariance becomes det ( E + A ( d ) ) = 0 . This equation only can be satisfied if the anomalous exponents of the vertex functions cooperate in some way. In fact, the vertex functions depend on each other hierarchically. One way to express this interdependence are the Dyson–Schwinger equations.