= Superficial degree of divergence
{title2=$D$}
{wiki}
The <superficial degree of divergence> is the ultraviolet power obtained by scaling all independent <loop momenta> of a <Feynman diagram> together. If a scalar graph in $d$ dimensions has $L$ loops, $I$ internal propagators behaving as <momentum> to the power $-2$, and $N_\partial$ powers of <loop momentum> from derivative vertices or numerators, then
$$
D=dL-2I+N_\partial.
$$
Each loop integration contributes $d$ powers, each propagator removes two, and each numerator power adds one. A scalar theory with no derivative vertices has the familiar specialization $D=dL-2I$.
For a connected graph in four-dimensional <quartic scalar field theory>, $L=I-V+1$ and $4V=2I+E$, where $V$ is the number of <interaction vertices> and $E$ the number of external lines. Eliminating $I,V$ gives
$$
D=4-E.
$$
Thus a two-point graph has superficial degree two, while a four-point graph has degree zero, corresponding to possible logarithmic ultraviolet behavior.
The word superficial matters: tensor or gauge cancellations can reduce the actual divergence, and a graph with negative overall $D$ can still contain divergent subgraphs. Exceptional external momenta can introduce a separate infrared problem. <Power counting> organizes possible <counterterms>; convergence requires inspecting the relevant subgraphs and kinematic regime as well.
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