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The central charge is the coefficient of the fourth-order pole in the stress-tensor operator-product expansion, $T(z)T(w)\sim c/[2(z-w{)}^4]+\cdots$. A free real boson contributes one and a free real chiral fermion contributes one half.

An operator product expansion rewrites nearby local local field operators as a sum of local local field operators with separation-dependent coefficient functions: ${\mathcal{O}}_i(x){\mathcal{O}}_j(0)={\sum }_k{C}_{ij}{}^k(x){\mathcal{O}}_k(0)$, in the appropriate short-distance or convergent correlation function sense. The coefficients can contain singular terms, but regular terms and descendants also carry information.

For example, the stress-tensor operator-product expansion in a two-dimensional conformal field theory contains the identity contribution fixed by the central charge, followed by the stress-energy tensor and its derivative. Substituting an operator product expansion into a correlation function reduces a nearby pair to a sum of simpler insertions. Its domain and convergence properties depend on the theory and operator configuration; a schematic short-distance equality is not an unrestricted multiplication rule at every separation.

The holomorphic part of the free-boson worldsheet propagator givesApply Wick theorem to $T(z)T(w)$. The two double contractions give $D/[2(z-w{)}^4]$, while the single contractions reconstruct $T$ and its derivative. Thus the stress-tensor operator-product expansion isand the $D$ embedding coordinates have central charge $\boxed{c=D}$.

The holomorphic stress-energy tensor generates an infinitesimal conformal transformation throughTaking the three residues givesThe third derivative is the anomalous term that prevents $T$ from transforming as an ordinary weight-two Virasoro primary operator.

With Virasoro algebra modes $L_n=(2\pi i{)}^{-1}\oint dz{z}^{n+1}T(z)$, a second contour calculation givesso $A(m)=c(m^3-m)/12=D(m^3-m)/12$. This Virasoro central extension is the quantum conformal anomaly. In string theory the matter and ghost contributions must cancel; $c_{\mathrm{matter}}+c_{\mathrm{ghost}}=D-26=0$ gives the critical dimension of string theory $D=26$ and makes the BRST operator nilpotent.

The free embedding fields haveIn $T_X(z)T_X(w)$, the double contraction of the two derivatives in each normal-ordered stress tensor has two pairings. It givesThe stress-tensor operator-product expansion therefore has central charge $c=D$. Contour integration against the infinitesimal holomorphic vector field givesso in the notation of the question

The universal stress-tensor operator-product expansion in a two-dimensional conformal field theory isIts fourth-order coefficient defines the central charge $c$.

Differentiating $X(z)X(w)\sim -({\alpha }^{\prime }/2)\log (z-w)$ gives the contractions needed for Wick theorem. The free-boson part of $T$ supplies $1/[2(z-w{)}^4]$, corresponding to central charge one. The cross-contractions between $:\partial X\partial X:$ and ${\partial }^{2}X$ produce the required lower poles but no fourth-order scalar term. Finally,so the product of the two improvement terms contributes $3{\alpha }^{\prime }{q}^2/(z-w{)}^4$. Matching this with $c/[2(z-w{)}^4]$ in the stress-tensor operator-product expansion gives the linear dilaton conformal field theory