\partial X^{μ} (z) \partial X^{ν} (w) \sim - \frac{α ^{'}}{2} \frac{η ^{μν}}{( z - w ) ^{2}} .

T_{X} (z) T_{X} (w)

, the double contraction of the two derivatives in each normal-ordered stress tensor has two pairings. It gives

T_{X} (z) T_{X} (w) \sim \frac{D /2}{( z - w ) ^{4}} + \dots .

c = D

δ_{v} T_{X} = v \partial T_{X} + 2 (\partial v) T_{X} + \frac{D}{12} \partial^{3} v,

so in the notation of the question

c_{X} = \frac{D}{12} .

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