For this normalization,

T (z) = - : \partial X (z) \partial X (z) :, \partial X (z) X (w, \overset{w}{ˉ}) \sim - \frac{1}{2 ( z - w )} .

Contracting once and twice with

O_{k} (w, \overset{w}{ˉ}) =: e^{ik X (w, \overset{w}{ˉ})} :

gives

T (z) O_{k} (w, \overset{w}{ˉ}) \sim \frac{k ^{2} /4}{( z - w ) ^{2}} O_{k} (w, \overset{w}{ˉ}) + \frac{1}{z - w} \partial O_{k} (w, \overset{w}{ˉ}) .

It is therefore a primary operator with

h = h = \frac{k ^{2}}{4} .

Solved by gpt-5.6-sol high.

ii

Solution

Because normal ordering removes contractions within either exponential, Wick theorem sums all cross-contractions into

G_{k, - k} = exp [(ik) (- ik) ⟨ X (z, \overset{z}{ˉ}) X (w, \overset{w}{ˉ})⟩] = exp [- \frac{k ^{2}}{2} lo g ∣ z - w ∣^{2}] = ∣ z - w ∣^{- k^{2}} .

Under

z \mapsto λ z

, this scales as

λ^{- k^{2}}

. Each operator has scaling dimension

Δ = h + h = k^{2} /2

, so a two-point function of two such primaries must scale as

λ^{- 2Δ} = λ^{- k^{2}}

. The two calculations agree.

Solved by gpt-5.6-sol high.

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