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Exactly marginal operator Created 2026-09-28 Updated 2026-11-05

An exactly marginal operator has scaling dimension equal to the spacetime dimension and a deformation coupling whose renormalization-group beta function vanishes to all orders. Varying that coupling moves along a family of conformal field theories without introducing a scale.

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Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 354 / 1 / b / ii / Solution 2026-09-28

With

Z [J] = \int D φ exp (i S + i \int d^{d} x J O),

the time-ordered vacuum correlator can be written, in this source convention, as

F_{0} (x_{1}, x_{2}) = - □_{x_{2}} \frac{1}{Z [ J ]} \frac{δ ^{2} Z [ J ]}{δ J ( x _{1} ) δ J ( x _{2} )}_{J = 0} .

Equivalent formulas in terms of the connected generating functional differ only by the standard factors of

i

.

A constant source deforms the action by

J \int d^{d} x O

. Since

[J] = d - Δ

, a nonzero

J

introduces no scale only if

Δ = d .

This marginality is necessary; for the deformed theory to remain a CFT for every

J

, the operator must additionally be exactly marginal, with vanishing beta function.

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