With
$$
Z[J]=\int\mathcal D\varphi\,
\exp\left(iS+i\int d^dx\,J\mathcal O\right),
$$
the time-ordered vacuum correlator can be written, in this source convention, as
$$
\boxed{F_0(x_1,x_2)=
-\left.\square_{x_2}
\frac1{Z[J]}
\frac{\delta^2Z[J]}{\delta J(x_1)\delta J(x_2)}
\right|_{J=0}.}
$$
Equivalent formulas in terms of the connected generating functional differ only by the standard factors of $i$.

A constant source deforms the action by $J\int d^dx\,\mathcal O$. Since $[J]=d-\Delta$, a nonzero $J$ introduces no scale only if
$$
\boxed{\Delta=d.}
$$
This marginality is necessary; for the deformed theory to remain a CFT for every $J$, the operator must additionally be <exactly marginal operator>[exactly marginal], with vanishing beta function.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-354/1/b/ii/solution