Conformal Contact Terms and Semi-local Terms

Abstract

We study conformal properties of local terms such as contact terms and semi-local terms in correlation functions of a conformal field theory. Not all of them are universal observables, but they do appear in physically important correlation functions such as (anomalous) Ward–Takahashi identities or Schwinger–Dyson equations. We develop some tools such as embedding space delta functions and effective action to examine conformal invariance of these local terms.

A Check of Special Conformal Invariance in Coordinate Space

In this appendix, we perform a check of the special conformal invariance of some two-point functions directly in the coordinate space for $d=1$. The generalization to higher dimension should be straightforward.

Let us begin with the usual non-local two-point functions

$$\begin{aligned} \langle O_{\Delta _1}(x_1) O_{\Delta _2}(x_2) \rangle = \frac{1}{(x_1-x_2)^{\Delta _1 + \Delta _2}} \ . \end{aligned}$$

(44)

Assuming that $O_{\Delta _1}$ and $O_{\Delta _2}$ are primary operators, the left-hand side transforms under the special conformal transformation $x \rightarrow x + \epsilon x^2$ as

$$\begin{aligned} (1+2\epsilon x_1)^{-\Delta _1} (1+2\epsilon x_2)^{-\Delta _2} \langle O_{\Delta _1}(x_1) O_{\Delta _2}(x_2) \rangle \ . \end{aligned}$$

(45)

On the other hand, the right-hand side transforms as

$$\begin{aligned} \frac{1}{(x_1 + \epsilon x_1^2 - x_2 - \epsilon x_2^2)^{\Delta _1 + \Delta _2}} = \frac{1-\epsilon (x_1 + x_2)(\Delta _1 + \Delta _2)}{(x_1 - x_2)^{\Delta _1 + \Delta _2}} \ . \end{aligned}$$

(46)

The equality holds only if $\Delta _1 = \Delta _2$, which is the well-known constraint on the two-point functions.

The constraint is weaker in the contact term without derivatives on the delta function. Let us consider

$$\begin{aligned} \langle O_{\Delta _1}(x_1) O_{\Delta _2}(x_2) \rangle = \delta (x_1-x_2) \end{aligned}$$

(47)

with $\Delta _1 + \Delta _2 = 1$ from the scale invariance. Assuming that $O_{\Delta _1}$ and $O_{\Delta _2}$ are primary operators, the left-hand side transforms under the special conformal transformation $x \rightarrow x + \epsilon x^2$ as

$$\begin{aligned}&(1+2\epsilon x_1)^{-\Delta _1} (1+2\epsilon x_2)^{-\Delta _2} \langle O_{\Delta _1}(x_1) O_{\Delta _2}(x_2) \rangle \nonumber \\&\quad = ( 1-2\epsilon x_1 (\Delta _1 + \Delta _2)) \delta (x_1 - x_2) \ . \end{aligned}$$

(48)

On the other hand, the right-hand side transforms as

$$\begin{aligned}&\delta (x_1 + \epsilon x_1^2 - x_2 - \epsilon x_2^2) \nonumber \\&\quad = (1-2\epsilon x_1) \delta (x_1 -x_2) \ . \end{aligned}$$

(49)

The equality between (48) and (49) holds as long as $\Delta _1 + \Delta _2 = 1$ and there is no further constraint such as $\Delta _1 = \Delta _2$.

Let us finally consider the contact term with a derivative acting on the delta function.

$$\begin{aligned} \langle O_{\Delta _1}(x_1) O_{\Delta _2}(x_2) \rangle = \partial _{x_1} \delta (x_1-x_2) \end{aligned}$$

(50)

with $\Delta _1 + \Delta _2 = 2$ from the scale invariance. Assuming that $O_{\Delta _1}$ and $O_{\Delta _2}$ are primary operators, the left-hand side transforms under the special conformal transformation $x \rightarrow x + \epsilon x^2$ as

$$\begin{aligned} (1+2\epsilon x_1)^{-\Delta _1} (1+2\epsilon x_2)^{-\Delta _2} \langle O_{\Delta _1}(x_1) O_{\Delta _2}(x_2) \rangle \\= (1+2\epsilon x_1)^{-\Delta _1} (1+2\epsilon x_2)^{-\Delta _2} \partial _{x_1} \delta (x_1 - x_2) \ . \end{aligned}$$

(51)

Note that unlike the case above, we cannot set $x_1=x_2$ in front of the derivative of the delta function.⁹ On the other hand, the right-hand side transforms as

$$\begin{aligned}&(1-2\epsilon x_1) \partial _{x_1} \delta (x_1 + \epsilon x_1^2 - x_2 - \epsilon x_2^2)\nonumber \\&\quad = (1-2\epsilon x_1) \partial _{x_1}((1-2\epsilon x_1) \delta (x_1 -x_2)) \ . \end{aligned}$$

(52)

The equality between (51) and (52) holds only if $\Delta _1 = 1$ and $\Delta _2=1$. This is consistent with our embedding space formalism.