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Is the following true in a conformal field theory (CFT)?

Suppose I have some observable $\mathcal O$ which I know nothing about (e.g. I don't know if it is a primary), except that I know the leading term in its two-point correlation function: $$\langle \mathcal O(x) \mathcal O(0) \rangle \sim 1/x^{2\alpha} \; (+\textrm{subleading} \cdots)$$ Is it then true that there must exist a primary field $\phi$ with scaling dimension $\Delta \leq \alpha$ ?

Equivalently, but worded differently: is it true that all descendant fields ''fall off quicker'' than the ''slowest decaying'' primary field?

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By translation and dilatation invariance, the two-point function of two scalar fields with conformal dimensions $\Delta_1$ and $\Delta_2$ is $$ \langle \mathcal O_1(x_1) \mathcal O_2(x_2) \rangle \propto |x_1-x_2|^{-\Delta_1-\Delta_2} $$ This holds whenever the two fields are eigenvalues of the dilatation operator, which is true in particular for descendents and primaries. (If the fields are primary you have additional conformal Ward identities that say that the two-point function vanishes unless $\Delta_1=\Delta_2$.)

By definition, descendents are obtained from primaries by acting with creation operators. Such operators increase conformal dimensions. So if you assume that all your fields are primary or descendent, and that conformal dimensions are bounded from below, then the field with the lowest conformal dimension must be primary.

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  • $\begingroup$ I guess I was being a bit too naive in my question, presuming that every field could be consider a descendant of a primary. What if I have a field which not even necessarily a descendant? $\endgroup$ – Ruben Verresen Mar 17 '17 at 10:43
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    $\begingroup$ If you have fields that are neither primary nor descendent, then you are in a quite exotic representation of the conformal algebra. In two dimensions such exotic representations occur for instance in logarithmic CFTs. Having only primaries and descendents is the most common and useful case. $\endgroup$ – Sylvain Ribault Mar 17 '17 at 14:42

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