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Given a four-point function in a CFT $$\langle\phi_1(x_1)\phi_2(x_2)\phi_3(x_3)\phi_4(x_4)\rangle,$$ the standard argument is that we can insert the $\phi_1\phi_2$ OPE and $\phi_3\phi_4$ OPE, $$\langle\phi_1(x_1)\phi_2(x_2)\phi_3(x_3)\phi_4(x_4)\rangle=\sum_{\text{Primaries }\mathcal{O}}C_{\Delta12}(x_{12},\partial_2)C_{\Delta34}(x_{34},\partial_4)\langle\mathcal{O}(x_2)\mathcal{O}(x_4)\rangle,$$ or we can insert the $\phi_1\phi_3$ OPE and $\phi_2\phi_4$ OPE, $$\langle\phi_1(x_1)\phi_2(x_2)\phi_3(x_3)\phi_4(x_4)\rangle=\sum_{\text{Primaries }\mathcal{O}}C_{\Delta13}(x_{13},\partial_3)C_{\Delta24}(x_{24},\partial_4)\langle\mathcal{O}(x_3)\mathcal{O}(x_4)\rangle,$$ and the resulting expansions should be equal, $$\sum_{\text{Primaries }\mathcal{O}}C_{\Delta12}(x_{12},\partial_2)C_{\Delta34}(x_{34},\partial_4)\langle\mathcal{O}(x_2)\mathcal{O}(x_4)\rangle=\sum_{\text{Primaries }\mathcal{O}'}C_{\Delta13}(x_{13},\partial_3)C_{\Delta24}(x_{24},\partial_4)\langle\mathcal{O}'(x_3)\mathcal{O}'(x_4)\rangle,$$ which is known as crossing symmetry or OPE associativity. However, the OPE is convergent only as long as there are no other operator insertions in between the operators we are contracting, so it seems that these two expansions should not be convergent for the same choices of $x_1$, $x_2$, $x_3$, and $x_4$ (ie, they should both be convergent, but not at the same time). So, why is OPE associativity valid?

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  • $\begingroup$ OPE associativity is an axiom in the bootstrap program, or are you asking for intuitive justification? $\endgroup$ Feb 23, 2023 at 21:39
  • $\begingroup$ Associativity follows simply from the state-operator map and radial quantization. $\endgroup$
    – Prahar
    Feb 23, 2023 at 23:23

2 Answers 2

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Your statement that the OPE in one channel converges if and only if the other diverges is incorrect. Choosing the points in your example to be \begin{equation} x_1 = (z, \bar{z}), \; x_2 = 0, \; x_3 = 1, \; x_4 = \infty \end{equation} we can see that $\phi_1 \phi_2 | \phi_3 \phi_4$ converges when $z$ is in the unit circle centered at $0$ while $\phi_1 \phi_3 | \phi_2 \phi_4$ converges when $z$ is in the unit circle centered at $1$. So there are overlapping regions of validity.

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In a CFT, we can replace $O_I O_J$ with its OPE $\sum_K c_{IJK}(x-y) O_K(y)$ if and only if the distance between $O_I$ and $O_J$ is smaller than the distance of $O_I$ to any other operator in the correlation function. This comes from the state-operator correspondence.

For instance, in the four-point function $\langle O_1(0)O_2(1)O_3(4)O_4(10)\rangle$, I can replace the product $O_1O_2$ with its OPE, but I cannot do the same for $O_1 O_3$ or $O_1O_4$.

Now, let us turn to the bootstrap programme. Here, we set three of the operators at $x_1=x$, $x_2=0$ and $x_3=1$ and $x_4=\infty$. Here, the distance between $O_1$ and $O_4$ is infinity which is greater than all other distances, so it is not valid for us to replace $O_1 O_4$ with its OPE.

On the other hand, we can replace $O_1O_2$ with its OPE as long as $|x|<1$ (because 1 is the distance between $O_2$ and $O_3$). Similarly, we can replace $O_1O_3$ with its OPE as long as $|x-1|<1$. In the overlapping region $|x|<1$ AND $|x-1|<1$, both the OPEs are valid and we can setup a bootstrap equation.

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