Title: The sphere free energy of the vector models to order $1/N$ Show Chinese title

Title: 矢量模型的球体自由能到$1/N$阶

Abstract: We calculate the large-$N$ expansion of the sphere free energy $F=-\log Z_{S^d}$ of the O(N) $\phi^4$ and the Gross-Neveu $(\bar{\psi} \psi)^2$ CFTs to order $1/N$. Analytical regularization of these theories requires consistently shifting the UV scaling dimension of the auxiliary field: this can only be done by modifying its kinetic term. This modification combines with the counterterms to give the result that matches the $\epsilon$-expansion, resolving a puzzle raised by Tarnopolsky in arXiv:1609.09113. These $F$s can be written compactly in terms of the anomalous dimensions, for both the short-range and the long-range versions of these CFTs. We also provide various technical results including a computation of the counterterms on the sphere and a neat derivation of the sphere free energy of a free conformal field. Finally, we observe that the long-range CFT becomes the short-range CFT at exactly the point where its $\tilde{F} =-\sin \tfrac{\pi d}{2} F$ is maximized as a function of the vector's scaling dimension. Show Chinese abstract

Abstract: 我们计算了O(N)$\phi^4$和Gross-Neveu$(\bar{\psi} \psi)^2$CFTs 的球面自由能$F=-\log Z_{S^d}$在大$N$展开的项到$1/N$阶。 这些理论的解析正则化需要一致地改变辅助场的紫外标度维数：这只能通过修改其动能项来实现。 这种修改与反常项结合，得到的结果与$\epsilon$展开相匹配，解决了Tarnopolsky在arXiv:1609.09113中提出的一个谜题。 这些$F$可以用奇异维度简洁地表示，适用于这些CFTs 的短程和长程版本。 我们还提供了各种技术结果，包括在球面上反常项的计算以及自由共形场的球面自由能的简洁推导。 最后，我们观察到，当向量的标度维数作为函数时，$\tilde{F} =-\sin \tfrac{\pi d}{2} F$在该点达到最大值，此时长程CFT变为短程CFT。