Seven-point functions have two inequivalent topologies or channels. The comb channel has been computed previously and here we compute scalar conformal blocks in the extended snowflake channel in $d$ dimensions. Our computation relies on the known action of the differential operator that sets up the operator product expansion in embedding space. The scalar conformal blocks in the extended snowflake channel are obtained as a power series expansion in the conformal cross-ratios whose coefficients are a triple sum of the hypergeometric type. This triple sum factorizes into a single sum and a double sum. The single sum can be seen as originating from the comb channel and is given in terms of a ${}_3F_2$-hypergeometric function, while the double sum originates from the snowflake channel which corresponds to a Kampe de Feriet function. We verify that our results satisfy the symmetry properties of the extended snowflake topology. Moreover, we check that the behavior of the extended snowflake conformal blocks under several limits is consistent with known results. Finally, we conjecture rules leading to a partial construction of scalar $M$-point conformal blocks in arbitrary topologies.

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扩展雪花通道及更远的七点共形块

七点函数有两个不等价的拓扑或通道。梳状通道之前已经计算过，这里我们在扩展的雪花通道中以 $d$ 维度计算标量共形块。我们的计算依赖于微分算子的已知动作，该动作在嵌入空间中设置算子乘积扩展。扩展雪花通道中的标量共形块作为共形交叉比的幂级数展开获得，其系数是超几何类型的三重和。这个三重和分解为一个单和和一个双和。单和可被视为源自梳状通道，并根据 ${}_3F_2$-超几何函数给出，而双和源自对应于 Kampe de Feriet 函数的雪花通道。我们验证我们的结果满足扩展雪花拓扑的对称性。此外，我们检查了扩展雪花保形块在几个限制下的行为与已知结果一致。最后，我们推测规则导致在任意拓扑中部分构造标量 $M$ 点保形块。