Building on the recent derivation of a bare factorization theorem for the b -quark induced contribution to the h → γγ decay amplitude based on soft-collinear effective theory, we derive the first renormalized factorization theorem for a process described at subleading power in scale ratios, where λ = m b / M h « 1 in our case. We prove two refactorization conditions for a matching coefficient and an operator matrix element in the endpoint region, where they exhibit singularities giving rise to divergent convolution integrals. The refactorization conditions ensure that the dependence of the decay amplitude on the rapidity regulator, which regularizes the endpoint singularities, cancels out to all orders of perturbation theory. We establish the renormalized form of the factorization formula, proving that extra contributions arising from the fact that “endpoint regularization” does not commute with renormalization can be absorbed, to all orders, by a redefinition of one of the matching coefficients. We derive the renormalization-group evolution equation satisfied by all quantities in the factorization formula and use them to predict the large logarithms of order αα s 2 L k $$ {\alpha \alpha}_s^2{L}^k $$ in the three-loop decay amplitude, where L = ln − M h 2 / m b 2 $$ L=\ln \left(-{M}_h^2/{m}_b^2\right) $$ and k = 6 , 5 , 4 , 3. We find perfect agreement with existing numerical results for the amplitude and analytical results for the three-loop contributions involving a massless quark loop. On the other hand, we disagree with the results of previous attempts to predict the series of subleading logarithms ∼ αα s n L 2 n + 1 $$ \sim {\alpha \alpha}_s^n{L}^{2n+1} $$ .

中文翻译：

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在 h → γγ 衰减中细分功率和端点发散。第二部分。重整化和尺度演化

基于最近基于软共线有效理论推导出 b 夸克对 h → γγ 衰变幅度的贡献的裸分解定理，我们推导出了第一个重整化分解定理，用于在标度比中以次领先幂描述的过程，在我们的例子中，λ = mb / M h « 1。我们证明了端点区域中匹配系数和算子矩阵元素的两个重构条件，其中它们表现出奇异性，导致发散卷积积分。重构条件确保衰减幅度对调节端点奇异性的快速调节器的依赖性与扰动理论的所有阶相抵消。我们建立分解公式的重整化形式，证明“端点正则化”不与重整化交换这一事实产生的额外贡献可以通过重新定义匹配系数之一来吸收到所有阶数。我们推导出分解公式中所有量都满足的重整化群演化方程，并用它们来预测阶 αα s 2 L k $$ {\alpha \alpha}_s^2{L}^k $$ in三环衰减幅度，其中 L = ln − M h 2 / mb 2 $$ L=\ln \left(-{M}_h^2/{m}_b^2\right) $$ 和 k = 6 , 5 , 4 , 3. 我们发现与现有的数值结果完全一致，包括无质量夸克环的三环贡献的振幅和解析结果。另一方面，