论文标题
Ansätze用于从$ p $ - adic数字和代数几何形状散射幅度
Ansätze for Scattering Amplitudes from $p$-adic Numbers and Algebraic Geometry
论文作者
论文摘要
已知特殊功能在散射幅度中的理性系数在奇异表面上简化，通常比幼稚期望的差异更大。为了系统地研究这些表面和对它们的合理功能，我们采用了代数几何形状的工具。我们展示了合理函数的差异如何限制其分子以属于与奇异表面相关的理想的符号力量。为了研究系数的差异，我们使用$ p $ adadic的数字，与有限的字段密切相关。这些使我们能够以稳定的方式执行接近奇异表面的数值评估，从而表征系数的差异。然后，我们使用此信息来构建低维Ansätze，以实现有理系数。作为我们算法的概念验证应用，我们重建了两环$ 0 \ rightarrow q \ bar q \ barqγγγ$五边形 - 功能系数，其数值评估少于1000。
Rational coefficients of special functions in scattering amplitudes are known to simplify on singular surfaces, often diverging less strongly than the naïve expectation. To systematically study these surfaces and rational functions on them, we employ tools from algebraic geometry. We show how the divergences of a rational function constrain its numerator to belong to symbolic powers of ideals associated to the singular surfaces. To study the divergences of the coefficients, we make use of $p$-adic numbers, closely related to finite fields. These allow us to perform numerical evaluations close to the singular surfaces in a stable manner and thereby characterize the divergences of the coefficients. We then use this information to construct low-dimensional Ansätze for the rational coefficients. As a proof-of-concept application of our algorithm, we reconstruct the two-loop $0 \rightarrow q\bar qγγγ$ pentagon-function coefficients with fewer than 1000 numerical evaluations.
