In this paper, the reduction of Feynman integrals in the parametric representation is considered. This method proves to be more efficient than the integration-by-part (IBP) method in the momentum space. Tensor integrals can directly be parametrized without performing tensor reductions. The integrands of parametric integrals are functions of Lorentz scalars, instead of four momenta. The complexity of a calculation is determined by the number of propagators that are present rather than the number of all the linearly independent propagators. Furthermore, the symmetries of Feynman integrals under permutations of indices are transparent in the parametric representation. Since all the indices of the propagators are nonnegative, an algorithm to solve those identities can easily be developed, which can be used for automatic calculations.

中文翻译：

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参数化表示中费曼积分的约简

在本文中，考虑了参数表示中费曼积分的约简。在动量空间中，这种方法被证明比逐部分集成 (IBP) 方法更有效。张量积分可以直接参数化而无需执行张量约简。参数积分的被积函数是洛伦兹标量的函数，而不是四个动量。计算的复杂度取决于存在的传播器的数量，而不是所有线性无关的传播器的数量。此外，费曼积分在指数排列下的对称性在参数表示中是透明的。由于传播子的所有索引都是非负的，因此可以轻松开发解决这些身份的算法，该算法可用于自动计算。