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<p>Energy Correlators (EC) are the simplest IR finite observables, which connect theories and experiments. In this paper, we provide a systematic algorithm to calculate the canonical differential equations for energy correlators at a generic angle in 𝒩 = 4 super Yang-Mills theory. The integrand is obtained from the 5-point form factor square for scalar half-BPS operators. Applying the algorithm, we obtain the canonical basis for three-point EC and the full set of master integrals for four-point EC. We analyze the function space for the four-point case. For multiple polylogarithmic (MPLs) integrals, we calculate their symbols, and for integrals beyond MPLs, we make further investigation by Picard-Fuchs operators. We find two elliptic curves and one genus 2 hyperelliptic curve. The results are achieved by means of integration by parts (IBP) reduction and differential equations powered by computational algebraic geometry methods. We provide a package that implements the algorithm. The data is a valuable reference for exploring the structure of physical observables in perturbation theories.</p>

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Differential equations for energy correlators in any angle

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年2月2日

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  • Rourou Ma,
  • Jianyu Gong,
  • Jingwen Lin,
  • Kai Yan,
  • Gang Yang,
  • Yang Zhang

摘要

Energy Correlators (EC) are the simplest IR finite observables, which connect theories and experiments. In this paper, we provide a systematic algorithm to calculate the canonical differential equations for energy correlators at a generic angle in 𝒩 = 4 super Yang-Mills theory. The integrand is obtained from the 5-point form factor square for scalar half-BPS operators. Applying the algorithm, we obtain the canonical basis for three-point EC and the full set of master integrals for four-point EC. We analyze the function space for the four-point case. For multiple polylogarithmic (MPLs) integrals, we calculate their symbols, and for integrals beyond MPLs, we make further investigation by Picard-Fuchs operators. We find two elliptic curves and one genus 2 hyperelliptic curve. The results are achieved by means of integration by parts (IBP) reduction and differential equations powered by computational algebraic geometry methods. We provide a package that implements the algorithm. The data is a valuable reference for exploring the structure of physical observables in perturbation theories.