AI 中文总结
研究有限QCD介质中三共线$q \to qc\bar{c}$分裂函数因式分解情况，通过重新表述不透明度展开，计算到不透明度一阶和$N_c$领头阶，证明因式分解可扩展到介质修正情况，还关联相关问题并讨论其应用。
AI 中文摘要
在强有序共线极限下，三共线$1\to 3$分裂函数可分解为$1\to 2$分裂函数的乘积。本文研究在有限QCD介质中这种因式分解是否以及在何种条件下仍然成立。为此，将不透明度展开重新表述为共线$1\to n$分裂函数的紧凑Catani - Grazzini表示的介质修正。计算了不透明度一阶和$N_c$领头阶的全微分介质修正三共线$q\to q c\bar{c}$分裂函数，表明它可表示为动量移位真空$q\to q c\bar{c}$分裂函数乘以简单干涉因子的和。真空中该分裂函数有单个强有序共线极限，而介质修正的分裂函数有三个不同的强有序极限。在所有三种情况下，三共线分裂在介质中也分解为$q\to qg$和$g\to c\bar{c}$分裂函数的张量积。在其中两个极限中，后者在不透明度一阶下被介质修正。这首次证明了因式分解扩展到介质修正的三共线分裂函数。最后，将结果与重叠形成时间的长期问题相关联，并讨论其对基于介质修正$1\to 2$分裂的喷注淬火部分子淋浴的影响。还概述了形式主义的进一步应用，并指出结果可能为致密QCD物质中末态辐射的自旋（去）关联的未来现象学研究提供信息。
英文摘要
In strongly ordered collinear limits, triple-collinear $1\to 3$ splitting functions factorize into products of $1\to 2$ splitting functions. Here, we investigate whether, and under which conditions, this factorization persists in a finite QCD medium. To this end, we reformulate the opacity expansion as a medium-modification of the compact Catani-Grazzini representation of collinear $1\to n$ splitting functions. We compute the fully differential medium-modified triple-collinear $q\to q c\bar{c}$ splitting function to first order in opacity and leading order in $N_c$, and show that it can be expressed as a sum of momentum-shifted vacuum $q\to q c\bar{c}$ splitting functions multiplied by simple interference factors. In vacuum, this splitting function has a single strongly ordered collinear limit, in which the invariant mass of the outgoing $c\bar{c}$ pair is much smaller than all other invariant masses. The medium-modified splitting function instead exhibits three distinct strongly ordered limits, depending on how the relevant invariant masses -- multiplied by a boost factor and interpretable as inverse formation times -- compare with the medium length. Remarkably, in all three cases the triple-collinear splitting factorizes also in the medium into a tensor product of $q\to qg$ and $g\to c\bar{c}$ splitting functions. In two of these limits, the latter are medium modified to first order in opacity. This first proof shows that factorization extends to medium-modified triple-collinear splitting functions. Lastly, we relate our results to the longstanding problem of overlapping formation times and discuss their implications for jet-quenching parton showers based on medium-modified $1\to 2$ splittings. We also outline further applications of our formalism and note that our results may inform future phenomenological studies of spin (de)correlations in final-state radiation in dense QCD matter.
