扩展R *-运算：具有任意分子的Feynman图的红外与紫外散度处理

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"这篇论文是关于在量子场论中处理Feynman图的R *-运算的扩展，特别是涉及到具有泛型分子（包括张量）的Feynman图。R *-运算由Chetyrkin、Tkachov和Smirnov提出，是对BPHZ R运算的改进，它能够同时消除具有非异常外部动量的欧氏Feynman图的紫外和红外发散。这种运算对于理解量子场论中的发散行为至关重要，因为它允许通过下环简单图的乘积来计算复杂图的发散部分。在本文中，作者Franz Herzog和Ben Ruijl首先介绍了R *-运算的背景和基本概念，然后展示了如何将其应用于更广泛的Feynman图类，包括那些具有任意分子结构的图。他们特别关注了张量分子，这是量子场论中常见且复杂的结构。此外，他们提出了一种新的红外反对称项的定义方法，这种方法在形式上与紫外反对称项的定义类似，简化了计算过程。论文中进一步阐述了红外和紫外反术语如何被表达为具有对数级发散度的无标度真空图。通过对这些无标度真空图的对称性、积和积分关系的深入分析，他们能够显著减少计算的复杂性和工作量。这种技术的实际应用体现在了其FORM实现中，用于计算量子色动力学（QCD）中规范群的五环β函数。为了清晰地展示这个过程，作者在四维空间的3理论中计算了所有五圈顶级传播图在维度正则化下的极点，作为具体示例。这个实例不仅验证了新方法的有效性，也为理解和应用R *-运算提供了实际的参考。论文的发布日期为2017年5月，由Springer为SISSA出版，并可在JHEP（Journal of High Energy Physics）上获取，标识为JHEP05(2017)037，且标记为Open Access，意味着公众可以免费访问。"

JHEP05(2017)037

The ﬁve UV subdivergences of this Feynman graph are given by

, γ

1234

, γ

1235

, γ

2345

(2.11)

with

ω(γ

) = ω(γ

1234

) = ω(γ

1235

) = ω(γ

2345

) = 0 . (2.12)

All of these subgraphs pairwise overlap: they share at least one common vertex or line.

Thus, no combination of these subgraphs can diverge independe ntly.

2.2 IR divergences in non-exceptional euclidean Feynman graphs

In analogy to the UV one can quantify the degree of IR divergence of a Feynman graph or

subgraph G by introduc i ng the notion of an IR superﬁcial degree of divergence ˜ω(G). Let

us consider the simple example

G =

p p

k + p

iπ

D/2

)

(k + p)

. (2.13)

We will use a small dot to indicate a vertex. A vertex which has only two edges in eﬀect

creates a squared propagator. Such squared propagators are the simplest instance of an

IR divergence i n non-exceptional Feynman graphs. We note that a graph can only have a

superﬁcial IR divergence if no external momenta ﬂow through it. To st udy the IR properties

of the line with a dot, we route the external momentum through the top l i ne and compute

its superﬁcial degree of divergence by rescaling, k → λk. This time we wish to extract the

leading power −˜ω of λ in t he limit λ → 0. Performing this rescal i ng and taking the limit

we ﬁnd

lim

λ→0

G ∼ λ

D−4

iπ

D/2

. (2.14)

And thus we get that ˜ω(G) = 0 for D = 4, indicating that the integral diverges loga-

rithmically in the IR. One interesting diﬀerence be tween IR and UV divergences is that

IR divergences are not usually associated to 1PI subgraphs, but they are (with minor ex-

ceptions) associated to connected subgr aphs , which, as it turns out, is why many of the

features of UV divergences explained above exte nd to the case of IR di vergences. The ex-

ception is always related to self-energy insertions, of which we will give an example further

below. This case was in fact missed in the original R

∗

-paper [

28] and was later corrected

in [

29].

We will use the notation γ

′

abc...

to identify a certain IR subgraph containing lines

a, b, c, . . . and those vertices whi ch connect only to a, b, c, . . ., but n o other lines . That

is, the graph γ

′

does not contain any vertices through which it is connected to the re-

maining graph; such vertices will be called external to the graph γ

′

. We de ﬁne the graph

¯γ = G\γ

′

to be the remaining graph where the IR subgraph is deleted from G. Further

we deﬁne the contra cted vac u u m graph ˜γ = G/¯γ by contracting ¯γ to a poi nt in G. We

remark that the UV construction is diﬀerent: the contracted vacuum graph ˜γ contains the

– 6 –

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