四维抗德西特空间中的一环散射振幅分析

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"Anti-de Sitter空间中的循环：四维欧几里德抗Sitter空间的相互作用标量场理论的四点幅度与两点函数的解析计算及其在共形边界临界指数的应用" 本文由Igor Bertan和Ivo Sachs发表，他们在四维欧几里德抗Sitter(Anti–de Sitter, AdS)空间中研究了相互作用的标量场理论，主要关注一环处的四点幅度和两点函数。他们强调在整个过程中没有依赖共形场理论的知识，这提供了一种独立的方法来处理这类问题。在四维AdS空间中，他们成功地得到了四点幅度的一环解析结果。这一成就对于理解高维量子场论中的散射振幅尤其重要，因为在通常的平坦空间时空中，计算高阶循环图是极具挑战性的。AdS空间因其特殊的几何特性，如其负曲率，为量子场论的研究提供了新的视角。对于两点函数，研究者给出了最多两个循环的解析表达式。这一点尤其重要，因为两点函数是理解场的性质和相互作用的基础。这些解析表达式将有助于更深入地理解AdS空间中的场动力学。此外，文章提出了一个新颖的观点，即在AdS空间的共形边界附近的相关函数的临界指数可以用于确定重正化条件。通常，量子场论中的重正化条件是通过满足“壳条件”来设定的，即物理量在特定能量尺度（例如，粒子的动量）上的行为。然而，这里的作者主张，AdS空间的边界行为可能提供一种替代方法，这可能简化或深化对量子场论重正化的理解。这个工作不仅为AdS/CFT对应（Anti-de Sitter/Conformal Field Theory correspondence）提供了新的洞察，也可能对黑洞物理、弦理论以及宇宙学等领域有潜在的影响，因为这些领域都涉及到量子场在弯曲时空中的行为。这项研究展示了在AdS空间中进行精确计算的潜力，并且为理解量子场论在非平坦背景下的行为提供了新的工具。它为未来的理论研究和可能的实验验证开辟了新的路径，尤其是在量子引力和高能物理的交叉领域。

Loops in Anti–de Sitter Space

Igor Bertan

and Ivo Sachs

†

Arnold Sommerfeld Center for Theoretical Physics, Ludwig Maximilian University of Munich,

Theresienstraße 37, D-80333 München, Germany

(Received 3 May 2018; published 4 September 2018)

We obtain analytic results fo r the four-point amplitude, at one loop, of an interacting scalar field theory in

four-dimensional, Euclidean anti–de Sitter space without exerting any conformal field theory knowledge.

For the two-point function, we provide analytic expressions up to two loops. In add ition, we argue that the

critical exponents of correlation functions near the conformal boundary of anti–de Sitter space provide the

necessary data for the renormalization conditions, thus replacing the usual on-shell condition.

DOI: 10.1103/PhysRevLett.121.101601

Introduction.—Over the past sixty years, there has been

tremendous progress in the calculation of scattering ampli-

tudes in quantum field theory—in particular, concerning

higher loop amplitudes in Yang-Mills theory and (super)

gravity.

At the same time, we have very few analytic results for

loop amplitudes in curved space-times. More precisely,

while their short-distance expansion and, in particular, the

structure of counterterms they give rise to is rather well

known [1,2], we know little about their dependence on

coordinates beyond that (e.g. Ref. [3] and references

therein). Even in de Sitter (dS) or anti–de Sitter (AdS)

space, which are maximally symmetric, admitting the same

number of isometries as Minkowski space, little is known

about such amplitudes; see Refs. [4–10] for recent progress.

The reason for this is that, while in Minkowski space, the

momentum representation leads to a hierarchy of elementary

integrals; in dS or AdS, this is not the case, and the coordinate

representation generally leads to integral expressions that

are more manageable in (A)dS. Still, except for some

special cases, we lack the technical tools for performing

the integrations completely.

In this Letter we report on some progress considering the

simplest interacting renormalizable field theory. Concretely,

we compute the two- and four-point functions for λϕ

theory

[11] to the second order in the coupling λ on the Poincar´e

patch of Euclidean AdS

by explicitly evaluating the

corresponding one- and two-loop integrals in coordinate

representation. Working on AdS, we avoid complications

that arise from IR effects on dS, for instance [3]. It turns out

that even this simplified setting is beyond reach for external

legs at generic points in AdS, but for insertions on the

conformal boundary we are able to get explicit expressions.

Being able to go beyond the short-distance expansion,

we encounter an interesting complication concerning the

renormalization conditions: For distances that are small

compared to the curvature scale, the problem reduces to

that in flat space, and the physical masses provide the right

boundary conditions for the renormalized propagator, for

instance. At scales of the order of the curvature radius,

however, there is no meaningful definition of the mass of

a scalar field, and one needs to identify a reasonable

renormalization condition. In the present case, we will

find that the critical exponents of correlation functions at

the conformal boundary of AdS provide just that. Indeed,

the bulk amplitudes on AdS with external legs inserted

at the boundary define a crossing symmetric point corre-

lation function of some hitherto unknown primary operator

of a conformal field theory (CFT) on the conformal

boundary by construction, and therefore a consistent

CFT. Of course, we do not know what is the microscopic

realization of this CFT, nor do we need it. What matters is

that a primary operator has a well-defined dimension which

is given by the critical exponent of its correlation function

near the conformal boundary. This is what replaces the

physical mass at large distances (see also Ref. [12]).

Concretely, let us consider a scalar field with classical

action [13]

S ¼ −

dμ



ð∂ϕÞ



ð1Þ

on the Poincar´e patch of hyperbolic space of radius 1=a

with the metric

ðdz

þ dx

Þ; ðz; x

Þ ∈ ðR

≥0

; R

Þ: ð2Þ

There are two admissible boundary conditions for the

classical scalar field corresponding to the asymptotic

Published by the American Physical Societ y under the terms of

the Creative Commons Attribution 4.0 International license.

Further distribution of this work must maintain attribution to

the author(s) and the published article’s title, journal citation,

and DOI. Funded by SCOAP

PHYSICAL REVIEW LETTERS 121, 101601 (2018)

0031-9007=18=121(10)=101601(6) 101601-1 Published by the American Physical Society

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