高维场论解析：Jost-Lehmann-Dyson表示与振幅性质

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"这篇论文是关于高维场论中的Jost-Lehmann-Dyson (JLD) 表示的研究，作者Jnanadeva Maharana在印度物理研究所和NISER工作。文章发表于Physics Letters B 764(2017)，详细探讨了在时空维度D大于4的四点散射振幅的大规模标量场理论中的JLD表示。" Jost-Lehmann-Dyson表示是量子场论中的一个重要概念，它在高维空间的场论计算中具有核心地位。这个表示形式主要关注的是四点散射振幅，即两个粒子相互作用后转化为另外两个粒子的过程。在高维空间（D>4）中，这样的过程变得更加复杂，因为更多的自由度和可能的交互模式增加了理解和计算的难度。 JLD表示的推导基于莱曼-席曼齐克-齐默尔曼（Lehmann-Symanzik-Zimmermann, LSZ）的公理方法。LSZ公理是量子场论的基础之一，它们提供了一种从基本场的绿函数推导出散射振幅的框架。通过应用这些公理，作者证明了在高维空间中存在JLD表示的可能性。这种表示对于分析散射振幅的解析性质至关重要，因为解析性质反映了物理过程的基本连续性和可逆性。 JLD表示的存在不仅证实了散射振幅的数学结构，而且允许研究人员深入探究其物理特性。例如，通过这个表示，可以证明散射振幅的解析性质，这在量子场论中是非常重要的，因为它确保了理论的稳定性和一致性。此外，JLD表示还揭示了振幅的渐近行为，这在高能物理中特别重要，因为粒子在高能量下通常表现出不同的行为模式。文章指出，从JLD表示的推导中会得出一系列有趣的物理结果。这些结果可能包括对散射振幅在特定能量范围内的精确描述，以及与单位arity、因果性和重整化等基本原理的联系。此外，这种表示还可以帮助理解高维场论中的粒子相互作用，这对于理解和预测粒子碰撞实验中的现象至关重要。最后，由于文章是开放获取的，意味着任何对高维场论感兴趣的读者都可以免费访问并研究其内容，这对于学术交流和知识传播具有积极意义。通过CC BY许可，作者赋予了公众复制、分发和适应作品的权利，只要适当引用原始来源即可。

Physics Letters B 764 (2017) 212–217

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Jost–Lehmann–Dyson representation in higher dimensional ﬁeld

theories

Jnanadeva Maharana

Institute of Physics and NISER, Bhubaneswar, 751005, India

a r t i c l e i n f o a b s t r a c t

Article history:

Received

15 August 2016

Received

in revised form 26 October 2016

Accepted

28 October 2016

Available

online 23 November 2016

Editor: M.

Cveti

The Jost–Lehmann–Dyson representation is derived for massive scalar ﬁeld theories in higher spacetime

dimensions, D > 4, for the four point scattering amplitude. The representation is very crucial to

investigate the analyticity properties of the amplitude. The axiomatic approach of Lehmann–Symanzik–

Zimmermann

is adopted to show the existence of such a representation. Consequently, a host of

interesting results will follow from derivation of JLD representation such as proof of analyticity properties

and asymptotic behavior of the amplitude.

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

The purpose of this letter is to derive the Jost–Lehmann–Dyson

(JLD) representation [1,2] for the four point scattering amplitude

of higher dimensional, D > 4, scalar, neutral, massive ﬁeld theo-

ries.

It plays a central role in the investigation of the analyticity

properties of the amplitude as is elaborated in sequel. We work in

the Lehmann–Symanzik–Zimmermann (LSZ) [3] formalism of ax-

iomatic

ﬁeld theory to derive the representation.

There

exist rigorous bounds on experimentally measurable pa-

rameter

in the context of four dimensional theories. These have

been derived from the study of the analyticity properties of scat-

tering

amplitudes. The most important attribute of these rigorous

results is that their derivations are based on the results proved

in the axiomatic ﬁeld theoretic formalisms without resorting to

any speciﬁc model. Moreover, there are no observed experimen-

tal

violation of exacts results on total cross sections, width of

the diffraction peak in elastic scattering to mention a few. The

Froissart–Martin [4,5] bound on the total cross section, σ

, is the

most celebrated of all

≤

4π

ln(

)

(1)

where t

=4m

for most of the hadronic processes; s is the center

of mass (c.m.) energy squared and s

is a scale introduced to make

the argument of the logarithm dimensionless. The crucial ingredi-

ents

leading to the derivation of the bound (1) are the unitarity

of S-matrix, crossing symmetry and analyticity of the amplitude

in s and t, the momentum transfer squared. The afore mentioned

results follow from the axiomatic ﬁeld theoretic formalism [6–14].

E-mail address: maharana@iopb.res.in.

The analyticity of scattering amplitude in s and t facilitates writing

of the ﬁxed t dispersion relation in s; t lying within the Lehmann

ellipse [15]. The partial wave expansion of the scattering amplitude

converges inside this ellipse.

The

role of the (JLD) representation is very fundamental in this

context. The proof of the existence of Lehmann ellipse rests on

the works of JLD. Moreover, the edge-of-the-wedge theorem [16]

proved by using this construction.

There

is growing consensus that theories in higher spacetime

dimensions, D > 4, are important to study fundamental interac-

tions

and geometric attributes of spacetime. There is increasing

interest to investigate properties of the scattering amplitude afresh

from various perspectives [18,19] in D > 4theories. The super-

symmetric

theories and supergravity theories have opened up the

possibilities to explore such theoretical constructions. Superstring

theories hold the promise of unifying the fundamental forces of

Nature. Moreover, the ﬁve perturbatively consistent string theories

live in D = 10. Of course, the standard model is deﬁned in D = 4

and

its predictions have been experimentally tested with great ac-

curacy.

Thus the extra spatial dimensions of the D-dimensional

theories must be compactiﬁed. In the context of string theories,

most of the phenomenological analyses argue that the energy scale

of compactiﬁcation is very high and therefore, the present acces-

sible

accelerators cannot reveal the existence of the compact extra

dimensions. However, in recent years elaborate models have been

constructed where the compactiﬁcation scale is quite low i.e. the

presence of extra dimensions can be explored in LHC energy range.

There have been considerable phenomenological activities to ex-

amine

and propose the signatures for the spectra of new particles

[20,21]. Moreover, the present data does not rule out existence of

low compactiﬁcation scales.

http://dx.doi.org/10.1016/j.physletb.2016.10.080

0370-2693/

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