Mellin变换与α'-展开：开放字符串圆盘积分解析
161 浏览量
更新于2024-07-16
收藏 302KB PDF 举报
"α'-通过Mellin变换扩展开弦盘积分"
这篇论文详细探讨了如何利用Mellin变换来处理开放弦圆盘积分，并将其转换为Beta函数乘积的轮廓积分形式。这一方法使计算场论极限附近的α'展开与四维极限附近的费曼环积分中的ϵ-展开在数学处理上具有相似性。α'是弦理论中的一个关键参数，它描述了弦的长度尺度，而开放弦圆盘积分在弦理论中扮演着重要角色，尤其是在研究弦相互作用时。
作者Ellis Ye Yu指出，从标准的Koba-Nielsen表示法直接得到的Mellin空间公式适用于α'不等于0的区域。然而，当α'趋向于0时，积分的轮廓会被极点夹紧，这就需要进行解析连续，以确保积分的正确性。解析连续是一种数学技术，用于扩展函数的定义域，使其能够穿过具有奇异点的区域。
在解析连续过程中，积分路径的变形会导致极点的截断，形成（n-3）!的多维残差。这些残差包含了全部的场论信息。这意味着通过对这些残差的计算和分析，可以提取出关于弦相互作用的重要物理特性。这与通过散射方程直接推导场论公式的方法形成了对比，同时也提供了一种可能的互补方法。
文章最后部分讨论了通过这种方法得到的场论公式与从散射方程式推导的公式之间的类比，这可能揭示了两种方法之间的深层联系，有助于进一步理解弦理论中的物理过程。论文的发表对于促进弦理论和场论的研究，以及深化对高维物理现象的理解具有重要意义。
这项工作展示了Mellin变换在处理复杂弦理论计算中的潜力，特别是对于理解和展开α'参数的重要性。它不仅提供了新的计算工具，还促进了理论物理学家对弦理论基本概念的深入探索。
E.Y. Yuan / Nuclear Physics B 891 (2015) 296–311 299
as the leading terms at order α
−3
in its Laurent expansion (2) around α
= 0. There are alto-
gether four spurious poles at s
1,2
− s
1:3
= 0, s
3:5
− s
4,5
= 0, s
1:3
− s
5,6
= 0 and s
1,2
− s
5,6
= 0
respectively, but they cancel away in the summation, resulting in 14 terms corresponding to the
14 planar scalar diagrams with the given ordering (123456), as expected.
It is interesting that Mellin transformation has been an essential tool in the wo
rk of Stieberger
and Taylor [15,16]. There it was motivated from the analogous structure of superstring and su-
pergravity amplitudes. After switching the disk boundary positions z
a
to n(n − 3)/2 cross-ratios
u
a,b
= (z
a
− z
b
)(z
a−1
− z
b+1
)/(z
a
− z
b+1
)(z
a−1
− z
b
) in one-to-one correspondence to the inde-
pendent kinematic invariants s
a,b
, Mellin transformation draws an exact correspondence between
the two sides. In this paper Mellin transformation is used as the tool to disentangle z variables
and to manifest the singularity structure of general disk integrals in a different space.
The paper is or
ganized as follows. In Section 2 we illustrate this method by discussing in great
detail its application to the simplest non-trivial case at five points. Following that, we generalize it
to any multiplicity and any orderings in Section 3, and we show how to apply multi-dimensional
contour deformation to e
xtract the leading terms from the result in 2 in the zero-slope limit, and
describe the general structure of these leading field-theory pieces obtained from this calculation.
Explicit example at six points is provided in Section 4. In the end we point out several possible
future explorations.
2. Five-point amplitude: illustrating the technique
In this section we perform our analysis at five points, which is the first non-trivial case. We
choose the canonical ordering I = (12 ...n) on the boundary of the disk, and fix the SL(2, R)
redundancy by setting {z
1
, z
2
, z
5
} ={∞, 0, 1}. As the first example we study Z[I |I ], where the
Parke–Taylor factor shares the same ordering as that on the boundary of the disk. In this case the
amplitude is
dz
3
dz
4
z
α
s
2,3
−1
3
z
α
s
2,4
4
(1 − z
3
)
α
s
3,5
(1 − z
4
)
α
s
4,5
−1
(z
4
− z
3
)
α
s
3,4
−1
, (8)
where the integration is carried out over the region 0 ≤ z
3
≤ z
4
≤ 1. We do a change of variables
as
z
3
= y
3
y
4
,z
4
= y
4
, (9)
so that both y
3
and y
4
are integrated over [0, 1], and the amplitude becomes
1
0
dy
3
dy
4
y
α
s
2,3
−1
3
y
α
s
1,5
−1
4
(1 − y
3
y
4
)
α
s
3,5
(1 − y
4
)
α
s
4,5
−1
(1 − y
3
)
α
s
3,4
−1
. (10)
Our purpose is to fully integrate out the moduli y
3
and y
4
, and so we need to decompose the factor
(1 − y
3
y
4
) so that different moduli get separated. This is done using Mellin transformation (6).
In order to avoid the appearance of any undesirable factor of the form (−1)
w
when applying (6),
we first choose to rewrite this factor as
(1 − y
3
y
4
) = (1 − y
3
) + y
3
(1 − y
4
), (11)
i.e., setting A
1
= 1 − y
3
and A
2
= y
3
(1 − y
4
) in (6). Then Mellin transformation yields
(1 − y
3
y
4
)
α
s
3,5
=
dw
2πi
B
−α
s
3,5
+ w, −w
(1 − y
3
)
α
s
3,5
−w
y
3
(1 − y
4
)
w
. (12)
剩余15页未读，继续阅读
2021-02-13 上传
2021-07-26 上传
2010-10-14 上传
2024-09-03 上传
2023-05-11 上传
2023-07-08 上传
2024-05-20 上传
2024-03-22 上传
2023-08-07 上传
weixin_38594252
- 粉丝: 7
- 资源: 920
我的内容管理
展开
最新资源
- 探索数据转换实验平台在设备装置中的应用
- 使用git-log-to-tikz.py将Git日志转换为TIKZ图形
- 小栗子源码2.9.3版本发布
- 使用Tinder-Hack-Client实现Tinder API交互
- Android Studio新模板：个性化Material Design导航抽屉
- React API分页模块：数据获取与页面管理
- C语言实现顺序表的动态分配方法
- 光催化分解水产氢固溶体催化剂制备技术揭秘
- VS2013环境下tinyxml库的32位与64位编译指南
- 网易云歌词情感分析系统实现与架构
- React应用展示GitHub用户详细信息及项目分析
- LayUI2.1.6帮助文档API功能详解
- 全栈开发实现的chatgpt应用可打包小程序/H5/App
- C++实现顺序表的动态内存分配技术
- Java制作水果格斗游戏：策略与随机性的结合
- 基于若依框架的后台管理系统开发实例解析
