Nakanishi权重函数的束缚态方程与Light-Front波函数
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"这篇论文详细探讨了中西权函数在束缚态方程中的应用，特别是在Bethe-Salpeter振幅和Light-Front波函数表示中的角色。作者J. Carbonella, T. Frederico和V.A. Karmanov分别来自法国巴黎南大学核物理研究所、巴西航空技术学院和俄罗斯列别捷夫物理研究所。文章发表于《Physics Letters B》期刊，讨论了如何通过复平面中的Light-Front波函数来获取Nakanishi权函数，并为束缚态情况推导出新的积分方程。"
在量子场论中，Bethe-Salpeter方程（BS方程）是一种描述多粒子束缚态的非相对论量子场论方程。它是由Bethe和Salpeter在1951年提出的，用来计算两粒子系统的相对运动波函数，考虑了相对论效应。BS振幅是这个方程的核心，它包含了系统动力学的所有信息。
Nakanishi权重函数g是一个关键的概念，它将复杂的动力学信息编码在一个二维积分形式中，使得分析更为简便。论文指出，同样的权重函数g也可以用于Light-Front（LF）波函数的表示，这是一种在LF量子化条件下的量子场论方法，特别适合处理高能问题。
通过使用广义的Stieltjes变换，研究者能够从LF波函数出发得到Nakanishi权重函数g。这一变换允许他们在复平面上处理波函数，从而揭示更多关于系统动力学的细节。对于束缚态，他们推导出了一种新的积分方程，该方程以标准形式g = Ng表示，其中N是一个二维积分算子。这个算子N可以从BS方程的核K中得到，而这个核K描述了粒子间相互作用的强度。
论文提供了一个通用的方法来处理任何由不可约的费曼振幅给出的核。这意味着这个框架不仅适用于简单的模型，还适用于更复杂的真实物理系统。这种方法的适用性扩展了我们理解束缚态性质的能力，特别是在LF量子化中，这有助于更深入地理解高能量物理过程。
关键词包括：Bethe-Salpeter方程、Nakanishi表示以及Light-Front方法，表明这篇论文集中于将这些理论工具应用于束缚态问题的研究。通过这样的工作，科学家们能够更有效地解析复杂的多粒子系统，尤其是在高能物理和核物理领域，这对于理解强相互作用和粒子的性质至关重要。
Physics Letters B 769 (2017) 418–423
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Bound state equation for the Nakanishi weight function
J. Carbonell
a
, T. Frederico
b
, V.A. Karmanov
c,∗
a
Institut de Physique Nucleaire, Université Paris-Sud, IN2P3-CNRS, 91406 Orsay Cedex, France
b
Instituto Tecnológico de Aeronáutica, DCTA, 12228-900, S. José dos Campos, Brazil
c
Lebedev Physical Institute, Leninsky Prospekt 53, 119991 Moscow, Russia
a r t i c l e i n f o a b s t r a c t
Article history:
Received
19 December 2016
Received
in revised form 4 April 2017
Accepted
10 April 2017
Available
online 13 April 2017
Editor:
J.-P. Blaizot
Keywords:
Bethe–Salpeter
equation
Nakanishi
representation
Light-Front
The bound state Bethe–Salpeter amplitude was expressed by Nakanishi using a two-dimensional integral
representation, in terms of a smooth weight function g, which carries the detailed dynamical information.
A similar, but one-dimensional, integral representation can be obtained for the Light-Front wave function
in terms of the same weight function g. By using the generalized Stieltjes transform, we first obtain g
in
terms of the Light-Front wave function in the complex plane of its arguments. Next, anew integral
equation for the Nakanishi weight function g is derived for a bound state case. It has the standard form
g = Ng, where N is a two-dimensional integral operator. We give the prescription for obtaining the
kernel N starting with the kernel K of the Bethe–Salpeter equation. The derivation is valid for any kernel
given by an irreducible Feynman amplitude.
© 2017 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP
3
.
1. Introduction
The Bethe–Salpeter (BS) equation [1] for a few-body system
takes into account the relativity and, implicitly, all the possible
intermediate states. As demonstrated in [2] for the case of the lad-
der
kernel – which displays the strongest (pole) singularities – it
is possible to find the bound states and the scattering solutions in
Minkowski space by a direct numerical calculations, in spite of the
many singularities appearing in the integral and inhomogeneous
terms.
There
exist however another efficient approach to solve this
equation, proposed in [3] and developed in a series papers [4–9],
which relies on the Nakanishi representation [10] of the BS ampli-
tude
(k, p) in terms of a non-singular weight function g(γ , z) as
follows:
(k, p) =
1
−1
dz
∞
0
dγ
g(γ
, z
)
γ
+κ
2
−k
2
− p·kz
−i
3
, (1)
where κ
2
=m
2
−
M
2
4
, m is the constituent mass and M is the total
mass of the bound state (p
2
= M
2
). Once g is known, one can
compute the BS amplitude by means of (1) and calculate other
observables, like the electromagnetic form factors (see [11]).
*
Corresponding author.
E-mail
address: karmanov@sci.lebedev.ru (V.A. Karmanov).
The weight function g determines also, via another integral rep-
resentation,
the light-front (LF) wave function [4]:
ψ
LF
(γ , z) =
1 − z
2
4
∞
0
g(γ
, z)dγ
γ
+γ + z
2
m
2
+
1 − z
2
κ
2
2
. (2)
In this one-dimensional representation, variable z plays the role of
a parameter, varying in the limits −1 ≤ z ≤ 1. The relations be-
tween
variables (γ , z) and the standard LF variables (k
⊥
, x) are
given by γ = k
2
⊥
and z = 2x − 1. By Eq. (2) one can obtain the
LF wave function and calculate its contribution to the form factors
as well as the momentum distributions.
In
a series of recent papers [4–7,12,13], two independent meth-
ods
for finding g have been developed. In the first one [4–7] the
Nakanishi weight function g is determined by solving the double
integral equation, previously established in [4]:
∞
0
g(γ
, z)dγ
γ +γ
+ z
2
m
2
+(1 − z
2
)κ
2
2
=
∞
0
dγ
1
−1
dz
V (γ , z;γ
, z
)g(γ
, z
), (3)
where the kernel V is expressed via the BS kernel. The solution g
of
equation (3) was found in the case of an OBE ladder BS kernel
in [4] and for the ladder plus cross-ladder one in [5,15].
http://dx.doi.org/10.1016/j.physletb.2017.04.016
0370-2693/
© 2017 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by
SCOAP
3
.
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