ρ-介子的动态共振：Bethe-Salpeter框架新计算

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"这篇研究论文探讨了在Bethe-Salpeter框架中矢量介子，特别是ρ-介子作为动态共振的理论计算方法。在交互核中，除了传统的彩虹梯模型中的T通道单胶子交换，作者还考虑了两点交换的贡献。这是一项重要的改进，因为它能更准确地描述粒子的相互作用。文章详细阐述了如何通过积分的奇异结构，通过适当的路径变形来确定共振极的宽度，这是一种处理虚部的方法。根据计算结果，ρ-介子的质量和宽度分别被估计为640 MeV和100 MeV，这对应于一个耦合常数gρππ= 5.7。值得注意的是，这种排斥性的修正可能有助于将理论预测与实验观测相吻合。此外，作者还讨论了这些参数如何随介子质量变化，以及这些结果如何与晶格量子色动力学（Lattice QCD）的研究相对比。Lattice QCD是一种强大的数值方法，用于在量子色动力学的非微扰领域研究粒子性质。通过这种方式，研究者能够进一步理解强相互作用下的粒子行为，特别是在考虑不稳定态时。这项工作为理解和模拟强相互作用下的介子行为提供了新的见解，并为未来更精确的理论预测和实验比较奠定了基础。其关键词包括束缚和不稳定的态、介子、Bethe-Salpeter方程以及Dyson-Schwinger方程，这些都是量子色动力学中的核心概念。" 在本文中，作者Richard Williams利用了Bethe-Salpeter方程和Dyson-Schwinger方程的结合，这是一种处理场论中束缚态和不稳定性粒子的重要工具。他们展示了如何在这个框架内处理ρ-介子的动态性质，不仅考虑了胶子交换，还引入了两介子交换，从而更好地捕捉了物理系统的复杂性。这种方法的创新性和实用性对于深入理解基本粒子的性质和相互作用具有重要意义。

Physics Letters B 798 (2019) 134943

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Vector mesons as dynamical resonances in the Bethe–Salpeter

framework

Richard Williams

Institut für Theoretische Physik, Justus-Liebig–Universität Giessen, 35392 Giessen, Germany

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

Article history:

Received

28 January 2019

Received

in revised form 17 June 2019

Accepted

15 August 2019

Available

online 17 September 2019

Editor: A.

Ringwald

Keywords:

Bound

and unstable states

Mesons

Bethe–Salpeter

equations

Dyson–Schwinger

equations

We present the ﬁrst dynamical calculation of the ρ-meson as a resonance in the Dyson–Schwinger/Bethe–

Salpeter

framework by including explicit two-pion exchange in addition to the t-channel one-gluon

exchange of rainbow-ladder in the interaction kernel. The width is determined from the imaginary part

of the resonance pole and is generated by the singularity structure of the integrand, treated by means

of suitable path deformations. We ﬁnd the resonant mass and width to be 640 MeV and 100 MeV

respectively, corresponding to a coupling constant g

ρππ

= 5.7; repulsive corrections beyond rainbow-

ladder

are expected to bring these in line with experiment. Furthermore we present the dependence of

these parameters on the pion mass and compare with results from lattice QCD.

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

1. Introduction

One of the long-standing goals of hadron physics is understand-

ing

the dynamics of the strong interaction. Below the strong-decay

threshold, a number of bound-states e.g. π , K , D, B, etc. serve as

a valuable playground for non-perturbative studies of QCD. Most

hadrons, however, lie above these thresholds and manifest as reso-

nances

whose decay channels subsequently dominate in the decay

width. Therefore any study is rendered incomplete when hadrons

are described as stable bound-states instead of their ephemeral na-

ture

being embraced.

Many

approaches have sought to describe the resonant struc-

ture

of bound-states, e.g. in constitutent quark models [1–4], lat-

tice

QCD [5–10] and chiral perturbation theory [11,12]. In Lattice

QCD there has been signiﬁcant progress recently in the deter-

mination

of resonant properties of bound-states. Naively one ex-

pects

no continuum of states due to ﬁnite volume—and hence no

branch cut(s)—entailing that there is a single Riemann sheet. Con-

sequently,

unitarity dictates that all poles fall on the real-axis and

one would conclude that resonances are inaccessible. However,

decay properties may be inferred by performing a volume study

of the spectrum [13–15], with recent techniques such as distilla-

tion [16] contributing

to improved signals and hence statistics.

the continuum, where the Bethe–Salpeter equation (BSE)

is formulated, resonances appear as poles in unphysical Riemann

E-mail address: richard.williams@theo.physik.uni-giessen.de.

sheets (see e.g. Refs. [17,18]for a discussion). However, they

only arise dynamically when the interaction kernel explicitly in-

corporates

decay processes, requiring truncations beyond that of

rainbow-ladder [19,20]. The exception to this is in systems where

for structural reasons a resonance would be generated even with

nothing more than one-gluon exchange, such as for tetraquarks

e.g. [21]; where however no resonance has been directly resolved

yet due to limitations of the numerical methods thus far em-

ployed.

To describe resonances dynamically in the BSE we require

two things: a (chiral-symmetry preserving) truncation that pro-

vides

access to its decay channel; and a robust means by which

the inevitable intermediate particle poles can be treated numeri-

cally.

It is the purpose of this letter to address both of these points

for the explicit example of a ρ-meson. Generalization to e.g. the

scalar meson is straightforward and the approach may be applied

to other hadrons such as baryons and—in particular—tetraquarks.

2. Framework

We employ an approach based on the Dyson–Schwinger equa-

tions

(DSEs), which have been successfully applied to a wide range

of topics in particle physics and in particular QCD [22,23]. In order

to accommodate for decay channels, as inspired by Ref. [24], we

employ the natural extension of the hadronic unquenching effects

explored in Refs. [25–28]by complementing t-channel one-pion

exchange with two-pion s- (and u-via crossing symmetry) chan-

nel

exchange, see Fig. 1; detailed discussions of its origins can be

found in Refs. [24,26]. Such an extension has no deleterious impact

https://doi.org/10.1016/j.physletb.2019.134943

0370-2693/

SCOAP

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