J.-E. Bourgine, D. Fioravanti / Nuclear Physics B 906 (2016) 408–440 413
Fig. 2. General structure of sub-minimal clusters. G-links (dashed) form a necklace involving six dressed meta-vertices.
since removing a G-link produces a minimal cluster. We conclude that the sub-minimal clusters
contain a single G1-cycle, they have the form of a necklace of meta-vertices dressed by G-trees
and p-clusters (see Fig. 2). They produce the contribution F
(1)
B
to the free energy.
2.2. Brief reminder of the results obtained at leading order
A functional equation characterising the leading order Y
0
(x) of the dressed vertex has been
established in [20] by exploiting the G-tree structure of minimal clusters. This structure implies
the factorisation property Y(x) = Y
p
(x)Y
G
(x) of the generating functions. It is due to the fact
that the descendants of vertices related to the root by a G-link form independent sub-clusters.
The same fact is further responsible for the property
Y
G
(x) =qQ(x)exp
G(x − y)Y(y)
dy
2iπ
. (2.8)
Considering Y
p
(x), the G-tree structure also implies the independence of the clusters with roots
attached to x via a path of p-links, which provides a third relation,
Y
p
(x) =
∞
l=1
x
l
ε
−(l−1)
2
σ(
x
l
)
i∈V(
x
l
){x}
Y
G
(φ
i
)
dφ
i
2iπ
<ij >∈E(
x
l
)
ε
2
p(φ
ij
), (2.9)
where
x
l
denotes the rooted p-cluster that consists of the vertices connected to the root x with
a path of p-links, and the p-links between them that form the set E(
x
l
). In short, it describes
the inner structure of a meta-vertex where p-links renders the strong interaction of the instantons
that constitute the bound state. It has been shown that at first order the mean field approximation
Y
G
(φ
i
) Y
G
(x) holds. The remaining series of integrals
I
l
(α) = ε
−(l−1)
2
x
l
1
σ(
x
l
)
l−1
i=1
dφ
i
2iπ
<ij >∈E(
x
l
)
ε
2
p(φ
ij
) (2.10)
gives the self-energy of the hadrons, i.e. the energy coming from the p-interactions of the in-
stantons in the hadron. Due to the invariance under translation of the variable φ
i
, this quantity is
actually independent of the mean position x of the constituents. It has been evaluated at α = 1,
I
l
(1) = 1/l, which provides the first order in ε
2
as I
l
(α) = I
l
(1) + O(ε
2
). Using the mean field
approximation, and the value 1/l for the self-energy, we find after multiplication of formula (2.9)
by Y
(0)
G
(x) the following relation:
剩余32页未读，继续阅读
weixin_38507121
- 粉丝: 10
- 资源: 928
我的内容管理
展开
最新资源
- 计算机人脸表情动画技术发展综述
- 关系数据库的关键字搜索技术综述：模型、架构与未来趋势
- 迭代自适应逆滤波在语音情感识别中的应用
- 概念知识树在旅游领域智能分析中的应用
- 构建is-a层次与OWL本体集成：理论与算法
- 基于语义元的相似度计算方法研究：改进与有效性验证
- 网格梯度多密度聚类算法：去噪与高效聚类
- 网格服务工作流动态调度算法PGSWA研究
- 突发事件连锁反应网络模型与应急预警分析
- BA网络上的病毒营销与网站推广仿真研究
- 离散HSMM故障预测模型：有效提升系统状态预测
- 煤矿安全评价：信息融合与可拓理论的应用
- 多维度Petri网工作流模型MD_WFN：统一建模与应用研究
- 面向过程追踪的知识安全描述方法
- 基于收益的软件过程资源调度优化策略
- 多核环境下基于数据流Java的Web服务器优化实现提升性能
资源上传下载、课程学习等过程中有任何疑问或建议，欢迎提出宝贵意见哦~我们会及时处理！
点击此处反馈
