非相对论夸克模型中Laplace变换法下的重介子特性研究
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更新于2024-09-04
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本文主要探讨了非相对论夸克模型中，利用拉普拉斯变换方法(Laplace Transformation Method, LTM)来求解矢量和标量势结合的N维径向薛定谔方程的理论研究。作者M.Abu-Shady和E.M.Khokha针对重轻介子(Highly Heavy-Light Mesons, HLMs)的研究，特别关注了B、Bs、D和Ds这类介子。他们首先通过LTM得到了一个解析解，这个解考虑了自旋超精细、自旋轨道以及张量相互作用，这使得方程能更准确地描述这些强相互作用粒子的行为。
在N维空间中，研究者成功计算出了一系列特征值和对应的特征函数，这些是理解HLM系统的基本构建块。他们发现，随着维度空间的增加，重介子的质量也随之提升，这反映了量子力学中的空间维度效应。这种对空间维度的深入分析对于理解和预测真实世界中的介子性质至关重要。
除了质量计算，论文还涉及了伪标量子和矢量介子的衰减常数的测定，这对于了解介子的稳定性和衰变过程有着重要意义。研究者进一步讨论了B +、D +和Bs +介子的轻子衰变宽度和分支比，这涉及到粒子间的弱相互作用，以及其在物理过程中的表现。
值得注意的是，本文所采用的拉普拉斯变换方法提供了与实验数据相当良好的结果，相较于其他理论研究，这种方法显示出更高的精度和改进。由于该研究采用了开放获取的方式，学术界可以自由访问和利用这些研究成果，促进了科学知识的共享和进步。
这篇研究通过数学工具揭示了非相对论夸克模型中重介子行为的新见解，不仅深化了我们对基本粒子性质的理解，也为后续的实验验证和理论探索提供了有价值的基础。
Advances in High Energy Physics 3
suciently strong singularity in the function ()as →0.
In particular
L
𝛼
Γ
(
+1
)
=
1
𝛼+1
,>−1,
(2)
where Γis the gamma function. e Laplace transform has
the derivative properties
L
(𝑛)
(
)
=
𝑛
L
(
)
−
𝑛−1
𝑘=0
𝑛−1−𝑘
(𝑘)
(
0
)
,
(3)
L
𝑛
(
)
=
(
−1
)
𝑛
(𝑛)
(
)
,
(4)
where the superscript ()stands for the -th derivative with
respect to for
(𝑛)
()and with respect to for
(𝑛)
().If
0
is
the singular point, the Laplace transform behaves near →
0
as
(
)
=
1
−
0
𝜐
,
(5)
andthenfor→∞
(
)
=
1
Γ
(
)
𝜐−1
𝑧
0
𝑡
.
(6)
On the other hand, if near origin ()behaves like
𝛼
with
>−1,then()behaves near →∞as
(
)
=
Γ
(
+1
)
𝛼+1
.
(7)
3. Analytic Solution of the -Dimensional
Radial Schrödinger Equation
e N-dimensional radial Schr
¨
odinger equation that de-
scribes the interaction between quark-antiquark systems
takes the form [41]
2
2
+
(
−1
)
−
(
+−2
)
2
+2−
𝑞𝑞
(
)
Ψ
(
)
=0,
(8)
where ,represent the angular quantum number and the
dimensional number greater than one, respectively, and =
𝑞
𝑞
/(
𝑞
+
𝑞
)is the reduced mass of the quark-antiquark
system.
In the nonrelativistic quark model, the quark-antiquark
potential
𝑞𝑞
() consists of the spin independent potential
()and the spin dependent potential
𝑆𝐷
(),respectively:
𝑞𝑞
(
)
=
(
)
+
𝑆𝐷
(
)
.
(9)
e spin independent potential is taken as a combination of
vector and scalar parts [93]:
(
)
=
𝑉
(
)
+
𝑆
(
)
,
(10)
𝑉
(
)
=
2
+−
,
(11)
𝑆
(
)
=1−
2
+, (12)
where
𝑉
() and
𝑆
() are the vector and scalar parts,
respectively, and stands for the mixing coecient. , b,
and c are arbitrary parameters where a, b,andc >0which
are tted with experimental data. e harmonic and linear
terms represent the conning part at long distance and the
Coulomb term stands for the quark-antiquark interactions
through one gluon exchange at short distances which gives
better description of quark-antiquark interaction.
e spin dependent potential is extended to three types
of interaction terms as [94]
𝑆𝐷
(
)
=
𝐿𝑆
(
)(
L .S
)
+S
12
𝑇
(
)
+
𝑆𝑆
(
)
S
1
.S
2
,
(13)
while the spin-orbit
𝐿𝑆
()and tensor
𝑇
()terms give the
ne structure of the states, the spin-spin
𝑆𝑆
()interaction
term describes the hyperne splitting of the state, and L is an
angular quantum operator, and S is a spin operator (for detail,
see [94]).
𝐿𝑆
(
)
=
1
2
𝑞
𝑞
3
𝑉
−
𝑆
,
(14)
𝑇
(
)
=
1
12
𝑞
𝑞
1
𝑉
−
2
𝑉
2
,
(15)
𝑆𝑆
(
)
=
2
3
𝑞
𝑞
∇
2
𝑉
,
(16)
where ∇
2
is radial Laplace operator.
S
1
.S
2
=
1
2
(
+1
)
−
3
2
,
(17)
L .S
=
1
2
[
(
+1
)
−
(
+1
)
−
(
+1
)
]
,
(18)
S
12
=2S
2
−3
(
S .
r
)(
S .
r
)
. (19)
e diagonal elements of the S
12
are dened.
S
12
=
4
(
2+3
)(
2−1
)
2
2
−3
L .S
2
−
3
2
L .S
.
(20)
Substituting (11)-(16) into (9) then the nonrelativistic quark-
antiquark potential
𝑞𝑞
()takes the form
𝑞𝑞
(
)
=
2
+++
+
3
,
(21)
where
=
2
𝑞
𝑞
2S
1
.S
2
+2−
1
2
(
L .S
)
,
(22)
=
𝑞
𝑞
4
3
S
1
.S
2
+
1
12
S
12
+2−
1
2
(
L .S
)
−,
(23)
=
3
2
𝑞
𝑞
1
6
S
12
+
(
L .S
)
.
(24)
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