NRQCD校准下的康奈尔模型分析

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"使用NRQCD校准朴素的康奈尔模型" 本文主要探讨了康奈尔模型(Cornell model)与非相对论量子色动力学(NRQCD)之间的关系，尤其是在描述重味强子现象学中的应用。康奈尔模型在无法直接应用QCD有效理论的场景下展现了极高的成功性。作者们首次尝试通过NRQCD的预测来校准康奈尔模型，以更好地理解其参数与量子色动力学(QCD)基本常数之间的联系。 NRQCD是一种处理重夸克系统的有效理论，它在高能和低能区域之间提供了一个过渡，特别适用于分析重介子的性质。而康奈尔模型则是一个简单的强相互作用模型，包含库仑和线性势，用于解释重粒子的结合。在文章中，研究人员基于之前对重介子谱的研究，利用NRQCD的预测对康奈尔模型进行了校准，重点关注最低的底层bottom态。他们发现康奈尔模型的质量参数可以在微扰不确定性的范围内，用尺度为$$ R = 1 \, \hbox {GeV} $$的MSR（跑动质量）来确定，这个结果独立于$$ \alpha_s $$（强耦合常数）的值、底部夸克的质量，以及考察的所有扰动阶。此外，研究揭示了两个关键点：(a) "弦张力"参数与底部夸克的质量无关，这意味着康奈尔模型中的线性势与夸克质量独立，(b) 康奈尔模型的库仑强度$$ \kappa $$可以与QCD的强耦合常数$$ \alpha_s $$在非相对论尺度上关联。这一发现意味着康奈尔模型的参数能够与更基础的QCD理论相联系。为了进一步消除静态效应，文章还讨论了如何通过切换到更小的尺度，即短距离的MSR质量来实现这一目标。这些发现为理解和统一康奈尔模型与QCD提供了新的视角，有助于增强我们在非微扰QCD领域的理解，特别是对于那些不能直接用QCD理论处理的现象。参考文献： Eur.Phys.J.C(2019)79:323 https://doi.org/10.1140/epjc/s10052-019-6808-2 作者：Vicent Mateu, Pablo G. Ortega, David R. Entem, Francisco Fernández 发布日期：2019年1月24日，接受日期：2019年3月24日，在线发表日期：2019年4月10日开放获取文章 - 版权归作者所有。

Eur. Phys. J. C (2019) 79 :323 Page 5 of 23 323

functions, so a numerical approach must be employed. There

are many numerical methods to solve the Cornell potential

in the literature. For its simplicity and robustness, we will

use the Numerov algorithm [22,23] (also called Cowell’s

method). Further details can be found in Appendix A. We

have implemented the Numerov algorithm and the matrix

element computations in a Fortran 2008 code [24], which

contains both public and in-house routines, and is used to

predict the mass of the bound states as well as to perform ﬁts

to data.

The potential in Eq. (1) by itself is nonetheless unable

to describe the Υ(1S) − η

(1S) mass splitting, as it does

not depend on the spin of the quarkonium state. Further-

more, it provides degenerate masses for the χ

multiplet

(with J ={0, 1, 2}). Hence, in order to describe with higher

accuracy the low-lying bottomonium spectrum, one has to

add 1/m

terms to the Cornell static potential that take

into account the spin–spin, spin–orbit and tensor interac-

tions, breaking the present degeneracy [25]. Such contribu-

tions arise from the leading relativistic corrections of the

t-channel gluon and conﬁnement interactions, giving the fol-

lowing terms [5,26]:

3,4

OGE

(r) =

8α

Cornell

· S

)δ(r),

OGE

(r) =

2α

Cornell

(L · S ),

CON

(r) =−

(L · S ),

OGE

(r) =

Cornell

, (13)

where S = S

+ S

is the total spin, L the relative orbital

momentum and S

the tensor operator of the Q Q bound

state, deﬁned as

= 2 (S

·ˆr)(S

·ˆr) − (S

· S

), (14)

with ˆr = r/r. Analytical expressions can be found for the

spin–spin, spin–orbital and tensor operators,

S

· S

=

[2 s (s + 1) − 3] ,

(15a)

L · S=

[ j ( j + 1) − (+ 1) − s (s + 1)],

(15b)

S

=

2 [2 (+ 1) s (s + 1) − 3 L · S−6 L · S

]

(2  − 1)(2  + 3)

(15c)

Here the LS term from conﬁnement is obtained by the exchange of a

scalar particle.

We denote with the superscript “OGE” the terms arising from gluon

exchanges and with “CON” those coming from the conﬁnement inter-

action.

with ( j,,s) the quantum numbers of the Q Q state. The

speciﬁc values for the quantum numbers considered in this

article are given in Table 1.

Given the assumed large mass of the heavy quark, the

above terms are expected to be small. Therefore, their con-

tribution to the bottomonium mass will be incorporated to

the model using ﬁrst-order perturbation theory. For consis-

tency, if perturbation theory is to be used to second order, one

needs to include the 1/m

potential. In practice one needs

to take matrix elements of the operators in Eq. (13). To that

end we use the angular decomposition in Eq. (3) and write

the three-dimensional integration in spherical coordinates

r = r

dr dΩ such that the angular integrations are carried

out with ordinary angular momentum algebra [see Eq. (15)]

and the radial integral becomes a one-dimensional matrix ele-

ment of the reduced wave function u

n

(r), given that the Jaco-

bian factor r

cancels when writing |R

n

(r)|

=|u

n

(r)|

as in Eq. (6).

3 Fitting the Cornell model to experimental data

In this section we present the results from χ

ﬁtsofthe

Cornell model parameters to charmonium and bottomonium

experimental data. We will restrict ourselves to n

≤ 2, since

for higher values of the principal quantum number one needs

to include string-breaking effects, as can be seen in Fig. 4.

The case of bottomonium will serve as a proof of concept for

our calibration, that is, it will show that the three parameters

of the model can be determined from ﬁts to the 8 lowest-lying

bottomonium bound states. This is crucial since in QCD we

can only reliably predict these within perturbation theory, as

argued in Ref. [27]. In many phenomenological applications

of the quark model approach, in which more sophisticated

Hamiltonians are used, the parameters of the potential are

not obtained by a full ﬂedged χ

minimization, but sim-

ply adjusted by a rough comparison to data plus physically

motivated priors. This procedure is in many cases justiﬁed,

since there is a large amount of data that the model aims

to describe, but its accuracy might vary widely for different

observables. Given that it is very hard to add a theoretical

covariance matrix to the χ

(the matrix does not provide a

method to quantify its “modeling” error), such a ﬁt could

lead to biased model parameter values. For the simpler situa-

tion of the naïve Cornell model, which seeks to describe only

the low-lying quarkonium spectrum, we show that the ﬁt is

indeed possible, and we shall see that no bias is observed.

The minimization of the χ

is carried out using the For-

tran 77 package MINUIT [28]. We have checked that the

algorithm is very effective in ﬁnding the minimum, but due

to the strong correlation between the various ﬁt parameters

[see Eq. (17)] it does not estimate the covariance matrix cor-

rectly. To solve this problem we compute a three-dimensional

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