Nf=2+1动态QCD下D与Ds介子衰减常数的连续极限计算

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本文主要探讨了在连续极限下的Nf = 2 +1范畴壁晶格量子色动力学(QCD)中D和Ds介子的衰减常数f_D和f_{D_s}的数值计算。这项研究由RBC/UKQCD合作团队进行，他们利用了先进的领域壁技术在动态包含两种轻夸克（u和d）和一种重夸克(s)的QCD模拟上进行了细致的工作。模拟中，轻夸克的质量设置在物理和非物理范围，包括了0.11至0.07 femtometer (fm)的晶格间距，确保了结果具有较高的精度和可比性。领域壁技术被应用于所有价夸克的离散化，这是一种高效且能保持chiral symmetry的方案，对于计算轻子的衰减常数这类涉及轻夸克性质的物理量至关重要。衰减常数是衡量强相互作用下粒子衰变为更稳定状态时释放的能量强度的物理量，对于理解D和Ds介子的内部结构以及它们与其他粒子的相互作用有着重要意义。研究结果显示，在连续极限下，即去除所有有限体积效应和晶格间距的影响后，计算得到的D介子的衰减常数f_D为208.7(2.8) stat + 2（括号内的值分别代表统计误差和系统误差）。这个结果相较于先前的研究有所提升，为理论与实验之间的比较提供了更为精确的数值基础。同时，对于Ds介子的衰减常数f_{D_s}，文章并未直接给出数值，但可以推测其计算方法类似，也是基于同样的理论框架和数据处理流程。这篇发表于《Journal of High Energy Physics》(JHEP) 2017年第12期的文章，对于理解QCD中的强相互作用性质，尤其是轻味强子的性质有重要贡献，且由于是开放访问资源，使得更多的科学家可以获取并验证这些计算结果。此外，研究团队通过精密的数值模拟和数据分析，展示了在现代高精度的 lattice QCD 方法下，如何逼近真实的量子场论物理，为后续的理论研究和实验验证提供了宝贵的数据支持。

JHEP12(2017)008

0 20 40 60 80

t/a

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

res

(

)

Figure 1. The eﬀective residual mass on the C0 ensemble for all simulated masses.

Name M

bare

C0 1.6 12 0.3, 0.35, 0.4, 0.45

C1 1.6 12 0.3, 0.35, 0.4

C2 1.6 12 0.3, 0.35, 0.4

M0 1.6 8 0.22, 0.28, 0.34, 0.4

M1 1.6 12 0.22, 0.28, 0.34, 0.4

M2 1.6 12 0.22, 0.28, 0.34, 0.4

F1 1.6 12 0.18, 0.23, 0.28, 0.33, 0.4

Table 3. M¨obius domain wall parameters for the heavy quarks on all ensembles. All quoted values

for am

are bare quark masses in lattice units. As described in the text, the value indicated in red

was only used to verify our assumptions about the applicability of the quenched pilot study to the

dynamical case.

3 Data analysis

In this section we describe how we make predictions for the decay constants starting from

the evaluation of Euclidean two-point correlation functions on all ensembles and for all

parameters discussed above. In particular, from ﬁts to the simulated data we determine

pseudoscalar masses m

and decay constants f

and ratios thereof, which depend on

the simulation parameters (a, m

, m

). Of particular relevance for the analysis are

P = π, K, D, D

and η

(the latter one corresponding to an unphysical c¯c

state made of

two distinct ﬂavours of quarks having the same charm quark mass). We extrapolate the

data for each observable O to physical light, strange and charm-quark masses as well as to

vanishing lattice spacing and inﬁnite volume, O(a = 0, m

phys

, m

phys

, m

phys

Besides the decay constants and ratios thereof, for reasons that will become clear later,

we will carry out the analysis of the decay constants in terms of the quantity f

√

and

only remove the factor

√

in the ﬁnal step.

– 6 –

JHEP12(2017)008

3.1 Correlation functions

In practice we determine the matrix element in equation (1.2) from the time dependence

of Euclidean QCD zero-momentum two-point correlation functions,

i,MN

(t) ≡

x,y

i,M

(t, y) (O

i,N

)

†

(0, x) i =

(k)

i,M



(k)

i,N



∗

(k)



−E

(k)

± e

−E

(k)

(T −t)



(3.1)

We consider the cases i = π, K, D, D

or η

. The operators O

i,M

are interpolating operators

with the quantum numbers of the desired mesons, e.g. O

= ¯c Γ

s , where we consider

= γ

and Γ

= γ

, respectively. The sum on the r.h.s. of eq. (3.1) is over excited

states k and in practice we will consider the ground and ﬁrst excited state in the data

analysis. The constants Z

(k)

i,M

are deﬁned by Z

(k)

i,M

= hP

(k)

| (O

i,M

)

†

|0 i where P

(k)

is the

corresponding kth excited meson state.

When computing the quark propagators we use Z(2)×Z(2) stochastic wall sources [48–

50] on a large number of time planes. Details of how many diﬀerent source planes are used

for the various ensembles are listed in the column “hits/conf” in table 1. The results on a

given gauge conﬁguration are averaged into one bin.

The calculation of the light and strange quark propagators were performed using the

HDCG algorithm [51], reducing the numerical cost and hence making this computation

feasible. For the heavy quark propagators a CG inverter was used and we monitored

satisfactory convergence using the time-slice residual introduced in ref. [52].

Masses and decay constants have been determined by simultaneous multi-channel ﬁts

to the two-point correlation functions C

i,AP

and C

i,P P

for a given choice of i. We attempted

the use of correlated ﬁts, but found the correlation matrix to be too poorly estimated for

a reliable inversion. We therefore carry out uncorrelated ﬁts, i.e. assume the correlation

matrix to be diagonal (compare ref. [37]).

The statistical precision of the ground state mass and matrix elements is improved

by ﬁtting the ground state as well as the ﬁrst excited state, allowing for earlier time slices

(with smaller statistical errors) to contribute to the ﬁt. This was done for i = D, D

, where

we are interested in the matrix elements and not merely in the masses as in the case for

i = π, η

. During all these ﬁts the χ

/d.o.f. were monitored. Tables with the bare results

in lattice units on all ensembles can be found in appendix B.

Figure 2 illustrates the correlator ﬁt for the charm-quark mass closest to the physical

value on the M0 ensemble. The ﬁt ranges were chosen by systematically varying t

min

and

max

, ﬁxing them in a region where no dependence is observed. Figure 3 shows the ground

state results of varying t

min

and t

max

for the case of the mass point closest to the physical

charm quark mass on the medium ensemble M0.

A ﬁrst impression of the range of ensembles (lattice spacing, light sea quark mass,

charm-quark mass) for which we generated data is given in ﬁgure 4, representatively for

the ratio of decay constant f

plotted against the inverse of the measured η

mass.

– 7 –

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