蒋绍周，蒋杰臣

(广西大学物理学院，广西大学-国家天文台天体物理和空间科学研究中心，广西南宁　530004)



次领头阶低能常数的改进*

蒋绍周，蒋杰臣

(广西大学物理学院，广西大学-国家天文台天体物理和空间科学研究中心，广西南宁530004)

【目的】通过合适的处理，减少低能赝标介子手征微扰理论中出现的输入参数，得到符合实验的低能常数理论值，提高理论的预言性。【方法】将已有方法中出现的Schwinger-proper time方法引入的Λ趋于无穷，并通过在介子质量770 MeV处对领头阶的低能常数进行重整化。借助Schwinger-Dyson方程，得到所有的次领头阶低能常数。【结果】通过参数的调节，以及低能常数和耦合常数参数的关系，找到一组符合实验的参数值；减少Λ和F0两个输入参数可以得到三味和两味的低能常数。【结论】减少Λ和F0两个输入参数处理低能常数的方法是可行的。两味的低能常数对耦合常数中参数的依赖比较大，而三味的对其依赖相对较小。

手征微扰理论耦合常数低能常数

0　引言

【研究意义】量子电动力学的计算值和实验值符合得很好，主要原因在于其可以通过耦合常数(α≈1/137)做展开，进行微扰计算[1]。但是，对于量子色动力学(QCD)来说，微扰理论不再适用，一般需要进行非微扰计算。非微扰现象的原因在于QCD的耦合常数随着动量的减小而增大，因此，对于低能QCD的相互作用，不能采用原来比较成熟的微扰论来处理。手征微扰理论是一种有效处理低能赝标介子相互作用的方法[2-4]，但该理论中起重要作用的低能常数还未能通过解析的办法得到。目前一种常用的求解方式是借助耦合常数，将耦合常数和夸克自能联系起来[5-6]，进而求解出低能常数，且该方法已经取得了一些进展[7]。【前人研究进展】目前，低能常数与夸克自能、夸克自能与耦合常数的关系都已有一定精度下的理论结果[5-7]。文献[7]较精确地计算到次次领头阶低能常数；文献[8]研究耦合常数对次领头阶低能常数的影响；更有Maris-Tandy模型和Qin-Chang模型[9-10]不需要额外输入参数，就能计算出低能耦合常数。【本研究切入点】已有的低能常数求解方法需要输入的参数相对较多，除耦合常数之外，还需要Schwinger-proper time方法引入的截断参数Λ和领头阶的低能常数F0。因此，理论的预言性不是很强。为提高理论的预言性，本研究取消Λ和F0两个输入参数，仅保留耦合常数作为输入。【拟解决的关键问题】利用Maris-Tandy模型和Qin-Chang模型的耦合常数，通过Schwinger-Dyson(SD)方程求出夸克自能，并计算出所有次领头阶低能常数。在输入参数减少之后，已有的耦合常数并不适用于现在问题，需对已有的耦合常数参数进行适当修改，以期其与实验值相符。

1　夸克自能的求解

对于以前的耦合常数形式，主要以分段函数的形式为主[11]，该形式连接点的二阶导数不连续，会对低能常数的全微分关系产生影响，并且还包含待定参数ΛQCD。文献[8]采用的耦合常数同样存在待定常数，计算时需要调节待定常数，理论预言性低。因此我们采用下面两种有比较确定形式的耦合常数进行讨论[9-10]：

(1)

(2)

利用上面的耦合常数就可以通过SD方程求出夸克自能Σ(p2)。经过大NC展开，做梯形近似和角近似之后的SD方程为[5]

(3)

(4)

其中，γ=9NC/2(33-2Nf)。方程(3)没有解析解，但是可以通过数值方法求出，而(4)式可以检验数值求解夸克自能的正确性。

2　低能常数的计算

(5)

表1两种不同耦合常数得到的低能常数

Table 1LECs obtained from the two kinds of the coupling constants

耦合常数Couplingconstants低能常数Lowenergyconstantsl1l2l3l4l5l6αs1(p2)-4.737.70-0.684.0822.1221.86αs2(p2)-6.998.82-0.704.0826.3426.33文献[3]Reference[3]-2.3±3.76.0±1.32.9±2.44.3±0.913.9±1.316.5±1.1文献[12]Reference[12]-0.4±0.64.3±0.12.9±2.44.4±0.212.24±0.2116.0±0.5±0.7

表2调节耦合常数αs1(p2)得到的低能常数

Table 2LECs obtained by adjusting the coupling constant αs1(p2)

序号No.ΛQCD(MeV)ω(GeV)(Dω)1/3(GeV)l1l2l3l4l5l6-<ψψ>1/3r(MeV)11500.400.72-4.747.70-1.094.1122.2421.88526.4822000.400.72-4.747.70-1.144.1322.1921.87394.4333000.400.72-4.737.70-1.264.1822.0721.83261.8744000.400.72-4.717.69-1.424.2621.9221.76194.6655000.400.72-4.697.68-1.674.3721.7521.69152.9166000.400.72-4.667.67-2.014.5721.5621.60122.3272340.310.72-9.6910.12-1.184.0631.8331.76277.8682340.350.72-7.298.95-1.194.0927.0826.93305.0192340.400.72-4.747.70-1.184.1522.1521.86336.76102340.450.72-2.616.67-1.124.2118.1417.71365.73112340.500.72-0.865.85-1.024.2914.9314.42391.43122340.600.721.524.82-0.614.5110.4110.35429.94132340.400.401.194.96-0.294.3610.8410.84284.11142340.400.70-4.407.54-1.154.1521.5021.20335.04152340.400.75-5.237.94-1.214.1423.1022.82339.15162340.400.80-6.018.32-1.274.1324.6224.38342.73172340.400.85-6.768.68-1.314.1226.0825.87345.88182340.400.90-7.469.03-1.364.1127.4727.29348.67

表3调节耦合常数αs2(p2)得到的低能常数

Table 3 LECs obtained by adjusting the coupling constant αs2(p2)

序号No.ΛQCD(MeV)ω(GeV)(Dω)1/3(GeV)l1l2l3l4l5l6-<ψψ>1/3r(MeV)12300.400.72-6.998.82-1.044.1426.3526.34305.8322500.400.72-6.988.82-1.064.1526.3126.31281.3833000.400.72-6.968.81-1.114.1726.2126.25234.4043600.400.72-6.938.79-1.184.2026.1026.18200.6254000.400.72-6.908.78-1.264.2425.9926.11175.0062340.400.72-6.998.82-1.054.1426.3526.33300.6172340.450.72-5.127.91-1.014.1922.6522.63324.2082340.500.72-3.557.16-0.954.2419.5919.58345.1092340.550.72-2.246.54-0.854.3017.0417.09363.12102340.600.72-1.166.05-0.724.3614.9315.08378.00112340.700.720.445.35-0.344.4711.7612.33396.98122340.400.201.744.881.404.259.0510.51162.02132340.400.30-0.035.550.134.2712.5613.08220.52142340.400.40-1.906.39-0.414.2316.2516.44253.09152340.400.50-3.677.22-0.704.1919.7319.79273.70162340.400.60-5.277.99-0.894.1622.9122.92288.09172340.400.70-6.728.69-1.024.1425.8025.79298.78

可见，只有表3中的第13组参数得到的低能常数符合文献[3，12]所给出的结果，该组参数得到的夸克凝聚结果也大致与文献[13-14]给出-(250 MeV)3相符。对大部分的参数来说，参数的调整对夸克凝聚能量的影响都不是很大，其结果都在-(250 MeV)3左右。

类似地，我们可以得到三味夸克的结果。一般来说，需要重新调整耦合常数相应的参数来求解三味的夸克自能。但是，在实际的数值计算中发现，这些参数对三味的低能常数的影响并不大。另外，四味的结果包含的c夸克较重，和手征微扰论适用的低能范围差距较远，因此我们不考虑四味夸克的结果。由表4可见，三味的低能常数和其他通过实验得到的结果[4，12]基本符合。因此，本研究采取的处理是合理的。

表4Nf=3,(Dω)1/3=0.3 GeV时对应的低能常数

Table 4LECs in Nf=3,(Dω)1/3=0.3 GeV

项目Item低能常数Lowenergyconstants103L1103L2103L3103L4103L5103L6103L7103L8103L9103L10结果Result0.561.13-2.9501.360-0.531.335.10-4.83文献[4]Reference[4]0.9±0.31.7±0.7-4.4±2.50±0.52.2±0.50.0±0.3-0.4±0.151.1±0.37.4±0.7-6.0±0.7文献[12]Reference[12]0.53±0.060.81±0.04-3.07±0.20≡0.31.01±0.060.14±0.05-0.34±0.090.47±0.105.93±0.43-3.8±0.4

3　结论

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(责任编辑：米慧芝)

Improvement of the Next Leading Order Low-energy Constants

JIANG Shaozhou，JIANG Jiechen

(Department of Physics，GXU-NAOC Center for Astrophysics and Space Sciences，Guangxi University，Nanning，Guangxi，530004，China)

【Objective】There is not an effective way to calculate the low-energy constants (LECs) for the pseudoscalar meson chiral perturbation theory.There are too many input parameters,which reduce the predictability of the theory.Reducing the input parameters in the calculation was carried out through the appropriate treatment,in order to get a set of theoretical values that are consistent with the experiments.【Methods】The Schwinger-proper time method was introduced into the Λ to infinity,and a leading order of the LECs was renormalized in the meson mass at 770 MeV.All the next leading order LECs can be obtained through Schwinger-Dyson equation.【Results】A set of parameters that match the experiments was obtained through measuring parameters and comparing the relationships between the LECs and coupling constants.Finally the input parameters Λ and F0were reduced in the three-flavor and two-flavor quark.【Conclusion】The new method that treats LECs by reducing the input parameters is feasible.Two-flavor LECs are sensitive to the coupling constants,but three-flavor LECs are not.

chiral perturbation theory,coupling constants,low-energy constants

2016-05-13

2016-06-20

蒋绍周(1982-)，男，博士，副教授，硕士生导师，主要从事粒子物理理论方向的研究。

O412.3

A

1005-9164(2016)03-0202-04

*国家自然科学基金项目(11565004)资助。

广西科学Guangxi Sciences 2016,23(3):202～205

网络优先数字出版时间：2016-07-13【DOI】10.13656/j.cnki.gxkx.20160713.007

网络优先数字出版地址：http://www.cnki.net/kcms/detail/45.1206.G3.20160713.0857.014.html