次領頭階低能常數的改進*

蔣紹周，蔣杰臣

(廣西大學物理學院，廣西大學-國家天文臺天體物理和空間科學研究中心，廣西南寧　530004)

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次領頭階低能常數的改進*

蔣紹周，蔣杰臣

(廣西大學物理學院，廣西大學-國家天文臺天體物理和空間科學研究中心，廣西南寧530004)

【目的】通過合適的處理，減少低能贗標介子手征微擾理論中出現的輸入參數，得到符合實驗的低能常數理論值，提高理論的預言性?！痉椒ā繉⒁延蟹椒ㄖ谐霈F的Schwinger-proper time方法引入的Λ趨于無窮，并通過在介子質量770 MeV處對領頭階的低能常數進行重整化。借助Schwinger-Dyson方程，得到所有的次領頭階低能常數?！窘Y果】通過參數的調節，以及低能常數和耦合常數參數的關系，找到一組符合實驗的參數值；減少Λ和F0兩個輸入參數可以得到三味和兩味的低能常數?！窘Y論】減少Λ和F0兩個輸入參數處理低能常數的方法是可行的。兩味的低能常數對耦合常數中參數的依賴比較大，而三味的對其依賴相對較小。

手征微擾理論耦合常數低能常數

0　引言

【研究意義】量子電動力學的計算值和實驗值符合得很好，主要原因在于其可以通過耦合常數(α≈1/137)做展開，進行微擾計算[1]。但是，對于量子色動力學(QCD)來說，微擾理論不再適用，一般需要進行非微擾計算。非微擾現象的原因在于QCD的耦合常數隨著動量的減小而增大，因此，對于低能QCD的相互作用，不能采用原來比較成熟的微擾論來處理。手征微擾理論是一種有效處理低能贗標介子相互作用的方法[2-4]，但該理論中起重要作用的低能常數還未能通過解析的辦法得到。目前一種常用的求解方式是借助耦合常數，將耦合常數和夸克自能聯系起來[5-6]，進而求解出低能常數，且該方法已經取得了一些進展[7]?！厩叭搜芯窟M展】目前，低能常數與夸克自能、夸克自能與耦合常數的關系都已有一定精度下的理論結果[5-7]。文獻[7]較精確地計算到次次領頭階低能常數；文獻[8]研究耦合常數對次領頭階低能常數的影響；更有Maris-Tandy模型和Qin-Chang模型[9-10]不需要額外輸入參數，就能計算出低能耦合常數?！颈狙芯壳腥朦c】已有的低能常數求解方法需要輸入的參數相對較多，除耦合常數之外，還需要Schwinger-proper time方法引入的截斷參數Λ和領頭階的低能常數F0。因此，理論的預言性不是很強。為提高理論的預言性，本研究取消Λ和F0兩個輸入參數，僅保留耦合常數作為輸入?！緮M解決的關鍵問題】利用Maris-Tandy模型和Qin-Chang模型的耦合常數，通過Schwinger-Dyson(SD)方程求出夸克自能，并計算出所有次領頭階低能常數。在輸入參數減少之后，已有的耦合常數并不適用于現在問題，需對已有的耦合常數參數進行適當修改，以期其與實驗值相符。

1　夸克自能的求解

對于以前的耦合常數形式，主要以分段函數的形式為主[11]，該形式連接點的二階導數不連續，會對低能常數的全微分關系產生影響，并且還包含待定參數ΛQCD。文獻[8]采用的耦合常數同樣存在待定常數，計算時需要調節待定常數，理論預言性低。因此我們采用下面兩種有比較確定形式的耦合常數進行討論[9-10]：

(1)

(2)

利用上面的耦合常數就可以通過SD方程求出夸克自能Σ(p2)。經過大NC展開，做梯形近似和角近似之后的SD方程為[5]

(3)

(4)

其中，γ=9NC/2(33-2Nf)。方程(3)沒有解析解，但是可以通過數值方法求出，而(4)式可以檢驗數值求解夸克自能的正確性。

2　低能常數的計算

(5)

表1兩種不同耦合常數得到的低能常數

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

耦合常數Couplingconstants低能常數Lowenergyconstantsl1l2l3l4l5l6αs1(p2)-4.737.70-0.684.0822.1221.86αs2(p2)-6.998.82-0.704.0826.3426.33文獻[3]Reference[3]-2.3±3.76.0±1.32.9±2.44.3±0.913.9±1.316.5±1.1文獻[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調節耦合常數αs1(p2)得到的低能常數

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

序號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調節耦合常數αs2(p2)得到的低能常數

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

序號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

可見，只有表3中的第13組參數得到的低能常數符合文獻[3，12]所給出的結果，該組參數得到的夸克凝聚結果也大致與文獻[13-14]給出-(250 MeV)3相符。對大部分的參數來說，參數的調整對夸克凝聚能量的影響都不是很大，其結果都在-(250 MeV)3左右。

類似地，我們可以得到三味夸克的結果。一般來說，需要重新調整耦合常數相應的參數來求解三味的夸克自能。但是，在實際的數值計算中發現，這些參數對三味的低能常數的影響并不大。另外，四味的結果包含的c夸克較重，和手征微擾論適用的低能范圍差距較遠，因此我們不考慮四味夸克的結果。由表4可見，三味的低能常數和其他通過實驗得到的結果[4，12]基本符合。因此，本研究采取的處理是合理的。

表4Nf=3,(Dω)1/3=0.3 GeV時對應的低能常數

Table 4LECs in Nf=3,(Dω)1/3=0.3 GeV

項目Item低能常數Lowenergyconstants103L1103L2103L3103L4103L5103L6103L7103L8103L9103L10結果Result0.561.13-2.9501.360-0.531.335.10-4.83文獻[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文獻[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　結論

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(責任編輯：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

蔣紹周(1982-)，男，博士，副教授，碩士生導師，主要從事粒子物理理論方向的研究。

O412.3

A

1005-9164(2016)03-0202-04

*國家自然科學基金項目(11565004)資助。

廣西科學Guangxi Sciences 2016,23(3):202～205

網絡優先數字出版時間：2016-07-13【DOI】10.13656/j.cnki.gxkx.20160713.007

網絡優先數字出版地址：http://www.cnki.net/kcms/detail/45.1206.G3.20160713.0857.014.html