两个夸克之间的强相互作用势
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摘要
本工作中借助已成熟的量子场论知识,推导出了在树图阶情况下不同费米子相互作用势势函数在动量空间中的解析表达式,对其进行傅里叶(Fourier)变换,将其转换到坐标空间.考虑到色因子的相互作用和电荷的符号,可以得到两个夸克(正反夸克对)之间强相互作用的库仑势能的解析表达式.通过解两个粒子的薛定谔方程计算出了两夸克系统的能级公式,与氢原子的能级公式相互比较,并以三味重夸克(c、b、t)为例,结合氢原子的电离能得到树图情况下三味重夸克基态能的大小.
考虑到单圈图的修正对物理理论结果具有重要的影响,因此我们利用高能物理做图软件(FeynArts)做出在双夸克系统中有两个或多个胶子交换的有效单圈图,利用微扰量子色动力学的知识,对正反夸克对强相互作用的单圈图做了详细的分析推导。最后选取适当的参数利用LoopTools软件包对多点格林函数进行计算,采用合理的近似对其中的结构表达式进行化简,最后得出了有效单圈图对树图的结果的修正值.本文的计算结果显示,双夸克系统中的强相互作用势中库仑势能部分并非无穷大,即使计算了最主要的单圈图修正之后,库仑势能依然为一有限大小,因此单纯有夸克的强相互作用并不能引起“夸克禁闭”现象,但是同时也证明了在双夸克系统中,存在着很大的库仑势能,因此很难从强子内部轰出自由态的夸克.
此外,本工作的意义还在于,对目前最为流行的势模型具有非常重要的意义,因为势模型只考虑树图的结果,本文考虑了圈图的修正；该结果还有助于禁闭势中唯象参数的选取,因为禁闭势中所包含的库仑势和禁闭势之间有一个函数关系.
In this paper, with the aid of the ripe theory of quantum field, and with the same way we detruded the analytic expression of interaction potential functions between different fermions for the tree diagram in the momentum space, then transit it into expression in the spatial space via Fourier transform. Considering the interaction between color factors and the symbol of the electrons, we can get the analytic Coulomb potential expression for the strong interactions between the two quarks (quark-antiquark pairs). We obtain the energtic formula of quark and antiquark system through calculating the Schrodinger equation for two quarks. Taking the three heavy flavor quarks (c, b, and t).as example and comparing with the ionization potential of the H atom, we obtain the ground state energy for the three heavy flavor quarks for the tree diagram.
Considering that the one loop diagram correction has important influence to theoretical results, the strong interactions potential of the dominant one-loop diagrams for quark-antiquark pairs must to be computed. Using the FeynArts software, we drawn all the effective one-loop diagrams with two or more gluons exchanging. According to the perturbative QCD, the one-loop diagrams of strong interactions for quark-antiquark pairs to be analyzed detailedly. At last, with the help of LoopTools, we simplify expressions and calculate the functions under the apt parameters, and finally get the numerical result for contributions of the potential from the tree diagram and one-loop diagrams in quark and antiquark system.
In this paper, it can be found that the obtained Coulomb potential of strong interactions is not infinite in the quark-antiquark systems, even if putting the corrections of the dominate one-loop diagrams, the Coulomb potential is still limited. So, it can not cause the "quark confinement" only by the strong interactions. However, by the same time there is a deep Coulomb potential in the quark-antiquark pairs which makes us to discover the free quark quite difficult.
This work have a deep sense for the potential model that is very popular at present, which results only think over the tree diagram, our result given the correction of the dominate one-loop diagrams. Besides, our work helps to select the parameters in confinement potential which has a connection with Coulomb potential.
考虑到单圈图的修正对物理理论结果具有重要的影响,因此我们利用高能物理做图软件(FeynArts)做出在双夸克系统中有两个或多个胶子交换的有效单圈图,利用微扰量子色动力学的知识,对正反夸克对强相互作用的单圈图做了详细的分析推导。最后选取适当的参数利用LoopTools软件包对多点格林函数进行计算,采用合理的近似对其中的结构表达式进行化简,最后得出了有效单圈图对树图的结果的修正值.本文的计算结果显示,双夸克系统中的强相互作用势中库仑势能部分并非无穷大,即使计算了最主要的单圈图修正之后,库仑势能依然为一有限大小,因此单纯有夸克的强相互作用并不能引起“夸克禁闭”现象,但是同时也证明了在双夸克系统中,存在着很大的库仑势能,因此很难从强子内部轰出自由态的夸克.
此外,本工作的意义还在于,对目前最为流行的势模型具有非常重要的意义,因为势模型只考虑树图的结果,本文考虑了圈图的修正；该结果还有助于禁闭势中唯象参数的选取,因为禁闭势中所包含的库仑势和禁闭势之间有一个函数关系.
In this paper, with the aid of the ripe theory of quantum field, and with the same way we detruded the analytic expression of interaction potential functions between different fermions for the tree diagram in the momentum space, then transit it into expression in the spatial space via Fourier transform. Considering the interaction between color factors and the symbol of the electrons, we can get the analytic Coulomb potential expression for the strong interactions between the two quarks (quark-antiquark pairs). We obtain the energtic formula of quark and antiquark system through calculating the Schrodinger equation for two quarks. Taking the three heavy flavor quarks (c, b, and t).as example and comparing with the ionization potential of the H atom, we obtain the ground state energy for the three heavy flavor quarks for the tree diagram.
Considering that the one loop diagram correction has important influence to theoretical results, the strong interactions potential of the dominant one-loop diagrams for quark-antiquark pairs must to be computed. Using the FeynArts software, we drawn all the effective one-loop diagrams with two or more gluons exchanging. According to the perturbative QCD, the one-loop diagrams of strong interactions for quark-antiquark pairs to be analyzed detailedly. At last, with the help of LoopTools, we simplify expressions and calculate the functions under the apt parameters, and finally get the numerical result for contributions of the potential from the tree diagram and one-loop diagrams in quark and antiquark system.
In this paper, it can be found that the obtained Coulomb potential of strong interactions is not infinite in the quark-antiquark systems, even if putting the corrections of the dominate one-loop diagrams, the Coulomb potential is still limited. So, it can not cause the "quark confinement" only by the strong interactions. However, by the same time there is a deep Coulomb potential in the quark-antiquark pairs which makes us to discover the free quark quite difficult.
This work have a deep sense for the potential model that is very popular at present, which results only think over the tree diagram, our result given the correction of the dominate one-loop diagrams. Besides, our work helps to select the parameters in confinement potential which has a connection with Coulomb potential.
引文
[1]张启仁,周月梅.原子核理论-它的深化与发展[M].北京：北京大学出版社.1999,11-32.
[2]M. Gell-Mann. A Schematic Model of Baryons and Mesons[J]. Phys. Lett.1964,8:214-215.
[3]J. J. Aubert, U. Becker and P. J. Biggs et al.Experimental Observation of a Heavy Particle J[J]. Phys.Rev.Lett,1974,33:1404-1406.
[4]J. E. Augustin et al. Discovery of a Narrow Resonance in e+e-Annihilation[J]. Phys. Rev.Lett,1974,33: 1406-1408.
[5]G. S. Abrams et al.Discovery of a Second Narrow Resonance in e+e-Annihilation[J]. Phys. Rev.Lett, 1974,33:1453-1455.
[6][美]基本粒子物理专门小组.基本粒子物理学[M].北京：科学出版社.1992,26-31.
[7]C.Amsler, M.Doser and W. M.Yao, et al.. PARTICLE PHYSICS BOOKLIET[M]. Berkeley.2006, 21-66.
[8]王凡.核力和重子相互作用[J].物理学进展,2004,24(1)：1-17.
[9]黄涛.高能物理学所面临的两大难题[J].现代物理知识,2000,12(5)：6-9.
[10]宁平治：核力研究新进展及前沿问题[J].物理学进展,2008,28(4)：432-455.
[11]John H. Schwarz. Dual resonance theory[J]. Phys.Rep.C,1973,8:269-335.
[12]S. Mandelstam. Dual-resonance models[J]. Phys.Rep.C,1974,13:259-353.
[13]W. A. Bardeen, M. S. Chanowitz and S. D. Drell et al.. Heavy quarks and strong binding:A field theory of hadron structure[J]. Phys.Rev.D,1975,11:1094-1136.
[14]A. Chodos, R. L. Jaffe and K. Johnson, et al.. New extended model of hadrons[J]. Phys. Rev.D,1974, 9:3471-3495.
[15]A.Chodos, R.L.Jaffe and K.Johnson et al..Baryon structure in the bag theory[J]. Phys. Rev.D,1974,10: 2599-2604.
[16]李政道.场论和粒子物理学(下册)[M].北京：科学出版社.1981,23-30.
[17]Jihn E. Kim.θ Bag and Quark Confinement[J]. Phys.Lett.B,1992,283:107-112.
[18]Kurt Langfeld. Quark Confinement in a Random Background Gross-Neveu Model[J]. Nucl.Phys.A, 1996,602:375-387.
[19]H.P. Pavel, D. Blaschke and G. Ropke et al.. Squeezed Gluon Condensate and Quark Confinement in the Global Color Model of QCD[J]. Int.J.Mod.Phys.A,1999,14:205-224.
[20]V. Gogokhia. The mass gap and solution of the quark confinement problem in QCD I[J], arXiv:0704.1745; The mass gap and solution of the quark confinement problem in QCD II[J], arXiv:0704.3189.
[21]Theodore J. Allen, M. G. Olsson and Sinisa Veseli et al.. On quark confinement dynamics[J]. Phys.Rev.D,1997,55:5408-5413.
[22]V. N. Gribov. The theory of quark confinement[J]. Eur.Phys.J.C,1999,10:91-105.
[23]C. C. Barros Jr. Quark confinement and curved spaces[J]. Eur.Phys.J.C,2006,45:421-425.
[24]C. B. Compean, M. Kirchbach. Trigonometric quark confinement potential of QCD traits[J]. Eur.Phys.J.A,2007,33:1-4.
[25]Qiao Cong-Feng, Huang Han-Wen and Chao Kuang-Ta. Large Possible retardation effects of quark confinement on the meson spectrum Ⅱ[J]. Phys.Rev.D,1999,60:094004.
[26]H.J. Wanng, W. T. Geng. Quark Confinement and the Fractional Quantum Hall Effect[J], Chinese Physics C,2008,32:705—709.
[27]C.S.Fischer, D. Nickel and R. Williams. On Gribov's supercriticality picture of quark confinement[J]. Eur.Phys.J.C,2008,60:1434-6052.
[28]T.T. Takahashi, H. Matsufuru and Y. Nemoto et al.. Three-Quark Potential in SU(3) Lattice QCD[J]. Phys.Rev.Lett,2001,86:18-21.
[29]H. Matsufuru, Y. Nemoto and H. Suganuma et al.. Static three quark potential in the quenched lattice QCD[J]. Nucl.Phys.B (Proc. Suppl.),2001,94:554-557.
[30]G.S. Bali, C. Schlichter and K. Schilling. Observing Long Colour Flux Tubes in SU(2) Lattice Gauge Theory[J]. Phys.Rev.D,1995,51:5165-5198.
[31]高崇寿,曾谨言.粒子物理与核物理讲座[M].北京：高等教育出版社.1990,30-37.
[32]H.Yukawa. On the Interaction of Elementary Particles, Ⅰ [J]. Proc.Phys.Math. Soc,1935,17:48-57.
[33]H.Yukawa and S.Sakata. On the Interaction of Elementary Particles, Ⅱ [J]. Proc.Phys.Math. Soc,1937, 19:1084-1093.
[34]C.N.Yang and R.L.Mills. Conservation of Isotopic Spin and Isotopic Gauge Invariance[J]. Phys.Rev, 1954,96:191-195.
[35]薛晓舟.粒子物理学初步[M].郑州：河南科技出版社.1983,6-66.
[36]李双九,赵素英.电荷禁闭和夸克禁闭的相似性与对偶性[J].河北大学学报(自然科学版),2004,24(5)：477-481.
[37]裘忠平.现代量子场论导引[M].武汉：华中师范大学出版社.1992,64-76.
[38]刘健恒译,杨炳麟著.量子场论导引(下册)[M].北京：科学出版社.1988.
[39]L.MICU. Decay rates of meson resonance in a quark model[J]. Nucl,Phys.B,1969,10:521-526.
[40]DING YIBING, CHAO KUANGTA,QIN DANHUA. Possible Effects of Color Screening and Large String Tension in Heavy Quarkonium Spectra[J]. Phys.Rev. D,1995,51:5064-5068.
[41]FRANK E. CLOSE, PHILIP R.PAGE. Do ψ(4040),ψ(4160) signal hybrid charmonium?[J]. Phys.Lett. B,1996,366:323-328.
[42]ZHAO Shu-Min,LIU Li-Quan et al. Study ofψ(4415)in the QPC model[J].Chinese Physics C.2008,32 (10):793-796.
[43]A.Le Yaoua, L.Oliver, O.Pene and J.Raynal. Hadron Transitions in the Quark Model[M]. New York: Gordon and breachscience publishers,1987,99-122.
[44]刘立全,赵树民.ψ(4415)的谐振子状态[J].河北大学学报(自然科学版).2009,29(1)：29-32.
[45]张肇西.强相互作用量子色动力学的渐进自由—2004年诺贝尔物理奖成果简介[J].科技导报,2005,23(2)：17-22.
[46]刘耀阳,江向东.标准模型及其扩充[J].现代物理知识,1997,9(5)：2-5.
[47]魏安赐.弱电统一理论的标准模型.现代物理知识[J],2003,15(6)：6-8.
[48][德]K-Grotz H.V.Klapdor著；朱允伦 裘照明 译；庆承瑞 校.核、粒子和天体物理中的弱相互作用[M].济南：山东科学技术出版社.1997,154-188.
[49]赵树民,刘素玲.标量电子形成的束缚态[J].河北师范大学学报(自然科学版)。2008,32(5)：603-607.
[50]周世勋.量子力学教程[M].北京：高等教育出版社,1979.
[51]曾谨言.量子力学卷2[M].第3版.北京：科学出版社,2000.
[52]Harald Fritzsch, Yu-Feng Zhou. The gluonic B and J/psi decays into the eta' meson[J]. Phys.Rev. D, 2003,68:034015.
[53]Li Gang, Li Tong and Li Xue-Qian. Revisiting the OZI-forbidden Radiative Decays of Orthoquarkonia[J]. Nucl.Phys. B,2005,727:301-317.
[54]Thomas Hakn. LoopTools user's guide [DB/OL]. http://www. feynarts. dellooptools, October 30, 2001.
[55]李佟,赵树民.多点格林函数数值积分费曼参数积分的程序分析及应用[J].南开大学学报(自然科学版).2004,37(4)：75-79.
[56]Barnes T. Charmonium at BES and CLEO-c[J]. arXiv:hep-ph/0406327.
[57]Li Xue-Qian, Liu Xiang, Wei Zheng-Tao. Charm Physics-A Field Full with Challenges and Opportunities[J]. Front.Phys.China,2009,4:49-74.
[2]M. Gell-Mann. A Schematic Model of Baryons and Mesons[J]. Phys. Lett.1964,8:214-215.
[3]J. J. Aubert, U. Becker and P. J. Biggs et al.Experimental Observation of a Heavy Particle J[J]. Phys.Rev.Lett,1974,33:1404-1406.
[4]J. E. Augustin et al. Discovery of a Narrow Resonance in e+e-Annihilation[J]. Phys. Rev.Lett,1974,33: 1406-1408.
[5]G. S. Abrams et al.Discovery of a Second Narrow Resonance in e+e-Annihilation[J]. Phys. Rev.Lett, 1974,33:1453-1455.
[6][美]基本粒子物理专门小组.基本粒子物理学[M].北京：科学出版社.1992,26-31.
[7]C.Amsler, M.Doser and W. M.Yao, et al.. PARTICLE PHYSICS BOOKLIET[M]. Berkeley.2006, 21-66.
[8]王凡.核力和重子相互作用[J].物理学进展,2004,24(1)：1-17.
[9]黄涛.高能物理学所面临的两大难题[J].现代物理知识,2000,12(5)：6-9.
[10]宁平治：核力研究新进展及前沿问题[J].物理学进展,2008,28(4)：432-455.
[11]John H. Schwarz. Dual resonance theory[J]. Phys.Rep.C,1973,8:269-335.
[12]S. Mandelstam. Dual-resonance models[J]. Phys.Rep.C,1974,13:259-353.
[13]W. A. Bardeen, M. S. Chanowitz and S. D. Drell et al.. Heavy quarks and strong binding:A field theory of hadron structure[J]. Phys.Rev.D,1975,11:1094-1136.
[14]A. Chodos, R. L. Jaffe and K. Johnson, et al.. New extended model of hadrons[J]. Phys. Rev.D,1974, 9:3471-3495.
[15]A.Chodos, R.L.Jaffe and K.Johnson et al..Baryon structure in the bag theory[J]. Phys. Rev.D,1974,10: 2599-2604.
[16]李政道.场论和粒子物理学(下册)[M].北京：科学出版社.1981,23-30.
[17]Jihn E. Kim.θ Bag and Quark Confinement[J]. Phys.Lett.B,1992,283:107-112.
[18]Kurt Langfeld. Quark Confinement in a Random Background Gross-Neveu Model[J]. Nucl.Phys.A, 1996,602:375-387.
[19]H.P. Pavel, D. Blaschke and G. Ropke et al.. Squeezed Gluon Condensate and Quark Confinement in the Global Color Model of QCD[J]. Int.J.Mod.Phys.A,1999,14:205-224.
[20]V. Gogokhia. The mass gap and solution of the quark confinement problem in QCD I[J], arXiv:0704.1745; The mass gap and solution of the quark confinement problem in QCD II[J], arXiv:0704.3189.
[21]Theodore J. Allen, M. G. Olsson and Sinisa Veseli et al.. On quark confinement dynamics[J]. Phys.Rev.D,1997,55:5408-5413.
[22]V. N. Gribov. The theory of quark confinement[J]. Eur.Phys.J.C,1999,10:91-105.
[23]C. C. Barros Jr. Quark confinement and curved spaces[J]. Eur.Phys.J.C,2006,45:421-425.
[24]C. B. Compean, M. Kirchbach. Trigonometric quark confinement potential of QCD traits[J]. Eur.Phys.J.A,2007,33:1-4.
[25]Qiao Cong-Feng, Huang Han-Wen and Chao Kuang-Ta. Large Possible retardation effects of quark confinement on the meson spectrum Ⅱ[J]. Phys.Rev.D,1999,60:094004.
[26]H.J. Wanng, W. T. Geng. Quark Confinement and the Fractional Quantum Hall Effect[J], Chinese Physics C,2008,32:705—709.
[27]C.S.Fischer, D. Nickel and R. Williams. On Gribov's supercriticality picture of quark confinement[J]. Eur.Phys.J.C,2008,60:1434-6052.
[28]T.T. Takahashi, H. Matsufuru and Y. Nemoto et al.. Three-Quark Potential in SU(3) Lattice QCD[J]. Phys.Rev.Lett,2001,86:18-21.
[29]H. Matsufuru, Y. Nemoto and H. Suganuma et al.. Static three quark potential in the quenched lattice QCD[J]. Nucl.Phys.B (Proc. Suppl.),2001,94:554-557.
[30]G.S. Bali, C. Schlichter and K. Schilling. Observing Long Colour Flux Tubes in SU(2) Lattice Gauge Theory[J]. Phys.Rev.D,1995,51:5165-5198.
[31]高崇寿,曾谨言.粒子物理与核物理讲座[M].北京：高等教育出版社.1990,30-37.
[32]H.Yukawa. On the Interaction of Elementary Particles, Ⅰ [J]. Proc.Phys.Math. Soc,1935,17:48-57.
[33]H.Yukawa and S.Sakata. On the Interaction of Elementary Particles, Ⅱ [J]. Proc.Phys.Math. Soc,1937, 19:1084-1093.
[34]C.N.Yang and R.L.Mills. Conservation of Isotopic Spin and Isotopic Gauge Invariance[J]. Phys.Rev, 1954,96:191-195.
[35]薛晓舟.粒子物理学初步[M].郑州：河南科技出版社.1983,6-66.
[36]李双九,赵素英.电荷禁闭和夸克禁闭的相似性与对偶性[J].河北大学学报(自然科学版),2004,24(5)：477-481.
[37]裘忠平.现代量子场论导引[M].武汉：华中师范大学出版社.1992,64-76.
[38]刘健恒译,杨炳麟著.量子场论导引(下册)[M].北京：科学出版社.1988.
[39]L.MICU. Decay rates of meson resonance in a quark model[J]. Nucl,Phys.B,1969,10:521-526.
[40]DING YIBING, CHAO KUANGTA,QIN DANHUA. Possible Effects of Color Screening and Large String Tension in Heavy Quarkonium Spectra[J]. Phys.Rev. D,1995,51:5064-5068.
[41]FRANK E. CLOSE, PHILIP R.PAGE. Do ψ(4040),ψ(4160) signal hybrid charmonium?[J]. Phys.Lett. B,1996,366:323-328.
[42]ZHAO Shu-Min,LIU Li-Quan et al. Study ofψ(4415)in the QPC model[J].Chinese Physics C.2008,32 (10):793-796.
[43]A.Le Yaoua, L.Oliver, O.Pene and J.Raynal. Hadron Transitions in the Quark Model[M]. New York: Gordon and breachscience publishers,1987,99-122.
[44]刘立全,赵树民.ψ(4415)的谐振子状态[J].河北大学学报(自然科学版).2009,29(1)：29-32.
[45]张肇西.强相互作用量子色动力学的渐进自由—2004年诺贝尔物理奖成果简介[J].科技导报,2005,23(2)：17-22.
[46]刘耀阳,江向东.标准模型及其扩充[J].现代物理知识,1997,9(5)：2-5.
[47]魏安赐.弱电统一理论的标准模型.现代物理知识[J],2003,15(6)：6-8.
[48][德]K-Grotz H.V.Klapdor著；朱允伦 裘照明 译；庆承瑞 校.核、粒子和天体物理中的弱相互作用[M].济南：山东科学技术出版社.1997,154-188.
[49]赵树民,刘素玲.标量电子形成的束缚态[J].河北师范大学学报(自然科学版)。2008,32(5)：603-607.
[50]周世勋.量子力学教程[M].北京：高等教育出版社,1979.
[51]曾谨言.量子力学卷2[M].第3版.北京：科学出版社,2000.
[52]Harald Fritzsch, Yu-Feng Zhou. The gluonic B and J/psi decays into the eta' meson[J]. Phys.Rev. D, 2003,68:034015.
[53]Li Gang, Li Tong and Li Xue-Qian. Revisiting the OZI-forbidden Radiative Decays of Orthoquarkonia[J]. Nucl.Phys. B,2005,727:301-317.
[54]Thomas Hakn. LoopTools user's guide [DB/OL]. http://www. feynarts. dellooptools, October 30, 2001.
[55]李佟,赵树民.多点格林函数数值积分费曼参数积分的程序分析及应用[J].南开大学学报(自然科学版).2004,37(4)：75-79.
[56]Barnes T. Charmonium at BES and CLEO-c[J]. arXiv:hep-ph/0406327.
[57]Li Xue-Qian, Liu Xiang, Wei Zheng-Tao. Charm Physics-A Field Full with Challenges and Opportunities[J]. Front.Phys.China,2009,4:49-74.