夸克物质色超导的复能隙函数和角动量混合效应研究
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摘要
库珀理论告诉我们,只要费米面附近有净吸引相互作用,系统将不再稳定,两个费米子将形成束缚对,即库珀对。在渐近自由的密度下,QCD的单胶子交换相互作用可以被分成反对称的三重态与对称的六重态,其中前者提供吸引相互作用。在中等密度下,交换瞬子的相互作用也是吸引相互作用,因此冷密夸克物质的基态不再是简单的Fermi球分布,极有可能发生夸克配对从而形成所谓的色超导态,由此导致冷密夸克物质的一系列非Fermi流体行为。本文利用渐近密度下QCD的微扰性质,探讨了夸克物质色超态的复能隙函数以及一味夸克色超导中的角动量混合效应。因此在第一章,我们简要地介绍了QCD的拉氏量及其基本对称性,并就我们目前所了解的信息对QCD的相图作了简单的描述。
在第二章,作为了解色超导理论的基础,我们介绍了电磁超导的BCS理论,其中所涉及的许多概念在色超导理论中同样会被使用。夸克不同于电子,除了带有电荷之外还有非阿贝尔的色和昧自由度,因此色超导具有非常丰富的相结构。在这一章,我们主要介绍了2SC、CFL以及一味夸克的色超导态,分析了它们的配对类型以及对称性。除此之外,我们还介绍了其它一些非BSC型的色超导态,如gapless态、LOFF态和BP态等。在自然界中,中子星的内部极有可能出现色超导态。因此,本章的最后一节简单要介绍了与色超导研究有关的中子星的质量半径关系和冷却率。
第三章我们计算了色超导的复能隙函数。硬密圈(HDL)胶子传播子的色电部分被Deybe屏蔽,其相互作用长度与Debye质量成反比,但磁相互作用只是被动力学屏蔽(Landau屏蔽),因此仍然是长程相互作用。考虑到长程的磁相互作用,色超导能隙函数会不同于普通BCS理论的预言,由正比于e~(-c/g~2)变成正比于e~(-c/g)。另外,Landau屏蔽也将导致HDL胶子传播子在复能量平面的实轴上没有定义。因此,色超导的能隙函数在实轴附近将产生一个非零的虚部,也就意味着能隙函数是复数。我们采用电磁超导的Eliashberg理论,自洽地求解了实能量的能隙函数实部和虚部。在具体计算之前,我们考查了宇称和时间反演不变的色超导系统的复能隙函数对能量和动量的依赖关系,并总结得到了分别适用于实能量和虚能量公式的两个定理。该定理在解释有关准粒子极点的问题时有很强的指导作用。由于QCD的微扰性质,在我们所需要的精度范围内,并不用像真正的Eliashberg理论一样通过解一组耦合方程而同时得到实部和虚部。精确到次领头阶时,我们可以忽略虚部对实部的贡献,单独解实部的能隙方程就足够了。在第二节中,我们从一个常数的能隙方程出发利用迭代法求解非线性的实部的能隙方程,其中每一次迭代都转化成一个本征值问题,并利用QCD的微扰性质求解其本征值与本征函数。我们发现只需要迭代一次就足于得到精确到次领头阶的实能量的能隙函数的实部,并且我们的结果与文献中得到的虚能量的实部或者准粒子能量在质壳上的实部都一致。领头阶的能隙函数的虚部也同样只需要一个能隙方程即可得到,这就是虚能量能隙方程的解析延拓。我们发现能隙函数虚部比实部压低一个耦合常数因子,因此对实部的贡献只是次次领头阶。最后,我们讨论了几个与能隙函数虚部有关的问题,尤其是准粒子的极点。
第四章我们讨论了一味夸克色超导中的角动量混合效应。由于泡利不相容原理的限制,一味夸克库珀对的角动量必须大于零(J＞0),类似于~3He超流理论选取库珀对的角动量J=1的态即为文献中的Spin-1色超导。它有四种主要的相结构,即Polar,Planar,A以及CSL态。如果只考虑纵向与横向配对,那么这四种态中仅有CSL态的能隙函数是各向同性,其它三种态的能隙函数都是非各向同性,我们把前者叫球形态而后者被称为非球形态。QCD的单胶子交换势却包含所有分波的贡献,并且由于夸克之间的单胶子散射振幅存在向前散射奇异性,所有分波的贡献在领头阶都相等,只是在次领头阶才逐渐减小。因此,原则上来说,非球形的能隙函数也应该包含所有的分波,并且由于QCD能隙方程的非线性性质,所有的分波都会混合在一起,我们把这个现象叫做角动量混合效应。我们首先利用一个非相对论的简单模型讨论了角动量混合的机制,结果表明当配对势中包含所有分波时,能隙函数不能只限制在一个分波,并且由于能隙方程的非线性性质,所有分波发生了混合。接着在第二节利用CJT有效作用量,我们详细考查了非球形的Polar,Planar以及A态。我们发现角动量混合确实在非球形态中发生了,非球形态的能隙函数将要被修正,在其分波展开中所有角动量都有贡献,不过同时我们也发现由于单胶子交换势的次领头阶效应,高角动量的分波对能隙函数的贡献非常小,而J=1的分波占据了绝大部分的贡献。不仅如此,角动量混合还降低了Spin-1色超导中非球形态的自由能,但是角动量混合使自由能降低的程度太小,不足于使非球形态的自由能比球形CSL态的自由能更低。即使是自由能与CSL态相差非常小的Planar态(仅相差两个百分点),角动量混合也不足于使其自由能低于CSL态,由此我们猜测,不论是横向还是纵向配对中,角动量混合的非球形态都不会与球形的CSL态更稳定。在第二节的最后部分,我们通过简单的但却严格的证明证实了我们的猜测。最后一节我们初步地分析了有可能改变一味夸克色超导基态的两种情况,即夸克的质量和强磁场。
第五章是对本论文主要工作的总结,并对未来可能涉及到的工作进行了展望。为了避免论文主体内容的烦琐,我们将一些公式的推导细节和证明放在了附录中以方便查阅。
Cooper's theorem implies that if there is an attractive interaction in a cold Fermi sea,the system is unstable with respect to the formation of a particle-particle condensate.In QCD case at asymptotically high density,single-gluon exchange can be decomposed into a antisymmetic color antitriplet and a symmetric color sextet,the former one provides the attractive interaction.As for the moderate density,the interaction between quarks induced by instanton is also attractive. Thus,it is unavoidable that cold dense quark matter is a color superconductor,which in turn will generate a lot of non-Fermi liquid behaviors.In this dissertation,we employed the perturbative properties of QCD in weak coupling to investigate the complex gap function and the angular momentum mixing in color superconductivity.In the first chapter,we briefly introduced the basics of QCD lagrangian and its main symmetries as well as the QCD phase diagram of our present knowledges.
In chapter two,as a first step of understanding color superconductivity(CSC),we reviewed the BCS theory in ordinary superconductivity,many definitions of which will also be used in CSC.Since quarks are not only electrically charged but also carry non-abelian color and flavor, CSC has a more rich phase structure than that of the ordinary superconductivity.Within this chapter,2SC,CFL phase and the single flavor CSC have been discussed in detail.We also briefly reviewed some exotic states regarding the presence of the Fermi momentum mismatch,such as gapless CSC,LOFF and BP state.In nature,the core of compact stars is very likely to be a color superconductor.Therefore,the mass-radius relation and the cooling rate of neutron stars have been discussed as applications of CSC at the end of this chapter.
In chapter three,we calculated the complex gap function of CSC.The HDL gluon propagator is effectively screened by a Debye mass m_D in its electric part,while the magnetic part is poorly screened via Landau damping and thus it is still a long range interaction.Taking into account the long-range magnetic interaction,the leading order of gap is proportional to exp(-c/g).This behavior is different from the naive expectation from BSC theory,which predicts exp(-c/g~2). In addition,the general pairing potential containing landau damping has a branch cut along the real axis of the complex energy plane.Consequently,the gap function acquires a nontrivial imaginary part along the axis of real energy,which means the gap is complex.Before embarking on the calculation of the complex gap,we clarified some general properties of the gap regarding its functional dependence on the energy and momentum dictated by the invariance under a space inversion or a time reversal.With these properties,we solved some confusion regarding the analytic continuation of the quasi-particle pole from Matsubara energy to real energy.Eliashberg theory formulated for an electronic superconductor of strong pairing force regards both the complex gap and the quai-particle weight.They are determined at equal footing from a pair of self consistent equations of the electron self energy.For QCD at asymptotic density,however, the full complexity of the Eliashberg equations is unnecessary.In the leading approximation,we may ignore the imaginary part in the gap equation of the real part.In section two,we solved the nonlinear gap equation by iterations starting with a constant gap.In each step of iterations,the integral equation defines an eigenvalue problem,which can be analyzed with the perturbation method developed in literatures.We found only the first iteration is required for our purpose. To determine the imaginary part in leading order,we also need only one Eliashberg equation, which is the analytic continuation from the gap equation of Euclidean energy.We found that the imaginary part of the gap is down by g relative to the real part and is therefore corresponds to the sub-sub-leading contribution to the complex gap function.At the end of this chapter,we discussed some questions regarding the imaginary part,especially the pole of the quasi-particle.
Chapter four was devoted to the angular momentum mixing in single flavor CSC.Cooper pair in single flavor pairing should be implemented at a higher total angular momentum as required by Pauli principle.The obvious choice is the p-wave pairing analogous to ~3He superfluidity,which had been called Spin-1 CSC in literatures.There are four main phases in single flavor pairing correspond to different pairing patterns,except for the color-spin-locked(CSL) phase,the other gaps including polar,A and planar are non-spherical.In QCD,the pairing potential mediated by one-gluon exchange contains all partial waves and they are equal to the leading order because of the forward singularity.Therefore,in principle,the non-spherical gaps can not be restricted in a single non-s-wave channel and the mixing among different channels will occur because of the nonlinearity of the gap equation.This is what we called angular momentum mixing.In the first section,we clarified the concept of the angular momentum mixing with a toy model of non-relativistic and spinless fermion.In the next section,we started from the CJT effective action to examine the angular momentum mixing in non-spherical polar,A and planar phases in single flavor CSC.We found that the mixture of angular momenta indeed occured and all non-spherical phases would be modified by including all partial waves,although the contribution from higher angular momentum was small because the pairing strength of each partial wave fell off with increasing J in sub-leading order.The free energy would be brought down by angular momentum mixing compared with that contained p-wave only.However,the drop amount of the free energy was too small to favor the non-spherical pairing.Even the transverse planar phase,which exhibited high potential to be able to compete CSL phase since the gain of the condensation energy from the former to the latter was only two percent falling within the range of the percentage increment by angular momentum mixing,can not become the favored state by including the mixing.Therefore,we conjecture that angular momentum mixing of various nonspherical CSC is not sufficient to compete with the CSL state energetically in ultra relativistic limit.A rigorous proof was presented at the end of this section.In the last section,we proposed two possible mechanisms,i.e.s quark mass and the strong magnetic field,which may offset the energy balance between non-spherical states and CSL.
Chapter five is our concluding remarks and outlooks.Some technical details in the calculations had been deferred to the appendix in order to avoiding the complexity of the main part of this thesis.However,it is convenient for the interesting readers to refer to.
在第二章,作为了解色超导理论的基础,我们介绍了电磁超导的BCS理论,其中所涉及的许多概念在色超导理论中同样会被使用。夸克不同于电子,除了带有电荷之外还有非阿贝尔的色和昧自由度,因此色超导具有非常丰富的相结构。在这一章,我们主要介绍了2SC、CFL以及一味夸克的色超导态,分析了它们的配对类型以及对称性。除此之外,我们还介绍了其它一些非BSC型的色超导态,如gapless态、LOFF态和BP态等。在自然界中,中子星的内部极有可能出现色超导态。因此,本章的最后一节简单要介绍了与色超导研究有关的中子星的质量半径关系和冷却率。
第三章我们计算了色超导的复能隙函数。硬密圈(HDL)胶子传播子的色电部分被Deybe屏蔽,其相互作用长度与Debye质量成反比,但磁相互作用只是被动力学屏蔽(Landau屏蔽),因此仍然是长程相互作用。考虑到长程的磁相互作用,色超导能隙函数会不同于普通BCS理论的预言,由正比于e~(-c/g~2)变成正比于e~(-c/g)。另外,Landau屏蔽也将导致HDL胶子传播子在复能量平面的实轴上没有定义。因此,色超导的能隙函数在实轴附近将产生一个非零的虚部,也就意味着能隙函数是复数。我们采用电磁超导的Eliashberg理论,自洽地求解了实能量的能隙函数实部和虚部。在具体计算之前,我们考查了宇称和时间反演不变的色超导系统的复能隙函数对能量和动量的依赖关系,并总结得到了分别适用于实能量和虚能量公式的两个定理。该定理在解释有关准粒子极点的问题时有很强的指导作用。由于QCD的微扰性质,在我们所需要的精度范围内,并不用像真正的Eliashberg理论一样通过解一组耦合方程而同时得到实部和虚部。精确到次领头阶时,我们可以忽略虚部对实部的贡献,单独解实部的能隙方程就足够了。在第二节中,我们从一个常数的能隙方程出发利用迭代法求解非线性的实部的能隙方程,其中每一次迭代都转化成一个本征值问题,并利用QCD的微扰性质求解其本征值与本征函数。我们发现只需要迭代一次就足于得到精确到次领头阶的实能量的能隙函数的实部,并且我们的结果与文献中得到的虚能量的实部或者准粒子能量在质壳上的实部都一致。领头阶的能隙函数的虚部也同样只需要一个能隙方程即可得到,这就是虚能量能隙方程的解析延拓。我们发现能隙函数虚部比实部压低一个耦合常数因子,因此对实部的贡献只是次次领头阶。最后,我们讨论了几个与能隙函数虚部有关的问题,尤其是准粒子的极点。
第四章我们讨论了一味夸克色超导中的角动量混合效应。由于泡利不相容原理的限制,一味夸克库珀对的角动量必须大于零(J＞0),类似于~3He超流理论选取库珀对的角动量J=1的态即为文献中的Spin-1色超导。它有四种主要的相结构,即Polar,Planar,A以及CSL态。如果只考虑纵向与横向配对,那么这四种态中仅有CSL态的能隙函数是各向同性,其它三种态的能隙函数都是非各向同性,我们把前者叫球形态而后者被称为非球形态。QCD的单胶子交换势却包含所有分波的贡献,并且由于夸克之间的单胶子散射振幅存在向前散射奇异性,所有分波的贡献在领头阶都相等,只是在次领头阶才逐渐减小。因此,原则上来说,非球形的能隙函数也应该包含所有的分波,并且由于QCD能隙方程的非线性性质,所有的分波都会混合在一起,我们把这个现象叫做角动量混合效应。我们首先利用一个非相对论的简单模型讨论了角动量混合的机制,结果表明当配对势中包含所有分波时,能隙函数不能只限制在一个分波,并且由于能隙方程的非线性性质,所有分波发生了混合。接着在第二节利用CJT有效作用量,我们详细考查了非球形的Polar,Planar以及A态。我们发现角动量混合确实在非球形态中发生了,非球形态的能隙函数将要被修正,在其分波展开中所有角动量都有贡献,不过同时我们也发现由于单胶子交换势的次领头阶效应,高角动量的分波对能隙函数的贡献非常小,而J=1的分波占据了绝大部分的贡献。不仅如此,角动量混合还降低了Spin-1色超导中非球形态的自由能,但是角动量混合使自由能降低的程度太小,不足于使非球形态的自由能比球形CSL态的自由能更低。即使是自由能与CSL态相差非常小的Planar态(仅相差两个百分点),角动量混合也不足于使其自由能低于CSL态,由此我们猜测,不论是横向还是纵向配对中,角动量混合的非球形态都不会与球形的CSL态更稳定。在第二节的最后部分,我们通过简单的但却严格的证明证实了我们的猜测。最后一节我们初步地分析了有可能改变一味夸克色超导基态的两种情况,即夸克的质量和强磁场。
第五章是对本论文主要工作的总结,并对未来可能涉及到的工作进行了展望。为了避免论文主体内容的烦琐,我们将一些公式的推导细节和证明放在了附录中以方便查阅。
Cooper's theorem implies that if there is an attractive interaction in a cold Fermi sea,the system is unstable with respect to the formation of a particle-particle condensate.In QCD case at asymptotically high density,single-gluon exchange can be decomposed into a antisymmetic color antitriplet and a symmetric color sextet,the former one provides the attractive interaction.As for the moderate density,the interaction between quarks induced by instanton is also attractive. Thus,it is unavoidable that cold dense quark matter is a color superconductor,which in turn will generate a lot of non-Fermi liquid behaviors.In this dissertation,we employed the perturbative properties of QCD in weak coupling to investigate the complex gap function and the angular momentum mixing in color superconductivity.In the first chapter,we briefly introduced the basics of QCD lagrangian and its main symmetries as well as the QCD phase diagram of our present knowledges.
In chapter two,as a first step of understanding color superconductivity(CSC),we reviewed the BCS theory in ordinary superconductivity,many definitions of which will also be used in CSC.Since quarks are not only electrically charged but also carry non-abelian color and flavor, CSC has a more rich phase structure than that of the ordinary superconductivity.Within this chapter,2SC,CFL phase and the single flavor CSC have been discussed in detail.We also briefly reviewed some exotic states regarding the presence of the Fermi momentum mismatch,such as gapless CSC,LOFF and BP state.In nature,the core of compact stars is very likely to be a color superconductor.Therefore,the mass-radius relation and the cooling rate of neutron stars have been discussed as applications of CSC at the end of this chapter.
In chapter three,we calculated the complex gap function of CSC.The HDL gluon propagator is effectively screened by a Debye mass m_D in its electric part,while the magnetic part is poorly screened via Landau damping and thus it is still a long range interaction.Taking into account the long-range magnetic interaction,the leading order of gap is proportional to exp(-c/g).This behavior is different from the naive expectation from BSC theory,which predicts exp(-c/g~2). In addition,the general pairing potential containing landau damping has a branch cut along the real axis of the complex energy plane.Consequently,the gap function acquires a nontrivial imaginary part along the axis of real energy,which means the gap is complex.Before embarking on the calculation of the complex gap,we clarified some general properties of the gap regarding its functional dependence on the energy and momentum dictated by the invariance under a space inversion or a time reversal.With these properties,we solved some confusion regarding the analytic continuation of the quasi-particle pole from Matsubara energy to real energy.Eliashberg theory formulated for an electronic superconductor of strong pairing force regards both the complex gap and the quai-particle weight.They are determined at equal footing from a pair of self consistent equations of the electron self energy.For QCD at asymptotic density,however, the full complexity of the Eliashberg equations is unnecessary.In the leading approximation,we may ignore the imaginary part in the gap equation of the real part.In section two,we solved the nonlinear gap equation by iterations starting with a constant gap.In each step of iterations,the integral equation defines an eigenvalue problem,which can be analyzed with the perturbation method developed in literatures.We found only the first iteration is required for our purpose. To determine the imaginary part in leading order,we also need only one Eliashberg equation, which is the analytic continuation from the gap equation of Euclidean energy.We found that the imaginary part of the gap is down by g relative to the real part and is therefore corresponds to the sub-sub-leading contribution to the complex gap function.At the end of this chapter,we discussed some questions regarding the imaginary part,especially the pole of the quasi-particle.
Chapter four was devoted to the angular momentum mixing in single flavor CSC.Cooper pair in single flavor pairing should be implemented at a higher total angular momentum as required by Pauli principle.The obvious choice is the p-wave pairing analogous to ~3He superfluidity,which had been called Spin-1 CSC in literatures.There are four main phases in single flavor pairing correspond to different pairing patterns,except for the color-spin-locked(CSL) phase,the other gaps including polar,A and planar are non-spherical.In QCD,the pairing potential mediated by one-gluon exchange contains all partial waves and they are equal to the leading order because of the forward singularity.Therefore,in principle,the non-spherical gaps can not be restricted in a single non-s-wave channel and the mixing among different channels will occur because of the nonlinearity of the gap equation.This is what we called angular momentum mixing.In the first section,we clarified the concept of the angular momentum mixing with a toy model of non-relativistic and spinless fermion.In the next section,we started from the CJT effective action to examine the angular momentum mixing in non-spherical polar,A and planar phases in single flavor CSC.We found that the mixture of angular momenta indeed occured and all non-spherical phases would be modified by including all partial waves,although the contribution from higher angular momentum was small because the pairing strength of each partial wave fell off with increasing J in sub-leading order.The free energy would be brought down by angular momentum mixing compared with that contained p-wave only.However,the drop amount of the free energy was too small to favor the non-spherical pairing.Even the transverse planar phase,which exhibited high potential to be able to compete CSL phase since the gain of the condensation energy from the former to the latter was only two percent falling within the range of the percentage increment by angular momentum mixing,can not become the favored state by including the mixing.Therefore,we conjecture that angular momentum mixing of various nonspherical CSC is not sufficient to compete with the CSL state energetically in ultra relativistic limit.A rigorous proof was presented at the end of this section.In the last section,we proposed two possible mechanisms,i.e.s quark mass and the strong magnetic field,which may offset the energy balance between non-spherical states and CSL.
Chapter five is our concluding remarks and outlooks.Some technical details in the calculations had been deferred to the appendix in order to avoiding the complexity of the main part of this thesis.However,it is convenient for the interesting readers to refer to.
引文
[1]D.J.Gross and F.Wilczek,Phys.Rev.Lett.30,1343(1973);H.D.Politzer,Phys.Rev.Lett.30,1346(1973)
[2]J.C.Collins and M.J.Perry,Phys.Rev.Lett.34,1353(1975)
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