对称性与中微子混合模型
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摘要
中微子是Pauli为了解释β衰变连续谱而引入的,最初被认为是一种没有质量的中性费米子。然而Pontecorvo和Gribov提出,如果中微子是有质量的,并且不同质量的中微子之间存在混合,那么中微子的传播就会带来味振荡。一系列中微子实验,如Homestake、Kamiokande等,证实了中微子存在味振荡现象,并且测定了中微子混合参数θ12与θ23以及质量参数△m2Aσm与△m2sol。非零的混合角θ13也逐步得到了实验的确认。
中微子味振荡现象的发现表明中微子具有微小的质量,这在标准模型的框架内是无法解释的。作为一种唯象学假设,seesaw机制引入了大质量的右手中微子来压低中微子的质量标度。但是目前还没有发现大质量右手中微子的显著证据。即使我们接受seesaw机制对中微子微小质量标度的解释,由于中微子具有质量以及不同代中微子之间存在混合,我们需要对标准模型中涉及中微子的概念和定律进行新的检查。味态、能量-动量守恒、含时微扰等概念、定律和方法在中微子混合、中微子振荡的背景下变得不再简单、平庸。首先,由于中微子味荷的时间依赖性,普适的中微子味态是难以定义的,我们只能在有限的、近似的条件下使用味态这一概念。为了避免理论争议,在讨论中微子振荡问题时人们甚至可以避开味态,将中微子视为中间传播子。其次,为了协调弱作用中代轻子数守恒与中微子传播中代轻子数破坏,通行的含时微扰论在应用到涉及中微子反应的概率计算时会受到限制。在涉及中微子反应的概率幅计算中,含时微扰的时间标度必须远小于1S,否则会带来弱作用过程中显著的味破坏,从而违反微观因果律。再次,由于中微子能量、动量存在不确定度,通常的能量-动量守恒约束并不带来中微子与伴随粒子的运动学纠缠,运动学纠缠只是特定实验条件和观测的结果；另一方面,对称性分析表明,味振荡对中微子能量、动量的不确定度下限也提出了要求,不仅是中微子动量的不确定度,中微子能量的不确定度也存在下限,提高中微子动量或能量的测量精度会破坏中微子的味振荡。
标准模型中的规范对称群无法解释中微子振荡模式与中微子混合模式,分立的非阿贝尔味对称群因此被引入了中微子模型,这些分立对称群如A4、S4、T’等可以解释典型的中微子混合模式,如三双最大中微子混合。随着实验上对非零混合角013的确证,特别是大亚湾中微子实验的显著结果,这些分立味对称群模型需要进行修正和微扰处理。然而,即使这些模型有着较好的理论解释能力,并且可以对实验上不确定的中微子混合参数与质量参数给出限制,但是模型中的标量场真空期望值需要人为的微调,因而其自然性受到了质疑。更为经济的模型构造方案是,避开希格斯机制与额外的标量场,直接定义满足特定分立对称性的中微子质量矩阵,通过质量矩阵的对角化导出期望的中微子混合矩阵。这样的策略虽然避开了标量场真空期望值的人为微调,但是得出同一混合模式的分立对称群并不是唯一的,我们无法确认哪一种对称群才是真正的味对称群。因此,对中微子常规混合模式的理论解释还有待于更为深入的研究。
除了常规的中微子混合模式,一系列中微子振荡实验发现了反常的振荡信号。有些实验信号,如MINOS实验,表明中微子的振荡可能不同于反中微子的振荡。另一些实验信号,如LSND、MiniBooNE实验,表明除了太阳中微子与大气中微子振荡质量标度,还可能存在更大的中微子振荡质量标度：Δmew～1eV2。作为一种可能的理论解释,通过引进Lorentz对称破缺和CPT破坏,中微子与反中微子的振荡差异可以得到初步解释；而且由于Lorentz对称破缺模型中混合矩阵元的能量依赖性,可以对部分的反常振荡提供试探性的解释。然而,由于Lorentz对称破缺的引入,不仅使得混合矩阵元是能量依赖的,同时中微子能量-动量色散关系也需要作出修正,即可能出现超光速中微子。除了MINOS合作组与OPERA合作组的报道以外,目前各种中微子实验结果表明并不存在显著的中微子超光速事例。特别是,最近OPERA小组自身也声明了其实验中的缺陷。因此,理论上可行的Lorentz对称破缺模型对中微子能量-动量色散关系的修正必须是不显著的。在无超光速中微子的框架下,检验Lorentz对称破缺模型的方法主要集中在中微子混合矩阵元的能量依赖关系上。目前实验对Lorentz对称破缺中微子混合模型的约束是非常严格的,但是鉴于这些约束的模型依赖性,构造满足实验约束的Lorentz对称破缺混合模型仍然是可能的。
总的说来,从对称性角度讨论中微子振荡问题是十分重要的研究方法,对于中微子混合模型的构造有着很好的启发作用。相信在以后的中微子唯象学研究中,对称性分析仍将是一种有效的方法。
In order to interpret the continuous energy spectra of the β decay, Pauli proposed the existence of the neutrino, which was considered as the neutral massless fermion. However, Pontecorvo and Gribov proposed that if the neutrino is massive and there is the mixing of the different massive neutrinos the propagation of the neutrinos may bring the flavor oscillation. A variety of the neutrino experiments, such as the Homestake, Kamiokande and etc, have confirmed the flavor oscillation of the neutrinos and determined the mixing parameters θ12,θ23and the mass parameters ΔmAm2, ΔmSlo2. The nonzero mixing angle θ13is being confirmed by the experiments.
The flavor oscillation of the neutrinos reveals that the mass scale of the neutrino is tiny, which can not be interpreted in the standard model of the particle physics. As a phenomenological assumption, the seesaw mechanism introduces the right-handed neutrino with the large mass scale to depress the mass scale of the neutrino. There is no obvious evidences for the existence of the right-handed neutrino with the large mass scale. Even though we accept the seesaw mechanism as the interpretation of the tiny mass of the neutrino, the introductions of the neutrino mass and the mixing of the massive neutrinos ask for the new examinations of the familiar concepts and laws in the standard model. The concept of the flavor state, the conservation law of the energy-momentum and the time-dependent perturbation theory become non-simple and nontrivial in the background of the neutrinos mixing and the neutrinos oscillation. First, because of the time-dependence of the flavor charge of the neutrino, it is difficult to define the universal flavor state. We can only employ the flavor state with the limited and the approximate conditions. One can even avoid the flavor state for less theoretical controversies and treat the neutrino as the intermedial particle in the discussion of the neutrinos oscillation. Second, in order to coordinate the flavor conservation in the weak interaction and the flavor violation in the neutrino propagation, the conventional time-dependent perturbation theory is limited in the calculation of the reaction probability where the neutrino is involved. In the calculation of the probability involving the neutrino, the time scale in the time-dependent perturbation theory should be much less than Is. Otherwise, it will bring the obvious flavor violation in the weak interaction which is in contradiction with the micro-causality. Third, because of the uncertainty of the energy-momentum of the neutrino, the conventional constraint of the conservation of the energy-momentum does not mean the kinetic entanglement of the neutrino and the recoiling particle. The kinetic entanglement is the result of the special experiment design and the special detection. On the other hand, the analysis of the symmetry demonstrates that the flavor oscillation involves the limit of the uncertainties of the energy and the momentum of the neutrino. Both the momentum and the energy are uncertain for the flavor neutrino. The improvement of the resolution of the measurement of the momentum or the energy can destroy the flavor oscillation of the neutrino.
The pattern of the neutrino oscillation and the neutrino mixing can not be interpreted with the conventional symmetry group in the standard model. Thus, the discrete non-Abel flavor symmetry group is introduced in the neutrino model. These discrete group such as the A4, S4, T'and so on can interpret the classical mixing pattern, namely the tribimaximal mixing pattern. With the confirmation of the nonzero mixing angle θ13by the experiments, especially the convincing results of the neutrinos experiments at the Daya Bay, the modification or the perturbation is needed in these models with the discrete flavor group. However, even though these models can interpret the conventional mixing pattern and give the constraints on the undetermined mixing parameters and the mass parameters, there is the special fine-tuning of the vacuum expected value of the scalar field in these models. The naturalness of these model is not convincing. As a more economical procedure of the model construction, one can introduce the mass matrix of the neutrinos directly without the employment of the Higss mechanism and the extra scalar fields. The mass matrix satisfies the special discrete flavor symmetry. By diagonalization of the mass matrix, we can obtain the expected mixing matrix of the neutrinos. Although this method avoids the fine-tuning of the scalar field, the discrete symmetry group that induces the same mixing matrix is not unique. We can not discriminate these different symmetry groups. So lots of work is needed to deepen the interpretation of the neutrino mixing pattern.
Besides the conventional mixing pattern, some anomalous oscillation signals were reported in the experiments. Some experiments signals, such as that of the MINOS, reveal that the oscillation of the neutrino is different to that of the anti-neutrino. Other signals, such as that of the LSND and MiniBooNE, reveal that there is the extra large mass scale of the order leV2besides the tiny mass scales in the solar neutrino and the atmosphere neutrino. As a plausible interpretation, the introduction of the Lorentz invariance violation and the CPT violation can illustrate the difference of the oscillation of the neutrinos and the anti-neutrinos. Further more, because of the energy-dependence of the mixing matrix element in the model of the Lorentz invariance violation, some of the other anomalous oscillation can be interpreted. However, because of the introduction of the Lorentz invariance violation, we should modify the dispersion relation of the neutrino besides the energy-dependence mixing element, which means there may be superluminal neutrinos. Except the MINOS experiment and the OPERA experiment, kinds of the neutrinos experiments by far reveal that there is no obvious event of the superluminal neutrinos. In particular, very recently, the OPERA collaboration has claimed the flaw in their experiments. Therefore, the plausible neutrino model with the Lorentz violation should satisfies the condition that the modification of the dispersion relation is not obvious compared with the conventional relation. In the regime without the superluminal neutrinos, the method testing the Lorentz-violation models focuses on the energy-dependence of the mixing element of the neutrinos. The constraints on the Lorentz-violation models by far are very strict. However, because of the model-dependence of the constraints, it is still possible to construct the viable Lorentz-violation models of the neutrinos.
Generally speaking, the symmetry analysis is one of the most important methods in the studying of the phenomenology of the neutrino oscillation, which is enlightening in the construction of the mixing models of the neutrinos. It is believed that the symmetry analysis will be still an effective method in the future phenomenological researches of the neutrinos.
中微子味振荡现象的发现表明中微子具有微小的质量,这在标准模型的框架内是无法解释的。作为一种唯象学假设,seesaw机制引入了大质量的右手中微子来压低中微子的质量标度。但是目前还没有发现大质量右手中微子的显著证据。即使我们接受seesaw机制对中微子微小质量标度的解释,由于中微子具有质量以及不同代中微子之间存在混合,我们需要对标准模型中涉及中微子的概念和定律进行新的检查。味态、能量-动量守恒、含时微扰等概念、定律和方法在中微子混合、中微子振荡的背景下变得不再简单、平庸。首先,由于中微子味荷的时间依赖性,普适的中微子味态是难以定义的,我们只能在有限的、近似的条件下使用味态这一概念。为了避免理论争议,在讨论中微子振荡问题时人们甚至可以避开味态,将中微子视为中间传播子。其次,为了协调弱作用中代轻子数守恒与中微子传播中代轻子数破坏,通行的含时微扰论在应用到涉及中微子反应的概率计算时会受到限制。在涉及中微子反应的概率幅计算中,含时微扰的时间标度必须远小于1S,否则会带来弱作用过程中显著的味破坏,从而违反微观因果律。再次,由于中微子能量、动量存在不确定度,通常的能量-动量守恒约束并不带来中微子与伴随粒子的运动学纠缠,运动学纠缠只是特定实验条件和观测的结果；另一方面,对称性分析表明,味振荡对中微子能量、动量的不确定度下限也提出了要求,不仅是中微子动量的不确定度,中微子能量的不确定度也存在下限,提高中微子动量或能量的测量精度会破坏中微子的味振荡。
标准模型中的规范对称群无法解释中微子振荡模式与中微子混合模式,分立的非阿贝尔味对称群因此被引入了中微子模型,这些分立对称群如A4、S4、T’等可以解释典型的中微子混合模式,如三双最大中微子混合。随着实验上对非零混合角013的确证,特别是大亚湾中微子实验的显著结果,这些分立味对称群模型需要进行修正和微扰处理。然而,即使这些模型有着较好的理论解释能力,并且可以对实验上不确定的中微子混合参数与质量参数给出限制,但是模型中的标量场真空期望值需要人为的微调,因而其自然性受到了质疑。更为经济的模型构造方案是,避开希格斯机制与额外的标量场,直接定义满足特定分立对称性的中微子质量矩阵,通过质量矩阵的对角化导出期望的中微子混合矩阵。这样的策略虽然避开了标量场真空期望值的人为微调,但是得出同一混合模式的分立对称群并不是唯一的,我们无法确认哪一种对称群才是真正的味对称群。因此,对中微子常规混合模式的理论解释还有待于更为深入的研究。
除了常规的中微子混合模式,一系列中微子振荡实验发现了反常的振荡信号。有些实验信号,如MINOS实验,表明中微子的振荡可能不同于反中微子的振荡。另一些实验信号,如LSND、MiniBooNE实验,表明除了太阳中微子与大气中微子振荡质量标度,还可能存在更大的中微子振荡质量标度：Δmew～1eV2。作为一种可能的理论解释,通过引进Lorentz对称破缺和CPT破坏,中微子与反中微子的振荡差异可以得到初步解释；而且由于Lorentz对称破缺模型中混合矩阵元的能量依赖性,可以对部分的反常振荡提供试探性的解释。然而,由于Lorentz对称破缺的引入,不仅使得混合矩阵元是能量依赖的,同时中微子能量-动量色散关系也需要作出修正,即可能出现超光速中微子。除了MINOS合作组与OPERA合作组的报道以外,目前各种中微子实验结果表明并不存在显著的中微子超光速事例。特别是,最近OPERA小组自身也声明了其实验中的缺陷。因此,理论上可行的Lorentz对称破缺模型对中微子能量-动量色散关系的修正必须是不显著的。在无超光速中微子的框架下,检验Lorentz对称破缺模型的方法主要集中在中微子混合矩阵元的能量依赖关系上。目前实验对Lorentz对称破缺中微子混合模型的约束是非常严格的,但是鉴于这些约束的模型依赖性,构造满足实验约束的Lorentz对称破缺混合模型仍然是可能的。
总的说来,从对称性角度讨论中微子振荡问题是十分重要的研究方法,对于中微子混合模型的构造有着很好的启发作用。相信在以后的中微子唯象学研究中,对称性分析仍将是一种有效的方法。
In order to interpret the continuous energy spectra of the β decay, Pauli proposed the existence of the neutrino, which was considered as the neutral massless fermion. However, Pontecorvo and Gribov proposed that if the neutrino is massive and there is the mixing of the different massive neutrinos the propagation of the neutrinos may bring the flavor oscillation. A variety of the neutrino experiments, such as the Homestake, Kamiokande and etc, have confirmed the flavor oscillation of the neutrinos and determined the mixing parameters θ12,θ23and the mass parameters ΔmAm2, ΔmSlo2. The nonzero mixing angle θ13is being confirmed by the experiments.
The flavor oscillation of the neutrinos reveals that the mass scale of the neutrino is tiny, which can not be interpreted in the standard model of the particle physics. As a phenomenological assumption, the seesaw mechanism introduces the right-handed neutrino with the large mass scale to depress the mass scale of the neutrino. There is no obvious evidences for the existence of the right-handed neutrino with the large mass scale. Even though we accept the seesaw mechanism as the interpretation of the tiny mass of the neutrino, the introductions of the neutrino mass and the mixing of the massive neutrinos ask for the new examinations of the familiar concepts and laws in the standard model. The concept of the flavor state, the conservation law of the energy-momentum and the time-dependent perturbation theory become non-simple and nontrivial in the background of the neutrinos mixing and the neutrinos oscillation. First, because of the time-dependence of the flavor charge of the neutrino, it is difficult to define the universal flavor state. We can only employ the flavor state with the limited and the approximate conditions. One can even avoid the flavor state for less theoretical controversies and treat the neutrino as the intermedial particle in the discussion of the neutrinos oscillation. Second, in order to coordinate the flavor conservation in the weak interaction and the flavor violation in the neutrino propagation, the conventional time-dependent perturbation theory is limited in the calculation of the reaction probability where the neutrino is involved. In the calculation of the probability involving the neutrino, the time scale in the time-dependent perturbation theory should be much less than Is. Otherwise, it will bring the obvious flavor violation in the weak interaction which is in contradiction with the micro-causality. Third, because of the uncertainty of the energy-momentum of the neutrino, the conventional constraint of the conservation of the energy-momentum does not mean the kinetic entanglement of the neutrino and the recoiling particle. The kinetic entanglement is the result of the special experiment design and the special detection. On the other hand, the analysis of the symmetry demonstrates that the flavor oscillation involves the limit of the uncertainties of the energy and the momentum of the neutrino. Both the momentum and the energy are uncertain for the flavor neutrino. The improvement of the resolution of the measurement of the momentum or the energy can destroy the flavor oscillation of the neutrino.
The pattern of the neutrino oscillation and the neutrino mixing can not be interpreted with the conventional symmetry group in the standard model. Thus, the discrete non-Abel flavor symmetry group is introduced in the neutrino model. These discrete group such as the A4, S4, T'and so on can interpret the classical mixing pattern, namely the tribimaximal mixing pattern. With the confirmation of the nonzero mixing angle θ13by the experiments, especially the convincing results of the neutrinos experiments at the Daya Bay, the modification or the perturbation is needed in these models with the discrete flavor group. However, even though these models can interpret the conventional mixing pattern and give the constraints on the undetermined mixing parameters and the mass parameters, there is the special fine-tuning of the vacuum expected value of the scalar field in these models. The naturalness of these model is not convincing. As a more economical procedure of the model construction, one can introduce the mass matrix of the neutrinos directly without the employment of the Higss mechanism and the extra scalar fields. The mass matrix satisfies the special discrete flavor symmetry. By diagonalization of the mass matrix, we can obtain the expected mixing matrix of the neutrinos. Although this method avoids the fine-tuning of the scalar field, the discrete symmetry group that induces the same mixing matrix is not unique. We can not discriminate these different symmetry groups. So lots of work is needed to deepen the interpretation of the neutrino mixing pattern.
Besides the conventional mixing pattern, some anomalous oscillation signals were reported in the experiments. Some experiments signals, such as that of the MINOS, reveal that the oscillation of the neutrino is different to that of the anti-neutrino. Other signals, such as that of the LSND and MiniBooNE, reveal that there is the extra large mass scale of the order leV2besides the tiny mass scales in the solar neutrino and the atmosphere neutrino. As a plausible interpretation, the introduction of the Lorentz invariance violation and the CPT violation can illustrate the difference of the oscillation of the neutrinos and the anti-neutrinos. Further more, because of the energy-dependence of the mixing matrix element in the model of the Lorentz invariance violation, some of the other anomalous oscillation can be interpreted. However, because of the introduction of the Lorentz invariance violation, we should modify the dispersion relation of the neutrino besides the energy-dependence mixing element, which means there may be superluminal neutrinos. Except the MINOS experiment and the OPERA experiment, kinds of the neutrinos experiments by far reveal that there is no obvious event of the superluminal neutrinos. In particular, very recently, the OPERA collaboration has claimed the flaw in their experiments. Therefore, the plausible neutrino model with the Lorentz violation should satisfies the condition that the modification of the dispersion relation is not obvious compared with the conventional relation. In the regime without the superluminal neutrinos, the method testing the Lorentz-violation models focuses on the energy-dependence of the mixing element of the neutrinos. The constraints on the Lorentz-violation models by far are very strict. However, because of the model-dependence of the constraints, it is still possible to construct the viable Lorentz-violation models of the neutrinos.
Generally speaking, the symmetry analysis is one of the most important methods in the studying of the phenomenology of the neutrino oscillation, which is enlightening in the construction of the mixing models of the neutrinos. It is believed that the symmetry analysis will be still an effective method in the future phenomenological researches of the neutrinos.
引文
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