非对易标准模型中若干散射过程的研究
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摘要
近十多年来,人们对超出标准模型的新物理的探索经历了一个重大的转变,将注意力从单纯扩充四维标准模型更多地转移到探索时空结构的奇异性上来。额外维理论的提出使得人们意识到超出四维的时空可能是解决标准模中能量等级问题的有效手段。与此同时,上世纪九十年代后期人们在弦论研究中发现如果将开弦末端限制在D-brane上并与膜上的恒定的NS-NS B场相互作用,那么在低能极限下的开弦理论将退化为一个定义在非对易时空流形上的量子场论。黑洞和量子引力方面的其他研究也指出时空存在一个最小尺度是任何量子引力理论的共同特性。自此,非对易时空上的量子场论及其TeV能标上现象学成为近十多年来备受关注的热点课题。经过多年的努力,一个重要的进展是人们已经建立起了一个基于Weyl-Moyal乘积的非对易时空上的标准模型(NCSM),这个模型的最大特点是它不是在SU(3)*SU(2)*U(1)李代数下封闭,而是在李代数的覆盖代数(Enveloping Algebra)下封闭的,这是规范场在Weyl-Moyal乘积的自恰性所要求的。另一方面Seiberg和Witten指出非对易规范场和对易规范场存在着一个映射,即Seiberg-Witten映射,覆盖代数的系数可以表达成对易时空中规范场的函数,这就避免了覆盖代数下必须要引入无穷多个场的困难,也导致了非对易标准模型中存在着大量对易时空中所禁止的新相互作用顶点,如破坏Lorentz不变性的三光子顶点,Z玻色子-光子-光子顶点,中微子和光子的耦合等。这些新相互作用在唯象上有丰富的内涵。
本文的研究是在基于Weyl乘积和Seiberg-Witten映射所构建的非对易标准模型(NCSM)下,探讨在下一代正负电子对撞机上探测Tev能标上非对易效应。目前为止,现有的研究大部分是在Seiberg-Witten映射下对非对易参数θ作有限阶展开(大多是一阶或二阶),利用微扰的方法得到相关Feynman图的一阶或二阶非对易修正以作为进一步计算的基础。这在低能过程下是有效的,然而如果假设非对易的能标是TeV,那么下一代对撞机的能量很可能会达到甚至超过非对易能标,这样大多数文献所用的对展开的低阶微扰就不准确了。为了克服这一困难,本文找到了一个包含任意阶的Seiberg-Witten映射并利用它研究了正负电子对撞机上高能现象学。我们首先研究了两个Higgs粒子的产生过程e+e-→ZH和e+e-→HH过程,发现非对易效应对总散射截面和角分布有着显著的影响。值得注意的是e+e-→HH在树图下是被标准模型所禁止的,因此相关新物理的信号对探测非对易效应有重要意义。研究发现,对于不同的非对易能标,存在一个最佳的碰状能量使得反应截面在这个碰撞能量下产生最大的非对易修正。通过数值拟合可以给出了ANG和Eoc线性关系。规范场论在非对易时空中存在着多余的自由度,这些自由度一部分来自于Seiberg-Witten映射的齐次解,另一部分源自U(1)规范群在Weyl-Moyal积乘积下本身的自由度,即混合规范变换(Hybrid gauge transfomation).本文通过考察e+e-→μ+μ-散射过程来研究混合规范变换自由度对非对易散射过程的影响。计算表明,在未考虑混合规范变换的非对易标准模型中,非对易效应对散射振幅只贡献一个相因子,最后导致散射截面没有任何的非对易修正。而在新的模型下,混合规范变换会对物理过程有重要的贡献。文中在地球坐标系下计算了总散射截面和角分布。并讨论了自发性Lorentz破坏效应。
In the recent decades, there is a big change for people's exploration of new physics beyond the standard model. More consideration is taken from only extending the standard model in four dimensions to probing the singularity of space-times. People believe that the possible existence of extra dimension is an effective solution to overcome the hierarchy problem of standard model. At the mean time, it is founded that when the open string is fixed at D-brane and interact with NS-NS B field, the string theory at low energy limit will lead to a quantum field theory in noncommutative space-time. Black hole physics and quantum gravity also strongly suggest the existence of noncommutative space-time. By this reason, the quantum field theory in noncommutative space-time and its implication to high energy phenomenology at TeV scale become a focus in recent ten years. After some effort, people have build a noncommutative standard model(NCSM) based on Weyl-Moyal product where the gauge group is not closed in SU(3)*SU(2)*U(1) Lie algebra but in the enveloping algebra. This property is necessity for the consistence of noncommutative gauge theory under Weyl-Moyal product. On the other hand, Seiberg and Witten point out that there is a map between the field in noncommutative space and ordinary space-time. i,e. Seiberg-Witten map. Using Seiberg-Witten map, the coefficient of enveloping algebra become the function of field variable in ordinary space-time. In this sense, the infinity of field is avoid. The NCSM induce many new particle vertex beyond the ordinary Sm, such as γ-γ-γ vertex, Z-γ-γ vertex, neutrino-photon vertex etc, leading to interesting phenomenological implication.
In this paper, we discuss the possibility of testing the TeV noncommutative effect at electron collision machine in the frame work of noncommutative standard model. By this time, most of the work is based on the Seiberg-Witten map by expanding to θ order or θ2order and getting the noncommutative correction of Feynman diagram up to θ or θ2order correspondingly. This method is effective for low energy processes. If We assume the non- commutative effect is located at TeV scale, however, the collision energy of LHC or ILC may access or even exceed the noncommutative scale. Then the first or second perturbative method is not accurate. To overcome this problem, we find a Seiberg-Witten map involv-ing any order of noncommutative parameter θ and apply it to the high energy collision process. We first study the Higgs production process e+e-→ZH and e+e-→HH. It is shown that the noncommutative effect have extensive impact on the cross section and angular distribution. It is noted that the the process e+e-→HH is forbid in the ordinary standard model, so any signal of the Higgs pair production has an important meaning for detecting the noncommutative space-time. For each noncommutative scle ANC, there exist a optimal collision energy where the noncommutative correction get the most. We give a linear relation between the noncommutative scle and optimal collision energy. There are also many degrees of freedom in noncommutative gauge theory. Some of them come from the solution of Seiberg-Witten map and some of them are derived from the Weyl-Moyal product of U(1) gauge theory. i.e. the hybrid gauge transformation. We study the process e+e-→μ+μ-and discuss the impact of hybrid gauge transformation. It is shown that without the hybrid gauge transformation the noncommutative effect only contribute a exp phase to the magnitude, leading to no noncommutative correction. When the hybrid gauge transformation is considered, the process get a large noncommutative correction. We cal-culate the total cross section and angular distribution at rotational reference on earth. The intrinsic Lorentz breaking effect is also discussed.
本文的研究是在基于Weyl乘积和Seiberg-Witten映射所构建的非对易标准模型(NCSM)下,探讨在下一代正负电子对撞机上探测Tev能标上非对易效应。目前为止,现有的研究大部分是在Seiberg-Witten映射下对非对易参数θ作有限阶展开(大多是一阶或二阶),利用微扰的方法得到相关Feynman图的一阶或二阶非对易修正以作为进一步计算的基础。这在低能过程下是有效的,然而如果假设非对易的能标是TeV,那么下一代对撞机的能量很可能会达到甚至超过非对易能标,这样大多数文献所用的对展开的低阶微扰就不准确了。为了克服这一困难,本文找到了一个包含任意阶的Seiberg-Witten映射并利用它研究了正负电子对撞机上高能现象学。我们首先研究了两个Higgs粒子的产生过程e+e-→ZH和e+e-→HH过程,发现非对易效应对总散射截面和角分布有着显著的影响。值得注意的是e+e-→HH在树图下是被标准模型所禁止的,因此相关新物理的信号对探测非对易效应有重要意义。研究发现,对于不同的非对易能标,存在一个最佳的碰状能量使得反应截面在这个碰撞能量下产生最大的非对易修正。通过数值拟合可以给出了ANG和Eoc线性关系。规范场论在非对易时空中存在着多余的自由度,这些自由度一部分来自于Seiberg-Witten映射的齐次解,另一部分源自U(1)规范群在Weyl-Moyal积乘积下本身的自由度,即混合规范变换(Hybrid gauge transfomation).本文通过考察e+e-→μ+μ-散射过程来研究混合规范变换自由度对非对易散射过程的影响。计算表明,在未考虑混合规范变换的非对易标准模型中,非对易效应对散射振幅只贡献一个相因子,最后导致散射截面没有任何的非对易修正。而在新的模型下,混合规范变换会对物理过程有重要的贡献。文中在地球坐标系下计算了总散射截面和角分布。并讨论了自发性Lorentz破坏效应。
In the recent decades, there is a big change for people's exploration of new physics beyond the standard model. More consideration is taken from only extending the standard model in four dimensions to probing the singularity of space-times. People believe that the possible existence of extra dimension is an effective solution to overcome the hierarchy problem of standard model. At the mean time, it is founded that when the open string is fixed at D-brane and interact with NS-NS B field, the string theory at low energy limit will lead to a quantum field theory in noncommutative space-time. Black hole physics and quantum gravity also strongly suggest the existence of noncommutative space-time. By this reason, the quantum field theory in noncommutative space-time and its implication to high energy phenomenology at TeV scale become a focus in recent ten years. After some effort, people have build a noncommutative standard model(NCSM) based on Weyl-Moyal product where the gauge group is not closed in SU(3)*SU(2)*U(1) Lie algebra but in the enveloping algebra. This property is necessity for the consistence of noncommutative gauge theory under Weyl-Moyal product. On the other hand, Seiberg and Witten point out that there is a map between the field in noncommutative space and ordinary space-time. i,e. Seiberg-Witten map. Using Seiberg-Witten map, the coefficient of enveloping algebra become the function of field variable in ordinary space-time. In this sense, the infinity of field is avoid. The NCSM induce many new particle vertex beyond the ordinary Sm, such as γ-γ-γ vertex, Z-γ-γ vertex, neutrino-photon vertex etc, leading to interesting phenomenological implication.
In this paper, we discuss the possibility of testing the TeV noncommutative effect at electron collision machine in the frame work of noncommutative standard model. By this time, most of the work is based on the Seiberg-Witten map by expanding to θ order or θ2order and getting the noncommutative correction of Feynman diagram up to θ or θ2order correspondingly. This method is effective for low energy processes. If We assume the non- commutative effect is located at TeV scale, however, the collision energy of LHC or ILC may access or even exceed the noncommutative scale. Then the first or second perturbative method is not accurate. To overcome this problem, we find a Seiberg-Witten map involv-ing any order of noncommutative parameter θ and apply it to the high energy collision process. We first study the Higgs production process e+e-→ZH and e+e-→HH. It is shown that the noncommutative effect have extensive impact on the cross section and angular distribution. It is noted that the the process e+e-→HH is forbid in the ordinary standard model, so any signal of the Higgs pair production has an important meaning for detecting the noncommutative space-time. For each noncommutative scle ANC, there exist a optimal collision energy where the noncommutative correction get the most. We give a linear relation between the noncommutative scle and optimal collision energy. There are also many degrees of freedom in noncommutative gauge theory. Some of them come from the solution of Seiberg-Witten map and some of them are derived from the Weyl-Moyal product of U(1) gauge theory. i.e. the hybrid gauge transformation. We study the process e+e-→μ+μ-and discuss the impact of hybrid gauge transformation. It is shown that without the hybrid gauge transformation the noncommutative effect only contribute a exp phase to the magnitude, leading to no noncommutative correction. When the hybrid gauge transformation is considered, the process get a large noncommutative correction. We cal-culate the total cross section and angular distribution at rotational reference on earth. The intrinsic Lorentz breaking effect is also discussed.
引文
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