暗物质直接探测中核效应的研究
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摘要
随着观测的手段和设备越来越先进，人类对宇宙的认识越来越深入，相应的理论也在逐步改进和完善。由于对宇宙的背景辐射，氢，氦丰度等的预言完全被实验观测证实，大爆炸理论已作为成功的理论被普遍被接受。然而在庆祝该理论成功的同时，许许多多的新问题又展现在科学家面前。正反物质不对称的起源，大爆炸后的inflation时期的物理等等都是亟需解决的困难问题。然而，最迫切要理解的是暗能量和暗物质的存在。现代的天文观测肯定了构成我们宇宙中可直接观测的可见物质仅占宇宙全部物质的4%左右，而大部分的物质是“暗”的。2011年的物理学诺贝尔奖表彰了三位天文物理学家对超新星观测得到宇宙在加速膨胀的结论，从而进一步激发科学家探讨暗能量的热情。通过综合分析，我们可以确认宇宙能量的72%以上是暗能量，24%是暗物质，其他4%是发光物质。暗能量的讨论目前还在一个朦胧的阶段，因为我们暂时还没有什么手段去观测暗能量。然而对暗物质的理论研究和实验观测却已经开始走到实质性的阶段了。
暗物质是二十世纪和二十一世纪最有挑战性的课题，是粒子物理和天体物理两个看似不同领域研究的热点。物理学家共同的认识是我们天文观测可以给我们提供建立理论模型的基础，但不能最终确认模型的正确或合理程度，只有在我们的探测器（无论是在卫星上还是在高山或地下）直接观测到它们，检验是否与相应的理论预言一致，我们才可以得到结论。这就不可避免地涉及了天文和粒子物理研究的交叉。事实上，这两个领域的交叉可以追溯到牛顿时代，甚至到更早（如果我们说古希腊，或中国古代，不知道是否会太过分）。对暗物质的研究肯定成为21世纪天文和粒子物理的理论研究和实验观测的重点，它不仅是挑战，而且提供了最有生命力的机遇。
早在1933年加州理工学院的瑞士天文学家Fritz Zwicky就发现了Coma星系团反常的旋转速度，由此提出星系团周围存在暗物质的假设。如果星系团只存在发光物质的话，那么天文观测到的旋转星系的速度不符合万有引力定律计算出来的速度，即被观测恒星的速度应该与中心之间的距离平方根成反比，但是观测到的数据指出随着与星系中心距离的加大，速度趋近常数。这就意味着星系团中存在着我们看不见的物质，即暗物质。除此之外，星团对撞以及引力透镜的观测也确认了暗物质的存在。作为实验物理学家和理论家的首要课题是在实验上直接寻找暗物质，当然暗物质候选者的确认也就成了理论家最重要的任务。
目前的理论认为可能存在三类暗物质:冷暗物质(Cold Dark Matter)，热暗物质(Hot Dark Matter)和温暗物质(Warm Dark Matter)。根据目前的分析，冷暗物质很可能占据暗物质的主要成分。显然暗物质不参加强相互作用和电磁相互作用（所以是暗的）。冷暗物质的构成包括一种或多种弱相互作用重粒子WIMP(Weakly Interacting Massive Particle)的贡献。WIMPs最可能的候选者就是超对称理论预言的最轻的中性粒子neutralino。它是两个Higgsino和photino，bino混合后的最轻的本征态。即使存在很小的R-宇称守恒破坏，neutralino的寿命也会足够长，可以和今天宇宙年龄（135亿年左右）相比。当然如果它可以衰变，就理所当然地为宇宙中正电子超出提出可能的解释，是间接探测暗物质的依据之一（当然还有其他的机制）。
当然目前最主要的实验研究是直接探测宇宙空间到来的暗物质流。这是建立在我们期望暗物质粒子真是WIMPs，除了引力作用（这是一定有的，因为就是根据对引力的观测得到暗物质存在的证据），它（们）还参加弱相互作用。事实上，对此我们并没有得到任何实验上确切的支持，但如果暗物质真的只参加引力作用而不参加弱作用的话，我们将不可能用现有的任何手段去探知，从而得到相关信息。这当然是很悲惨的事，但不是没有可能。我们希望它还参加弱相互作用，neutralino就是我们期望的候选者，当然除它之外还有darkon，technipion等候选者。我们的研究是基于这种priori的假定，暗物质（不论它是什么）会通过弱相互作用和探测器中标准模型物质相互作用，从而产生可观测的信号。
直接探测是在地下安装探测器来直接探测暗物质与探测器物质中核子碰撞时原子核反冲能量形成的电，光或热信号，不同的探测器采用不同的探测物质和探测手段，所观测的信号也不完全相同。除此之外，物理学家期望在加速器上产生暗物质粒子，但它们既不参加电磁作用，又不衰变（长寿命），因而只能对应丢失的能量(missing energy)，需要通过其他看得见的粒子的能动量分析来确定丢失的能量和猜测它们的携带者，从而判断missing粒子的性质，以决定它是否就是我们寻找的暗物质粒子。和宇宙学观测相对照，最终找到结论。这是一个漫长但充满机遇和挑战的路！
本论文的主要研究内容围绕对暗物质与原子核碰撞的散射截面的研究而展开，暗物质相对地球的的速度约为220～600km/s，即使暗物质的质量在几十到几百GeV，它的动能也就在100keV范围内，一般来讲只能产生核的弹性碰撞，即核不可能跃迁到激发态，因而只能通过核的反冲动能和动量来研究暗物质粒子的性质。暗物质与原子核的相互作用分自旋相关（SD）和自旋无关（SI）两种方式。
在本论文中，我们研究了：
第一，暗物质与原子核的相互作用与自旋无关时，球形核的形状因子F(q)。由于暗物质粒子与原子核间的相互作用与自旋无关，原子核的贡献可以用它的形状因子反映出来。即形式因子的计算可以具体化为原子核密度的傅里叶变换。
描述原子核密度最简单的模型有2PF模型(two-Parameter-Fermi model)。另一种模型为折叠模型(Folding model)，这种模型假设原子核内部的电荷密度为均匀分布，同时引进一个高斯函数用来处理原子核边缘密度，通过卷积定理将两个函数‘折叠’在一起，结果与2PF密度模型给出的形状因子很相似，但是有一些差别。2PF的傅里叶变换没有解析式，折叠模型的傅里叶变换有解析式，称为Helm模型。文中也给出了由Sick提出的高斯模型(Gaussians)密度和形状因子，以及Fourier-Bessel-expansion模型的密度和形状因子。2PF和Helm model是目前暗物质探测理论中广泛使用的模型。作为an alternative，我们使用了描述原子核多体理论的相对论平均场(Relativistic Mean-Filed)来计算原子核的密度。相对论平均场模型(简称RMF模型)认为核子之间交换σ，ω，ρ，π介子(玻色子)来传递相互作用，同时引入了介子场的期望值代替相应介子场方程(通过欧拉方程Euler-Equation得到)的场源项。核子场的Dirac方程与介子场方程之间通过互相耦合的自洽迭代，可以计算出原子核的一些基态性质如原子核密度，核子结合能，均方根半径等。本文计算得出了球形核16O，40Ca，72Ge，132Xe，208Pb等原子核的密度，再进行傅里叶变换的数值积分，得到了相应的形状因子。40Ca，72Ge，132Xe都是暗物质探测器广泛使用的元素材料。我们给出了2PF模型，Helm模型，RMF模型三种形状因子的数值结果，并进行了比较。相对论平均场模型计算出的原子核密度和其他模型的结果在能量-动量依赖性上略有不同，但总的趋势是类似的。它的优点在于充分考虑了各种原子核的独特性质，因而更为接近实际，结果也更可信。
第二，研究了暗物质与原子核的相互作用与自旋无关时，原子核形变对形状因子的影响。原子核的多极形变效应有：零极形变为球形核，四极形变为椭球形状，以及多极的高阶效应。对我们所涉及的暗物质探测器采用的原子核来说，四极形变已经足够精确，所以我们只对四级核形变进行研究。2PF模型可以描述球形原子核的质量密度或者电荷密度分布，当考虑可能的核变形时，可以对模型进行修正。早期有人通过修正2PF，使之半径参数与角度有关，这样的修正模型可以用来描述变形核的密度。类似地，我们对折叠模型进行了修正：根据椭球方程，我们引入不相等的半长轴和半短轴，从而参数化了椭球表面半径。我们仍认为在椭球内部的核子密度为常数，得到了变形核的密度。在第三种计算方法中，导出变形核密度的方法是基于Nilsson平均场模型。Nilsson平均场是引入轴对称的谐振子势，自旋轨道耦合项，以及对较大轨道角动量起压低作用的平方项。在此基础上，从两个方面出发：一方面考虑了原子核价核子对的临近轨道对力相互作用，一方面没有考虑临近轨道对力相互作用，这一部分当中，我们计算了变形核73Ge，131Xe的密度。我们用这三种模型计算了四极形变原子核在不同极角方向(10o，30o，45o，60o，90o)的密度，并对得到的密度并且进行了比较。我们对变形核密度进行傅里叶变换来计算动量不同方向上的形状因子，此时形状因子F(q,cosθ)也是依赖于角度的。结果表明，对一般用来做暗物质探测的探测器物质来说，相应的原子核形变较小，原子核形变对形状因子的影响可以忽略不计。
第三，本论文研究了自旋相关的散射振幅。事实上，大部分暗物质与标准模型物质的相互作用都会导致与自旋相关的散射，例如通过交换Z或Z或其他超越标准模型的新物理模型粒子。暗物质粒子与原子核的相互作用本质是暗物质粒子与核子内的夸克或者胶子发生相互作用，这是基本相互作用，相应的Lagrangian是在所有理论模型中给出的，也正是我们要真正了解的。强子内的胶子在领先阶不参与弱相互作用，因此我们只需关心暗物质粒子与夸克的相互作用。自旋相关的散射截面比自旋无关的散射截面计算上要复杂很多，因为这时粒子物理过程与核物理过程不能分离开来，也就是不能因子化，无法分开暗物质与原子核碰撞元过程和原子核的集体效应。实际上，自旋相关的散射截面计算过程与原子核的β衰变有很大相似之处。J.Engel等人利用Walecka的多极算符展开方法计算了暗物质与原子核碰撞的散射振幅，在冲量近似和非相对论近似的前提下，将相互作用流用矢量球谐函数展开，利用C-G系数耦合，得到所有与角度相关的部分，最后归结到简单的约化矩阵元计算。在转移动量q=0和q=0的情况下计算了散射振幅，其中约化矩阵元的计算依赖于核物理的具体模型。本文从另一个角度，根据通常的量子场论方法，重新推导了散射振幅的表达式，最后结果和Engel等人的相同。但我们的优越性在于，具体计算中，我们可以只考虑最外层壳的贡献（价核子）。在碰撞过程中，假设原子核满壳层内部的核子对散射矩阵元总的贡献为0，我们只计算价核子的贡献，这样计算可以大大简化。我们利用了最近Luo等人推导出的原子核结构模型，对几个特征的原子核的自旋相关的散射截面做了计算。由于在这个原子核模型中考虑了自旋轨道角动量的耦合，相应的波函数是用(j，m，l，s)来描写的。为了能合理地计算，波函数要用C-G系数展开到以球谐函数和自旋波函数为基的表示中，虽然计算略为繁琐，但并没有原则上的困难。
论文的最后我们对暗物质探测的理论计算中的原子核效应做了一些讨论，希望本论文的结果对暗物质探测实验的设计，以及从暗物质探测的数据中提取与基本相互作用相关的信息有所帮助。
With the rapid development of observational method and getting better appara-tus, the human’s understanding for the universe become more profound, at the sametime the relevant theory are gradually improving and comprehensive. As a successfultheory, Big Bang hypothesis has been wildly accepted by human owing to the cosmicmicrowave background (CMB) radiation and thermal relic density of helium and hydro-gen and some other predictions have been confirmed by the experimental observation.However, in the mean time for celebrating the theory success, many more new prob-lems have been put in the front of the scientists. Such as origin of the matter-antimatterasymmetry, the physics of inflation era after the Big Bang and so on are all problemsto be resolved. Among the awkward subjects the dark matter and dark energy becomeour first thing to be understood. contemporary observations has verified that luminousmatter that can be observed directly constitute only4%of the matter in the universe,and most of the matter are’dark’. The2011Nobel Prize in Physics is awarded to threeastrophysicists for their discovery of the accelerating of the universe, the conclusionresulted from the observation for the Supernova. Thus it strike scientists’ enthusiasmto explore the dark energy. Via comprehensive analysis, we have confidence that72%of the whole energy is dark energy,24%is dark matter, and the rest4%of the matteris luminous. The discussion for dark energy is still a obscure stage, we temporarilyhave no efficient method to detect it. Fortunately, the theory study and experimentaldetection has entered a substantial process.
Dark matter is the most challenging subject of the20th and21st centuries. Itis also a seemingly different hot topic between particle physics and astrophysics. Thecommon sense of the physicist is astro observations can provide enough foundations forestablishing theoretical models, but can not confirm the correction of property of thosemodels. Until our detectors(no matter in the satellites, mountains, or underground)ob-serve the dark matter, and test if it is consistent with the relevant theory model, we candraw the conclusion. It inevitably involves with interdisciplinary between astronomyand particle physics. In fact the interdisciplinary between the two fields can trace backto the Newton times even earlier(we don’t known whether it is excessive if saying an-cient Greece,ancient China). Undoubtly the research for dark matter surly becomes thefocal point of theoretical research and experimental observations in21centry, both byastrophysics and particle physics. It is not only a challenge, but also providing the most attractive opportunities.
As a matter of fact, the conjecture about existence of dark matter was proposedquite a long time ago in1933by Zwicky to explain the anomalously large velocitynear the Coma star clusters astronomically observed at that time. The astronomicalobservation shows that the rotational curves of the test stars in the galaxies did not obeythe gravitational law if only the luminous matter which clusters at the center of thegalaxies existed. Namely, the velocities of a test stars were supposed to be inverselyproportional to square root of their distances from the center of the galaxy, instead, therotational curve turns flat. It means that there must be large amount of matter exist inthe cluster of galaxies that can not be seen, namely dark matter. Besides, the galaxyclusters collision and gravitational lens also confirm the existence of dark matter. Theprimary task for the theorist and experimental physicist is to search for the dark matterdirectly in experiments, meanwhile, identifying the candidate of dark matter becomesthe most important mission.
The present theories divide dark matter into three types: Cold dark matter(CDM),Hot dark matter(HDM),Warm dark matter(WDM). On the basis of analysis so far, theCDM occupies the most part of the whole dark matter. It is obviously that dark mat-ter don’t participates strong and electromagnetic interactions. The component of thedark matter my be include one or multiple WIMP(Weakly interacting Massive Parti-cle)that contribute. The most promising candidate of WIMPs should be the lightestsuper-symmetric particle(LSP)neutralino, which is a linear combinationof the SUSYpartners of the photon,Z0, and Higgs bosons. Even there exist small R-parity broken,the neutralino’s lifetime long enough and comparable with the tody’s age of the uni-verse(13.5billon yeas). Of course if it decays, the excess of the positron in the universecan be expained and will be one of the evidence for indirect detection of dark matter.
Currently the leading experimental investigation is directly detect the dark matterflux coming from the outer space. This is based on the assumption that dark matterconsist of WIMPs, except the gravitational interaction(it surely exist because it is grav-itation of the dark matter that induced the evidence)between them, dark matter alsoparticipate in weak interaction, actually, we haven’t any support from the experimentsabout this. If the dark matter only take part in gravitational interaction other than weakinteraction, it is impossible for us to ascertain them. It sounds miserable but may likelybe happen to some of extent. Thus we expect dark matter interact weakly with nor-mal materials and neutralino will be our expected candidate. Certainly there are other candidates including darkon, techipion and so on. Our work is based on this assump-tion, dark matter(no matter what kind of is)will interact weakly with general matter ofstandard model in detectors, and produce the observational signals.
The direct detection is carried out via collisions between WIMPs and nuclei in de-tectors installed underground. The recoil energy can form electron, photon, or thermalsignals, which is determined by various detections. Besides, physicists hope to producedark matter particles in colliders, but dark matter don’t participate electromagnetic in-teractions nor decay(long lifetime), we have to find other visible particle with explicitenergy and momentum, then deduce and judge the missing particle’s properties, andfinally to determine if it is dark matter. This is long time but full of opportunities andchallenge.
The research of our work this thesis centers on the scattered amplitude andform factors between dark matter and nucleus. The velocity of dark matter isabout220～600km/s in the earth frame coordinate. Even the mass of dark matter ar-range from dozens of GeV to hundreds of GeV, is kinetic energy is only a few tens ofKeV, which is too small to cause an inelastic transition for the nucleus. As a result wehave to investigate he dark matter’s property from nucleus recoil energy and transferredmomentum. The interaction between dark matter and nuclei can also be divided twoparts: spin-independent(SI) interaction and spin-dependent interaction. In this thesiswe have studied:
first of all, we studied form factors of spherical nucleus in the case of spin-independent interaction between dark matter and nucleus. In this condition, contri-bution from the nucleus can be reflected with form factor, namely the calculation of theform factors can be transformed to the nucleus density’s fourier transformation.
The most simplest nucleus density model is2PF(two-parameter-Fermi)model.Another model is Folding model. This model assumes the uniform density in the nu-clear inner part, and introduce a Gaussian’surface smearing’ density function, convolvethe two together to describe the nucleus density. The result is similar to that of the2PF, but also have some difference. There is no analytic expression for fourier trans-formation of2PF model, but Folding model’s fourier actions can be written, it alsonamed as Helm form factor. The text also give some other density model such asGaussian model,which is proposed by Sick, and Fourier-Bessel-expansion model fordensity and relevant form factor.2PF and Helm model are widely used in the theoryof dark matter detection. As an alternative, we discuss Relativistic Mean field(RMF)of nucleus many-body theory to calculate the nuclear density. RMF theory assumes theinteraction between nucleons happened by exchange bosonic meson such as σ, ω, ρ, π,as a approximation, introducing the expectation value of meson field to replace thefield source terms in the Clein-Gordon equation. Using self-consistent iteration be-tween Dirac equation(nucleon field)and Clein-Gordon equations, we can obtain theground states properties such as nuclear density, binding energy, square root meanand so on. We calculate spherical nucleus density(16O,40Ca,72Ge,132Xe,208Pb), thenperform fourier transformation to the density to get the corresponding form factors.Elements40Ca,72Ge,132Xe are widely used in detectors for dark matter detection. Wecompared2PF, Helm, RMF three types of models. The results from RMF model arelittle different from the other two models, but has the same tendency. Its advantage isRMF theory completely considered nuclear uniqueness, thus more approach the realityand believable.
second, we studied effects of nuclear deformation on the form factor in the case ofscalar interaction(spin-independent). Nuclear multiple deformation include: zero orderdeformation(that is spherical shape), quadruple deformation(corresponding ellipsoid),octupole deformation and higher order deformations. For the nucleus in the detectorsin dark matter detections, it is precise enough only consider the quadruple deformation.Therefore we primary address ourselves to the quadruple deformation.2PF model isused to describe the nuclear density distribution and form factor. When the nucleardeformation is considered, this model can be revised. In early period, one revised thehalf-radius to be related with angle in order to adjust to deformed nucleus. Similarly, wemodified the Folding model: beginning from the ellipsoidal equation, we introduce longsemi-axes and short semi-axes, accordingly the surface radius is parameterized. Westill assume the density in nucleus is homogeneous distributed, and convolved it with’smearing surface’ Gaussian function referred aforementioned. The third method tocalculate deformed nuclear density is based on Nilsson mean field, which is introducinga axially symmetric harmonic oscillator, also the considering the spin-orbit couplingterm and, and Dl2which is used to supress the potential well. We start from twoaspects, one is consider both Nilsson mean field and nearest orbital paring interaction ofthe nucleus, another is only Nilsson mean field included. In this paper we calculate thedeformed nucleus densities of73Ge,131Xe. We used three models above to obtain thedensities from different polar angles(10o,30o,45o,60o,90o) and compared with eachmodel. The relevant form factors were carried out by taking fourier transformation to the densities. The results show us that in the detectors for dark matter, the effects ofsmall deformation can be neglected for the nuclear form factors.
third, we discuss the scattering amplitude in the spin dependent interaction. Infact most of the dark matter interact with particles of standard model will lead to spin-dependent scattering, which exchange Z or Z’ some other new particles beyond stan-dard model. The WIMP particle interacts with quarks or gluons inside the nucleon.Gluons in the hadron do not participate in the weak interaction at the leading order,so that the fundamental processes concern only the interaction between DM particleand quarks. The cross section of spin-dependent is more complex than that of spin-independent condition. For the spin-independent cross section, the particle-physics andnuclear-physics contributions can be separated, namely the nuclear effects can be fac-tored out and included in a form factor F(q). Actually the calculation process is similarto the nuclear beta decay. J.Engel et al gave scattering amplitude for dark matter andnucleus collision, employing Walecka’s multi-pole expansion method, with the impulseapproximation, expanding the interaction current with vector spherical harmonics, withC-G coefficients including, and get the result related to angles. Final aim is to deal withthe reduced matrix element of the scattering. The amplitude is divided into two situ-ations: momentum transfer q=0and q=0. The detailed expression of reduced matrixelement relies on concrete nuclear models. We obey the quantum field principle andstart from another aspect to deduce renewedly the scattering amplitude. Our advantageis we only consider the valence nucleons outside the full shell,because contributionsfrom the nucleons in the inner part of full shell cancels each other, this make calcula-tion more simplify. We use nucleus structure model derived from Luo, calculate crosssection of spin dependent interaction. The spin-orbital coupling is included, the rele-vant wave-function is written with(j,m,l,s). For the sake of reasonable calculation, thewave-function will be expand with C-G coefficients in the representation which is us-ing spherical harmonic function and spin wave-function as basis, may be some of bittedious in the process but no principle difficulty.
The final part in our job discussed the nuclear effect in the theoretical calculationfor the dark matter detection. We hope our scenario can be helpful for the design ofexperiments of dark matter detection, and be some of help to extract any informationabout the fundamental interaction from the experimental data.
暗物质是二十世纪和二十一世纪最有挑战性的课题，是粒子物理和天体物理两个看似不同领域研究的热点。物理学家共同的认识是我们天文观测可以给我们提供建立理论模型的基础，但不能最终确认模型的正确或合理程度，只有在我们的探测器（无论是在卫星上还是在高山或地下）直接观测到它们，检验是否与相应的理论预言一致，我们才可以得到结论。这就不可避免地涉及了天文和粒子物理研究的交叉。事实上，这两个领域的交叉可以追溯到牛顿时代，甚至到更早（如果我们说古希腊，或中国古代，不知道是否会太过分）。对暗物质的研究肯定成为21世纪天文和粒子物理的理论研究和实验观测的重点，它不仅是挑战，而且提供了最有生命力的机遇。
早在1933年加州理工学院的瑞士天文学家Fritz Zwicky就发现了Coma星系团反常的旋转速度，由此提出星系团周围存在暗物质的假设。如果星系团只存在发光物质的话，那么天文观测到的旋转星系的速度不符合万有引力定律计算出来的速度，即被观测恒星的速度应该与中心之间的距离平方根成反比，但是观测到的数据指出随着与星系中心距离的加大，速度趋近常数。这就意味着星系团中存在着我们看不见的物质，即暗物质。除此之外，星团对撞以及引力透镜的观测也确认了暗物质的存在。作为实验物理学家和理论家的首要课题是在实验上直接寻找暗物质，当然暗物质候选者的确认也就成了理论家最重要的任务。
目前的理论认为可能存在三类暗物质:冷暗物质(Cold Dark Matter)，热暗物质(Hot Dark Matter)和温暗物质(Warm Dark Matter)。根据目前的分析，冷暗物质很可能占据暗物质的主要成分。显然暗物质不参加强相互作用和电磁相互作用（所以是暗的）。冷暗物质的构成包括一种或多种弱相互作用重粒子WIMP(Weakly Interacting Massive Particle)的贡献。WIMPs最可能的候选者就是超对称理论预言的最轻的中性粒子neutralino。它是两个Higgsino和photino，bino混合后的最轻的本征态。即使存在很小的R-宇称守恒破坏，neutralino的寿命也会足够长，可以和今天宇宙年龄（135亿年左右）相比。当然如果它可以衰变，就理所当然地为宇宙中正电子超出提出可能的解释，是间接探测暗物质的依据之一（当然还有其他的机制）。
当然目前最主要的实验研究是直接探测宇宙空间到来的暗物质流。这是建立在我们期望暗物质粒子真是WIMPs，除了引力作用（这是一定有的，因为就是根据对引力的观测得到暗物质存在的证据），它（们）还参加弱相互作用。事实上，对此我们并没有得到任何实验上确切的支持，但如果暗物质真的只参加引力作用而不参加弱作用的话，我们将不可能用现有的任何手段去探知，从而得到相关信息。这当然是很悲惨的事，但不是没有可能。我们希望它还参加弱相互作用，neutralino就是我们期望的候选者，当然除它之外还有darkon，technipion等候选者。我们的研究是基于这种priori的假定，暗物质（不论它是什么）会通过弱相互作用和探测器中标准模型物质相互作用，从而产生可观测的信号。
直接探测是在地下安装探测器来直接探测暗物质与探测器物质中核子碰撞时原子核反冲能量形成的电，光或热信号，不同的探测器采用不同的探测物质和探测手段，所观测的信号也不完全相同。除此之外，物理学家期望在加速器上产生暗物质粒子，但它们既不参加电磁作用，又不衰变（长寿命），因而只能对应丢失的能量(missing energy)，需要通过其他看得见的粒子的能动量分析来确定丢失的能量和猜测它们的携带者，从而判断missing粒子的性质，以决定它是否就是我们寻找的暗物质粒子。和宇宙学观测相对照，最终找到结论。这是一个漫长但充满机遇和挑战的路！
本论文的主要研究内容围绕对暗物质与原子核碰撞的散射截面的研究而展开，暗物质相对地球的的速度约为220～600km/s，即使暗物质的质量在几十到几百GeV，它的动能也就在100keV范围内，一般来讲只能产生核的弹性碰撞，即核不可能跃迁到激发态，因而只能通过核的反冲动能和动量来研究暗物质粒子的性质。暗物质与原子核的相互作用分自旋相关（SD）和自旋无关（SI）两种方式。
在本论文中，我们研究了：
第一，暗物质与原子核的相互作用与自旋无关时，球形核的形状因子F(q)。由于暗物质粒子与原子核间的相互作用与自旋无关，原子核的贡献可以用它的形状因子反映出来。即形式因子的计算可以具体化为原子核密度的傅里叶变换。
描述原子核密度最简单的模型有2PF模型(two-Parameter-Fermi model)。另一种模型为折叠模型(Folding model)，这种模型假设原子核内部的电荷密度为均匀分布，同时引进一个高斯函数用来处理原子核边缘密度，通过卷积定理将两个函数‘折叠’在一起，结果与2PF密度模型给出的形状因子很相似，但是有一些差别。2PF的傅里叶变换没有解析式，折叠模型的傅里叶变换有解析式，称为Helm模型。文中也给出了由Sick提出的高斯模型(Gaussians)密度和形状因子，以及Fourier-Bessel-expansion模型的密度和形状因子。2PF和Helm model是目前暗物质探测理论中广泛使用的模型。作为an alternative，我们使用了描述原子核多体理论的相对论平均场(Relativistic Mean-Filed)来计算原子核的密度。相对论平均场模型(简称RMF模型)认为核子之间交换σ，ω，ρ，π介子(玻色子)来传递相互作用，同时引入了介子场的期望值代替相应介子场方程(通过欧拉方程Euler-Equation得到)的场源项。核子场的Dirac方程与介子场方程之间通过互相耦合的自洽迭代，可以计算出原子核的一些基态性质如原子核密度，核子结合能，均方根半径等。本文计算得出了球形核16O，40Ca，72Ge，132Xe，208Pb等原子核的密度，再进行傅里叶变换的数值积分，得到了相应的形状因子。40Ca，72Ge，132Xe都是暗物质探测器广泛使用的元素材料。我们给出了2PF模型，Helm模型，RMF模型三种形状因子的数值结果，并进行了比较。相对论平均场模型计算出的原子核密度和其他模型的结果在能量-动量依赖性上略有不同，但总的趋势是类似的。它的优点在于充分考虑了各种原子核的独特性质，因而更为接近实际，结果也更可信。
第二，研究了暗物质与原子核的相互作用与自旋无关时，原子核形变对形状因子的影响。原子核的多极形变效应有：零极形变为球形核，四极形变为椭球形状，以及多极的高阶效应。对我们所涉及的暗物质探测器采用的原子核来说，四极形变已经足够精确，所以我们只对四级核形变进行研究。2PF模型可以描述球形原子核的质量密度或者电荷密度分布，当考虑可能的核变形时，可以对模型进行修正。早期有人通过修正2PF，使之半径参数与角度有关，这样的修正模型可以用来描述变形核的密度。类似地，我们对折叠模型进行了修正：根据椭球方程，我们引入不相等的半长轴和半短轴，从而参数化了椭球表面半径。我们仍认为在椭球内部的核子密度为常数，得到了变形核的密度。在第三种计算方法中，导出变形核密度的方法是基于Nilsson平均场模型。Nilsson平均场是引入轴对称的谐振子势，自旋轨道耦合项，以及对较大轨道角动量起压低作用的平方项。在此基础上，从两个方面出发：一方面考虑了原子核价核子对的临近轨道对力相互作用，一方面没有考虑临近轨道对力相互作用，这一部分当中，我们计算了变形核73Ge，131Xe的密度。我们用这三种模型计算了四极形变原子核在不同极角方向(10o，30o，45o，60o，90o)的密度，并对得到的密度并且进行了比较。我们对变形核密度进行傅里叶变换来计算动量不同方向上的形状因子，此时形状因子F(q,cosθ)也是依赖于角度的。结果表明，对一般用来做暗物质探测的探测器物质来说，相应的原子核形变较小，原子核形变对形状因子的影响可以忽略不计。
第三，本论文研究了自旋相关的散射振幅。事实上，大部分暗物质与标准模型物质的相互作用都会导致与自旋相关的散射，例如通过交换Z或Z或其他超越标准模型的新物理模型粒子。暗物质粒子与原子核的相互作用本质是暗物质粒子与核子内的夸克或者胶子发生相互作用，这是基本相互作用，相应的Lagrangian是在所有理论模型中给出的，也正是我们要真正了解的。强子内的胶子在领先阶不参与弱相互作用，因此我们只需关心暗物质粒子与夸克的相互作用。自旋相关的散射截面比自旋无关的散射截面计算上要复杂很多，因为这时粒子物理过程与核物理过程不能分离开来，也就是不能因子化，无法分开暗物质与原子核碰撞元过程和原子核的集体效应。实际上，自旋相关的散射截面计算过程与原子核的β衰变有很大相似之处。J.Engel等人利用Walecka的多极算符展开方法计算了暗物质与原子核碰撞的散射振幅，在冲量近似和非相对论近似的前提下，将相互作用流用矢量球谐函数展开，利用C-G系数耦合，得到所有与角度相关的部分，最后归结到简单的约化矩阵元计算。在转移动量q=0和q=0的情况下计算了散射振幅，其中约化矩阵元的计算依赖于核物理的具体模型。本文从另一个角度，根据通常的量子场论方法，重新推导了散射振幅的表达式，最后结果和Engel等人的相同。但我们的优越性在于，具体计算中，我们可以只考虑最外层壳的贡献（价核子）。在碰撞过程中，假设原子核满壳层内部的核子对散射矩阵元总的贡献为0，我们只计算价核子的贡献，这样计算可以大大简化。我们利用了最近Luo等人推导出的原子核结构模型，对几个特征的原子核的自旋相关的散射截面做了计算。由于在这个原子核模型中考虑了自旋轨道角动量的耦合，相应的波函数是用(j，m，l，s)来描写的。为了能合理地计算，波函数要用C-G系数展开到以球谐函数和自旋波函数为基的表示中，虽然计算略为繁琐，但并没有原则上的困难。
论文的最后我们对暗物质探测的理论计算中的原子核效应做了一些讨论，希望本论文的结果对暗物质探测实验的设计，以及从暗物质探测的数据中提取与基本相互作用相关的信息有所帮助。
With the rapid development of observational method and getting better appara-tus, the human’s understanding for the universe become more profound, at the sametime the relevant theory are gradually improving and comprehensive. As a successfultheory, Big Bang hypothesis has been wildly accepted by human owing to the cosmicmicrowave background (CMB) radiation and thermal relic density of helium and hydro-gen and some other predictions have been confirmed by the experimental observation.However, in the mean time for celebrating the theory success, many more new prob-lems have been put in the front of the scientists. Such as origin of the matter-antimatterasymmetry, the physics of inflation era after the Big Bang and so on are all problemsto be resolved. Among the awkward subjects the dark matter and dark energy becomeour first thing to be understood. contemporary observations has verified that luminousmatter that can be observed directly constitute only4%of the matter in the universe,and most of the matter are’dark’. The2011Nobel Prize in Physics is awarded to threeastrophysicists for their discovery of the accelerating of the universe, the conclusionresulted from the observation for the Supernova. Thus it strike scientists’ enthusiasmto explore the dark energy. Via comprehensive analysis, we have confidence that72%of the whole energy is dark energy,24%is dark matter, and the rest4%of the matteris luminous. The discussion for dark energy is still a obscure stage, we temporarilyhave no efficient method to detect it. Fortunately, the theory study and experimentaldetection has entered a substantial process.
Dark matter is the most challenging subject of the20th and21st centuries. Itis also a seemingly different hot topic between particle physics and astrophysics. Thecommon sense of the physicist is astro observations can provide enough foundations forestablishing theoretical models, but can not confirm the correction of property of thosemodels. Until our detectors(no matter in the satellites, mountains, or underground)ob-serve the dark matter, and test if it is consistent with the relevant theory model, we candraw the conclusion. It inevitably involves with interdisciplinary between astronomyand particle physics. In fact the interdisciplinary between the two fields can trace backto the Newton times even earlier(we don’t known whether it is excessive if saying an-cient Greece,ancient China). Undoubtly the research for dark matter surly becomes thefocal point of theoretical research and experimental observations in21centry, both byastrophysics and particle physics. It is not only a challenge, but also providing the most attractive opportunities.
As a matter of fact, the conjecture about existence of dark matter was proposedquite a long time ago in1933by Zwicky to explain the anomalously large velocitynear the Coma star clusters astronomically observed at that time. The astronomicalobservation shows that the rotational curves of the test stars in the galaxies did not obeythe gravitational law if only the luminous matter which clusters at the center of thegalaxies existed. Namely, the velocities of a test stars were supposed to be inverselyproportional to square root of their distances from the center of the galaxy, instead, therotational curve turns flat. It means that there must be large amount of matter exist inthe cluster of galaxies that can not be seen, namely dark matter. Besides, the galaxyclusters collision and gravitational lens also confirm the existence of dark matter. Theprimary task for the theorist and experimental physicist is to search for the dark matterdirectly in experiments, meanwhile, identifying the candidate of dark matter becomesthe most important mission.
The present theories divide dark matter into three types: Cold dark matter(CDM),Hot dark matter(HDM),Warm dark matter(WDM). On the basis of analysis so far, theCDM occupies the most part of the whole dark matter. It is obviously that dark mat-ter don’t participates strong and electromagnetic interactions. The component of thedark matter my be include one or multiple WIMP(Weakly interacting Massive Parti-cle)that contribute. The most promising candidate of WIMPs should be the lightestsuper-symmetric particle(LSP)neutralino, which is a linear combinationof the SUSYpartners of the photon,Z0, and Higgs bosons. Even there exist small R-parity broken,the neutralino’s lifetime long enough and comparable with the tody’s age of the uni-verse(13.5billon yeas). Of course if it decays, the excess of the positron in the universecan be expained and will be one of the evidence for indirect detection of dark matter.
Currently the leading experimental investigation is directly detect the dark matterflux coming from the outer space. This is based on the assumption that dark matterconsist of WIMPs, except the gravitational interaction(it surely exist because it is grav-itation of the dark matter that induced the evidence)between them, dark matter alsoparticipate in weak interaction, actually, we haven’t any support from the experimentsabout this. If the dark matter only take part in gravitational interaction other than weakinteraction, it is impossible for us to ascertain them. It sounds miserable but may likelybe happen to some of extent. Thus we expect dark matter interact weakly with nor-mal materials and neutralino will be our expected candidate. Certainly there are other candidates including darkon, techipion and so on. Our work is based on this assump-tion, dark matter(no matter what kind of is)will interact weakly with general matter ofstandard model in detectors, and produce the observational signals.
The direct detection is carried out via collisions between WIMPs and nuclei in de-tectors installed underground. The recoil energy can form electron, photon, or thermalsignals, which is determined by various detections. Besides, physicists hope to producedark matter particles in colliders, but dark matter don’t participate electromagnetic in-teractions nor decay(long lifetime), we have to find other visible particle with explicitenergy and momentum, then deduce and judge the missing particle’s properties, andfinally to determine if it is dark matter. This is long time but full of opportunities andchallenge.
The research of our work this thesis centers on the scattered amplitude andform factors between dark matter and nucleus. The velocity of dark matter isabout220～600km/s in the earth frame coordinate. Even the mass of dark matter ar-range from dozens of GeV to hundreds of GeV, is kinetic energy is only a few tens ofKeV, which is too small to cause an inelastic transition for the nucleus. As a result wehave to investigate he dark matter’s property from nucleus recoil energy and transferredmomentum. The interaction between dark matter and nuclei can also be divided twoparts: spin-independent(SI) interaction and spin-dependent interaction. In this thesiswe have studied:
first of all, we studied form factors of spherical nucleus in the case of spin-independent interaction between dark matter and nucleus. In this condition, contri-bution from the nucleus can be reflected with form factor, namely the calculation of theform factors can be transformed to the nucleus density’s fourier transformation.
The most simplest nucleus density model is2PF(two-parameter-Fermi)model.Another model is Folding model. This model assumes the uniform density in the nu-clear inner part, and introduce a Gaussian’surface smearing’ density function, convolvethe two together to describe the nucleus density. The result is similar to that of the2PF, but also have some difference. There is no analytic expression for fourier trans-formation of2PF model, but Folding model’s fourier actions can be written, it alsonamed as Helm form factor. The text also give some other density model such asGaussian model,which is proposed by Sick, and Fourier-Bessel-expansion model fordensity and relevant form factor.2PF and Helm model are widely used in the theoryof dark matter detection. As an alternative, we discuss Relativistic Mean field(RMF)of nucleus many-body theory to calculate the nuclear density. RMF theory assumes theinteraction between nucleons happened by exchange bosonic meson such as σ, ω, ρ, π,as a approximation, introducing the expectation value of meson field to replace thefield source terms in the Clein-Gordon equation. Using self-consistent iteration be-tween Dirac equation(nucleon field)and Clein-Gordon equations, we can obtain theground states properties such as nuclear density, binding energy, square root meanand so on. We calculate spherical nucleus density(16O,40Ca,72Ge,132Xe,208Pb), thenperform fourier transformation to the density to get the corresponding form factors.Elements40Ca,72Ge,132Xe are widely used in detectors for dark matter detection. Wecompared2PF, Helm, RMF three types of models. The results from RMF model arelittle different from the other two models, but has the same tendency. Its advantage isRMF theory completely considered nuclear uniqueness, thus more approach the realityand believable.
second, we studied effects of nuclear deformation on the form factor in the case ofscalar interaction(spin-independent). Nuclear multiple deformation include: zero orderdeformation(that is spherical shape), quadruple deformation(corresponding ellipsoid),octupole deformation and higher order deformations. For the nucleus in the detectorsin dark matter detections, it is precise enough only consider the quadruple deformation.Therefore we primary address ourselves to the quadruple deformation.2PF model isused to describe the nuclear density distribution and form factor. When the nucleardeformation is considered, this model can be revised. In early period, one revised thehalf-radius to be related with angle in order to adjust to deformed nucleus. Similarly, wemodified the Folding model: beginning from the ellipsoidal equation, we introduce longsemi-axes and short semi-axes, accordingly the surface radius is parameterized. Westill assume the density in nucleus is homogeneous distributed, and convolved it with’smearing surface’ Gaussian function referred aforementioned. The third method tocalculate deformed nuclear density is based on Nilsson mean field, which is introducinga axially symmetric harmonic oscillator, also the considering the spin-orbit couplingterm and, and Dl2which is used to supress the potential well. We start from twoaspects, one is consider both Nilsson mean field and nearest orbital paring interaction ofthe nucleus, another is only Nilsson mean field included. In this paper we calculate thedeformed nucleus densities of73Ge,131Xe. We used three models above to obtain thedensities from different polar angles(10o,30o,45o,60o,90o) and compared with eachmodel. The relevant form factors were carried out by taking fourier transformation to the densities. The results show us that in the detectors for dark matter, the effects ofsmall deformation can be neglected for the nuclear form factors.
third, we discuss the scattering amplitude in the spin dependent interaction. Infact most of the dark matter interact with particles of standard model will lead to spin-dependent scattering, which exchange Z or Z’ some other new particles beyond stan-dard model. The WIMP particle interacts with quarks or gluons inside the nucleon.Gluons in the hadron do not participate in the weak interaction at the leading order,so that the fundamental processes concern only the interaction between DM particleand quarks. The cross section of spin-dependent is more complex than that of spin-independent condition. For the spin-independent cross section, the particle-physics andnuclear-physics contributions can be separated, namely the nuclear effects can be fac-tored out and included in a form factor F(q). Actually the calculation process is similarto the nuclear beta decay. J.Engel et al gave scattering amplitude for dark matter andnucleus collision, employing Walecka’s multi-pole expansion method, with the impulseapproximation, expanding the interaction current with vector spherical harmonics, withC-G coefficients including, and get the result related to angles. Final aim is to deal withthe reduced matrix element of the scattering. The amplitude is divided into two situ-ations: momentum transfer q=0and q=0. The detailed expression of reduced matrixelement relies on concrete nuclear models. We obey the quantum field principle andstart from another aspect to deduce renewedly the scattering amplitude. Our advantageis we only consider the valence nucleons outside the full shell,because contributionsfrom the nucleons in the inner part of full shell cancels each other, this make calcula-tion more simplify. We use nucleus structure model derived from Luo, calculate crosssection of spin dependent interaction. The spin-orbital coupling is included, the rele-vant wave-function is written with(j,m,l,s). For the sake of reasonable calculation, thewave-function will be expand with C-G coefficients in the representation which is us-ing spherical harmonic function and spin wave-function as basis, may be some of bittedious in the process but no principle difficulty.
The final part in our job discussed the nuclear effect in the theoretical calculationfor the dark matter detection. We hope our scenario can be helpful for the design ofexperiments of dark matter detection, and be some of help to extract any informationabout the fundamental interaction from the experimental data.
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