I would like to express my endless gratitude to everyone who helped me with this project. A very special thanks to my supervisor Dr.Narendra Sahu who inspired me, encouraged me and believed in me during this entire year. I would like to express my deep gratitude to my parents for their love and constant support in every situation. The work for this thesis is twofold, first to understand the Standard Model (SM) and then to search for Dark Matter (DM) physics that is outside the Standard Model.
We discuss Standard Model as a gauge theory where abelian and non-abelian gauge theory are thoroughly discussed. Then we move to the electroweak sector where we discuss the interaction of intermediate vector bunches and fermions. Then we concentrate our work on the direct detection of DM where we calculate the DM nuclear scattering cross section, which is used in the terrestrial laboratories for the detection of DM.
Introduction
Gauge Theory
Abelian gauge theory
In electrodynamics, the gauge principle[6] provides a method to transform a Lagrangian that is invariant with respect to global symmetry from the U(1) group (abelian) to a Lagrangian that is invariant with respect to a local symmetry or gauge invariant. The hypothetical mass term for the gauge field (Aµ), Lm = 12m2AµAµ would not be invariant under the gauge transformation.
Non-Abelian gauge theory
Then we wish to find a field strength tensor that transforms under the local gauge transformation G as Fµν0 =GFµνG−1.
Spontaneous Symmetry Breaking
Higgs Mechanism
Abelian case
Non-Abelian case
Electroweak Theory Of Leptons
- Higgs Mechanism and the W and Z mass
- Lepton masses
- Evidence
- Thermodynamics in the Expanding Universe
- DM Relic Density
To obtain the right combination of fields coupling to the electromagnetic current, we perform the rotation in the neutral fields, defining the new fields A and Z by[6]. The neutrinos cannot acquire mass or couple to the H-field since there is no right-handed neutrino νR-field in the SM. Identifying the nature of dark matter (DM) remains one of the key open questions in physics. The Standard Model of particle physics alone cannot explain its nature (DM), suggesting that the model needs to be expanded.
A wide variety of experiments are currently underway to search for DM interactions in the laboratory and the sky. There is much evidence in support of the existence of invisible mass or DM. The most convincing and direct evidence for dark matter on galactic scales comes from observations of galactic rotation curves, namely the plot of the circular velocities of stars and gas as a function of their distance from the galactic center.
For distances beyond the galactic disk (r ≥ Rdisk), Gauss's law tells us that M must remain constant, assuming all mass is concentrated in the disk, and v ∝ r−1/2. In 1933, from measurements of the velocity dispersion of galaxies in the coma cluster, F. Zwicky deduced a mass-to-light ratio of about 400 solar masses per solar luminosity, exceeding the ratio in the solar neighborhood by two orders of magnitude. The mass of a cluster can be determined by several methods, including the application of the virial theorem to the observed distribution of radial velocities, through weak gravitational lensing, and by studying the X-ray profile that shows the distribution of hot emitting gas in rich regions follows. clusters.
One of the consequences of general relativity is that massive objects (such as a galaxy cluster) that lie between a more distant source and an observer must act as a lens to bend the light from that source. By measuring the distortion geometry, the mass of the intermediate cluster can be obtained. In dozens of cases where this has been done, the obtained mass-to-light ratios match the dynamical DM measurements of clusters. The key to understanding the thermal history is the comparison between the rate of interactions Γ and the rate of expansion H.
The evolution of the phase space density f(pµ, xµ) of a particle species is described by the Boltzmann equation which can be written as[4]. The kinetic equilibrium is maintained and the standard model particles in the interaction are in thermal equilibrium with the photon bath. Using the standard definition relating the cross section to the matrix element, we get pipj)2−(mimj)2 where, σij is the cross section for the scattering process.
Since σvm∅ varies slowly with changes in the number density of the initial and final state particles, it can be factored from the integrand to obtain. The DM number density n decreases as the universe expands (in addition to any number-changing effects due to the collision term), and this effect is usefully magnified by defining the quantity NX = ns, where s is the total entropy density of the universe and s= 2π4π2g?s(T)T3 ,g?s counts the number of relativistic degrees of freedom.
Direct Detection of Dark matter
- Introduction
- Ingredient
- Scattering Classifications
- Dark Matter Elastic Scattering off of a Nucleus
- Differential scattering cross section
- Spin dependent(SD) scattering
- Estimating the total Rate
The key ingredients for calculating the signal in direct detection experiments are the density and the velocity distribution of DM in the solar neighborhood and the DM nucleon scattering cross section. Elastic and Inelastic Scattering: The elastic scattering of a DM particle from a nucleus in a detector is simply the interaction of the DM with a nucleus as a whole, causing it to recoil enough to reflect the recoil energy spectrum in the target to measure. However, we will see for scalar (spin-independent) interaction, the increase in diameter dramatically dominates over spin-dependent scattering in present experiment using heavy atoms as targets.
2µ2v2σ0F2(q2) (2.9) We generally use the Helm form factor, which is approximately the Fourier transform of the nucleus mass distribution, given by [4]. We will see later that the spin-dependent scattering cross section plays the dominant role in direct detection. We now derive the differential distribution cross section for the DM-nucleus interaction, taking an effective operator approach.
The mass fractions fT qp,n are determined experimentally, so fp,n are constants of the theory once gφ is set. In this case, the DM is coherently coupled to the whole nucleus and the strength of the scattering interaction increases with the mass number of the nucleus. The effective interactions are called spin independent because the scattering cross section does not depend on the nuclear spin.
The scattering cross section is independent of the return energy and thus the differential rate is a decaying exponential. It arises due to the interaction of a DM particle with the spin of the nucleus. So it does not grow as fast with the size of the core as the SI interaction.
Let us consider only the spin-independent scattering process. We have our differential cross section. To compare data from different direct detection experiments that have different target nuclei, it is convenient to consider the DM-nucleon cross section σp. Note that the limits become weaker at masses Mx ~ 10GeV due to the energy thresholds of the experiments.
The thick orange line indicates the cross-section below which the experiments become sensitive to coherent neutrino scattering from nuclei. Now the question is, what WIMP nucleon cross section do you need to see 1 event/year for a 100GeV WIMP.
Conclusion
Applying this (leaving out the motion of the Sun and Earth) we would integrate to get Thus, we expect to see a smooth, exponentially decaying spectrum multiplied by the squared shape factor. For a more accurate calculation, the time dependence and the asymmetric velocity distribution as seen from Earth should be included.
The most sensitive experiments are currently beginning to probe DM nucleon cross sections ~10−45cm2, which is in the range expected for DM interacting with the nucleus through the exchange of a Higgs boson.
