*Three chapters: - Chapter 1: Introduction to particle physics - Chapter 2: A brief understanding about Dark Matter - Chapter 3: Neutrino-Electron scattering *Conclusion: Summary some re
Introduction to Particle physics
What is particles physics?
High-energy particle physics deals with the interactions of elementary particles and is a fundamental part of cosmological models of the early universe During the radiation-dominated era, when the average energy density was extremely high, these interactions shaped the evolution of the cosmos Consequently, processes such as pair production, particle scattering, and the decay of unstable particles play important roles in cosmology.
In cosmology, a scattering or decay process is cosmologically important during a specific epoch if its characteristic time scale is shorter than, or comparable to, the expansion time scale of the universe, which is given by the Hubble time 1/H at that epoch This Hubble time is roughly equal to the age of the universe at that time, providing a direct link between microphysical processes and the overall cosmic evolution.
Cosmological observations, including the cosmic microwave background and the cosmic abundance of elements, constrain the conditions of the early universe when compared with the predictions of the Standard Model of particle physics The success of the Standard Model in explaining these cosmological data confirms its validity beyond laboratory conditions However, phenomena inferred from cosmology, such as dark matter and CP-violation, point to physics beyond the Standard Model that is needed to fully account for the universe.
Protons, electrons, neutrons, neutrinos, and even quarks are often in the headlines of scientific discoveries, yet these subatomic particles are far too small to be seen, even with powerful microscopes While molecules and atoms form the basic elements of the substances we experience, real insight comes from looking inside atoms to study the elementary particles that compose matter By exploring these tiniest constituents, physicists uncover the fundamental forces and structures that shape the universe and explain the behavior of everything from everyday materials to cosmic phenomena.
K54-H.P.HUS Page 4 of this study is called Particle Physics, Elementary Particle Physics or sometimes High Energy Physics (HEP).
Discover ing the particle world
Democritus long ago postulated atoms as the indivisible building blocks of matter, a view that endured until the early 20th century when experiments, notably those of Rutherford, revealed that atoms are mostly empty space with electrons surrounding a dense central nucleus made up of protons and neutrons; from these findings, protons, neutrons, and electrons came to be regarded as the fundamental particles of nature.
Particle physics advanced dramatically after the invention of particle accelerators, devices that accelerate protons or electrons to exceptionally high energies and smash them into atomic nuclei In these high-energy collisions, researchers were surprised to observe a whole host of new particles emerging, broadening our understanding of the fundamental constituents of matter.
By the early 1960s, accelerators had reached higher energies and revealed a hundred or more particle types Was this the complete list of fundamental particles? Through a long program of experiments and theoretical work late in the 20th century, physicists uncovered a simple structure: two basic families of matter—quarks and leptons (the leptons include electrons and neutrinos)—and a small set of fundamental forces that govern their interactions These forces are transmitted by gauge bosons, particles that act as carriers, with the photon as the quantum of light and the transmitter of the electromagnetic force All these particle families have antiparticles—their complementary opposites—together forming matter and antimatter.
Today, the Standard Model is the framework that describes the role of fundamental particles and the interactions between them, while particle physics seeks to test this model in every conceivable way, aiming to uncover new phenomena and deepen our understanding of the forces that govern the universe.
K54-H.P.HUS Page 5 whether something more lies beyond it Below we will describe this Standard Model and its salient features.
The Standard Model
All known matter in the universe is composed of quarks and leptons, bound together by the fundamental forces These forces are mediated by gauge bosons—exchange particles that convey the interactions between particles.
(c) charm-quark mass = 1.5 (s) strange-quark mass = 0.16
(t) top-quark mass = 186 (b) bottom-quark mass = 5.2
(νà) muon-neutrino mass ~ 0 (à) muon mass = 0.11
(ντ) tau-neutrino mass ~ 0 (τ ) tau mass = 1.9
Among the fundamental forces that govern matter, gravity, the weak force, electromagnetism, and the strong force act on quarks and leptons In our everyday world, gravity and electromagnetism dominate, while the strong force binds quarks into protons and neutrons and holds nucleons together in atomic nuclei The weak force drives radioactive decay and governs how neutrinos and other leptons interact with matter Together, these four fundamental forces shape the behavior of matter from subatomic particles to the structure of atoms.
Dark Matter
A brief under standing about Dar k Matter
Today, dark matter is one of the biggest problems in cosmology The dark matter problem has been with us since 1930s, and was first postulated in
Dark matter was hypothesized in the 1930s by Dutch astronomer Jan Oort and Swiss astrophysicist Fritz Zwicky as a form of matter about five times more abundant than ordinary matter, yet it cannot be seen because it does not emit, absorb, or reflect light; its existence is inferred from the gravitational effects it exerts on surrounding matter By the early 1980s, improved cosmological observations of spiral galaxies’ rotation curves revealed orbital velocities that cannot be explained by visible mass alone, implying a substantial unseen mass component Thus, dark matter is needed to explain these dynamics, interacting with ordinary matter primarily through gravity and not via the strong or electromagnetic forces, and it has never been directly detected Current cosmology places dark energy at roughly 74% and dark matter at about 22% of the universe’s energy content, with ordinary matter comprising the remaining portion.
2.1.2 The evidence for dark matter
* Evidence for dark matter in spiral galaxies
In spiral galaxies like the Milky Way, we infer the gravitational mass by observing how stars and gas clouds orbit the galactic center The rotation curve shows orbital velocity as a function of radius, and in many spirals the velocities remain flat even where no stars are visible This flatness implies that the gravitational mass far exceeds the luminous mass, with the total mass more than ten times the luminous mass, pointing to a substantial dark matter component.
*Evidence for dark matter in clusters of galaxies
In clusters of galaxies, the gravitational mass is inferred from the orbital motions of the member galaxies Because these galaxies lie at roughly the same distance from us, the spread in their redshifts is interpreted as orbital velocity around the cluster’s center, often exceeding 1000 km/s By measuring the redshifts of many galaxies within a cluster, we obtain the velocity dispersion and use it to estimate the cluster’s dynamical mass.
In galaxy clusters, we calculate the gravitational mass required to keep the galaxies bound in orbit rather than allowing them to escape The inferred gravitational mass is then compared to the luminous mass contributed by the galaxies themselves and the mass contributed by the hot X-ray gas in the cluster.
The main three types of dark matter[8] are:
- Hot dark matter (HDM) – neutrinos with rest masses of the order of few eV
- Cold dark matter (CDM) – supersymmetric particles with masses ~ 100 GeV or axions with ma ~ 10 -5 eV
- Ultra – cold (vacuum – like) matter – the cosmology constant
2.1.4 Particles candidate for dark matter
A Weakly Interacting Massive Particle, or WIMP, is a hypothetical particle believed to have been produced in the Big Bang WIMPs are defined by two key properties: weak-scale interactions and a large mass Their interactions are comparable to the weak nuclear force, with couplings near the weak-scale, and their masses lie near m ~ 100 GeV in particle-physics units—an energy scale similar to that of heavy Standard Model particles This combination makes them difficult to detect directly, while also making them compelling dark-matter candidates, since they could achieve the right relic abundance through thermal processes in the early universe.
Axions arise as a consequence of solving the strong-CP problem in particle physics; the fluctuations about the broken theory's vacuum are axions—the pseudo-Nambu-Goldstone bosons of the broken symmetry They are somewhat less natural than WIMPs because arranging the comoving number density to match the observed dark-matter density is challenging There are several axion production mechanisms, but the preferred dark-matter scenario is non-thermal coherent oscillations of the axion field near the QCD phase transition, which yield light axions born with essentially zero momentum.
- Gravitinos: The supersymetric partner of the graviton, may be dark matter Depending on exactly how supersymmetry is broken, the gravitino
Gravitinos could lie anywhere in the mass range from eV to TeV, though masses below keV are disfavored because they erase too much small-scale structure Because these gravitinos are produced in decays with relatively high momentum, they can smear out primordial density perturbations on small scales Gravitinos are not as favored as WIMPs as dark matter candidates due to the difficulty of obtaining the correct relic abundance and because they are harder to detect with conventional methods There are other plausible dark matter candidates that address different problems in physics, though they do not offer quite the same level of intrigue as the gravitino and WIMP scenarios discussed.
Sterile neutrinos are neutrinos that do not interact via the electroweak force Because the mass eigenstates do not coincide with the electroweak flavor eigenstates (ν_e, ν_μ, ν_τ), sterile neutrinos can mix with active neutrinos and participate indirectly in electroweak processes They have been proposed in several contexts: as a mass-generation mechanism for active neutrinos, as the right-handed partners to the active species, or to explain certain neutrino-experiment anomalies As dark matter, sterile neutrinos can be produced in the early universe through a variety of mechanisms, and their effects on small-scale structure provide constraints Since sterile neutrinos mix with active neutrinos, they have a small decay probability to an active neutrino and a photon, which leads to X-ray constraints The simplest sterile-neutrino dark-matter scenario, the Dodelson–Widrow non-resonant production model, is excluded by a combination of small-scale structure observations and non-detections of X-rays from galaxies.
Methods of detecting dark matter
Dark matter cannot be observed directly because it does not emit radiation, so physicists rely on direct and indirect detection methods to search for it Direct detection includes three main approaches: simultaneous light and heat detection (as used by scintillating bolometers), simultaneous heat and ionization detection, and simultaneous light and ionization detection Indirect detection looks for products of dark matter annihilation or decay in space, providing complementary evidence to the direct search methods.
K54-H.P.HUS Page 11 signals (the most famous being the search for an annual modulation in the dark matter signal caused by the orbiting of the Earth
Indirect detection methods seek the byproducts of dark matter interactions rather than the particles themselves Researchers search for signals such as neutrinos and gamma rays—photons produced when dark matter particles annihilate or decay in the cosmos—which can reveal the presence and distribution of dark matter throughout the Universe.
Neutrinos
Neutrino physics is one of the most exciting areas of research today Neutrinos are extraordinarily abundant throughout the universe, and understanding these elusive particles is fundamental to deciphering cosmic processes By studying neutrinos, scientists aim to unlock deeper insights into how the universe works and reveal the nature of matter and energy at the smallest scales.
“Neutrinos, they are very small
They have no charge and have no mass
And do not interact at all.”
Neutrinos are neutral, elementary subatomic particles produced by radioactive decay and other nuclear processes; they carry no electric charge and interact only very weakly with matter, earning their reputation as ghostly particles The idea was proposed by Wolfgang Pauli in 1930 to preserve energy conservation in beta decay Because of their elusive nature, the first direct detection took decades, and in 1956 Clyde Cowan, Frederick Reines, F B Harrison, H W Kruse, and A D McGuire published “Detection of the Free Neutrino: A Confirmation” in Science, providing the experimental proof of neutrinos' existence.
In 1962, Lederman, Schwartz, and Steinberger showed that more than one neutrino flavor exists by detecting interactions of the muon neutrino, proving neutrino flavors are distinct The discovery of the tau lepton in 1975 at the Stanford Linear Accelerator Laboratory suggested it should have an associated neutrino as well First evidence for the tau neutrino came from missing energy and momentum observed in tau decays, signaling the presence of an unseen neutrino accompanying the tau.
The tau neutrino's discovery is analogous to the beta decay process that first revealed the neutrino, underscoring a parallel in how fundamental particles are uncovered In the summer of 2000, the DONuT collaboration at Fermilab announced the first direct detection of tau neutrino interactions, making it the latest particle in the Standard Model to be observed directly.
2.2.2 Where are they coming from?
Most of the neutrinos floating around today were born about 15 billion years ago, shortly after the Big Bang Since then the universe has continued to expand and cool, and these relic neutrinos have persisted as a pervasive cosmic background The neutrino population is so large that it forms a cosmic neutrino background with a temperature of about 1.9 Kelvin (-271.2 Celsius) In addition to this primordial stream, neutrinos are continually produced by human-made sources—nuclear power stations, particle accelerators, and nuclear detonations—and by natural atmospheric processes, as well as by the births, collisions, and deaths of stars, especially supernova explosions.
Several properties of neutrinos have been measured:
The classic Wu et al experiment (1957) demonstrated that the weak interaction maximally violates parity conservation If neutrinos are massless, this implies they are fully polarized with a helicity of +1 or −1 In 1958, Goldhaber et al measured the neutrino helicity and found that only left-handed neutrinos—spin anti-parallel to their direction of motion—participate in the weak interaction.
Experimental results indicate that neutrinos observe lepton number conservation, that is they are always associated with their like-flavour charged lepton;
Studies of the Z boson line width at LEP and SLC indicate that there are only three neutrino species with standard couplings to the Z and masses less than 45 GeV/c^2 [9], while neutrinos with non-standard (much weaker) couplings have not been observed in these measurements.
K54-H.P.HUS Page 13 couplings, so-called sterile’ neutrinos, could exist in addition to the three standard species
A neutral fermion may exist either as a Dirac particle (fermion antifermion) or as a Majorana particle (fermion antifermion) For a Dirac fermion (neutral or charged), the mass term is
vanish using The mass term always connects the opposite chiral components of the same field The absence of either, R or L , automatically leads to m == 0
If neutrinos are Majorana particles, a mass term can be constructed even in the absence of right-handed components by using the antiparticle that is identical to its charge conjugate but with opposite chirality Unlike charged fermions, neutrinos are neutral and can be self-conjugate, ν_M = ν_M^c These particles are known as Majorana neutrinos, denoted ν_M.
Each fermionic field ψ has a corresponding antiparticle field ψ^C, obtained via the charge-conjugation operator C = i γ^2 γ^0 The charge-conjugated field is defined by ψ^C ≡ C (ψ̄)^T, which in common notation reduces to ψ^C = i γ^2 ψ^*, so that the fermion field F is ψ and its antifermion field F is ψ^C.
While for a charged fermion m is the only possible mass term, for a neutral fermion there are other possibilities In addition to the standard term
, the terms C C , C and C are equally valid The first C C is equivalent to , but the last two, C and C , may be written respectively as C L L C R R and L L C R R C Indeed
If the neutrino is a Majorana fermion, we can always construct a mass term C L L L L C without the right-handed component R precisely because L C is right-handed with positive helicity
The existence of Majorana neutrinos implies lepton-number violation, since a Majorana field is its own antiparticle and its couplings contain both Le = +1 and Le = −1 components If v_M is written as (ψ ψ^c)/2, the weak charged current that links the electron to a Majorana neutrino carries both lepton-number conserving and violating terms The most dramatic manifestation of this is neutrinoless double beta decay (0νββ) of a nucleus: N(Z, A) → N(Z+2, A) + e− + e−, where the initial state has Le = 0 and the final state with two electrons has Le = 2 In 0νββ decay, the Majorana neutrino emitted by one neutron (n → p e− v_M) can be absorbed by a second neutron (n′ → p′ e−), because the Majorana neutrino has no well-defined lepton number; when emitted it carries Le = −1 and when absorbed it carries Le = +1, enabling the two-neutron system to convert into two protons with two electrons in the final state.
On the other hand, with the Dirac neutrino for which the leptonic number is conserved, double -decay
N Z Z e e v v (Fig 3b), referred to as 2v , can only occur with two antineutrinos v e emitted together with two electrons Unlike the VM, the Dirac V e emitted in n p e v cannot be absorb.ed by n to become p e
By energy-mornentum conservation, the energy spectrum of the two- electron system in 2v decay with Dirac neutrinos is continuous In
If neutrinos are Majorana particles, neutrinoless double beta decay would produce a sharp peak in the two-electron energy spectrum—ideally a delta function—serving as the distinctive signature of this decay mode The amplitudes of both the two-neutrino (ββ2ν) and neutrinoless (ββ0ν) channels are second order in the Fermi constant GF, so their rates are very small; nevertheless, observations of the standard two-neutrino double beta decay have been reported for nine isotopes, with half-lives ranging from 10^19 to 10^24 years Experiments searching for neutrinoless decays in 136Xe, 76Ge, and 48Ca have not yielded conclusive results If electron-neutrino mixing is small, and/or if the Majorana mass is too small (through the Majorana propagator effect), neutrinoless double beta decay may still escape observation.
Fig.1 (a) Double neutrino less 0v decay by Majorana neutrino; (b) double 2v decay by Dirac neutrino
2.2.5 Direct sear ches for neutrino mass
Direct measurement of neutrino mass can be pursued by studying decay processes that involve neutrinos If neutrinos have mass, the decay kinematics differ from the massless case, leading to observable signatures such as changes in the energy spectra and angular distributions of the decay products These deviations provide a pathway to infer the neutrino mass directly through precision measurements in beta decay, pion decay, or other neutrino-involved processes, with carefully controlled systematics and high-resolution detectors.
K54-H.P.HUS Page 16 observable effects Studies of the endpoint of tritium decay have been used to search for non-zero electron neutrino masses via the process:
If the electron neutrino has a non-zero mass, it would induce potentially measurable distortions near the endpoint of the electron energy spectrum Interpreting this signal requires corrections for nuclear screening effects and final-state interactions in the tritium itself Two experiments—one at the University of Mainz and the other at Troitsk—were conducted to explore these distortions.
Two Russian experiments are currently collecting data Their fits to the current data yield 95% confidence level upper limits on the electron antineutrino mass of 5.6 and 3.9 eV/c^2, respectively Nevertheless, both experiments report best-fit values for the mass.
Measurements of the beta-decay spectrum by two groups, including the Troitsk group, yield negative m^2_v values and reveal systematic effects, notably a bump-like structure near the end-point that shifts over time These uncertainties make it difficult to derive a definitive limit on the electron neutrino mass with this technique, although both groups say that if the electron neutrino mass were around 25 eV/c^2, a clear signal above the systematic effects should be observable.
Some interactions of neutrinos
Neutrino – Electron scattering
Fig 1: Tree level Feynman diagrams: e e
In the center of mass:
Considering neutrino as massless particle and mass of electron: me = m
*Applying Feynman rule, we have:
With k 1 ≠ k 2 , p 1 ≠ p 2 → using the Mandelstam variables s = (k 1 + p 1 ) 2 = (k 2 + p 2 ) 2 t = (k 1 – k 2 ) 2 = (p 2 – p 1 ) 2 u = (k 1 – p 2 ) 2 = (k 2 – p 1 ) 2
In the center of mass:
+In the laboratory system: electron at rest, p 1 = 0
We have: s m 2 2 mE ; E : incoming neutrino energy
We imply the following relations:
Because of massless neutrino, the product q u( ) k 2 (1 5 )u( ) k 1 vanishes
*Trace of an odd number of ' s vanishes
Trace of other components equal zero because it is set of an odd number of γ’s
Neglecting m 2 s Q , 2 the integrated cross-section becomes:
For sM z 2 the logarithm term of (8) in powers of s M z 2 , then in the first approximation, the cross-section depends linearly on s:
The Z 0 propagator effect through (1 Q 2 M z 2 ) 2 in (3.8) is very important at high energy, since for s M Z 2 the cross-section in (3.9) ceases to increase with s, it bends down and tends asymptotically towards a constant
In the laboratory frame, at low energy mE M Z 2 , we neglect
Q M z and use (3.2), then (3.8) and (3.9) can be written as
The y distrinution as well as the integrated cross-section enable us to extract gV,gA
For the antineutrino scattering ( ) k 1 e p ( ) 1 ( ) k 2 e p ( ) 2 , its cross-section can be deduced from ( ) k 1 e ( p 1 ) ( k 2 ) e ( p 2 ) by simply substitution g R 2 ( g V g A ) 2 g L 2 ( g V g A ) 2 This rule can be traced back to (3.5) for which the current u k( 2 ) (1 5 ) ( )u k 1 replace by
, i.e k 1 k 2 and the substitution g R 2 g L 2 comes from (3.8) in which the last term proportional to m 2 is neglected Thus:
Numerically, with G mE F 2 27.05 10 42 cm 2 E / GeV we get:
Neutrino-electron scattering with U-particle
Here, q 2 0 in s-channel Basing on Feynman diagram and applying Feynman’s rule:
Similarity to the calculation above, we have:
In the center of mass:
The differential cross-section is:
Dark matter remains invisible, but that doesn’t mean it isn’t there—it's all around us in the universe, and there may even be dark matter passing through your body right this second We know dark matter exists because we can observe its effects on other objects in the universe, including how neutrinos behave, which helps illuminate this mysterious component of the cosmos In short, something you can’t see can still have a real, measurable influence, and that’s why scientists study dark matter to understand the structure and evolution of the universe.
The thesis some process of dark matter is time to end After all, this thesis obtained some results:
First, it introduces in general to Particle physics as well as its trend finding new particles and discovering more and more the universe
Second, it introduces briefly to dark matter and neutrinos in its properties, sources and its interactions with ordinary matter
Finally, it supports the detail computing of the invariant amplitude of neutrino-electron scattering By this way, we can calculate the lost energy and prove the existence of neutrinos
In brief, these results are basic to carry out experiments, measurements and verify the truth of theory of neutrinos and dark matter
( ) ( ) with 1, 2: spin up, down along z axis. s s s u p u p p m s
Appendix B Trace of the product of matrices
* (ABC) Tr(CAB) Tr(BCA); A, B,C is abitrary matrices.
Trace of an odd number of ' vanishes.