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General

 

Quarks are elementary particles that form the foundational elements of matter. They join together to form complex particles called hadrons, with protons and neutrons, which are integral to atomic nuclei, being the most stable examples. Commonly observed matter is primarily composed of up quarks, down quarks, and electrons. Quarks are always found bound together due to a principle known as color confinement. Therefore, they can only be discovered within hadrons, including baryons like protons and neutrons, and mesons, or within quark-gluon plasmas. Our comprehension of quarks largely relies on studying hadrons.

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Quarks come with several intrinsic properties, such as electric charge, mass, color charge, and spin. They are distinctive within the Standard Model of particle physics since they undergo all four fundamental interactions or forces: electromagnetism, gravitation, strong interaction, and weak interaction. Moreover, they are the only identified particles having electric charges that are not integral multiples of the basic charge.

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Quarks are categorized into six types or "flavours": up, down, charm, strange, top, and bottom. Up and down quarks have the smallest masses among all quark variants. Heavier quarks rapidly transition into up and down quarks via a process known as particle decay, representing a shift from a higher mass state to a lower one. Consequently, up and down quarks are generally stable and most prevalent in the universe, whereas strange, charm, bottom, and top quarks are only produced in high-energy collisions, such as those seen in cosmic rays and particle accelerators. Corresponding to each quark type, there's a related antiparticle called an antiquark. Antiquarks mirror quarks in their properties, with the only difference being some of these properties (like the electric charge) are of equal magnitude but have an opposing sign.

First appearance of Quarks in the Universe.

 

Quarks, as we understand them, are primarily defined by their interaction via the strong force. Therefore, speaking of "quarks" in a time before the strong force existed as a separate entity is a bit tricky. If there was a time before the strong force existed in its current form, then the "quarks" of that time would likely have been different in some ways from the quarks we are familiar with today.

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Our current theoretical models postulate that quarks, as we understand them, started to exist after the strong force became distinct from the other forces, during the quark epoch that started around 10⁻¹² seconds after the Big Bang. Prior to this, we believe the universe was filled with a hot, dense "soup" of particles and forces, The Quark-Gluon Plasma (QGP) but how exactly quarks fit into this picture is not entirely clear.

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The key characteristic of QGP is that it is a phase of quantum chromodynamics (QCD, the theory describing strong interactions) where the strong interaction is not confining. In more familiar states of matter, the strong force binds quarks together into hadrons (such as protons and neutrons). In QGP, the temperature and energy density are so high that this confinement is overcome, and quarks and gluons move freely. This is unlike regular plasma, where nuclei and electrons are free, but quarks are still confined within nuclei.

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Despite our knowledge of QGP being mostly theoretical, scientists have been able to recreate conditions similar to the early universe using particle accelerators such as the Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC). These experiments use heavy ions (like gold or lead nuclei) accelerated to nearly the speed of light and then collided, creating a small region of very high temperature and energy density, potentially forming a brief QGP.

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The data from these experiments have helped scientists explore the properties of QGP. For instance, they have found that QGP behaves more like a liquid than a gas, exhibiting strong collective behaviour. It's also nearly frictionless, making it one of the most perfect liquids we know of. Moreover, it's opaque to most particles, except for photons and leptons, which can escape and provide direct information about the QGP.

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Literature

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  • P. Rosnet. Quark-Gluon Plasma: from accelerator experiments to early Universe. Writeup of a talk given at the Rencontres du Vietnam 2015 on Cosmology. arXiv:1510.04200. 2015.

  • S. M. Sanches Jr., F. S. Navarra, D. A. Fogaça. The quark gluon plasma equation of state and the expansion of the early Universe. Nuclear Physics A 937 (2015), 1. doi 10.1016/j.nuclphysa.2015.02.004. 2015

  • A. Tawfik, M. Wahba, H. Mansour, T. Harko. Viscous Quark-Gluon Plasma in the Early Universe.Annals Phys.523:194-207,2011

  • Joseph I Kapusta. Quark-Gluon Plasma in the Early Universe. Proceedings of the International School of Astrophysics D. Chalonge, 8th. 2001

  • B. Banerjee, R. V. Gavai.  Can the Quark-Gluon Plasma in the Early Universe be supercooled? Phys.Lett. B293 (1992) 157-160

Quantum states of quarks

 

In quantum mechanics, the term "quantum state" refers to the state of a quantum system, completely described by a wavefunction. For quarks, there are several quantum properties or "states" that are important:

 

Flavor: Quarks come in six "flavors": up, down, charm, strange, top, and bottom. These flavors are related to the different types of quarks and antiquarks.

 

Color Charge: Quarks carry what is called a "color charge". Unlike the term suggests, it has nothing to do with actual colors but represents a type of charge under the strong force. The colors come in three types: red, green, and blue, and their corresponding anticolors.

 

Spin: Quarks, being fermions, have a spin of 1/2. This can take on two values when measured along any axis, typically called up and down or +1/2 and -1/2.

 

Mass: Each flavor of quark has a different mass, but it's important to note that the mass of a proton or neutron (common particles made of quarks) is much greater than the sum of the masses of the constituent quarks, due to the binding energy from the strong force.

 

Electric Charge: Quarks carry electric charge. For instance, up, charm, and top quarks have a charge of +2/3e, while down, strange, and bottom quarks have a charge of -1/3e, where e is the elementary charge.

 

Baryon Number: Quarks have a baryon number of +1/3, and antiquarks have a baryon number of -1/3. The baryon number is conserved in interactions, which is why quarks are always found in combinations that make up a whole number baryon number (like baryons with baryon number 1, made of three quarks, or mesons with baryon number 0, made of a quark and an antiquark).

 

Each of these properties represents a different aspect of the quantum state of a quark, and these states are subject to change due to interactions and decays, according to the rules of the Standard Model of particle physics.

 

In quantum mechanics, elementary particles such as quarks exhibit properties of both particles and waves, a concept known as wave-particle duality.

 

As "particles", quarks can be thought of as discrete entities. They are fundamental constituents of matter and are the building blocks for composite particles like protons and neutrons. Each quark carries specific quantum properties like electric charge, colour charge, spin, and others.

 

As "waves", the behaviour of quarks is described by a wavefunction, a mathematical description that encapsulates all the possible states of a quantum system. It provides a probabilistic description of where the quark might be found in space, and this wavefunction can exhibit interference and diffraction, much like classical waves.

 

The wave function is defined by the Schrödinger Equation :

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Here, Ψ(x,t)  is a wave function, a function that assigns a complex number to each point x at each time t. The parameter m is the mass of the particle, and V(x,t) is the potential that represents the environment in which the particle exists. [1] The constant i is the imaginary unit, and ħ is the reduced Planck constant, which has units of action (energy multiplied by time).

 

When an observation is made, the quark appears at a specific point, which reveals its particle nature. But the pattern of its possible locations is determined by its wave nature. This is a key part of the Copenhagen interpretation of quantum mechanics, which says that a quantum particle exists in all possible states (described by the wavefunction) until it is observed, at which point it "collapses" into one state.

 

So, a quark, like all quantum entities, exhibits properties of both particles and waves, and which properties are evident depends on how the quark is observed or measured. This is a fundamental aspect of quantum mechanics and is very different from classical physics, where objects are either particles or waves, but not both.

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Dynamics of the quarks

 

The dynamics of the quarks and gluons are controlled by the quantum chromodynamics Lagrangian. The gauge invariant QCD Lagrangian is

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where Ψi(x) is the quark field, a dynamical function of spacetime, in the fundamental representation of the SU(3) gauge group, indexed by i and j running from 1 to 3; Dμ is the gauge covariant derivate; the ϒμ  are Gamma matrices connecting the spinor representation to the vector representation of the Lorentz group.

Herein, the gauge covariant derivative (Dμ)ij = ∂μδij – ig(Ta)ijAaμ couples the quark field with a coupling strength g to the gluon fields via the infinitesimal SU(3) generators Ta in the fundamental representation. An explicit representation of these generators is given by Ta = λa/2, wherein the λa (a=1….8) are the Gell-Mann matrices.

The symbol Gaμν  represents the gauge invariant gluon field strength tensor, analogous to the electromagnetic field strength tensor,  Fμν, in quantum electrodynamics. It is given by:

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Where Aaμ (x) are the gluon fields, dynamical functions of spacetime, in the adjoint representation of the SU(3) gauge group, indexed by a, b and c running from 1 ti 8; and fabc are the structure constants of SU(3).

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The variables m and g correspond to the quark mass and coupling of the theory, respectively, which are subject to renormalization.

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Creation of quarks

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Quarks can be created through high-energy processes or reactions, often involving particle accelerators. Here are some examples:

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Particle Collisions in Accelerators: Particle accelerators, such as the Large Hadron Collider (LHC), can accelerate particles to near the speed of light and then cause them to collide. These collisions are high-energy enough to create quarks and other particles. In fact, the top quark was first observed in 1995 at Fermilab's Tevatron accelerator in a proton-antiproton collision.

Cosmic Ray Interactions: High-energy cosmic rays from space can collide with particles in the Earth's atmosphere, leading to the creation of quarks and other particles.

Particle Decay: Some particles can decay into quarks. For instance, Z bosons (a type of force-carrying particle) can decay into a quark and an antiquark.

Big Bang: In the moments following the Big Bang, the universe was extremely hot and dense, creating a state of matter known as a quark-gluon plasma. As the universe expanded and cooled, these quarks combined to form protons and neutrons.

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It's important to note that quarks are always produced in pairs (a quark and an antiquark), to conserve quantum numbers such as baryon number, charge, and color charge. Quarks are also never observed individually because of a property known as color confinement, which means they're always found within composite particles like protons, neutrons, and mesons. This is due to the nature of the strong force, which binds quarks together.

In the next chapters, the creation processes of the individual quarks will be illustrated with some examples. The examples are just a limited list of all possibilities. 

Literature

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Online Library

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References

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[1]  Zwiebach, Barton (2022). Mastering Quantum Mechanics: Essentials, Theory, and Applications. MIT Press. ISBN 978-0-262-04613-8. OCLC 1347739457.

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