# QCD Phases and Transitions: From Hadrons to Quark-Gluon Plasma

## Foundations of Quantum Chromodynamics PDF 12: A Comprehensive Guide

Quantum chromodynamics (QCD) is the theory that describes how quarks and gluons interact to form the matter we see around us. It is one of the pillars of the standard model of particle physics, which explains the fundamental forces and particles in nature. QCD is also a fascinating subject that reveals the richness and complexity of quantum field theory, which is the mathematical language of modern physics.

## foundations of quantum chromodynamics pdf 12

In this article, we will explore the foundations of quantum chromodynamics in a comprehensive way. We will cover the basic concepts, the mathematical framework, the experimental tests, and the open questions and future directions of QCD. We will also provide a link to a PDF file that contains more details and references for those who want to dive deeper into this topic. Whether you are a student, a researcher, or a curious reader, we hope this article will help you understand and appreciate the beauty and power of quantum chromodynamics.

## The Basic Concepts of Quantum Chromodynamics

Before we delve into the technical aspects of QCD, let us first review some of the basic concepts that underlie this theory. These concepts are essential for grasping the physical meaning and intuition behind QCD.

### Quarks and Gluons: The Building Blocks of Matter

According to QCD, all matter is made up of quarks and gluons. Quarks are elementary particles that come in six types or flavors: up, down, charm, strange, top, and bottom. Each quark has an electric charge that is either +2/3 or -1/3 times the charge of an electron. Quarks also have another property called color charge, which we will explain shortly.

Gluons are also elementary particles that mediate the strong interaction between quarks. Gluons are massless and electrically neutral, but they also carry color charge. There are eight types or colors of gluons, which we can label as red-antired, blue-antiblue, green-antigreen, red-antiblue, red-antigreen, blue-antired, blue-antigreen, and green-antired.

Quarks and gluons are bound together by the strong interaction to form composite particles called hadrons. There are two types of hadrons: baryons and mesons. Baryons are made up of three quarks, such as the proton (uud) and the neutron (udd). Mesons are made up of a quark and an antiquark, such as the pion (ud) and the kaon (us). Hadrons are the observable particles that we can detect in experiments, while quarks and gluons are confined inside them.

### Color Charge and Confinement: The Fundamental Forces of QCD

Color charge is the property that distinguishes quarks and gluons from other elementary particles. It is analogous to electric charge, but it has three values instead of two: red, green, and blue. Antiquarks and antigluons have the opposite color charge: antired, antigreen, and antiblue. Color charge is conserved in all QCD processes, meaning that the total color charge before and after a reaction is the same.

The strong interaction between quarks and gluons is governed by color charge. Quarks can exchange gluons with other quarks or with themselves, changing their color in the process. Gluons can also interact with other gluons, creating or annihilating quark-antiquark pairs. The strength of the strong interaction depends on the distance between the particles: it is weak at short distances and strong at long distances.

This leads to a remarkable phenomenon called confinement, which means that quarks and gluons cannot exist as free particles. They are always bound together by gluons to form colorless hadrons. If we try to separate a quark from a hadron, the energy required to do so will be so large that it will create new quark-antiquark pairs, resulting in more hadrons. This is why we never observe isolated quarks or gluons in nature.

### Asymptotic Freedom and Renormalization: The Key Features of QCD

The opposite behavior of the strong interaction at short and long distances is known as asymptotic freedom. It means that quarks and gluons behave almost like free particles when they are very close to each other, but they become strongly coupled when they are far apart. Asymptotic freedom is a unique feature of QCD that makes it different from other quantum field theories.

Asymptotic freedom also implies that QCD has a running coupling constant, which is a parameter that measures the strength of the interaction. The coupling constant depends on the energy scale or momentum transfer of the process: it decreases as the energy increases and vice versa. This means that QCD is a theory that can describe both weakly and strongly interacting phenomena, depending on the energy regime.

However, asymptotic freedom also poses a challenge for calculating QCD processes. When the coupling constant is small, we can use a method called perturbation theory, which is based on expanding the result in powers of the coupling constant. This works well for high-energy processes, such as deep inelastic scattering or jet production. But when the coupling constant is large, perturbation theory breaks down and we need to use other methods, such as lattice QCD or Monte Carlo simulation. These methods are more complicated and computationally intensive than perturbation theory.

Another challenge for QCD is renormalization, which is a procedure that removes the infinities that arise in quantum field theory calculations. These infinities are due to the fact that quantum field theory treats particles as point-like objects with no size or structure. This leads to divergences when we try to calculate processes involving very short distances or very high energies.

Renormalization solves this problem by introducing some physical parameters, such as masses and charges, that absorb these infinities and make the results finite and observable. These parameters depend on the energy scale of the process and need to be determined experimentally. Renormalization also ensures that QCD is consistent with special relativity and quantum mechanics.

## The Mathematical Framework of Quantum Chromodynamics

Now that we have reviewed some of the basic concepts of QCD, let us turn our attention to the mathematical framework that describes this theory. We will introduce some of the tools and methods that are used to formulate and calculate QCD processes.

### The Lagrangian and the Feynman Rules: The Tools for Calculating QCD Processes

muon) collides with a hadron (such as a proton or a neutron) and transfers a large amount of momentum to it. DIS is a powerful tool for studying the quark structure of hadrons, as it reveals how the quarks are distributed and how they carry the momentum and spin of the hadron.

Parton distribution functions (PDFs) are functions that describe the probability of finding a quark or a gluon with a given fraction of the hadron's momentum and spin. PDFs are extracted from DIS experiments by comparing the measured cross sections with the theoretical predictions based on QCD. PDFs are universal and process-independent, meaning that they can be used to calculate other processes involving hadrons, such as hadron-hadron collisions or hadron production.

DIS experiments have provided strong evidence for the existence and properties of quarks and gluons. They have also revealed some surprising and intriguing features of QCD, such as: - The scaling and Bjorken limit: The cross sections of DIS depend only on the ratio of the momentum transfer and the hadron mass, not on the individual values. This implies that quarks and gluons behave like free particles at high energies, which is a manifestation of asymptotic freedom. - The Callan-Gross relation: The cross sections of DIS satisfy a simple relation that implies that quarks have spin 1/2 and that they do not carry orbital angular momentum inside the hadron. - The EMC effect: The cross sections of DIS for nuclei are different from those for nucleons, implying that the quark structure of nucleons is modified by the nuclear environment. - The Gottfried sum rule violation: The cross sections of DIS for protons and neutrons differ by more than expected, implying that there is an excess of down quarks over up quarks in the proton sea. - The spin crisis: The cross sections of DIS for polarized protons and neutrons show that the quarks carry only a small fraction of the proton spin, implying that there are other sources of spin in QCD, such as gluons or orbital angular momentum.

### Jet Production and Fragmentation Functions: The Signatures for Gluon Dynamics

Jet production is a process where two high-energy particles (such as leptons, photons, or hadrons) collide and produce two or more collimated sprays of particles (called jets) in opposite directions. Jet production is a sensitive probe for gluon dynamics, as it reveals how gluons are emitted and absorbed by quarks and how they interact with each other.

Fragmentation functions are functions that describe the probability of finding a hadron with a given fraction of the jet's momentum and spin. Fragmentation functions are extracted from jet production experiments by comparing the measured jet spectra with the theoretical predictions based on QCD. Fragmentation functions are universal and process-independent, meaning that they can be used to calculate other processes involving jets, such as jet-hadron collisions or jet decay.

Jet production experiments have provided strong evidence for the existence and properties of gluons and their interactions. They have also revealed some surprising and intriguing features of QCD, such as: - The Drell-Yan process: The production of lepton pairs from hadron-hadron collisions is dominated by quark-antiquark annihilation at low energies, but by gluon-gluon fusion at high energies, implying that gluons carry most of the hadron momentum at high energies. - The three-jet events: The production of three jets from electron-positron collisions is a clear signature of gluon emission from quarks, implying that gluons are spin 1 particles that have color charge. - The jet algorithm dependence: The number and shape of jets depend on how they are defined and identified, implying that jets are not physical objects but mathematical constructs that depend on arbitrary parameters. - The jet quenching: The production of jets from heavy ion collisions is suppressed compared to those from proton-proton collisions, implying that jets lose energy as they traverse the hot and dense medium created by the collision, which is called quark-gluon plasma. - The jet substructure: The production of jets from proton-proton collisions at very high energies shows that jets have internal structure and hierarchy, implying that jets can be decomposed into smaller subjets that reflect the QCD radiation pattern.

### Heavy Quarkonium and Quark-Gluon Plasma: The Windows for QCD Phases

Heavy quarkonium is a bound state of a heavy quark and its antiquark, such as the J/psi (cc) or the Upsilon (bb). Heavy quarkonium is a useful probe for QCD phases, as it reveals how the strong interaction changes with temperature and density.

Quark-gluon plasma (QGP) is a state of matter where quarks and gluons are deconfined and form a hot and dense fluid. QGP is expected to exist at very high temperature and density, such as in the early universe or in heavy ion collisions. QGP is a challenging and exciting subject for QCD, as it involves non-perturbative and collective phenomena that are difficult to understand and predict.

Heavy quarkonium and QGP experiments have provided strong evidence for the existence and properties of QCD phases and their transitions. They have also revealed some surprising and intriguing features of QCD, such as: - The J/psi suppression: The production of J/psi from heavy ion collisions is suppressed compared to those from proton-proton collisions, implying that J/psi is dissociated by the QGP, which is a signal of deconfinement. - The Upsilon survival: The production of Upsilon from heavy ion collisions is not suppressed as much as J/psi, implying that Upsilon is more bound and more resistant to QGP, which reflects the hierarchy of quarkonium states. - The charmonium regeneration: The production of J/psi from heavy ion collisions at very high energies is enhanced compared to those from proton-proton collisions, implying that J/psi is regenerated by the recombination of charm quarks and antiquarks from the QGP, which is a signal of thermalization. - The quarkonium polarization: The production of quarkonium from hadron-hadron collisions shows that quarkonium has a preferential orientation or polarization, implying that quarkonium is produced by different mechanisms at different energies, such as color-singlet or color-octet models. - The quarkonium correlation: The production of quarkonium pairs from hadron-hadron collisions shows that quarkonium pairs have a strong correlation or correlation, implying that quarkonium pairs are produced by double parton scattering or double gluon fusion.

## The Open Questions and Future Directions of Quantum Chromodynamics

After we have discussed the experimental tests of QCD, let us now look at some of the open questions and future directions that motivate and inspire further research in this field. We will highlight some of the most important and interesting challenges and opportunities for QCD.

### The Origin of Mass and the Higgs Mechanism: The Mystery of QCD Symmetry Breaking

The origin of mass is one of the most fundamental and puzzling questions in physics. According to QCD, most of the mass of hadrons comes from the kinetic energy and the interaction energy of quarks and gluons, not from their intrinsic masses. This is because the intrinsic masses of quarks are very small compared to the hadron masses, except for the top quark, which is very heavy.

The origin of the intrinsic masses of quarks (and other elementary particles) is explained by the Higgs mechanism, which is a process that breaks the electroweak symmetry and gives mass to particles through their interaction with the Higgs field. The Higgs mechanism is confirmed by the discovery of the Higgs boson at the Large Hadron Collider (LHC) in 2012.

However, the origin of the Higgs mechanism itself is still unknown. Why does the electroweak symmetry break? What is the nature of the Higgs field and the Higgs boson? How does the Higgs mechanism relate to QCD and other forces? These are some of the questions that remain unanswered and that require further theoretical and experimental investigation.

### The Proton Spin and the Axion: The Puzzle of QCD Anomalies

the proton spin is very small, about 30%, leaving a large gap that needs to be explained by the gluon spin and orbital angular momentum contributions.

The proton spin puzzle is related to another puzzle in QCD, called the strong CP problem. CP is a symmetry that combines charge conjugation (C) and parity (P), which are transformations that change the sign of electric charge and spatial coordinates, respectively. CP is violated by the weak interaction, but it is expected to be conserved by the strong interaction.

However, QCD allows for a term in the Lagrangian that violates CP, called the theta term. The theta term has a parameter called theta angle, which measures the strength of CP violation. If the theta angle is nonzero, QCD predicts that the neutron has an electric dipole moment, which is a measure of the separation of positive and negative charges inside it. However, experiments show that the neutron electric dipole moment is very small, implying that the theta angle is very close to zero.

The strong CP problem is why the theta angle is so small, or why CP is conserved by QCD. One possible solution to this problem is the existence of a hypothetical particle called the axion, which is a light and weakly interacting particle that can dynamically adjust the theta angle to zero and restore CP symmetry. The axion is also a candidate for dark matter, which is a mysterious form of matter that makes up most of the mass of the universe but does not interact with light or ordinary matter.

### The Dark Matter and the Strong CP Problem: The Challenge of QCD Beyond the Standard Model

Dark matter is one of the most important and intriguing questions in physics and cosmology. According to observations, dark matter accounts for about 85% of the matter in the universe, but its nature and origin are unknown. Dark matter does not emit or absorb light or any other electromagnetic radiation, but it interacts with ordinary matter through gravity and possibly other forces.

Dark matter is one of the main motivations for physics beyond the standard model, which is a quest for new theories and phenomena that can explain the unresolved puzzles and mysteries in physics. The standard model is a theory that describes all the known elementary particles and forces (except gravity), including QCD and the electroweak theory. However, the standard model is not complete or satisfactory, as it leaves many questions unanswered and unexplained.

One of the candidates for physics beyond the standard model is supersymmetry, which is a symmetry that relates fermions (such as quarks and leptons) and bosons (such as gluons and photons). Supersymmetry predicts the existence of new particles that are superpartners of the standard model particles, with the same quantum numbers but different spin. Supersymmetry can solve some of the problems of the standard model, such as stabilizing the Higgs mass and unifying the forces at high energies.

One of the superpartners predicted by supersymmetry is called the neutralino, which is a mixture of four particles: the bino, the wino, and two higgsinos. The neutralino is a candidate for dark matter, as it is electrically neutral, colorless, and stable. The neutralino can interact with ordinary matter through weak and gravitational forces, and it can be produced or detected in high-energy collisions or astrophysical processes.

the axion, which have similar properties but different origins. ALPs are also light and weakly interacting particles that can solve the strong CP problem and be candidates for dark matter. ALPs can interact with ordinary matter through photons, electrons, or nucleons, and they can be produced or detected in high-energy collisions or astrophysical processes.

## Conclusion

In this article, we have explored the foundations of quantum chromodynamics in a comprehensive way. We have covered the basic concepts, the mathematical framework, the experimental tests, and the open questions and future directions of QCD. We have also provided a link to a PDF file that contains more details and references for those who want to dive deeper into this topic.

Quantum chromodynamics is a fascinating and challenging subject that reveals the richness and complexity of quantum field theory and the nature of matter. QCD is also a successful and powerful theory that explains the fundamental forces and particles in nature. QCD is not only a theory of quarks and gluons, but also a theory of hadrons, nuclei, stars, and the universe.

We hope this article has helped you understand and appreciate the beauty and power of quantum chromodynamics. If you have any questions or comments, please feel free to contact us. Thank you for reading!

## FAQs

Here are some common questions and answers about quantum chromodynamics.

#### Q: What is quantum chromodynamics?

A: Quantum chromodynamics (QCD) is the theory that describes how quarks and gluons interact to form the matter we see around