Introduction to Hadron Physics and the Early Universe
Lecture number one will be dedicated to the introduction of hadron physics. I would like to start by explaining the origin of hadron physics and how it appeared at the earlier stages of the universe, where it plays a role in the evolution of our universe.
We will discuss the matter composition and how quark-gluon physics plays its part. The Standard Model will be briefly covered, followed by the equations describing the motion of fields. We will also talk about the \(SU(3)\) gauge group and the key property of strong interaction known as confinement, which is crucial for matter formation.
Basic equations will be provided to help solve exercises, and we will conclude with logistics. There is additional material prepared for the end of the lecture.
This course is being offered for the second time, and I will reuse material from last year’s lectures. Last year, we experimented with recording lectures and converting them into text. We are working on automating this process and gathering more material for testing.
I would like to ask if anyone objects to me recording this lecture using my iPhone. If there are no objections, the recordings will benefit you as well, as the plan is to provide transcribed lecture text. Additionally, if someone takes pictures of the board, those will be processed by a language model to generate formatted notes.
To begin, we will establish a timeline for the universe, marking key epochs and stages. The numbers involved are often extremely large or small, making them difficult to visualize. For reference, the current age of the universe is estimated at 14 billion years.
The Big Bang and Early Universe Timeline
Here was the Big Bang. The point where the sound reaches the first row (one meter away) corresponds to a time of \(0.003 \ \text{s}\), since the speed of sound is about \(300 \ \text{m/s}\). If my clap represented the Big Bang, by the time the sound reached you, most of the early universe’s stages had already passed.
We will discuss hadron physics, which becomes relevant around this time. The timeline scales we consider are \(10^{-6} \ \text{s}\), \(10^{-12} \ \text{s}\), and then up to one second. At the very beginning, matter is produced, though we do not yet know its exact form. We assume the seeds of structure were already formed during inflation, the rapid expansion of space and time.
By \(t \approx 10^{-12} \ \text{s}\), the electroweak scale and Planck scale have been crossed. The Higgs potential has developed its minimum, and the universe has collapsed into the lower energy state. At this point, matter exists as a quark-gluon plasma (QGP), where the dots represent quarks and the fields are gluons.
\[ \text{QGP: Quarks and gluons as free particles in a high-temperature "soup"} \]
Between \(10^{-6} \ \text{s}\) and \(10^{-12} \ \text{s}\), this plasma evolves into structured matter through hadronization. By the time the sound reaches you, the universe already has elementary matter blocks: mesons and baryons. Quarks are now confined, and gluon fields are mostly localized within these particles.
Confinement is the property of strong interaction that binds quarks into hadrons (e.g., protons, neutrons).
Beyond this point, the universe undergoes 14 billion years of further evolution. A notable milestone is the one-second mark, where nucleosynthesis begins.
\[ t \approx 1 \ \text{s}: \text{Onset of Big Bang Nucleosynthesis (BBN)} \]
The Cosmic Microwave Background (CMB) emerges much later, as the universe cools and becomes transparent to radiation.
\[ \text{CMB: Remnant radiation from the early universe, observed today} \]
This is the condensed picture of the early universe’s evolution. The rest is the 14 billion years of cosmic history that follow.
Atomic Structure and Electron Orbitals
Somewhere here I mark one second, which is a good mark, where the nuclear synthesis starts. Another visually pleasant indication is when radiation separates from the matter. Big Bang nuclear synthesis and then radiation separates from the matter is around 400,000 years after the Big Bang. We’ve got pioneering Kanes and baryons.
For the rest of the semester, what will happen with our universe, which we just created? There will be some nuclear synthesis starting, but we won’t get to the forming of atoms with electron shells until much later, around 400,000 billion years after.
Now, let’s discuss the atom. What is the most abundant element on the crust of the earth? By mass, the most abundant element is iron, but in the crust, it is oxygen. Oxygen is an element with eight protons and eight neutrons, forming a compact object in the center of the atom — the nucleus.
The size of a single proton or neutron is roughly one fermi ( \(1 \, \text{fm}\) ). When 16 of these are packed together, the nuclear radius is approximately:
\[ r_{\text{nuc}} \approx 3 \, \text{fm} \]
This gives a diameter of about six fermi.
The electrons occupy shells of the atom, described by the electron configuration:
\[ 1s^2 2s^2 2p^4 \]
- \(1s^2\): Two electrons in the 1s orbital (spherically symmetric).
- \(2s^2\): Two electrons in the 2s orbital (also spherical).
- \(2p^4\): Four electrons in the 2p orbital (dumbbell-shaped).
The energy of electrons increases with angular momentum, with the 1s orbital being the lowest in energy. The wave function of the 1s orbital is:
\[ \psi \propto e^{-r/a_0} \]
where \(a_0\) is the Bohr radius, representing the size of the atom. Solving the Schrödinger equation for an electron bound in the electromagnetic field of the nucleus gives:
\[ a_0 \approx 59,000 \, \text{fm} = 5.9 \times 10^{-11} \, \text{m} \]
The charge of the nucleus affects the radius. For an oxygen nucleus ( \(Z = 8\) ), the effective radius scales inversely with \(Z\), but outer electrons are screened by inner electrons, reducing the effective charge they experience.
The outer electron shells extend to several hundred picometers, much larger than the nucleus itself.
Scaling Atomic Radii and Effective Charge
The outer shells of an atom extend to roughly several hundred picometers, with a clear estimation of 100. To visualize this, I compared it to our solar system. Here is Earth, and if I scale the radius of the nucleus to the orbital shells, it matches the ratio of the sun’s radius to Earth’s orbit. The sun must be scaled down by a factor of 150 to match the distances in the atom.
The average radius for the 1s orbital is:
\[ \langle r \rangle_{1s} = \frac{3}{2} a_0 \]
For the 2s orbital, it is:
\[ \langle r \rangle_{2s} = 6 a_0 \]
When calculating these values, I divided by the effective charge. For the 2s orbital, the effective charge is roughly 4 due to screening effects (starting with \(Z = 8\) but reduced by inner electrons). This gives a radius close to \(a_0\), which is why \(a_0\) serves as a reference for atomic size.
The Bohr radius \(a_0\) represents the characteristic size of an atom, approximately \(5.9 \times 10^{-11} \ \text{m}\). The effective charge experienced by outer electrons is reduced due to electron shielding.
Introduction to the Standard Model and Its Accuracy
The Standard Model is the most accurate and precise theory in particle physics, describing everything we have observed so far. It is so accurate that for the past 10 years, experiments at CERN and elsewhere have tried to find deviations from its predictions, but none have been found.
However, there are unresolved problems with the Standard Model, particularly at large scales and in the context of cosmic evolution. Questions remain about the naturalness of certain couplings and parameters, which are measured precisely but lack a fundamental explanation.
The Standard Model can be thought of as the electroweak sector combined with QCD (quantum chromodynamics). Its success as a theory does not imply completeness.
The Standard Model’s parameters are empirically determined, but their origins remain unexplained, posing open questions for theoretical physics.
Quantum Chromodynamics and Strong Interaction
The fact that the Standard Model is a good theory and works well does not mean we fully understand it. This particularly applies to quantum chromodynamics (QCD), the theory describing the strong interaction. Everything discussed in hadron physics is governed by the strong interaction, and QCD is the theory of the color charge.
While QCD works very well for high-energy phenomena, it is so complex that until recently, direct predictions from its fundamental equations to real-world observables were not possible. The reason, which we will discuss later, is that the fundamental interaction describes the behavior of quarks and gluons—the theory’s building blocks.
However, at lower energies, the theory transitions into a different regime where particles like hadrons (e.g., protons, neutrons) emerge as color-neutral objects. These are not fundamental particles but composite states where the color charge is confined.
In this course, we will explore how QCD, as a fundamental theory, manifests in the form of hadrons—objects where the theory’s charge is confined and only neutral combinations are observable.
Introduction to Standard Model Particles and Charges
In this course, we will explore the relationship between fundamental particles and the effective interactions that emerge between them. The Standard Model can be broadly divided into two sectors: the electroweak sector and the strong interaction sector.
The electroweak sector consists of three parts:
1. Electromagnetic interaction
2. Weak interaction
3. Mass-energy sector (Higgs mechanism)
Let’s review the particles and their classifications:
- Higgs boson: Belongs to the Higgs sector (mass-energy).
- W boson: Carrier of the weak interaction.
- Z boson: Also part of the weak sector.
- Photon (\(\gamma\)): Mediator of electromagnetic interactions.
- Quarks (e.g., top, charm): Governed by quantum chromodynamics (QCD), the theory of strong interactions.
The classification can sometimes appear vague because quarks are discussed in QCD, while leptons are part of electroweak theory. Photons mediate electromagnetism, and W/Z bosons mediate weak interactions.
The Higgs boson stands as a separate field, but particles interact based on their charges:
- If a particle carries an electric charge, it couples to photons.
- If it has weak charge, it interacts via W/Z bosons.
- Color-charged particles (quarks, gluons) interact through the strong force.
Quarks are elementary particles that form hadrons (e.g., protons, neutrons) at low energies due to color confinement. Earlier in the universe, quarks existed freely before hadronization occurred.
The Standard Model’s structure ensures that interactions are mediated by force carriers corresponding to the charges of the particles involved.
Quark Generations and Charge Classification
The charge of a particle determines how it interacts with force carriers. If you forget the charge of a quark, refer to this diagram:
- Top row quarks: up, charm, top
- Bottom row quarks: down, strange, bottom
The quarks are organized into three generations:
1. First generation: up (\(u\)), down (\(d\))
2. Second generation: charm (\(c\)), strange (\(s\))
3. Third generation: top (\(t\)), bottom (\(b\))
The electric charge depends on whether the quark is in the top or bottom row:
- Top row: \(+\frac{2}{3}\)
- Bottom row: \(-\frac{1}{3}\)
For weak interactions, the weak isospin charge differs:
- Upper quarks: \(+\frac{1}{2}\)
- Lower quarks: \(-\frac{1}{2}\)
The location in the diagram (top/bottom row) directly determines the electric and weak charges of the quark.
Quark Interactions and Charge Coupling
The upper quark has a big charge of \(+1\) and a weak charge of \(+\frac{1}{2}\), while the lower quark has a weak charge of \(-1\).
Quarks are the most diverse particles in the Standard Model because they couple to all charges:
- Electromagnetic charge: Enables interaction with photons (e.g., light from a window).
- Weak charge (big charge): Allows interaction with \(W\) and \(Z\) bosons.
- Strong charge (color charge): Mediates interaction with gluons.
The terms “color charge” and “strong charge” are interchangeable, similar to how an electron’s charge can be positive or negative.
For example, a quark can interact with:
1. Light (via electromagnetic charge),
2. \(W/Z\) bosons (via weak charge),
3. Gluons (via color charge).
Introduction to Color Charge and Strong Interaction
Just like an electron can have a positive or negative charge, the color charge can also be positive or negative. However, we use the labels red, green, and blue for color charges. Similarly, antiparticles carry anti-red, anti-green, or anti-blue charges.
You will quickly get used to referring to the strong charge as a color charge. In the context of hadron physics, we discuss the strong interaction, which is also called the color interaction.
When we talk about interactions, we mean the interaction of an object with a field. These fields are represented by force carriers:
- \(W\) and \(Z\) bosons for the weak interaction,
- Photons for the electromagnetic interaction,
- Gluons for the strong (color) interaction.
Lagrangian Mechanics and Field Theory Introduction
The standard framework in field theory to describe fields and interactions between particles and carriers starts with the Lagrangian, an expression that describes interactions in a condensed form.
Here is an example: You might recall Lagrangian mechanics from your first or second semester, where the entire motion of a system was condensed into a single equation — the Lagrange equation. It consists of a kinetic term and a potential term. When you subtract them, you get an expression representing the system’s energy (kinetic minus potential), which determines the equation of motion.
The evolution of the system from an initial state is described by an equation derived from the Lagrangian. To obtain it, you:
1. Differentiate the Lagrangian with respect to velocity (\(\dot{q}\)),
2. Subtract the term where the Lagrangian is differentiated by the coordinate (\(q\)).
For example, consider a system where a point mass slides without friction, connected to a pendulum with another mass. The equation of motion is a differential equation found by applying the classical Euler-Lagrange equation:
\[ \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}} \right) - \frac{\partial L}{\partial q} = 0 \]
This equation will be crucial for field theory, where we extend these principles to describe fields and their dynamics.
Introduction to Quantum Electrodynamics (QED) Lagrangian
The QED Lagrangian describes how light interacts with charged particles, such as electrons, muons, or quarks. This is relevant for two reasons: first, quarks carry charge and thus interact with photons, and second, the QED Lagrangian is simpler than that of Quantum Chromodynamics (QCD), making it a good starting point.
The Lagrangian is a function of two fundamental fields:
- \(\psi\): The fermion field (e.g., electron, muon, quark).
- \(A_\mu\): The photon field (gauge boson).
The Lagrangian is a scalar quantity, meaning it evaluates to a single number at any point in spacetime. This is achieved by ensuring all indices are properly contracted.
For example, the QED Lagrangian includes terms like:
\[ \mathcal{L} = \bar{\psi}(i \gamma^\mu D_\mu - m)\psi - \frac{1}{4} F_{\mu \nu} F^{\mu \nu} \]
- \(D_\mu = \partial_\mu - i e A_\mu\): The covariant derivative (incorporates electromagnetic interaction).
- \(F_{\mu \nu} = \partial_\mu A_\nu - \partial_\nu A_\mu\): The electromagnetic field strength tensor.
- \(\gamma^\mu\): Dirac matrices.
The structure ensures gauge invariance under \(U(1)\) transformations.
The indices follow Einstein summation convention — repeated indices imply summation.
The Lagrangian mechanics framework extends to field theory, where the Euler-Lagrange equation governs dynamics:
\[ \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}} \right) - \frac{\partial L}{\partial q} = 0 \]
This principle applies to fields, replacing coordinates \(q\) with field configurations.
Matrix Operations and Indices in QED Lagrangian
We use the Einstein summation convention, where repeated indices imply summation. For example, in the term \(F_{\mu \nu} F^{\mu \nu}\), the indices \(\mu\) and \(\nu\) are contracted. Here, \(\mu\) is a Lorentz index spanning 4 dimensions (3 spatial coordinates \(x, y, z\) and time). The contraction ensures we sum over \(\mu\) and \(\nu\) from 1 to 4.
The object \(F_{\mu \nu}\) is a matrix — specifically, a 4x4 tensor — because \(\mu\) and \(\nu\) each range over 4 dimensions. When multiplying such matrices, we do not perform standard matrix multiplication. Instead, we multiply component-wise and then sum all elements (similar to broadcasting in Python and taking the trace).
The components of \(F_{\mu \nu}\) are computed as:
\[ F_{\mu \nu} = \partial_\mu A_\nu - \partial_\nu A_\mu \]
where \(\partial_\mu\) is the derivative in spacetime (e.g., \(\partial_t\) for time, \(\partial_x\) for spatial coordinates).
If \(F_{\mu \nu}\) were a vector, contracting indices would be straightforward (like a dot product). However, \(F_{\mu \nu}\) is a matrix, and the contraction involves additional suppressed indices. These hidden indices ensure the result is a scalar quantity, as required for the Lagrangian.
The summation convention simplifies notation but requires careful tracking of indices to avoid ambiguity.
The challenge arises when dealing with terms like \(F_{\mu \nu} F^{\mu \nu}\), where both objects are matrices. The contraction must account for all implicit indices to produce a scalar. This is achieved by ensuring every index is paired and summed over appropriately.
Spinor Fields and Matrix Indices in QED Lagrangian
There is another set of indices that I suppressed, and they are the matrix indices \(\tau\) and \(\rho\). The question is: in which space do they reside? Here, \(\mu\) is the Lorentz index, related to relativity and spanning four dimensions. These four dimensions arise because particles are not scalar—they have spin. Thus, \(\tau\) and \(\rho\) correspond to the dimensions of spin. This is why the fermion field \(\psi\) has four components.
These indices are not Lorentz indices; they are matrix indices. For matrix indices, the distinction between covariant and contravariant does not apply—we simply sum over them. However, there is still an issue in the Lagrangian because we are adding a matrix and a scalar, which is not mathematically consistent. The missing piece is the diagonal matrix, which resolves this inconsistency.
The field \(\psi\) is a four-component spinor, while \(\bar{\psi}\) is not a four-component spinor but rather a row vector.
The distinction between Lorentz indices and matrix indices is crucial. Lorentz indices (\(\mu, \nu\)) follow the rules of relativity, while matrix indices (\(\tau, \rho\)) are internal to the spinor structure.
The Lagrangian must account for these indices properly to ensure mathematical consistency, particularly when combining terms involving matrices and scalars. The diagonal matrix ensures that operations between spinors and other quantities are well-defined.
Matrix Operations and Indices in QCD Lagrangian
The fermion field \(\psi\) is not a four-component spinor here but a row vector. To construct it, we take the conjugate transpose (dagger) of \(\psi\) and multiply it by the gamma matrix \(\gamma^0\) from the left, ensuring it remains a row of numbers. This allows contraction with other matrices in the Lagrangian.
One of the exercises is to analyze the same structure for the QCD Lagrangian, focusing on the dimensions of the objects involved. The QCD case introduces additional indices, but the logic remains similar.
The field strength tensor in QCD is given by:
\[ G^{\mu \nu}_a = \partial^{\mu} A^{\nu}_a - \partial^{\nu} A^{\mu}_a + g f_{abc} A^{\mu}_b A^{\nu}_c \]
Terms:
- \(G^{\mu \nu}_a\): Field strength tensor for QCD (non-Abelian gauge theory).
- \(A^{\mu}_a\): Gauge field with color index \(a\).
- \(f_{abc}\): Structure constants of the SU(3) group.
- \(g\): QCD coupling constant.
The dimensionality of the objects is crucial. The new object \(\lambda\) (Gell-Mann matrices) is three-dimensional, and the indices \(i, j\) correspond to the fundamental representation of SU(3). These indices appear in the contraction over \(A\), and the trace is taken in the \(ij\) space.
The Lagrangian involves matrices in both Lorentz (\(\mu, \nu\)) and color (\(i, j\)) spaces. For example, the term \(\text{Tr}(G^{\mu \nu} G_{\mu \nu})\) requires tracing over the color indices while respecting the Lorentz structure.
The distinction between Lorentz indices (\(\mu, \nu\)) and color indices (\(i, j, a, b, c\)) is essential. Lorentz indices span spacetime, while color indices belong to the internal SU(3) symmetry group.
The exercise involves recovering the indices and counting the number of terms, which is straightforward once the structure is clear. The additional indices in QCD compared to QED arise from the non-Abelian nature of the theory, introducing terms like \(f_{abc} A^{\mu}_b A^{\nu}_c\) in the field strength tensor.
The fermion field in QCD also carries color indices, e.g., \(\psi_i\), where \(i = 1, 2, 3\). The covariant derivative becomes:
\[ D_{\mu} \psi_i = \partial_{\mu} \psi_i - i g A^{\mu}_a (T_a)_{ij} \psi_j \]
Terms:
- \((T_a)_{ij}\): Generators of SU(3) in the fundamental representation.
- \(g\): QCD coupling constant.
This structure ensures gauge invariance and proper contraction of color indices.
Introduction to Flavor and Color Indices in QCD
The index \(j\) runs up to 3, and I will clarify this equation shortly. All indices must be properly introduced, and it’s important to ensure the mathematical structure of the equations is correct. Everyone should be able to follow this.
The index \(F\) traces the flavors of quarks: \(U, D, S, C, T, B\). Since there are six quark flavors, \(F\) spans all six possibilities.
The index \(I\) tracks the color charge, which has three dimensions (red, green, blue). The spinor indices (related to spin projections) are not explicitly included here to avoid excessive complexity.
For example, consider a quark field:
- Fix the flavor to “up” (\(U\)).
- Fix the color to “red”.
- It still has four spinor components corresponding to spin projections.
The fermion field \(\psi\) is not just a four-component spinor but also carries flavor and color indices. The QCD Lagrangian introduces additional structure due to these indices, but the underlying logic remains similar to QED.
The field strength tensor in QCD is given by:
\[ G^{\mu \nu}_a = \partial^{\mu} A^{\nu}_a - \partial^{\nu} A^{\mu}_a + g f_{abc} A^{\mu}_b A^{\nu}_c \]
- \(G^{\mu \nu}_a\): Field strength tensor (non-Abelian).
- \(A^{\mu}_a\): Gauge field with color index \(a\).
- \(f_{abc}\): SU(3) structure constants.
- \(g\): QCD coupling constant.
The covariant derivative for the fermion field is:
\[ D_{\mu} \psi_i = \partial_{\mu} \psi_i - i g A^{\mu}_a (T_a)_{ij} \psi_j \]
- \((T_a)_{ij}\): SU(3) generators in the fundamental representation.
- The indices \(i, j\) correspond to color (SU(3)) and must be properly contracted.
Lorentz indices (\(\mu, \nu\)) describe spacetime, while color indices (\(a, i, j\)) belong to the internal SU(3) symmetry. The trace operations must respect both structures.
Deriving Equations of Motion from the Lagrangian
The Lagrangian contains terms for spin projections, and once you know the Lagrangian, you can derive the equations of motion by applying the Euler-Lagrange equation.
For example, consider the term involving the field \(\psi\). The partial derivative of the Lagrangian with respect to the derivative of the field (\(\partial_{\mu} \psi\)) is non-zero only where \(\partial_{\mu} \psi\) appears. Expanding this, you get terms like \(\bar{\psi} \gamma^{\mu} \partial_{\mu} \psi\).
The derivative of the Lagrangian with respect to \(\psi\) itself is non-zero from the mass term and other interaction terms. Combining these, you obtain the equations of motion, which describe how the field evolves in spacetime.
For the gauge field \(A_{\mu}\), the procedure is similar. The field strength term contributes to the equations of motion through derivatives of \(A_{\mu}\), while interaction terms introduce additional dependencies.
Applying this to QCD, the equations of motion involve the field strength tensor:
\[ G^{\mu \nu}_a = \partial^{\mu} A^{\nu}_a - \partial^{\nu} A^{\mu}_a + g f_{abc} A^{\mu}_b A^{\nu}_c \]
- \(G^{\mu \nu}_a\): Non-Abelian field strength tensor.
- \(f_{abc}\): SU(3) structure constants.
- \(g\): QCD coupling constant.
The covariant derivative for fermions introduces gauge field interactions:
\[ D_{\mu} \psi_i = \partial_{\mu} \psi_i - i g A^{\mu}_a (T_a)_{ij} \psi_j \]
- \((T_a)_{ij}\): SU(3) generators in the fundamental representation.
- The color indices \(i, j\) must be properly contracted.
Phase Ambiguity and Local Gauge Symmetry
In quantum mechanics, there is a phase ambiguity for the wave function: we can update the phase, and the absolute square of the wave function will not change because multiplying \(\psi\) by its conjugate \(\bar{\psi}\) causes the phase to drop out. This is a symmetry of the theory — we should be allowed to perform this phase transformation.
The problem arises when we demand the theory to be invariant under phase changes at all spacetime points simultaneously, but with different phases at each point. This is a local gauge transformation, as opposed to a global one.
For example, consider the Dirac Lagrangian term:
\[ \bar{\psi} i \gamma^{\mu} \partial_{\mu} \psi \]
If we apply a local phase transformation \(\psi \rightarrow \psi' = e^{i \alpha(x)} \psi\), the derivative term becomes:
\[ \partial_{\mu} \psi' = \partial_{\mu} (e^{i \alpha(x)} \psi) = e^{i \alpha(x)} (\partial_{\mu} \psi + i (\partial_{\mu} \alpha) \psi) \]
This introduces an extra term \(i (\partial_{\mu} \alpha) \psi\), which means the original free Dirac Lagrangian is not invariant under local gauge transformations. In other words, the equation of motion for a free Dirac particle does not remain the same if we adjust the phase independently at different spacetime points.
Local gauge symmetry is not a symmetry of the free Dirac Lagrangian — it requires introducing additional fields (gauge fields) to restore invariance.
The issue arises because the derivative \(\partial_{\mu}\) does not commute with the local phase transformation. To fix this, we replace the ordinary derivative with a covariant derivative \(D_{\mu}\), which includes a gauge field \(A_{\mu}\):
\[ D_{\mu} \psi = (\partial_{\mu} - i e A_{\mu}) \psi \]
Under a local gauge transformation, the gauge field must also transform as:
\[ A_{\mu} \rightarrow A'_{\mu} = A_{\mu} + \frac{1}{e} \partial_{\mu} \alpha \]
This ensures the covariant derivative transforms covariantly:
\[ D_{\mu} \psi \rightarrow e^{i \alpha(x)} D_{\mu} \psi \]
Thus, the modified Lagrangian becomes gauge-invariant.
The necessity of gauge fields (like the photon in QED) arises from enforcing local gauge symmetry — they “compensate” for the phase changes in the wave function.
This concept extends to non-Abelian gauge theories (like QCD), where the gauge fields themselves interact due to the non-commutativity of the group generators. The field strength tensor for QCD is:
\[ G^{\mu \nu}_a = \partial^{\mu} A^{\nu}_a - \partial^{\nu} A^{\mu}_a + g f_{abc} A^{\mu}_b A^{\nu}_c \]
- \(f_{abc}\): Structure constants of SU(3).
- \(g\): Coupling constant.
The fermion covariant derivative in QCD includes the generators \(T_a\):
\[ D_{\mu} \psi_i = \partial_{\mu} \psi_i - i g A^{\mu}_a (T_a)_{ij} \psi_j \]
Local gauge symmetry is foundational in field theory, dictating the structure of interactions and the necessity of gauge fields.
Gauge Symmetry and Interaction Dynamics
The free particle theory is incomplete because it does not account for interactions with radiation (photons) and other particles. The full Lagrangian must include these effects.
For example, when you update both the phase of the field \(\psi\) and the electromagnetic field \(A_\mu\) simultaneously, an additional term appears in the covariant derivative \(D_\mu\):
\[ D_\mu = \partial_\mu - i e A_\mu \]
This term cancels exactly the phase-dependent contribution from the transformation \(\psi \rightarrow e^{i \alpha(x)} \psi\), ensuring the Lagrangian remains invariant. The correct sign in the covariant derivative ( \(-i e A_\mu\) ) can be verified from the equations of motion.
The equations of motion reveal how fields couple to each other. In the case of fermions and photons, the motion of fermion fields is influenced by photons, and vice versa. Gauge symmetry dictates the specific form of this interaction, enforcing a fixed coupling strength \(g\).
The structure of gauge symmetry determines how particles interact — photons and fermions must couple with strength \(g\) as a consequence of local gauge invariance.
Introduction to SU(2) Group and Matrix Exponentiation
The gauge symmetry tells us how objects interact with each other, which is very important. Now we will take this idea and move to the quantum theory of weak interactions before jumping to QCD.
For weak interactions, we consider wavefunctions in the space of two coordinates (up and down components). The weak charge for quarks is \(\pm \frac{1}{2}\), and here we have an up-field and down-field, which are fermions with hidden four-spinor components.
The transformation that updates the phase must not change observables. The observable is given by \(\psi^\dagger \psi\), where \(\psi\) is a 2-component field. The transformation is represented by a unitary matrix \(G\):
\[ \psi \rightarrow G \psi \]
Since observables should not change, \(G\) must be unitary. These transformations form the U(2) group. If we further restrict \(G\) to have determinant 1, we obtain the SU(2) group (where “S” stands for “special”).
Any element of SU(2) can be represented as a matrix exponential:
\[ G = e^{i \alpha} \]
where \(\alpha\) is a 2x2 matrix with zero trace. This condition arises because:
\[ \text{det}(e^{i \alpha}) = e^{i \text{tr}(\alpha)} = 1 \implies \text{tr}(\alpha) = 0 \]
The SU(2) group is generated by three traceless matrices (Pauli matrices), which we will discuss later.
Matrix exponentiation can be computed via expansion:
\[ e^A = I + A + \frac{A^2}{2!} + \cdots \]
For example, for \(A = \begin{pmatrix} 1 & 1 \\ 0 & 0 \end{pmatrix}\), the exponential is non-trivial and requires careful computation.
The determinant condition ensures that SU(2) transformations preserve the norm of the wavefunction, which is crucial for maintaining physical observables. This structure is fundamental for describing weak interactions in quantum field theory.
Generators and Properties of SU(2) Group
The trace of \(\alpha\) is equal to zero, and there are only three matrices that can span the entire basis of such matrices, which are the Pauli matrices. These are called generators of the group because they generate any group element.
Given the three generators \(T_1, T_2, T_3\), and any three real numbers \(\alpha_1, \alpha_2, \alpha_3\), we can compute:
\[ G = e^{i (\alpha_1 T_1 + \alpha_2 T_2 + \alpha_3 T_3)} \]
This exponential spans the entire SU(2) group. The generators are fixed matrices, and once they are known, any group element can be constructed by specifying the three parameters \(\alpha_i\).
The determinant of SU(2) matrices is fixed to 1 because SU(2) is one of the standard groups we study extensively. If we instead considered U(2), it would have more generators and be more complicated.
The factorization \(U(2) = U(1) \times SU(2)\) separates a trivial phase (U(1)) from the non-trivial matrix structure (SU(2)). SU(2) is a fundamental group in physics, and we know its properties well — the number of generators, matrix representations, and other features make it a convenient object to work with.
The restriction to determinant 1 simplifies the analysis while retaining the physically relevant structure of the transformations. This is why we focus on SU(2) rather than the more general U(2) group.
Generators and Interaction Terms in SU(3) Group
The factorization of \(U(3)\) is given by \(U(3) = U(1) \times SU(3)\), where the \(U(1)\) phase is factored out, leaving the standard \(SU(3)\) group. This structure is essential in quantum chromodynamics (QCD), where we deal with three color charges: red, blue, and green. The transformation is represented by a \(3 \times 3\) matrix with an overall phase, which is not overly complicated.
The non-trivial contributions arise when updating fields, where the matrices must satisfy the condition \(\det = 1\). These matrices can be expressed as exponents of traceless matrices. The basis for traceless \(3 \times 3\) matrices consists of eight elements, which must also satisfy anti-commutation properties.
For example, multiplying two generators yields:
\[ e^{i \alpha} \cdot e^{i \beta} = e^{i (\alpha + \beta)} \]
In two dimensions, the basis for such matrices (including tracelessness) gives three generators (Pauli matrices), while in three dimensions, there are eight (Gell-Mann matrices). The number of generators corresponds to the number of charge carriers in the field.
Each generator matrix is associated with the action of the field, as they appear in the interaction term. When attaching the field \(\psi\) to the interaction field \(A\), the generator matrix accompanies it.
This connection originates from computing derivatives and identifying an additional phase derivative, which introduces an extended derivative of the field. This leads to an interaction term in Feynman diagrams. In higher dimensions, the same derivative will couple with the appropriate generator matrix.
For \(SU(2)\), the generators are the Pauli matrices (\(\sigma\)), while for \(SU(3)\), they are the Gell-Mann matrices (\(\lambda\)). These matrices directly influence the interaction vertex in quantum field theory.
In \(SU(2)\), there are three generators corresponding to three charge carriers (\(Z, W^+, W^-\)), while \(SU(3)\) has eight generators, corresponding to the eight gluons in QCD. Unlike the electroweak force, the gluons lack distinct names due to their more complex structure.
Gluon Interactions and Color Charge Dynamics
The gluons are identified by their matrices, similar to the electroweak interaction where the \(Z\) boson corresponds to a diagonal matrix in the space. Some gluon fields are diagonal, analogous to the \(Z\), while others are non-diagonal, like the \(W^\pm\) matrices.
In the interaction vertex, the gluon’s flavor determines how it couples to the quark, and this is governed by the structure of the matrices (Gell-Mann matrices for \(SU(3)\)). The interaction term depends on how the color charges contract, which is a key aspect of QCD dynamics.
Confinement is a fundamental property of the strong interaction, where the interaction strength increases with distance. Unlike electromagnetism, where the force between an electron and positron weakens with separation, QCD exhibits the opposite behavior.
The strong interaction grows stronger at larger distances, preventing quarks and gluons from being isolated. This is why we only observe color-neutral hadrons in nature. The underlying mechanism is tied to the non-Abelian nature of \(SU(3)\) and the self-interactions of gluons.
Confinement and Strong Interaction in QCD
The strong interaction between quarks behaves opposite to other forces: its strength increases with distance. This is why quarks are confined to small scales, such as inside mesons and baryons. Let’s visualize them again. Here is a meson, and here is a baryon.
Confinement means the strong interaction is confined within these “bubbles” — there is no strong interaction outside of mesons or baryons. If you attempt to pull quarks apart with immense force, they will eventually separate, but new confined objects (color-neutral particles) will form.
Color neutrality is essential for confinement. A particle with zero color charge does not interact via the strong force outside its bubble. If a particle carried color charge, gluons would interact with it, breaking confinement.
This is why matter organizes into these small, color-neutral bubbles where strong interactions dominate internally but are absent externally. Confinement is not derived from first principles in the Lagrangian, but there are hints of it. For example, the Lagrangian’s structure suggests confinement, though it is not explicitly visible.
Gluon Self-Interaction and Confinement in QCD
There are indications of confinement in the Lagrangian, and one of them is the gluon self-interaction term. For QCD, this is the \(G_{\mu\nu}^a\) term, which includes interactions like:
\[ G_{\mu\nu}^a = \partial_\mu A_\nu^a - \partial_\nu A_\mu^a + g f^{abc} A_\mu^b A_\nu^c \]
Here, \(f^{abc}\) are the structure constants of SU(3), and \(g\) is the coupling constant. This leads to terms with two, three, or four gluon fields, such as \(G_\nu G_\nu\) or \(f^{abc} G_\mu^b G_\nu^c\). These manifest in interaction vertices like the three-gluon and four-gluon vertices, which are called gluon self-interactions.
Gluon self-interaction is one indication of confinement, though it is not a proof. Some field theories exhibit confinement, while others do not. Confinement remains one of the unsolved problems in physics, with a prize of 1 million euros for its explanation.
Momentum-Dependent Effective Interaction Strength in QCD
The effective interaction strength in QCD depends on the momentum scale \(Q\) with which you probe the quark-gluon system. This is analogous to the electromagnetic case where screening effects modify the interaction. Here, the effective coupling \(\alpha_s(Q)\) describes the strength of the interaction when a gluon interacts with a quark, accounting for quantum corrections.
The interaction can be represented as an effective “quantum bubble,” where the gluon’s momentum \(Q\) determines the observed coupling strength. This leads to a momentum-dependent \(\alpha_s(Q)\), meaning gluons with different momenta will experience different interaction strengths.
Unlike electromagnetism, QCD exhibits antiscreening (asymptotic freedom) at high \(Q\), where \(\alpha_s(Q)\) decreases with increasing momentum.
The behavior contrasts with screening in QED, as the gluon self-interactions dominate the running of \(\alpha_s(Q)\).
Asymptotic Freedom and Confinement Regimes in QCD
The effective interaction depends on the momentum scale \(Q\) in the following way. If \(Q\) is very high, we are in the regime of asymptotic freedom. When \(Q\) is low, we enter the confinement regime, where the interaction strength becomes large.
The transition between these regimes is governed by the running coupling \(\alpha_s(Q)\), which decreases at high \(Q\) (asymptotic freedom) but grows at low \(Q\) (confinement).
Hadron Dynamics and Low-Momentum Gluon Interactions
Here, “low” refers to the regime of confinement, where the momentum scale is on the order of GeV. Hadrons exist in this region, interacting by exchanging gluons. The gluons couple to fermions, and the momentum of these gluons is very small when the particles are close to each other. This occurs below 1 GeV.
The confinement regime is characterized by low-momentum gluon exchanges, where the interaction strength becomes significant.
Asymptotic Freedom and High-Momentum Coupling
When the distance between quarks and gluons is very small, we are below \(1 \ \text{GeV}\), where the interaction is super strong. This theory becomes asymptotically free. At very high momentum transfer, the coupling constant decreases.
Asymptotic freedom explains why quarks behave almost like free particles at high energies, as the strong coupling weakens.