lecture-02

Introduction to Cross Section Estimation in Hadronic Reactions

Welcome everyone to today’s lecture. It will be mostly about classification of hadrons. I will walk you a little bit through the history of how important discoveries were made and what we learned from there about the hadrons and how we can group them together.

In the 50s and 60s, there were a lot of new big accelerators built, like the Bevatron. This came into operation in 1954. It was a proton accelerator with energies of up to 13 GeV. These high-energy protons were shot on a fixed target, producing and detecting many different particles. Over 100 new particles were found, leading to what was called the “particle zoo.” This term was first mentioned by Oppenheimer at a high-energy physics conference.

Physicists at the time had to think about how to organize these particles. Is there a pattern? Are these all fundamental particles? This is what we will discuss today.

Note

The periodic table groups atoms by their proton and neutron numbers, as well as their outermost electrons. Similarly, we aim to classify hadrons based on their properties.

In the context of particle physics, we group hadrons together according to their quantum numbers. Let’s start with isospin.

In 1932, the neutron was discovered by Chadwick. Experiments at the Bevatron showed that proton-proton, proton-neutron, and neutron-neutron interactions had very similar interaction strengths. The masses of the proton and neutron are also nearly identical (939 MeV). This led to the idea that the proton and neutron could be considered the same particle but in two different states, described by isospin.

Important

The strong interaction does not distinguish between protons and neutrons. The difference becomes apparent only in an electromagnetic field.

Mathematically, we treat isospin similarly to spin. For example, the proton and neutron can be represented as two states of a nucleon in an isospin doublet.

At the quark level, protons and neutrons are composed of up ($u$) and down ($d$) quarks. The strong interaction does not distinguish between these flavors.

The isospin assignments are:
- Up quark: $I = \frac{1}{2}, \ I_3 = +\frac{1}{2}$
- Down quark: $I = \frac{1}{2}, \ I_3 = -\frac{1}{2}$

Tip

While isospin is not a true spin, its mathematical treatment is analogous to spin, simplifying calculations in hadronic physics.

Understanding Cross Section and Its Physical Interpretation

The spin and isospin can be treated similarly in mathematical terms, both following the $SU(2)$ algebra. In the previous lecture, we discussed the $SU(2)$ algebra, which applies to matrices and can be described using rotation matrices. The generators of this algebra are the Pauli matrices.

For the isospin operator, the notation is analogous to spin, where $I$ replaces $J$. The general form of these operators is written with the isospin $I$ and its third component $I_3$. For example, applying the $I_3$ operator to an “up” state yields an eigenvalue of $+1$.

We also have lowering operators in this framework. For a given isospin $I$, there are $2I + 1$ projections, labeled as $I_3 = I, I-1, \ldots, -I$. The lowering operators allow transitions between these projections.

\[ I_- \ket{I, I_3} = \sqrt{I(I+1) - I_3(I_3-1)} \ket{I, I_3 - 1} \]

This is a reminder of how the $SU(2)$ algebra operates. Now, let’s consider some examples.

For instance, take the up ($u$) and down ($d$) quarks. As discussed earlier, mesons can be formed by combining a quark and an antiquark. We can examine the possible particles formed from combinations of $u$ and $d$ quarks with their antiquarks.

Note

The isospin assignments for quarks are:
- Up quark: $I = \frac{1}{2}, \ I_3 = +\frac{1}{2}$
- Down quark: $I = \frac{1}{2}, \ I_3 = -\frac{1}{2}$

By studying these combinations, we can classify the resulting mesons based on their isospin properties.

Defining Charge in QED and QCD

For the antiquark in metals, we examine the isospin properties. Both the up and down quarks have isospin $I = \frac{1}{2}$. When combining two particles with isospin $\frac{1}{2}$, the possible outcomes are isospin $0$ or $1$. This is analogous to combining spins, where the dimensions follow similar rules.

For isospin $\frac{1}{2}$, there are two projections: $I_3 = +\frac{1}{2}$ and $I_3 = -\frac{1}{2}$. The dimensions can be represented as $2 \times 2$, and for an antiquark, we denote this with a bar on top.

For isospin $1$, the number of projections is given by $2I + 1$, which yields three projections: $I_3 = +1, 0, -1$. For isospin $0$, there is only one projection: $I_3 = 0$. In terms of dimensions, this decomposition results in $3 + 1$.

This can be expressed in group theory as a $4 \times 4$ matrix decomposing into a $3 \times 3$ matrix (triplet state) and a $1 \times 1$ matrix (singlet state). The triplet represents the isospin-1 state, while the singlet represents the isospin-0 state.

To determine the coefficients in these combinations, we use Clebsch-Gordan coefficients, similar to those used in spin combinations. For example, when combining two isospin-$\frac{1}{2}$ particles, the resulting states are:

\[ \ket{I = 1, I_3 = +1} = \ket{\frac{1}{2}, \frac{1}{2}} \otimes \ket{\frac{1}{2}, \frac{1}{2}} \] \[ \ket{I = 1, I_3 = 0} = \frac{1}{\sqrt{2}} \left( \ket{\frac{1}{2}, \frac{1}{2}} \otimes \ket{\frac{1}{2}, -\frac{1}{2}} + \ket{\frac{1}{2}, -\frac{1}{2}} \otimes \ket{\frac{1}{2}, \frac{1}{2}} \right) \] \[ \ket{I = 1, I_3 = -1} = \ket{\frac{1}{2}, -\frac{1}{2}} \otimes \ket{\frac{1}{2}, -\frac{1}{2}} \] \[ \ket{I = 0, I_3 = 0} = \frac{1}{\sqrt{2}} \left( \ket{\frac{1}{2}, \frac{1}{2}} \otimes \ket{\frac{1}{2}, -\frac{1}{2}} - \ket{\frac{1}{2}, -\frac{1}{2}} \otimes \ket{\frac{1}{2}, \frac{1}{2}} \right) \]

These coefficients describe how the isospin states combine to form either a triplet or a singlet. The triplet corresponds to the isospin-1 state, while the singlet corresponds to the isospin-0 state.

Note

The isospin assignments for quarks are:
- Up quark: $I = \frac{1}{2}, \ I_3 = +\frac{1}{2}$
- Down quark: $I = \frac{1}{2}, \ I_3 = -\frac{1}{2}$

This framework allows us to classify mesons formed from quark-antiquark combinations based on their isospin properties.

Defining Charge in QCD and Group Theory Introduction

We have an isospin triplet with isospin $I = 1$, which has three projections: $I_3 = +1, 0, -1$. The singlet state has isospin $I = 0$ with projection $I_3 = 0$. The Clebsch-Gordan coefficients are used here, and you must always take the square root of these coefficients.

For antiquarks, the isospin assignments are:
- Anti-up quark: $I_3 = -\frac{1}{2}$
- Anti-down quark: $I_3 = +\frac{1}{2}$

The minus sign is chosen so that quarks and antiquarks behave consistently under SU transformations.

When constructing states from quark-antiquark combinations:
- For the singlet state, combining up and anti-up gives a vector with projections $+\frac{1}{2}$ and $-\frac{1}{2}$, weighted by a square root factor.
- Similarly, for down and anti-down, the projections are $-\frac{1}{2}$ and $+\frac{1}{2}$, but due to the minus sign in the antiquark assignment, the overall sign cancels out.

The triplet states follow analogous combinations.

The particles associated with these states are:
- The triplet corresponds to pions: $\pi^+$ ($I_3 = +1$), $\pi^0$ ($I_3 = 0$), and $\pi^-$ ($I_3 = -1$).
- Rho particles ($\rho$) also form a triplet with the same isospin structure but differ in spin.

Introduction to SU(2) Group and Its Fundamental Representation

The distinction between the two states lies in their spin: one has anti-parallel spin (spin zero), while the other has parallel spin. Otherwise, they are identical. For the singlets, we can have an omega and a meta $E'$. By choosing the isospin quantum number, we can group certain particles together.

Now, let’s consider an example: shooting a $\pi$ beam at a proton target. The $\pi$ has isospin $I = 1$, and the nucleon target has isospin $I = \frac{1}{2}$. When combining these, the possible resulting isospin states are $I = \frac{3}{2}$ or $I = \frac{1}{2}$.

The dimensions of these states are as follows:
- For $I = 1$, there are three projections ($I_3 = +1, 0, -1$).
- For $I = \frac{1}{2}$, there are two projections ($I_3 = +\frac{1}{2}, -\frac{1}{2}$).
- For the combined $I = \frac{3}{2}$, there are four projections ($I_3 = \pm \frac{3}{2}, \pm \frac{1}{2}$).
- For $I = \frac{1}{2}$, there are again two projections.

This decomposition is represented mathematically by a $6 \times 6$ matrix, which can be broken down into smaller matrices: a $2 \times 2$ matrix and a $4 \times 4$ matrix. These are called irreducible representations.

Note

Irreducible representations cannot be further decomposed into smaller representations under the group action.

For small numbers, this decomposition is straightforward, but for larger numbers, specific methods are needed to determine the structure. While we won’t delve deeper into these methods here, it’s important to recognize the underlying principle of reducible vs. irreducible representations.

If you’ve studied group theory (e.g., in advanced quantum mechanics), you may already be familiar with these concepts. Otherwise, the key takeaway is that we are decomposing larger representations into smaller, fundamental ones.

Matrix Exponentiation and Taylor Series in SU(2)

When combining three quarks, we have $2 \times 2 \times 2$ states, corresponding to three quarks with two possible projections each ($\pm \frac{1}{2}$). Using the previous decomposition, $2 = 3 + 1$, we now combine this with another $2$ to get:

\[ (3 + 1) \times 2 = 4 + 2 + 2 \]

This results in an octet (8-dimensional representation) consisting of:
- A spin-$\frac{3}{2}$ quartet (4 states)
- Two spin-$\frac{1}{2}$ doublets (2 states each).

In the context of isospin, this decomposition helps explain observed cross-section differences in particle collisions. For example, the delta particles ($\Delta^{++}, \Delta^{+}, \Delta^{0}, \Delta^{-}$) correspond to the quartet with masses around 1232 MeV.

The projections for these states are:
- $I_3 = \pm \frac{3}{2}, \pm \frac{1}{2}$ for the $I = \frac{3}{2}$ quartet.
- $I_3 = \pm \frac{1}{2}$ for the $I = \frac{1}{2}$ doublets.

Note

The irreducible representations here are the $4$ (quartet) and $2$ (doublet), which cannot be further decomposed under SU(2) transformations.

This structure is useful for predicting particle properties and interaction strengths, as seen in exercises analyzing cross-section measurements.

Spin and SU(2) Symmetry in Quantum Mechanics

We can use the isospin number to understand why certain measured cross sections differ in size. For example, if we plot mass against cross section, we observe the cross section for $\pi^+ p$ and $\pi^- p$ reactions. In the delta mass region, there is a factor of three difference between these cross sections. You can explore in exercises why these differences arise between the reactions.

The isospin concept was introduced for protons and neutrons in the 1950s, before the discovery of quarks. Experimentally, researchers observed patterns in particle production, such as pions and later rho mesons. They grouped particles into multiplets based on symmetries, such as pions and rho mesons forming triplets, and deltas forming an isospin quadruplet.

Note

The idea of isospin emerged from observing similar reaction rates for processes involving protons and neutrons, suggesting the strong interaction treats them identically.

By analyzing cross sections, it became clear that the strong interaction does not distinguish between protons and neutrons, leading to the isospin symmetry framework. This concept later extended to larger patterns, such as strangeness and hypercharge, which we will discuss further.

The question arises: why introduce isospin when protons and neutrons differ in charge? The answer lies in reaction rates — if $pp$, $pn$, and $nn$ interactions yield similar outcomes, it implies the strong interaction is blind to their charge differences. This observation motivated the isospin formalism.

Introduction to Isospin and Generators in SU(2)

When analyzing cross sections, certain reactions appear stronger, necessitating new concepts. For instance, the strong interaction does not distinguish between protons and neutrons, or between up and down quarks. This similarity led to grouping the proton and neutron into an isospin doublet, represented as:

\[ p = \left| \frac{1}{2}, +\frac{1}{2} \right> \quad n = \left| \frac{1}{2}, -\frac{1}{2} \right> \]

The quark-level interpretation came later, where up and down quarks were also placed in an isospin doublet. Initially, the focus was on grouping particles like pions and rho mesons into multiplets based on symmetries.

Note

The discovery of isospin predated the quark model, emerging from patterns in particle production observed in the 1950s.

With the advent of larger accelerators like the Bevatron in the 1960s, over 100 particles were discovered, including deltas. Initially, their classification was unclear, but later they were understood as composites of smaller particles. This led to broader patterns incorporating quantum numbers like strangeness.

The isospin formalism was motivated by the observation that reactions like $pp$, $pn$, and $nn$ yielded similar outcomes, indicating charge-blindness in the strong interaction. This symmetry framework extended to larger particle groupings, such as pions and rho mesons forming triplets and deltas forming quadruplets.

Properties and Rank of SU(2) Group

The isospin algebra is given by the commutation relation:

\[ [I_i, I_j] = i \epsilon_{ijk} I_k \]

where $I_i$ are the isospin operators and $\epsilon_{ijk}$ is the Levi-Civita symbol.

We can take a short break before continuing with strangeness. Strangeness was introduced because some particles detected in cosmic rays behaved unusually. For example, in a cloud chamber, tracks resembling a V-shape were observed, corresponding to particles like $\pi^+$, $\pi^-$, and a proton. These particles always appeared in pairs and had relatively long lifetimes.

This led to the introduction of a new quantum number called strangeness, which was postulated to be conserved. Additionally, isospin is a quantum number conserved in strong interactions.

For particles composed of up and down quarks, the strangeness value is 0, as seen in pions, neutrons, and protons. Particles with strangeness $+1$ include hyperons like the $\Lambda$. Hyperons come in various types, such as the lambda hyperon.

Note

The conservation of isospin in strong interactions reflects the symmetry between protons and neutrons, or up and down quarks, in such processes.

Spin Projections and Matrix Representations in SU(2)

The eigenvalue equation for the third component of isospin $I_3$ acting on a state $\left| \frac{1}{2} \mu \right>$ is given by:

\[ I_3 \left| \frac{1}{2} \mu \right> = \mu \left| \frac{1}{2} \mu \right> \]

Hyperons come in different types, such as the lambda ($\Lambda$), sigmas ($\Sigma$), and cascades ($\Xi$). Kaons can form isospin doublets, like $K^+$ and $K^0$, or $K^-$ and $\overline{K^0}$, with strangeness $-1$. Since these particles appear in pairs, strangeness must be conserved in strong interactions but not in weak interactions.

For example, in the decay $\Lambda \to p + \pi^-$, the final particles have no strangeness, yet the decay occurs. This indicates strangeness violation, which is allowed in weak interactions. The same reaction would not happen if the initial particle were a proton instead of a lambda, as strangeness conservation would forbid it.

The hypercharge $Y$ is defined as:

\[ Y = B + S \]

where $B$ is the baryon number and $S$ is the strangeness. For baryons, $B = +1$; for antibaryons, $B = -1$; and for mesons, $B = 0$.

Note

Baryon number is always conserved. For instance, a reaction like $p + \pi^-$ cannot produce only two mesons in the final state—there must be a baryon present.

Isospin multiplets correlate with charge. For example, the pion triplet has $I_3$ projections $+1$ ($\pi^+$), $0$ ($\pi^0$), and $-1$ ($\pi^-$). The Gell-Mann–Nishijima formula relates charge $Q$ to isospin and hypercharge:

\[ Q = I_3 + \frac{Y}{2} \]

This can be verified for particles like the proton.

Angular Momentum Operators and Conserved Quantities in Quantum Mechanics

Let’s quickly verify the Gell-Mann–Nishijima formula for a photon. If $I = \frac{1}{2}$ and the first component is $+\frac{1}{2}$, we can check the hypercharge. For the proton, strangeness is zero, but it’s a baryon, so $B = 1$. Using the formula:

\[ Q = I_3 + \frac{Y}{2} \]

where $Y = B + S$, we get $I_3 = \frac{1}{2}$ and $Y = 1$. Substituting these, the charge $Q$ is $\frac{1}{2} + \frac{1}{2} = +1$, which matches the proton’s charge. Similarly, for the neutron, $I_3 = -\frac{1}{2}$ and $Y = 1$, giving $Q = 0$, as expected.

Gell-Mann and Niemann discovered larger patterns for these particles by incorporating strangeness. This led to the “eightfold way,” where particles are arranged in bigger multiplets. Instead of just the up and down quarks, they included the strange quark, extending the analysis to $SU(3)$ flavor symmetry.

Here’s a summary of the quantum numbers for the up, down, and strange quarks:
- Baryon number ($B$): Each quark has $B = \frac{1}{3}$, so a proton (uud) has $B = 1$.
- Charge ($Q$): Up quark has $Q = +\frac{2}{3}$, down has $Q = -\frac{1}{3}$, and strange has $Q = -\frac{1}{3}$.
- Strangeness ($S$): Up and down quarks have $S = 0$, while the strange quark has $S = -1$.

Note

The eightfold way organizes particles based on $I_3$ (third component of isospin) and strangeness, revealing larger symmetry patterns.

For example, plotting $I_3$ against strangeness for the up, down, and strange quarks shows how these quantum numbers define particle multiplets. This framework was foundational before quarks were directly discovered.

Construction of Arbitrary Representations in SU(2)

For the down and strange quarks, the isospin components are $-\frac{1}{2}$ and $-3$, respectively. These are all fermions, meaning they are spin-$\frac{1}{2}$ particles. We previously discussed strangeness, which is $-1$ for the strange quark.

Now, let’s examine larger patterns, starting with baryons. Baryons are composed of three quarks, which can be up, down, or strange. The combination of three quarks gives us multiplets: a decuplet (10 particles) and two octets (8 particles each). These are referred to as the ground state baryons.

The particles can be arranged based on the third component of isospin ($I_3$) on the horizontal axis and strangeness ($S$) on the vertical axis. Some textbooks also use hypercharge ($Y$) instead of strangeness. The isospin multiplets are visible here:
- A doublet for the neutron and proton,
- A triplet for the sigmas,
- And another doublet for the cascades.

For example, the isospin values here are $\frac{1}{2}, -\frac{1}{2}, 0$, and the strangeness values are $0, -1, -2$. The quark content is also apparent: particles with one strange quark are grouped together, while those with two strange quarks form another set.

Looking at the masses, the proton and neutron have nearly identical masses, differing by only about 1 MeV. This small mass difference is also observed in other isospin multiplets, such as the sigmas. However, along the strangeness axis, the mass differences are much larger—around 250 MeV between certain particles and 130 MeV between others.

This indicates that isospin symmetry in $SU(2)$ is a good approximation, but when the strange quark is included, the symmetry is significantly broken. If the symmetry were exact, all particles in the multiplet would have the same mass.

In the upper row of the diagram, we have particles with strangeness $0$, while the lower rows have $-1, -2, -3$. The highest $I_3$ component is plotted on this axis.

All these particles share a common feature: their spin and parity. The baryons in the octet have spin $\frac{1}{2}$ and positive parity, while those in the decuplet have spin $\frac{3}{2}$ and also positive parity. The spin configurations can vary, including all possible permutations.

Note

The mass differences along the isospin axis are small, indicating approximate $SU(2)$ symmetry, while the larger differences along the strangeness axis show significant symmetry breaking when the strange quark is included.

The quark content is explicitly visible in these arrangements: particles with one strange quark are grouped together, and those with two strange quarks form another distinct set. This organization helps illustrate the underlying symmetry patterns and their breaking due to the strange quark’s mass.

Introduction to Xi and XIC Particles

The spin configurations can vary, with all possible permutations giving a total spin of $\frac{1}{2}$. In some cases, the spins are fully aligned, like this.

At the time when Gell-Mann identified this pattern, the Omega minus had not yet been discovered. This was a major success for Gell-Mann, as he predicted the existence of the Omega minus, which was experimentally confirmed two years later in bubble chamber experiments. The reaction observed was:

\[ K^- + p \rightarrow \Omega^- + K^+ + K^0 \]

To conserve strangeness, additional kaons ($K$) are required in the decay chain. The Omega minus decays further into a cascade ($\Xi$) and a pion ($\pi^-$), eventually producing a $\Lambda^0$, neutral pions ($\pi^0$), and photons, followed by the decay of the $\Lambda$ to a proton and a $\pi^-$.

Not only did Gell-Mann predict the Omega minus, but he also estimated its mass. Looking at the mass differences between the horizontal lines in the multiplet:
- The $\Delta(1232)$ has a mass of 1232 MeV,
- The sigmas follow,
- The Omega minus has a mass around 1680 MeV.

The mass differences between these states are roughly 150 MeV.

These predictions stem from group theory and symmetry considerations, which successfully described many particles mathematically.

Note

The mass spacing in the baryon decuplet follows a consistent pattern, supporting the underlying $SU(3)$ symmetry.

If you examine the multiplet structure, the mathematical framework of group theory provides a powerful tool for understanding particle classifications and their properties.

Conflict in Particle Observations and Strong Interaction Symmetry

The mass difference between particles in the octet and decuplet can be attributed to spin-spin or spin-orbit interactions, meaning it arises from dynamics. For baryons, the parity can be calculated using:

\[ P = (-1)^L \]

For the octet ground states with $L = 0$, this gives a parity of $+1$ for both $\frac{1}{2}^+$ and $\frac{3}{2}^+$. For mesons, the parity formula is:

\[ P = (-1)^{L + 1} \]

This results in a parity of $-1$ for both cases. Charge conjugation ($C$) is well-defined only for neutral mesons. For example, applying it to $\pi^0$ gives $C = +1$.

When analyzing particle decays, quantum numbers like parity and $C$-parity are multiplicative. For instance, consider the decay of an $\omega$ meson into two pions. The $C$-parity of $\pi^0$ is $+1$, so the product for two pions is $(+1) \times (+1) = +1$. However, the $\omega$ meson has $C = -1$, making this decay forbidden.

Note

Parity and $C$-parity are crucial for determining allowed decay channels, as they must be conserved in strong and electromagnetic interactions.

For mesons, the quark-antiquark pairs in the octet or nonet (including $\eta'$) have quantum numbers $0^-+$, indicating anti-parallel spin orientation. Their parity is $-1$, and charge conjugation is $+1$. For the vector mesons (spin $1$), the spins are aligned parallel, resulting in parity $-1$ and $C$-parity $-1$.

Charge conjugation transforms a particle into its antiparticle. For baryons, if the orbital angular momentum $L$ is known, parity can be computed as $(-1)^L$.

Lastly, the mass differences in the baryon decuplet follow a consistent pattern, supporting the underlying $SU(3)$ symmetry. For example, the $\Delta(1232)$ has a mass of 1232 MeV, while the $\Omega^-$ is around 1680 MeV, with mass differences of roughly 150 MeV between states.

Important

Group theory and symmetry considerations provide a powerful framework for predicting particle properties and classifications, as demonstrated by Gell-Mann’s successful prediction of the $\Omega^-$.

Isospin and SU(2) Symmetry in Quark Systems

Nobody complained when I wrote down the quark content for the omega minus or the delta plus plus. I mentioned that all of these have aligned spins (spin up), and their flavor content is also identical. Do you see a problem here? Quarks are fermions, and according to the Pauli exclusion principle, fermions in a combined state must be distinguishable by at least one quantum number. Here, everything is the same.

Let’s examine the total wave function of these baryons. For $L = 0$, the spatial wave function is symmetric. The spin wave function is also symmetric because all spins are aligned up — exchanging any quark pair leaves the state unchanged. The flavor wave function is symmetric as well, since all quarks are identical in flavor.

When physicists encounter such a problem, we introduce a new quantum number. What could it be? Yes, color charge. This was briefly discussed last time. The color part of the wave function must be antisymmetric to ensure the total wave function is antisymmetric. We assign each quark a distinct color (red, green, blue), which distinguishes them.

Note

The introduction of color charge resolves the symmetry conflict in baryon wave functions, ensuring compliance with the Pauli exclusion principle.

Now, let’s briefly discuss an experiment confirming the number of colors. Last time, I mentioned we have three colors, but how do we know it’s not six? Experiments involving $e^+e^-$ annihilation provided evidence. In these collisions, a virtual photon is exchanged, producing either quark-antiquark pairs or leptons (e.g., $\mu^+\mu^-$). The observed production rates for hadrons versus leptons confirmed the existence of exactly three color charges.

The mass of the produced systems depends on the quark-antiquark pairs formed. This experimental result aligns with the theoretical prediction of three colors in QCD.

Combining Spin Representations in SU(2)

The given expression involves the spin operator $\hat{I}^2$ acting on the state $\left| I, I_3 \right>$:

\[ \hat{I}^2 \left| I, I_3 \right> = (I_1^2 + I_2^2 - i[I_1, I_2]) \left| I, I_3 \right> = (I_1^2 + I_2^2 + I_3^2 - I_3^2 + I_3) \left| I, I_3 \right> \]

Here, $I_1, I_2, I_3$ are the components of the spin operator, and $[I_1, I_2]$ is their commutator. The term $-i[I_1, I_2]$ simplifies to $I_3$ due to the commutation relations of SU(2) algebra.

This expression helps determine the mass of the produced systems in particle interactions. Experimental measurements involve comparing cross sections for hadronic production (quark-antiquark pairs) against leptonic production (e.g., $\mu^+\mu^-$). The ratio of these cross sections is proportional to the number of colors in QCD, as leptons lack color charge.

The process involves a photon exchange at the vertex, with proportionality to the charge squared. Different quark flavors contribute to the spectrum, appearing as peaks for mesons like $\rho$, $\omega$, and $\pi$. At higher energies, heavier bound states such as charmonium ($J/\psi$) and bottomonium ($\Upsilon$) are produced.

The observed step-like increases in cross sections confirmed the number of colors to be three. This experimental result aligns with the quark model, where quarks come in six flavors and three colors. The historical context shows how patterns in particle detection led to the quark model’s development, resolving earlier puzzles like the Pauli exclusion principle in baryon wave functions.

The total wave function for baryons must be antisymmetric under quark exchange. For $L = 0$ states, the spatial part is symmetric, the spin part is symmetric (all spins aligned), and the flavor part is symmetric (identical quarks). Introducing color charge ensures antisymmetry, satisfying the Pauli principle.

Note

The color degree of freedom distinguishes otherwise identical quarks, resolving symmetry conflicts in baryon wave functions.

The $e^+e^-$ annihilation experiments provided direct evidence for three colors by measuring the ratio of hadronic to leptonic production rates. This ratio, combined with theoretical predictions, solidified the three-color framework of QCD.

The mass of produced quark-antiquark systems depends on the specific pairs formed, with heavier states appearing at higher energies. This behavior is consistent with the quark model and the SU(2) spin representations discussed earlier.

Representation Theory and Spin Preservation

The particles were called “strange” because their behavior was confusing—they appeared only in pairs and had longer lifetimes than expected.

In experiments, you can only measure the invariant mass by detecting particles like protons and pions and analyzing their spectra. The proton was discovered first, followed by the neutron in 1932. Later, in the 1950s, the antiproton and other charged particles were observed.

The concept of C-parity is introduced to explain why certain decays are forbidden. If a decay is not observed experimentally, it suggests a violation of a quantum number.

Note

The invariant mass spectrum helps identify particle production mechanisms, such as quark-antiquark pairs (hadronic) versus lepton pairs (leptonic).

The ratio of hadronic to leptonic production cross sections is key in confirming the number of colors in QCD. This ratio is proportional to the squared charge of the quarks involved.

Different quark flavors produce distinct peaks in the spectrum, corresponding to mesons like $\rho$, $\omega$, and $\pi$. At higher energies, heavier states such as charmonium ($J/\psi$) and bottomonium ($\Upsilon$) appear.

The step-like increases in cross sections experimentally confirmed three colors, aligning with the quark model. This resolved earlier puzzles, such as the Pauli exclusion principle in baryon wave functions.

The total baryon wave function must be antisymmetric under quark exchange. For $L = 0$ states, the spatial part is symmetric, the spin part is symmetric (all spins aligned), and the flavor part is symmetric (identical quarks). The color degree of freedom ensures overall antisymmetry.

Important

The $e^+e^-$ annihilation experiments provided direct evidence for three colors by comparing hadronic and leptonic production rates.

Introduction to Isospin and Charge Conjugation

The isospin of a nucleon is given by:

\[ I = \frac{1}{2} \]

The violation of charge conjugation was introduced early in particle physics, around the time of the positron’s discovery, predating the antiproton’s observation in the 1950s. Historically, concepts emerged both from theory and experiments. For instance, the $\Omega^-$ particle was predicted theoretically before its experimental detection in 1964.

Initially, quark substructure was postulated under the name “partons” rather than quarks. The observed mass differences in particles like the $\Delta$, $\Sigma$, and $\Xi$ cascades hinted at underlying quark content. These differences were used to predict masses, which later aligned with quark model predictions.

Note

The quark model was introduced in a concise two-page paper, marking an exciting period in particle physics. The $\Omega^-$ discovery (1964) validated these theoretical predictions.

The quark model resolved earlier puzzles, such as the Pauli exclusion principle in baryon wave functions. For $L = 0$ states, the spatial, spin, and flavor parts are symmetric, while the color degree of freedom ensures overall antisymmetry.

Important

Experimental observations, like step-like increases in cross sections, confirmed three quark colors, aligning with QCD predictions.

Since the provided lecture chunk after --- is empty, there is no content to transcribe or expand upon. The output would simply be the header with no additional text:

## Charge Conjugation and Particle Properties

(No further content to include, as the original chunk was blank.)

Spin and Parity Combinations for Particle Systems

Since the provided lecture chunk (after ---) is empty, there is no content to process or expand upon. If you provide the actual lecture content, I can format it according to the given guidelines.

For now, the output would simply be:

## Symmetry and Wave Functions in Particle Systems

Let me know if you’d like to share the lecture content so I can assist further!

Excitations and Quantum States in Particle Systems

## Angular Momentum and Partial Waves in Particle Decay

(No meaningful content was provided in the lecture chunk after --- to include in the response.)

Given the input provided, there is no meaningful content after the --- separator to include in the response. The lecture chunk does not contain any complete sentences or technical details about the Hydrogen Atom Excitation Spectrum in a Magnetic Field.

Thus, no output can be generated while adhering to the guidelines.

(If additional context or corrections are provided, I can refine the response accordingly.)

Given the input provided, there is no meaningful content after the --- separator to include in the response. The lecture chunk does not contain any complete sentences or technical details about Quantum Numbers and Angular Momentum in Hydrogen Atom.

Thus, no output can be generated while adhering to the guidelines.

(If additional context or corrections are provided, I can refine the response accordingly.)

Thus, no output can be generated while adhering to the guidelines.

(If additional context or corrections are provided, I can refine the response accordingly.)

Introduction to Hadron Classification and Historical Context

In the 50s and 60s, there were a lot of new big accelerators built, like for example, the Bevatron. This came into operation in 1954. It was a proton accelerator with energies of up to 13 GeV. These high-energy protons were then shot on a fixed target, producing and detecting many different particles. Over 100 new particles were found, leading to what was called the “particle zoo.”

Note

The term “particle zoo” was first mentioned by Oppenheimer at a high-energy physics conference.

Physicists of the time had to think about how to organize these particles. Is there some pattern? Are these all fundamental particles? This is what we will be talking about today.

If you think about the periodic table, for example, we had all these atoms and were able to group them together according to their proton and neutron numbers. We knew how many electrons were in the outermost shells, which helped us arrange them in the periodic table. This is similar to what we want to do now with hadrons.

Introduction to Isospin and Nucleon States

We are now looking at hadrons and their classification based on quantum numbers. These quantum numbers help us group particles together and understand their behavior. Let’s start with isospin.

In 1932, the neutron was discovered by Chadwick. Experiments at the Bevatron showed that the interaction strengths in photon-proton, proton-neutron, and neutron-neutron interactions were very similar. The rates of these interactions were practically identical. Additionally, the masses of the proton and neutron are nearly the same:

\[ m_p \approx 938.272 \ \text{MeV}, \quad m_n \approx 939.565 \ \text{MeV} \]

This led to the idea that the proton and neutron could be treated as two states of the same particle, described by isospin. The strong interaction does not distinguish between them — they are only differentiated in electromagnetic interactions.

Note

The proton and neutron are collectively referred to as nucleons, with isospin describing their two possible states.

This concept simplifies the description of nuclear interactions, treating the nucleon as a single entity with two states rather than two distinct particles.

Isospin and Quark Combinations

The isospin $I$ is $\frac{1}{2}$ for nucleons, allowing us to describe the proton and neutron as two states of the same particle in an isospin doublet. This is analogous to electron spin, where we have spin-up or spin-down states.

At the quark level, protons and neutrons are composed of up and down quarks. The strong interaction does not distinguish between these flavors, meaning it treats up and down quarks identically.

The up quark has an isospin of $\frac{1}{2}$ with a third component $I_3 = +\frac{1}{2}$, while the down quark has $I_3 = -\frac{1}{2}$. Although isospin is not identical to spin, it can be treated mathematically like spin, following the $SU(2)$ algebra.

The isospin operators are represented similarly to spin operators, with Pauli matrices acting as generators. For example, applying the $I_3$ operator to the up quark state yields the eigenvalue $+\frac{1}{2}$. The lowering and raising operators allow transitions between different projections of isospin, which range from $-I$ to $+I$ in steps of 1, giving $2I + 1$ possible projections.

Note

The $SU(2)$ algebra for isospin mirrors that of spin, with the same commutation relations and matrix representations.

Now, let’s consider quark combinations. Mesons are formed by pairing a quark and an antiquark. For up and down quarks (each with $I = \frac{1}{2}$), combining two isospin-$\frac{1}{2}$ particles can yield either a total isospin of $0$ (singlet) or $1$ (triplet). The resulting meson states can be classified based on their total isospin and its third component $I_3$.

Isospin Projections and Matrix Decomposition

The commutation relation for isospin operators is given by:

\[ [I_i, I_j] = i \epsilon_{ijk} I_k \]

When combining two isospin-$\frac{1}{2}$ particles, the possible total isospin states are $0$ or $1$. This is analogous to spin combinations. For isospin-$\frac{1}{2}$, there are two projections: $+\frac{1}{2}$ and $-\frac{1}{2}$.

For an isospin-$1$ state, the number of projections is determined by $2I + 1$, which gives three possible projections: $+1$, $0$, and $-1$. For an isospin-$0$ state, there is only one projection: $0$.

In terms of dimensions, combining two isospin-$\frac{1}{2}$ particles (e.g., a quark and an antiquark) results in a decomposition into a triplet (isospin-$1$) and a singlet (isospin-$0$). This corresponds to a four-dimensional space, which can be represented as a $4 \times 4$ matrix decomposed into a $3 \times 3$ block (triplet) and a $1 \times 1$ block (singlet).

The triplet represents the three isospin-$1$ states, while the singlet represents the single isospin-$0$ state. This decomposition follows group theory principles and will be further explored in terms of matrix representations later.

Combining Isospins and Glebsch-Gordan Coefficients

The singlet state can be expressed in terms of up and down quarks and their corresponding antiquarks. To understand the coefficients in these states, we use Glebsch-Gordan coefficients, analogous to combining spins or isospins.

For example, when combining two isospin-$\frac{1}{2}$ particles, the possible total isospin states are $1$ (triplet) or $0$ (singlet). The triplet has three projections: $+1$, $0$, and $-1$, while the singlet has only one projection: $0$.

The Glebsch-Gordan coefficients determine the signs and coefficients in front of these states. For instance, combining an up quark ($I_3 = +\frac{1}{2}$) and an anti-up quark ($I_3 = -\frac{1}{2}$) gives:

\[ \left| 0, 0 \right> = \frac{1}{\sqrt{2}} \left( \left| \frac{1}{2}, \frac{1}{2} \right> \left| \frac{1}{2}, -\frac{1}{2} \right> - \left| \frac{1}{2}, -\frac{1}{2} \right> \left| \frac{1}{2}, \frac{1}{2} \right> \right) \]

For antiquarks, the isospin projections are inverted: the anti-up quark has $I_3 = -\frac{1}{2}$, and the anti-down quark has $I_3 = +\frac{1}{2}$. This convention ensures quarks and antiquarks transform consistently under SU(2) operations.

In the singlet state, combining up and anti-up quarks gives a term with a $\frac{1}{\sqrt{2}}$ coefficient. Similarly, combining down and anti-down quarks introduces a minus sign, but due to the antiquark projection convention, the final term becomes positive.

The same logic applies to the triplet states. For example, the $I_3 = 0$ triplet state is:

\[ \left| 1, 0 \right> = \frac{1}{\sqrt{2}} \left( \left| \frac{1}{2}, \frac{1}{2} \right> \left| \frac{1}{2}, -\frac{1}{2} \right> + \left| \frac{1}{2}, -\frac{1}{2} \right> \left| \frac{1}{2}, \frac{1}{2} \right> \right) \]

This demonstrates how the Glebsch-Gordan coefficients structure the combined states. The method is essential for constructing isospin eigenstates from quark and antiquark combinations.

Isospin Combinations and Particle Groupings

Here, the triplet typically consists of pions: $\pi^+$, $\pi^0$, and $\pi^-$. We can also have rho particles, which are similar but distinguished by their spin. The pions have antiparallel spin (spin 0), while the rho particles have parallel spin. Otherwise, their isospin structure is the same. For the singlets, we can have the omega ($\omega$) and the eta prime ($\eta'$). By choosing the isospin quantum number, we can group these particles together.

Now, let’s consider an example where a $\pi$ beam is shot at a proton target. The pion has isospin 1, and the nucleon (proton or neutron) has isospin $\frac{1}{2}$. When combining isospin 1 and isospin $\frac{1}{2}$, the possible total isospin states are $\frac{3}{2}$ or $\frac{1}{2}$.

The dimensions of these states are determined as follows:
- For isospin 1, there are 3 projections ($I_3 = +1, 0, -1$).
- For isospin $\frac{1}{2}$, there are 2 projections ($I_3 = +\frac{1}{2}, -\frac{1}{2}$).
- For the combined $\frac{3}{2}$ state, there are 4 projections ($I_3 = \pm\frac{3}{2}, \pm\frac{1}{2}$).
- For the $\frac{1}{2}$ state, there are 2 projections ($I_3 = \pm\frac{1}{2}$).

This is how the isospin addition works in this scenario. The resulting states are determined by the Clebsch-Gordan coefficients, which dictate the weights and signs of the combined states.

Decomposition of Matrices into Irreducible Representations

We have here a $6 \times 6$ matrix, and we can decompose it into matrices of the subspaces. Specifically, this decomposition includes a $2 \times 2$ matrix and a $4 \times 4$ matrix. These smaller matrices are called irreducible representations.

Note

Irreducible representations are fundamental in group theory, as they cannot be further decomposed into smaller invariant subspaces under the group action.

If you’ve studied group theory, particularly in advanced quantum mechanics, you may already be familiar with reducible and irreducible representations. The decomposition here illustrates how larger matrices can be broken down into simpler, irreducible components.

For example, the $6 \times 6$ matrix splits into:
- A $2 \times 2$ block (corresponding to one subspace).
- A $4 \times 4$ block (corresponding to another subspace).

This structure is essential for understanding symmetry operations in particle physics.

Introduction to Isospin and Delta Particles

For small numbers, it is easy to write down reducible representations, but for larger numbers, there are methods to determine them. This will not be discussed in depth, but it is important to understand the meaning.

For isospin in $SU(2)$, you get four projections, which correspond to the delta particles. These particles have four charge states:
- $\Delta^{++}$
- $\Delta^{+}$
- $\Delta^{0}$
- $\Delta^{-}$

Their mass is approximately 1232 MeV.

Note

Delta particles are part of the baryon family and arise from isospin symmetry in particle physics.

The next example will illustrate this further.

Isospin Conservation and Cross Section Differences

We can now consider combining three quarks to form baryons, which corresponds to the tensor product $2 \otimes 2 \otimes 2$. Each quark has two possible isospin projections ($\pm \frac{1}{2}$).

From the previous discussion, $2 \otimes 2 = 3 \oplus 1$, and adding another quark gives:

\[ 2 \otimes 2 \otimes 2 = (3 \oplus 1) \otimes 2 = 4 \oplus 2 \oplus 2 \]

This results in an octet (8 states), including a spin-$\frac{3}{2}$ multiplet (the delta particles) and two spin-$\frac{1}{2}$ doublets.

Note

The isospin quantum number helps explain differences in measured cross sections. For example, the cross sections for $\pi^+ p$ and $\pi^- p$ reactions in the delta mass region (around 1232 MeV) differ by a factor of three.

This discrepancy arises from isospin symmetry and the Clebsch-Gordan coefficients governing the transitions. The exercises will explore this further.

Introduction to Isospin and Particle Groupings

The isospin concept was initially introduced based on observations of protons and neutrons. In the 1950s, experiments revealed patterns in particle production, such as pions and later rho mesons. Researchers used isospin to group particles into multiplets based on symmetries.

For example, pions and rho mesons form isospin triplets, while the delta baryons belong to an isospin quadruplet. These groupings were initially motivated by mathematical patterns rather than quark-level understanding (since quarks were not yet known).

The discussion naturally leads to larger symmetry patterns, which will be explored further in this lecture.

Note

Isospin helps categorize particles into families, such as triplets (pions) and quadruplets (deltas), revealing underlying symmetries in strong interactions.

Conservation of Strangeness and Hypercharge

The action of the isospin component $I_3$ on the up quark state is given by:

\[ I_3 \left| \frac{1}{2} \frac{1}{2} \right> = \frac{1}{2} \left| \frac{1}{2} \frac{1}{2} \right> \]

Bigger patterns emerged when studying particle reactions. If you look at protons and neutrons, you might think only spin and charge matter. However, experiments showed that reaction rates for proton-proton, proton-neutron, and neutron-neutron interactions were nearly identical. This led to the conclusion that the strong interaction does not distinguish between protons and neutrons.

On the quark level, this means the strong interaction does not care whether a quark is up or down. This is why both the proton-neutron pair and the up-down quark pair are placed into an isospin doublet. Initially, isospin was introduced for nucleons, but later it was extended to quarks as more particles like pions and rho mesons were discovered.

In the 1960s, experiments like the Bevatron revealed over 100 new particles. Researchers struggled to categorize them until they realized these particles could be decomposed into smaller constituents (quarks). This led to the search for larger symmetry patterns, incorporating quantum numbers like strangeness.

Strangeness was introduced to explain the behavior of certain particles. For example, cosmic ray experiments showed tracks in cloud chambers forming V-shapes, corresponding to pairs like $\pi^+ \pi^-$ or proton + $\pi^-$. These particles always appeared in pairs and had unusually long lifetimes. This “strange” behavior prompted the introduction of strangeness as a conserved quantum number.

Note

Isospin symmetry groups particles into multiplets (e.g., doublets, triplets), while strangeness explains the peculiar production and decay patterns of certain hadrons.

The next step is to explore how strangeness and hypercharge fit into these larger symmetry patterns.

Introduction to Strangeness and Hypercharge in Particle Physics

Isospin is a quantum number conserved in strong interactions. For particles composed of up and down quarks, the isospin range must be 0, as seen in pions, neutrons, and protons.

For particles with strangeness $+1$, examples include the lambda hyperon or kaons. These are classified as hyperons, which come in different types like sigmas and cascades. Kaons can form isospin doublets:

$K^+$ and $K^0$ (strangeness $+1$)
$K^-$ and $\overline{K^0}$ (strangeness $-1$)

Since these particles are always produced in pairs, strangeness must be conserved in strong interactions. However, it is not conserved in weak interactions. For example, in the decay of a lambda hyperon ($\Lambda \to p + \pi^-$), the final particles have no strangeness, yet the decay occurs.

Hypercharge ($Y$) is defined as:

\[ Y = B + S \]

where $B$ is the baryon number and $S$ is the strangeness. This quantum number helps categorize particles in larger symmetry patterns.

Conservation Laws and Baryon Number in Particle Reactions

The action of the isospin component $I_3$ on the down quark state is given by:

\[ I_3 \left| \frac{1}{2} -\frac{1}{2} \right> = -\frac{1}{2} \left| \frac{1}{2} -\frac{1}{2} \right> \]

The baryon number is straightforward:
- Baryons have $B = +1$
- Antibaryons have $B = -1$
- Mesons have $B = 0$

For example, a reaction like $p \to \pi^+ + \pi^-$ cannot occur because it violates baryon number conservation. Similarly, if you start with a proton beam and a $\pi^-$, the final state must include a baryon — you cannot end up with only mesons.

Strangeness is also conserved in strong interactions. For instance, a $\Lambda$ hyperon cannot decay into a proton and a $\pi^-$ via the strong interaction, as this would violate strangeness conservation. However, such decays are allowed in weak interactions, where strangeness is not conserved.

Important

Baryon number and strangeness must be conserved in strong interactions, but strangeness can be violated in weak decays.

Isospin, Charge, and the Gell-Mann–Nishijima Formula

In the isospin triplet, we have projections of $+1$, $0$, and $-1$, which correlate with the charge. For example, the pions have charges of $-1$ for $\pi^-$, $0$ for $\pi^0$, and $+1$ for $\pi^+$. This suggests a relationship between isospin and charge, which is formalized by the Gell-Mann–Nishijima formula:

\[ Q = I_3 + \frac{Y}{2} \]

Here, $Q$ is the charge, $I_3$ is the third component of isospin, and $Y$ is the hypercharge ( $Y = B + S$, where $B$ is baryon number and $S$ is strangeness).

Let’s verify this with the proton:
- Isospin $I_3 = +\frac{1}{2}$
- Hypercharge $Y = 1$ (since the proton has strangeness $S = 0$ and baryon number $B = +1$)
- Charge: $Q = \frac{1}{2} + \frac{1}{2} = +1$, as expected.

Similarly, for the neutron:
- Isospin $I_3 = -\frac{1}{2}$
- Hypercharge $Y = 1$
- Charge: $Q = -\frac{1}{2} + \frac{1}{2} = 0$.

Using strangeness and hypercharge alongside isospin, Gell-Mann and Nishijima identified larger patterns for these particles, allowing them to be arranged into bigger multiplets. This will be explored further in the next section.

Important

The Gell-Mann–Nishijima formula connects charge, isospin, and hypercharge, providing a framework for classifying particles into multiplets.

Baryon Multiplets and the Eightfold Way

The isospin squared operator acts on the isospin state as:

\[ \hat{I}^2 \left| I, I_3 \right> = I(I+1) \left| I, I_3 \right> \]

Particles can be arranged into larger multiplets, which was historically called the “Eightfold Way.” For these larger patterns, we consider not just the up and down quarks but also the strange quark, placing us in the context of $SU(3)$ flavor symmetry.

Here, we plot the third component of isospin ($I_3$) against strangeness ($S$) for the up, down, and strange quarks. The quantum numbers for these quarks are:
- Baryon number: Each quark has $B = \frac{1}{3}$, so a proton (uud) has $B = 1$.
- Charge: $Q = +\frac{2}{3}$ for up, $Q = -\frac{1}{3}$ for down, and $Q = -\frac{1}{3}$ for strange.
- Spin: All are spin-$\frac{1}{2}$ fermions.
- Strangeness: $S = 0$ for up/down, $S = -1$ for strange.

Baryons are composed of three quarks, which can be up, down, or strange. The combinations $3 \otimes 3 \otimes 3$ decompose into $10 \oplus 8 \oplus 8 \oplus 1$, forming a decuplet and two octets. We focus on the ground-state baryons, which arrange into an octet and a decuplet.

On a plot of $I_3$ vs. strangeness (or hypercharge $Y = B + S$), we observe:
- Isospin doublet: Proton ($I_3 = +\frac{1}{2}$) and neutron ($I_3 = -\frac{1}{2}$).
- Isospin triplet: Sigma baryons ($\Sigma^+, \Sigma^0, \Sigma^-$).
- Isospin doublet: Cascade baryons ($\Xi^0, \Xi^-$).

The quark content reveals:
- Particles with one strange quark (e.g., $\Lambda, \Sigma$).
- Particles with two strange quarks (e.g., $\Xi$).

Mass differences provide insights:
- Horizontal (same strangeness): Small mass differences (e.g., proton-neutron mass difference ~1 MeV).
- Vertical (increasing strangeness): Larger mass differences (e.g., $\Sigma$ to $\Xi$ ~130 MeV, $\Sigma$ to $\Lambda$ ~250 MeV).

This indicates that $SU(2)$ isospin symmetry is a good approximation, but $SU(3)$ flavor symmetry (including the strange quark) is significantly broken, as evidenced by the mass disparities.

Important

The mass splitting patterns reveal that $SU(3)$ flavor symmetry is only approximate, with isospin symmetry being more robust within subsets of particles.

Spin, Parity, and the Omega Minus Prediction

The spin and parity of the particles in the octet are $J^P = \frac{1}{2}^+$, while the decuplet has $J^P = \frac{3}{2}^+$. These states can be visualized as spin configurations: for the octet, the spins can be in permutations like $\uparrow \downarrow \uparrow$, resulting in a net spin of $\frac{1}{2}$, whereas in the decuplet, all spins are aligned (e.g., $\uparrow \uparrow \uparrow$), giving $\frac{3}{2}$.

When Gell-Mann identified this pattern, the $\Omega^-$ had not yet been discovered. Its prediction was a major success, and it was experimentally confirmed two years later in bubble chamber experiments via the reaction:

\[ K^- + p \to \Omega^- + K^+ + K^0 \]

To conserve strangeness, the decay chain proceeds as:

\[ \Omega^- \to \Xi^0 + \pi^- \quad \text{(or } \Xi^- + \pi^0\text{)} \]

followed by further decays like $\Xi^- \to \Lambda^0 + \pi^-$ and $\Lambda^0 \to p + \pi^-$.

Gell-Mann also predicted the mass of the $\Omega^-$ by observing the mass spacing in the decuplet. The mass differences between horizontal levels (e.g., $\Delta(1232)$, $\Sigma(1385)$, $\Xi(1530)$) are roughly 150 MeV, leading to an $\Omega^-$ mass prediction near 1680 MeV.

Important

The $\Omega^-$ discovery validated the $SU(3)$ flavor symmetry framework and the quark model, despite the symmetry being approximate due to mass splitting.

Meson Multiplets and Quantum Numbers

The action of the isospin component $I_3$ on the isospin state is given by:

\[ I_3 \left| I, I_3 \right> = I_3 \left| I, I_3 \right> \]

These considerations from group theory mathematically describe many particles using symmetry arguments. The mass differences between particles in the octet and decuplet arise from their spin configurations. Specifically, the mass difference can be attributed to spin-spin or spin-orbit interactions, which are dynamical effects.

For mesons, the multiplet structure is derived from the quark-antiquark combinations of up, down, and strange quarks. The product of three quarks and three antiquarks gives an octet plus a singlet ( $3 \times \bar{3} = 8 + 1$ ). The diagonal elements of the multiplet represent charge states:

The lower diagonal entries have a charge of $-1$.
The middle entries have a charge of $0$ or $+1$.
The upper diagonal entries can reach up to $+2$.

This octet (or nonet, including the $\eta'$) has quantum numbers $0^-$ or $1^-$. For the pseudoscalar mesons (e.g., pions, kaons), the quark-antiquark pair has antiparallel spins, resulting in total spin $0$, parity $-1$, and charge conjugation $+1$. For the vector mesons (e.g., $\rho$, $K^*$), the spins are aligned parallel, giving spin $1$, parity $-1$, and charge conjugation $-1$.

Note

Parity inverts spatial coordinates, while charge conjugation converts a particle into its antiparticle.

For baryons, the parity is calculated as $(-1)^L$, where $L$ is the orbital angular momentum. For ground-state baryons ( $L = 0$ ), the parity is $+1$, leading to designations like $\frac{1}{2}^+$ for the octet and $\frac{3}{2}^+$ for the decuplet. For mesons, the parity formula is $(-1)^{L+1}$, which results in $-1$ for both pseudoscalar and vector mesons.

Important

Charge conjugation is well-defined only for neutral mesons, where it flips the particle and antiparticle states.

Meson Decays and C Parity Conservation

Charge conjugation is only well-defined for neutral mesons. For example, applying it to the $\pi^0$ gives a C-parity eigenvalue of $+1$. When analyzing particle decays, it is important to consider how quantum numbers combine multiplicatively, such as parity or C-parity, similar to how spins combine to form a total spin.

For instance, consider the decay of an $\omega$ meson into two $\pi^0$ mesons. The C-parity of $\pi^0$ is $+1$, so the product for two pions is $(+1) \times (+1) = +1$. However, the $\omega$ meson has C-parity $-1$, meaning this decay violates C-parity conservation and cannot occur.

Important

C-parity is multiplicative, and decays must conserve it if the interaction respects charge conjugation symmetry.

This principle applies broadly when evaluating possible decay channels. Further exploration of such constraints will be covered in exercises.

Introduction of Color Quantum Number in Quark Model

The possible eigenvalues of the isospin component $I_3$ are given by:

\[ 2I + 1 \text{ possible eigenvalues of } I_3: -I, -I+1, \ldots, I-1, I \]

Now, consider the quark content of particles like the $\Omega^-$ or $\Delta^{++}$. In these cases, all quarks have their spins aligned (all spin up) and identical flavor content. This raises a problem: quarks are fermions and must obey the Pauli exclusion principle, meaning their total wave function must be antisymmetric under exchange.

Let’s examine the total wave function of these baryons. For $L = 0$, the spatial wave function is symmetric. The spin wave function is also symmetric because all spins are aligned. The flavor wave function is symmetric as well, since all quarks are of the same flavor.

To resolve this, physicists introduced a new quantum number: color charge. The color part of the wave function must be antisymmetric to ensure the total wave function is antisymmetric. We assign each quark a distinct color (red, green, or blue), making them distinguishable.

Important

The introduction of color solves the Pauli exclusion problem for identical quarks in baryons and is a fundamental aspect of Quantum Chromodynamics (QCD).

This concept was briefly discussed earlier, and we will now transition to discussing an experiment related to these ideas.

Experimental Evidence for Three Quark Colors

We will now discuss one last experiment to confirm the number of quark colors. Previously, I mentioned that we have three colors, but how do we know it’s three and not six, for example? Since we have six types of quarks, experiments were conducted to verify this.

The experiment involved studying $e^+ e^-$ annihilation, where a virtual photon is exchanged. This process can produce different types of particles, such as quark-antiquark pairs or leptons like $\mu^+ \mu^-$. The mass of the produced systems can be measured, and the cross sections for these processes were compared.

The key idea is to take the ratio of cross sections:

\[ R = \frac{\sigma(e^+ e^- \to \text{hadrons})}{\sigma(e^+ e^- \to \mu^+ \mu^-)} \]

For leptons like $\mu^+ \mu^-$, there is no color charge, but for quarks, the cross section is proportional to the number of colors. By dividing the hadronic cross section by the leptonic one, we obtain a value proportional to the number of colors.

The virtual photon at the vertex introduces a proportionality to the squared charge of the quarks ($Q^2$). At lower energies, lighter quarks (up, down, strange) dominate, and resonances like the $\rho$, $\omega$, and $\pi$ mesons appear as peaks in the spectrum. At higher energies, heavier quark-antiquark bound states such as charmonium ($J/\psi$) and bottomonium ($\Upsilon$) are produced.

The measured steps in $R$ correspond to the thresholds for producing these heavier quarks. The exact size of these steps could be calculated, and the results consistently indicated that the number of colors must be three.

Important

This experimental evidence strongly supports the existence of three quark colors, a fundamental aspect of Quantum Chromodynamics (QCD).

We will discuss this further in upcoming sessions.

Historical Development of the Quark Model

The isospin ladder operators are given by:

\[ I_+ = I_1 + iI_2, \quad I_- = I_1 - iI_2 \]

The number of colors should be three, which we will discuss later. This lecture provided a brief historical background on how the observed particles revealed patterns using concepts like strangeness and isospin. These patterns led to the formation of multiplets and eventually the quark model, where all detected particles were understood as composites of smaller entities called quarks.

We concluded that quarks must come in six different flavors and three distinct colors. This summary captures the key historical developments that shaped our understanding of the quark model.

Discovery and Properties of Strange Particles

This is a bit of historical background to the topic. Generally, what I find difficult with histories like this is understanding what was known and unknown at the time. For example, when strangeness was introduced in cosmic rays, the key question was: how do you distinguish chaotic behavior from invariant masses?

These particles were called “strange” because their behavior was puzzling — they appeared only in pairs and had longer lifetimes than expected. To study them, you analyze invariant masses by detecting decay products like protons and pions, then examine their spectra.

The known particles at the time were the proton (discovered earlier) and the neutron (1932), followed by pions shortly after. The confusion around strange particles stemmed from their unusual production and decay patterns, which later led to the introduction of strangeness as a quantum number.

Introduction of Charge Correlation and Quantum Numbers

The isospin squared operator is expressed in terms of ladder operators as:

\[ \hat{I}^2 \left| I, I_3 \right> = (I_1^2 + I_2^2 - i[I_1, I_2]) \left| I, I_3 \right> = (I_1^2 + I_2^2 + I_3^2 - I_3^2 + I_3) \left| I, I_3 \right> \]

Note

Here, $I_1, I_2, I_3$ are the components of isospin, and $[I_1, I_2]$ represents their commutator. The ladder operators help describe transitions between states with different $I_3$ values.

The proton was discovered first, followed by the neutron in 1932. Later, in the 1950s, the antiproton and other charged particles were observed. Charge correlation was introduced around the same time as the concept of particles and antiparticles, likely with the positron (predating the antiproton’s discovery).

The introduction of quantum numbers like charge correlation often stems from experimental observations. For example, if a decay is not observed, it suggests a violated quantum number. Historically, some concepts emerged from theory (e.g., the prediction of the $\Omega$ particle), while others were inferred from cross-section measurements.

Important

The interplay between theory and experiment is key — quantum numbers are sometimes postulated to explain missing decays or unexpected cross sections, rather than being derived purely from first principles.

Postulation and Discovery of Quarks

The historical development was mostly correct. Initially, the multiplets were assembled under the assumption of a smaller substructure, which was first called partons rather than quarks. The quark content was added later to explain these patterns.

Note

The term partons was used before quarks became the established name for the fundamental constituents of hadrons. The quark model emerged as a theoretical framework to describe the observed particle multiplets.

Mass Differences and Quark Predictions

The simplified expression for the isospin squared operator is given by:

\[ \hat{I}^2 \left| I, I_3 \right> = (I^2 - I_3^2 + I_3) \left| I, I_3 \right> \]

Initially, the substructure particles were called partons before the term quarks was established. The mass differences observed in particles like the deltas, sigmas, and cascades suggested a pattern. For instance, an undetected particle was predicted to have a mass 150 MeV higher than others, which turned out to be correct.

The quark model was introduced in a concise two-page paper. The prediction was made in 1962, and the Omega minus was discovered in 1964, confirming the theoretical framework.

Note

The quark model emerged to explain the observed particle multiplets, replacing the earlier parton terminology.