QUANTUM MECHANICS

The Glue Holding the Many Particles Together: A Journey Through Quantum Chromodynamics

Featured image: Volume rendering of constant topological charge density for a quantum field theory on a lattice. (Courtesy C. Rebbi, A. Avakian, R. Babich, R. Brower, M. Clark, J. Osborn, D. Schaich, R. Gasser)

The Strong Interaction is one of the Four Fundamental Forces and the strongest at the range it operates. It is responsible for holding much of the matter around us together, with its residual effects being essential in holding the Nucleus together. It’s a mysterious force that operates at the scale of quarks, and the residual effects of its force produce another force necessary in keeping the atom together: The Nuclear Force.

In this article, we’ll explore the role of this Fundamental Force, the particles that interact with and mediate this force, and some of the equations and corresponding theories that explain how it operates.

The Strong Interaction is mediated by Gluons, which are a massless vector gauge boson, and they “hold together the quarks that make up protons and neutrons” (Fermilab Today, 2013 [1]) What this means is that all matter that’s composed of quarks (Hadrons) is held together through the Strong Interaction. Additionally, the residual effects of the Strong Interaction at a greater distance result in what’s known as the Nuclear Force. The Nuclear Force is responsible for “binding protons and neutrons together inside the nucleus of an atom.” (Fermilab Today, 2013 [1])

In stable nuclides, this attractive force is balanced with the electrostatic forces of repulsion, which act between the nucleons in the atom and keep nucleons within an optimum range of 1 fm (1/10¹⁵m). In nuclides that are either much larger, or have an imbalance of nucleons (say an excess of protons), the nuclide becomes unstable as these forces are no longer balanced. As a result, it can undergo radioactive decay due to the Weak Nuclear Force.

Introduction to Color Charge

Figure 1: A diagram representing the different ‘colors’ and ‘anticolors’, from The Open University, and the resulting color from varying combinations of colors.

All the Fundamental Forces will interact differently, depending on the value of some quantum number, which varies between forces. For Electromagnetic Interaction, the force experienced depends on the electric charge of the bodies, while for Gravity, the force experienced depends on the mass of the bodies in a system.

Likewise, a fundamental property of subatomic particles dictates how they are affected by the Strong Interaction, known as Color Charge.

Color Charge is simply a property that subatomic particles can have, and the possible individual values include red, green, and blue, with anti-values as well which are anti-red, anti-green, and anti-blue. Don’t mistake them for the colors that we observe in our daily lives, however, as the colors observed in our daily lives are just parts of the EM Spectrum, resulting from photons with specific frequencies, whilst Color Charge isn’t something observed in the macroscopic world. Also, just like all quantum numbers, or particle properties, it will be conserved during any interactions that take place in an isolated system.

Particles that don't interact through Strong Interaction include Leptons, and this is because they are colorless. For something to be colorless, it must have a white color charge which we can see, using Figure 1, requires an equal amount of each color, meaning an equal number of quarks with red, blue, and green color charges. Alternatively, if you don’t have each of those three colors, but an equal amount of said colors and anticolors, say an equal number of quarks with a red color charge, and anti-red color charge, then the system would still be considered colorless.

All the quarks which exist can take on any of the three color values (Red, Green, Blue), and their anti-particle counterparts can take on any of the ‘anti-Colors’.

As for the Hadrons which these quarks make up, there are two main subcategories of consideration: Baryons and Mesons.

Baryons

Baryons are particles made up of three quarks, with the main known examples being the proton and neutron. Just as there are Baryons, there are also anti-baryons, made up of three anti-quarks, with the combination of quarks resulting in a colorless end-particle.

Using Figure 1, we can see that a combination of all colors results in a colorless particle forming, which is the case for Baryons, as the combination of three quarks is such that one is red, one is blue, and one is green. As for Anti-Baryons, the combination of Anti-Quarks is such that one is anti-red, one is anti-blue, and one is anti-green.

For baryons, there’s a conserved value known as the Baryon Number which, as indicated by its name, determines the number of baryons in a system, where any individual Baryon has a Baryon number of 1, whilst Anti-Baryons have a Baryon number of –1, and other particles having a Baryon number of 0.

Mesons

Mesons are made up of two quarks: One quark, and one anti-quark. The combination of quark and anti-quark is such that the resulting Meson is colorless. Say we have a Neutral-Kaon (K⁰), made up of a Down-Quark and an Anti-Strange Quark (or vice versa): The Down-Quark would either have a color charge of Red, Green, or Blue, whilst the antistrange Quark would either have Antired, Antigreen, or Antiblue.

This idea that they can’t be found with independent color charges, where the combination of them always results in a colorless particle, is part of a larger phenomenon known as Color Confinement.

Gluons as the Mediators of the Strong Interaction

Gluons are the particles that are exchanged between quarks and are responsible for mediating this force. What makes them unique is that unlike photons, which solely mediate the Electromagnetic Interaction, Gluons are also affected by this interaction and take part.

The reasoning for this is that whilst Photons have a net-zero electric charge, meaning they’re unaffected by the EM Force, Gluons do have some resultant color charge.

When a quark emits a Gluon, the Quark’s color will change as the emitted Gluon has a specified color charge. Unlike quarks, however, the possible color charges aren’t just red, green, or blue, but rather a combination of the different colors and anti-colors, with an octet of possible combinations.

Figure 2: A Feynman Diagram representing a change in the Color Charge of Quarks as a result of the emission/absorption of a Gluon.

Now, say this Up-Quark emitted a Green-Antiblue Gluon: It would lose its ‘Greenness’, and as the Blue Color Charge between both of them must be 0 (Started with no blue in that quark), it will change to have a Color Charge of Blue.

As for the Down-Quark in this theorized system, it will absorb the Gluon, and when the Anti-Blue and Blue Charges combine, the resulting product is colorless. The only remaining component of color charge for the Down-Quark would thus be the color Green, hence its green color charge.

On this Feynman Diagram, the horizontal axis represents the energy of the particles, with the vertical axis representing time, where the diagonals above the horizontal line (which represents the transferred Gluon, which is our ‘interaction’ or ‘event’) are the system after our event, and the diagonals pointing in the direction of the Gluon’s line representing the system before the event.

The Up-Quark going backward after emitting the Gluon is due to a transfer of energy, with part of it being emitted via the Gluon. As the Gluon is absorbed by the Down-Quark, the Down-Quark must’ve gained energy; hence its movement towards the left (forward on the horizontal axis) in Figure 2.

Color Confinement

Color Confinement is a phenomenon that explains why quarks are confined, and can only be found combined with other quarks in Hadrons, as opposed to being found independently/free-roaming. As a result, only “color-single states can exist as free particles” (Cambridge High Energy Physics [2])

• Simply put, color-charged particles (meaning non-colorless particles), can’t be found independently and are instead colorless.
• The only free particles that can be observed in nature must have a “color-singlet state”, meaning the particle must be colorless, or part of a larger system of particles with this resulting system being colorless.

For a given set of particles, it will always have a combination of particles such that it will be colorless. The Hadrons that we observe in nature can exist as free particles because “these composites are color neutral” (Pittsburgh University [3])

• One of the reasons we know there aren’t Color-Singlet Gluons is that, if they existed, then they’d exist as free particles, with each essentially acting like a “strongly interacting photon” (Cambridge High Energy Physics [2]) What this would mean is that the Strong Interaction, as opposed to acting on the scale of quarks, would extend to an infinite range, which contradicts what we know about the Strong Interaction.

The specifics of why Color Confinement prevents non-colorless particles from being observed in nature can be explained by a phenomenon known as Asymptotic Freedom. For now, let’s redirect momentarily and mention what affects the strength of the Strong Interaction, which we can determine using the Strong Coupling Constant.

Coupling Constants: The Strong Coupling Constant

Coupling Constants are values that determine the strength of the force exerted in a given interaction. In Electromagnetism, the Coupling Constant is provided by a value known as the Fine-Structure Constant (α), which has a value of 1/137. The Fine-Structure constant is a value that determines the strength of the Electromagnetic (EM) interaction between elementary particles.

1) α = (1 / 4πε₀) × (e² / ħc)
Here, the term (1 / 4πε₀) is a value known as Coulomb’s Constant (k), used to calculate the Force acting between two charged bodies that interact via the EM Interaction. The above constant can be re-expressed as such:

2) α = ke² / ħc

e² here is equal to e×e, with e denoting the elementary charge, or the amount of electric charge within a proton/electron

The final term, or product of terms, is ħc, where c is the speed of light in a vacuum, and ħ is the Reduced Planck Constant, which is a term that relates to the Angular Momentum (L) of an Electron, at some energy level n.

• The Angular Momentum for an Electron about a given nucleus can be expressed as nħ, where moving to higher Orbitals will increase the Angular Momentum by a multiple of ħ.

Note that this is the Coupling Constant for the Electromagnetic Interaction, with the Strong Coupling Constant appearing slightly different. The value for the Strong Coupling Constant depends on the 4-momentum of the system.

• 4-momentum involves taking the classic 3-momentum (Momentum of a body by taking into consideration the sum of the Momenta in the x, y, and z directions) and “mapping them to a four-dimensional spacetime” (Wikipedia)

The components of 4-momentum can be expressed as such:

p = (E/c, Px, Py, Pz), where E is the energy of the particles in question, and the other 3 coordinates represent the momentum of the particle/body in a given direction (Px could represent a leftward/rightward momentum, Py could represent vertical momentum, and Pz could represent forward/backward momentum)

For a 4D spacetime, we use one time coordinate (which is calculated as E/c where E is the energy of the body in question), and 3 spatial coordinates. By spatial coordinates, we refer to values that describe a body’s motion through space. To make my point clearer, let’s describe the position of a body in a 4D spacetime, as such:

s = (t, x, y, z) where x, y, and z are all the positions of a body, compared to some point, in 3 different directions. If we had a 3D graph, then imagine each of our 3 spatial coordinates as being mapped to one of those axes.

• These coordinates each represent our position, based on that quantity, or how far we’ve moved in each direction. The faster we moved, the greater the value of our spatial coordinates relative to our time coordinates.
• Imagine someone running a 100m sprint in a single direction, with only one of the spatial coordinates changing: The faster they run, the less time it takes to cover this distance, so the relative change in position is greater with respect to time.

Now, a key point about the 4-momentum is that the time coordinates value is dependent on the energy of the system, meaning that if there’s a greater amount of energy, then there’s a greater value for the body’s 4-momentum. The big takeaway with it though is that 4-momentum relates to a body’s energy, so when we see it, think of it as being related to the energy.

The Strong Coupling constant (αₛ) is an example of a Running Coupling Constant, where its value is dependent on the energy level of individual particles, but for the most part, we use the value associated with the Z-Boson, which has a value of 0.1189 ± 0.0010.

If we don’t use this, then the generalized equation used looks like this:

3) αₛ(Q²) = 12π ÷ ((22–2nf) • ln(Q²/ Λ²) (National Superconducting Cyclotron Laboratory [4])

Λ: The QCD Scale, which is a parameter (input value) that our 4D Momentum tends towards. The maths associated with calculating it are very complicated, so we’ll treat it as a constant that’s approximately equal to 0.22GeV.

nf: A value which is calculated as “6 – the number of quark flavors” (National Superconducting Cyclotron Laboratory [4])

Q²: The squared value of our 4-Dimensional Momentum.

As for the separation, or distance between Quarks, it can be determined using the following equation:

4) λ = (ħ ÷ √Q²) = (ħ ÷ |Q|) (National Superconducting Cyclotron Laboratory [4])

λ represents the Quark separation here, and we can see that this separation is inversely proportional to the 4-momentum, where a greater value for the 4-momentum, and as such energy, means quarks are positioned closer to each other. Simply put, when there’s more energy supplied, or our Hadron is put in an excited state, the Quarks are positioned closer to one another.

The final term to look at is the Strong Potential, a term used to describe the strength of the Strong Force that holds quarks together.

First, what is Potential? Potential here represents the position of a body/particle, per given unit. The unit will describe what affects the particles’ interactions. Two examples include the Gravitational Potential, which represents the Gravitational Force per Unit Mass in kilograms, and the Electric Potential, which represents the Electric Potential per Unit Charge in Coulombs. The most important note for it is that it can help us describe the strength of a given interaction, based on this given unit, with the Gravitational Force being proportional to mass, and Electric Forces being proportional to the electric charge of the bodies.

Likewise, there’s a potential for QCD, which though different in appearance, serves a similar function: Describes the strength of a given interaction, here for the Color Force, based on a given factor, here being Color.

The Strong Potential can be calculated as such:

5) V = -4αₛ/3r + kr (University of Cambridge [5])

k here represents a physical constant, with an approximate value of 1GeV/fm, with r here representing the range. At a very small distance (below 1fm), the linear term (kr) will approach a near-zero value, with the reciprocal term (-4αₛ/3r) growing linearly. Equally, as our r value increases, our linear term (kr) will increase whilst our reciprocal term (-4αₛ/3r) falls down to 0. In either case, trying to change our distance from 1fm between quarks, then the amount of energy needed will increase, growing linearly the further we try to pull our quarks apart. The strength of the Strong Interaction (and thus the amount of force needed to separate quarks) can be expressed using our Potential, as such:

6) F = dV/dr = 4αₛ/3r² + k (University of Cambridge [5])

This idea of quarks having an ideal distance, and growing more difficult to separate as we pull them apart, links nicely to the idea of Asymptotic Freedom.

Figure 3: A diagram from the National Superconducting Cyclotron Laboratory relating the Strong Coupling Constant to the 4-momentum of a body.

The Strong Interaction’s Asymptotic Freedom

Unlike the other Fundamental Forces, the Strong Interaction has a unique attribute to it known as Asymptotic Freedom. First, we need to define what is meant by an Asymptote, as it helps us understand what we mean by Asymptotic Freedom. Simply put, an Asymptote is a value that we can forever tend to, without reaching: A numerical value that we cannot have.

• A simple mathematical example would be to have a mathematical function with a reciprocal of x, as such: f(x) = 1/x
• Here, our function can take any input values except for 0, where x≠0, with x tending to 0.

Asymptotic Freedom builds on this definition, in a sense, with the Strong Interaction between Quarks growing weaker at higher-energy levels, yet remaining present with Quarks, even at high-energy values, being unable to exist as free particles. To describe this, I’ll use an analogy I saw from the Physics Department at the College of Alameda, where the Strong Interaction is modeled like a spring, holding quarks together.

• If put closer together, the force acting on the spring is weaker, as can be determined using Hooke’s Law (F = –kx). With this analogy, we can say that the magnitude of the Strong Interaction is lower the closer quarks are together.
• To get these Quarks together, you need some amount of input energy, which is greater the closer your quarks are together. In our model, think of this as the force required to compress the spring, which is also given by Hooke’s Law. This spring analogy also seems to imply an ideal distance between Quarks.

Now, we’ll use the idea of Asymptotic Freedom to build on what we explained so far for the Strong Coupling Constant. The value of Q is inversely proportional to the separation between Quarks, meaning as Q² increases, the distance between Quarks decreases and, using Figure 4, we can see that as the value of Q² increases, the Strong Coupling Constant decreases in value asymptotically as for growing values of Q², Figure 3: A diagram from the National Superconducting Cyclotron Laboratory relating the Strong Coupling Constant to the 4-momentum of a body. grows and gets infinitely small, tending to 0, where “quarks can be considered free (Asymptotic Freedom)” [National Superconducting Cyclotron Laboratory].

• By contrast, an increasing separation of Quarks results in a smaller value for Q², which results in an increasing value for αₛ. What this means is that the further Quarks get apart, the stronger the Strong Interaction holding them together, making it impossible for Quarks to break free. This ability for the Strong Separation to grow increasingly strong as the Quark separation increases.

We can also use this to build on the idea of Confinement, as further extending the ‘spring’ beyond what it is at rest requires a large amount of input energy, which grows in magnitude the more the spring is extended, though, beyond a certain point, you’ll need to input more energy than can be supplied.

• Based on what we’ve said before, the higher the input energy, and as such the 4-momentum, the lower the Strong Coupling Constant and thus the weaker the Strong Interaction. This is one of the reasons why if sufficient energy is supplied, we can force Hadrons to decay, and change which Quarks are present in our system, resulting in a different end-product, though these Quarks cannot be observed as free-particles.

Now that we have an elementary understanding of how quarks are held together, and why we don’t observe them as free particles, we can now delve into the artificial ‘production’ of particles made up of quarks. Hadrons are produced in a process known as Hadronization.

Hadronization

Now that we’ve described the different properties that affect how particles interact via the Strong Interaction and the nature of free particles, we should discuss the process by which Hadrons are formed, known as Hadronization. This process involves creating high-energy collisions within particle accelerators, producing free quarks. However, we know that quarks/gluons cannot exist as free particles due to Color Confinement, and several theories describe what happens.

We’ll be using the Lund-String Model, which “treats all but the highest energy gluons as field lines” (Wikipedia [6]). Like with Electromagnetism, we can think of the Strong Interaction’s influence as acting on a given region due to the influence of a field, which stems from the exchange of virtual bosons, like photons or gluons. However, while Electromagnetic Fields can permeate through space freely, Gluons will be attracted to each other by the Strong Interaction, and as such, produce a “narrow tube (or string) of strong color field” (Wikipedia [6]).

• Simply put, the field that mediates the Strong Interaction (per the Lund-String Model) takes on a string-like structure and acts in a more limited range. These strings exert an influence that attracts quarks as well, causing them to bind and form Hadrons, with the series of Hadrons being observed in a jet (a conical structure consisting of Hadrons which we then observe).

Now that we’ve introduced what QCD is, and some of the governing principles and ideas on which it’s built, let’s address why it’s so important.

Real-World Significance of Quantum Chromodynamics

With an understanding of what QCD is, it’s only fit to understand its real-world use, and what we can understand about particles through its use.

1. It explains the likelihood of certain particle interactions occurring, using Color Factors.
2. It explains why Quarks don’t exist as free particles and the difficulty in temporarily observing them. The strength of the Strong Potential and its mathematical relationship with the quark-separation r allows us to understand the strength of the Strong Interaction and the nature of this interaction. Color Confinement also plays a role, as only colorless particles can exist as free particles.
3. Due to Asymptotic Freedom, when we try to separate quarks, it results in the production of a like particle-pair. Let’s say we have a Meson consisting of a Quark (Q) and an Anti-Quark (Ǭ): If the input/stored energy to separate the quarks has a value equal to the rest mass of both particles added, then a new QǬ Meson will be produced.
4. At a larger scale, all of Nuclear Physics revolves around manipulating/bombarding atomic nuclei, and trying to overcome or change the Strong Nuclear Force, brought about as a byproduct of the Strong Interaction that QCD explores.

To sum up, QCD is a fascinating field of physics and studies some of the underlying fundamentals that define how the matter around us is composed, and our exploration of this field has aided in developing our understanding of the world around us.

References

Reference 1: Fermilab today. (2013). https://www.fnal.gov/pub/today/archive/archive_2013/today13-04-12.html

Reference 2: Handout 8: Quantum Chromodynamics. (2009). Cambridge High Energy Physics. https://www.hep.phy.cam.ac.uk/~thomson/lectures/partIIIparticles/Handout8_2009.pdf

Reference 3: Color and confinement. (n.d.). Pittsburgh University https://fafnir.phyast.pitt.edu/particles/color.html

Reference 4: Strong coupling constant. (n.d.). National Superconducting Cyclotron Laboratory. https://people.nscl.msu.edu/~witek/Classes/PHY802/QCD2.pdf

Reference 5: Potter, T. (n.d.). QCD. https://www.hep.phy.cam.ac.uk/~chpotter/particleandnuclearphysics/Lecture_07_QCD.pdf

Reference 6: Wikipedia contributors. (2022, July 24). Lund string model. Wikipedia. https://en.wikipedia.org/wiki/Lund_string_model#References