commit 21eab252fb1b2ba0829c58f73f74300088750ab3
parent 5b2d01f4173c72298b84e8e6ad71fb3cc38ca734
Author: Ed van Bruggen <edvb@uw.edu>
Date: Thu, 25 Nov 2021 21:46:17 -0500
Add lattice QCD article
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diff --git a/content/posts/lattice-qcd.md b/content/posts/lattice-qcd.md
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+---
+title: "Intro to Lattice Quantum Chromodynamics"
+date: 2020-03-20
+tags: physics simulations
+categories: phys
+---
+
+At the start of the 20th century two new revolutionary and counterintuitive areas of
+physics were being developed, quantum mechanics and special relativity. While quantum mechanics
+describes the strange world of the subatomic with discrete energies, special relativity describes
+the effects of moving incredibly fast: time dilation and space contraction (general relativity is
+needed for a compete picture with mass warping spacetime).
+
+In order to understand the dynamics of subatomic particles moving near the speed of light these
+two fields had to be united into quantum field theory. This was first done to describe the
+electromagnetic force, quantizing it into the theory of quantum electrodynamics (QED) which
+satisfies both quantum mechanics and special relativity. However to fully understand the atom, the
+strong force also needs to be quantized into quantum chromodynamics (QCD).
+
+
+
+The strong force is what overcomes electric repulsion to hold protons and neutrons together
+in the nucleus, and binds quarks together to make up the protons and neutrons themselves. As the
+name implies, it is extremely strong compared to the electromagnetic force, which is why nuclei
+are able to be stable and not fly apart from the protons repelling each other (within certain
+limits).
+
+
+
+Similar to how the electromagnetic force only interacts with things that have electric charge
+(like protons and electrons, not neutrons), the strong force only interacts with things that have
+color charge. In the standard model of particle physics, which describes all currently known
+particles, the only particles that have color charge are called quarks and gluons. Gluons
+mediate the exchange of the strong force, and up and down quarks are what make up protons and
+neutrons, being "glued" together by gluons.
+
+
+
+However, unlike electric charge which comes in 2 kinds (positive and negative), color charge
+comes in 6 varieties: red, green, and blue, as well as their creatively named opposites anti-red,
+anti-green, and anti-blue (sometimes referred to instead as cyan, magenta, and yellow
+respectively). While electric charge can be thought of as being on one axis, with negative being
+the opposite of positive, color charge has 3 axes, each with its own color and anti-color.
+The standard non-antimatter quarks can at anytime have any of the 3 colors, while antimatter
+quarks can have any of the 3 anti-colors, and a gluon has one color and one anti-color at all
+times.
+
+When a particle is made up a quark and a anti-quark (called mesons), the anti-quark must have the
+anti-color of the quark. Likewise in a particle made up of three quarks (baryons) the quarks must
+have different colors, so a proton's up and down quarks must be red, green, and blue, and a
+anti-proton must have anti-red, anti-green, and anti-blue anti-quarks. This is due to a concept
+called color confinement, quarks can not be observed on their own, and any particle made up of
+quarks must have a net zero (also called white) color charge.
+
+
+
+Color confinement is one reason why we do not see the strong force play a role in everyday life,
+at sizes bigger than a proton it looks like there is no color charge. Another reason is that the
+gluon particle has mass unlike the photon which mediates electromagnetism, meaning that it has a
+limited lifetime and so can only act on short distances very close to quarks.
+
+It is also important to note that calling the strong force's charge "color charge" is only an
+analogy and visual aid, it has no relation or connection to the physics of color.
+
+{{<img "https://upload.wikimedia.org/wikipedia/commons/d/d0/Neutron_QCD_Animation.gif" 300 300>}}
+
+Because of the probabilistic nature of quantum mechanics, to calculate quantum field
+theory problems you need to account for all the possible ways an interaction could happen.
+This usually entails summing over an infinite number of possible configurations, which is
+impossible. Quantum electrodynamics solves this problem using perturbation theory, it can
+approximate a calculation by only including the most likely configurations. The more increasingly
+less-likely configurations you include the more accurate the calculation is.
+
+This works in QED because the electromagnetic coupling constant is less than 1, so more
+complicated configurations with more interactions contribute less and less to the final answer.
+However the strong force's coupling constant is 1, meaning that every possible way an interaction
+can happen is equally as likely. This makes it impossible to approximate the most likely
+scenarios using this method.
+
+In order remove this infinity and allow QCD problems to be calculated, space and time are instead
+discretized into a finite 4D grid. In this grid each point represents the quark field, and the
+lines connecting them represent the gluon field.
+
+
+
+Although this gets around the infinite possible configurations, it is still an extremely large
+problem. To reduce the complexity further so it can be computed in a reasonable time a Monte Carlo
+algorithm is utilized. Monte Carlo is a method of using random sampling to approximate a problem,
+it has many applications to problems in biology (see here for it used for protein folding),
+math, and physics. Monte Carlo is best demonstrated as seen below to approximate the value of pi
+by seeing if a random point lies within the circle or not, with the more random points the closer
+to the true value you get.
+
+
+
+Now with lattice QCD it is possible to preform calculations and make predictions about quarks and
+nuclei, such as the mass of mesons and baryons. The graph on the left below shows two different
+lattice QCD methods converging to the same mass for a pion, which very closely matches the
+experimentally measured mass. However the graph on the right shows what happens when this is
+applied to predict the mass of a neutron, they converge to a mass before diverging again.
+This divergence is due errors in the Monte Carlo approximation compounding together, creating
+a small golden window with the correct mass before error takes over.
+
+
+
+Lattice QCD can be used to solve countless problems in particle and nuclear physics, such as
+the very accurate mass predictions for more hadrons and quarks as seen below. There has also been
+a lot of progress in the last 10 years utilizing it to understand QCD decay constants, resonances,
+deep inelastic scattering, QCD phase transitions, and investigating color confinement.
+
+
+
+
+This article is based on a presentation I gave in 2020, see the full slides [here](/docs/lattice-qcd.pdf).
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