The Theory of Almost Everything: The Standard Model, the Unsung Triumph of Modern Physics - Robert Oerter (2006)

Appendix C. Interactions of the Standard Model

The Standard Model can be written either in mathematical form, using the Lagrangian function in Chapter 10, or in pictorial form, using Feynman diagrams. The two forms are completely equivalent if one knows the Feynman rules for translating between diagrams and mathematics. This appendix will display all of the Feynman diagrams for the Standard Model, thereby giving a complete list of all the fundamental interactions of the particles. These diagrams summarize the rules by which the universe runs. After giving all of the diagrams, I’ll indicate how they relate to the mathematical formulation of the theory.

First, there are the propagators. The propagator gives the probability for a given particle to go from one place to another in the absence of any interactions. The mathematical form of the propagator depends on the particle’s spin; this is indicated in the language of diagrams by using a straight line for a fermion, a squiggly line for a spin-one intermediate particle, and a dashed line for the spin-zero Higgs, as in the examples here:

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There is a propagator for each particle in the Standard Model, of course. Since they all look alike, I won’t bother to list them all separately. The labels on the ends specify which specific particle is involved.

Now, the interactions. All of the interactions of the gluons among themselves, and of the W, Z°, and photon with each other, are determined by the Yang-Mills symmetry of the theory. For gluons, there are the three-gluon and four-gluon interactions from Chapter 8:

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The electroweak intermediate particles interact among themselves as follows:

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(Here, and everywhere in this appendix, I leave out diagrams that can be obtained from these by exchanging all particles with their antiparticles.)

Next, we need the interactions between the fermions and the intermediate particles. The diagrams involving electrons and neutrinos are these:

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The diagrams for the muons and taus look exactly the same. Just replace all the electrons with muons (or taus) and all the electron neutrinos with mu neutrinos (or tau neutrinos). The interactions involving quarks are similar to the lepton interactions. We have, for the first quark family:

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The last diagram actually involves a quark from the second family, the strange quark (s), as well as a quark from the first family, the up quark (u). This is an example of quark mixing, in which quarks of different families get jumbled up. The amount of mixing is controlled by a parameter known as the Cabibbo angle. It’s one of the eighteen Standard Model parameters mentioned in Chapter 10. For the other quark families, the Feynman diagrams look just the same as the first three diagrams above. There are additional quark mixing diagrams as well, along with appropriate additional parameters.

The weak interactions of the fermions described by the diagrams above give a deeper understanding of beta decay. In beta decay, a neutron decays into a proton plus an electron plus an antineutrino. We can now describe this process using the quark model of the neutron, together with the interactions of the Standard Model. The picture looks like this:

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Here we see a neutron (dud) that emits a (virtual) W , turning into a proton (duu). The W- then decays into an electron (e-) and an antineutrino (124e )—exactly what’s needed for beta decay.

The gregarious Higgs boson interacts with just about everybody. It interacts with the intermediate particles:

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and with the fermions:

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Once again, only the diagrams for the first family are given; there are similar-looking diagrams for the other fermion families. These are the interactions that give the fermions their masses after spontaneous symmetry breaking. Note that there is no interaction between the neutrinos and the Higgs; thus, the neutrinos remain massless in the Standard Model.

Finally, the Higgs interacts with itself:

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These self-interactions generate the Mexican hat potential so crucial for spontaneous symmetry breaking.

The interactions just listed reveal how the Higgs gives mass to the fermions. Picture the Mexican hat potential with its central hump. The Higgs field is zero at the center of the potential—at the top of the hump, that is. When symmetry breaks, the Higgs field rolls off the hump and ends up in the trough, where it is no longer zero. As we saw in Chapter 9, this shift of the Higgs field occurs at every point in the entire universe. So there is a background of Higgs field throughout the universe. Picture yourself as a fermion: wherever you go, there is a Higgs field. You cannot escape it. Like an annoying little brother constantly plucking at your sleeve, it’s always there pulling on you, thanks to the interactions in the diagrams above. It is this constant tug of the background Higgs field that gives mass to the quarks and leptons.

Now you know everything that physicists have learned about the fundamental processes of the universe! Any (non-gravitational) interaction of the known fundamental particles involves some combination of the basic processes listed here. In order to generate actual numbers from the diagrams, though, we need a more mathematical description.

The mathematical description of the Standard Model starts from the Lagrangian function.

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As we learned in Chapter 10, G describes the gluon field and its interactions, W stands for the SU(2) field (describing the W+, W-, and Z° particles and their interactions with each other), F is the U(1) field, and H is the Higgs. The symbol Σ here means “add the following things together,” while each fj (for different values of the index j) stands for one of the fermions: the electron and its kin, the neutrinos, and the quarks. The last term in the Lagrangian therefore involves two fermion fields and the Higgs field. The Cjk are the coupling constants for the fermion-Higgs interactions. These numbers determine the masses of all the quarks and leptons. The actual numerical values of the coupling constants are not determined by the theory and must be learned from experiments. The symbol V represents a mathematical operation called a “covariant derivative.”

There is a direct relationship between each of the Feynman diagrams given above and some piece of the Lagrangian. Look, for instance, at the term -λH4. The symbol λ stands for a numerical parameter, while H4 represents four Higgs fields multiplied together. In terms of Feynman diagrams, this is exactly the piece that gives us the four-Higgs interaction shown above. The parameter λ tells the likelihood of the four-Higgs process: it is the coupling constant that must be associated with that particular Feynman diagram. Since no one has ever detected a Higgs particle, let alone measured the probability for Higgs particles to scatter off each other, no one knows the numerical value of λ. It is one of the eighteen parameters that determine the Standard Model of which, as yet, we have no experimental knowledge.

Every term in the Lagrangian corresponds to a Feynman diagram in just the same way. Not all of the interactions are as easy to pick out as the four-Higgs interaction: some interactions are hidden by the notation. The interactions of the gluons among themselves are hidden inside the G2 term, while the interactions between the fermions and the intermediate particles are hidden inside the V symbol. The purpose of all this sleight of hand is to make it possible to write the Lagrangian in a very compact form. To go from the Lagrangian to the Feynman diagrams, we need to make all of the hidden terms explicit. Like a Swiss army knife, the Lagrangian is only useful when you pull out the bits that are folded up inside.

Naturally, neither the Lagrangian nor the Feynman diagrams given here are enough for you to start calculating probabilities for particle interactions. There are other details (most of them having to do with the spin of the different particles) that I have left out of this simplified Lagrangian. Even beyond such simplifications, there is the whole structure of relativistic quantum field theory, which tells how to take a Lagrangian, or a set of Feynman diagrams, and extract actual numerical predictions. If you want to learn all of the details, you’ll just have to go to graduate school in physics!