Hello everyone! Recently we’ve been looking at the Feynman rules for the Higgs boson. Last time I posted, we started to make a very suggestive connection between the Higgs and the origin of mass. We noted that the Higgs has a special trick up it’s sleeve: it has a Feynman rule that allows a Higgs line to terminate:

This allowed us to draw diagrams with two fermions or two gauge bosons attached to a terminated Higgs:

We made the bold claim that these diagrams should be interpreted as masses for the particles attached to the Higgs. We’ll explore this interpretation in a later post, but for now let’s better understand why we should have this odd Feynman rule and what it means.

# Quantum Fields Forever

Before we get into that, though, we need to get back to one of the fundamental ideas of quantum physics: **wave-particle duality**. Douglas Hofstadter’s famous ambrigram summarizes the well-known statement in pop-science:

Wave-particle duality is one of those well-known buzz words of quantum mechanics. Is light a particle or a wave? Is an electron a particle or a wave? If these things are waves, then what are they waves *of? *

In high energy physics we’re usually interested in small things that move very quickly, so we use the framework of **quantum field theory** (QFT) which is the marriage of quantum mechanics (which describes “small” things) and special relativity (which describes “fast” things). The quantum ‘waves’ are waves in the quantum field associated with a particle. The loose interpretation of the field is the probability that you might find a particle there.

*particles,*but a local description of quantum particles is naturally packaged in terms of

*fields.*For a nice discussion (at the level of advanced undergrads), see the first half hour of this lecture by N. Arkani-Hamed at Perimeter.

So the quantum field is a mathematical object which we construct which tells us how likely it is that there’s a particle at each point in space and time. Most of the time the quantum field is pretty much zero: the vacuum of space is more-or-less empty. We can imagine a particle as a ripple in the quantum field, which former US LHC blogger Sue Ann Koay very nicely depicted thusly:

Sometimes ripples can excite others ripples (perhaps in other quantum fields), this is precisely what’s happening when we draw a Feynman diagram that describes the interaction of different particles.

# The vacuum and the ‘Higgs phase’

Now we get to the idea of the **vacuum**—space when there isn’t any stuff in it. Usually when you think of the vacuum of empty space you’re supposed to think of nothingness. It turns out that the vacuum is a rather busy place on small scales because of quantum fluctuations: there are virtual particle–anti-particle pairs that keep popping into existence and then annihilating. Further still, vacuum is also filled with cosmic microwave background radiation at 2.725 Kelvin. But for now we’re going to ignore both of these effects. It turns out that there’s something *much* more surprising about the vacuum:

It’s full of Higgs bosons.

The quantum field for normal particle species like electrons or quarks is zero everywhere except where there are particles moving around. Particles are wiggles on top of this zero value. The Higgs is different because the value of its quantum field in the vacuum is *not *zero. We say that it has a **vacuum expectation value**, or “**vev**” for short. It is precisely this Higgs vev which is represented by the crossed out Higgs line in our Feynman rules.

A loose interpretation for the Higgs vev is a background probability for there to be a Higgs boson at any given point in spacetime. These “background” Higgs bosons carry no momentum, but they can interact with other particles as we saw above:

The cross means that instead of a ‘physical’ Higgs particle, the dashed line corresponds to an interaction with one of these background Higgses. In this sense, we are swimming in a sea of Higgs. Our interactions with the Higgs are what give us mass, though this statement will perhaps only make sense after we spend some time in a later post understanding what mass really is.

A good question to ask is *why* the Higgs has a vacuum expectation value. This is the result of something called **electroweak symmetry breaking** and is related to the unification of the electromagnetic force and the weak force, i.e. somehow the Higgs is part of a broader story about unification of the fundamental forces.

Often people will say that the universe is in a ‘Higgs phase,’ a phrase which draws on very elegant connections between the quantum field theory of particles and the statistical field theory of condensed matter systems. Just as we can discuss phase transitions between liquid and gas states (or more complicated phases), we can also discuss how the universe underwent an electroweak phase transition which led to the Higgs vev that lends masses to our favorite particles.

# Next time…

When we continue our story of the Higgs, we’ll start to better understand the relation of the Higgs vev with the mass of the other Standard Model particles and will learn more about electroweak symmetry breaking.