# Understanding the proton’s mass – and then the universe’s

You are taught in school that protons and neutrons are particles. However, unless you get into physics research later in life, the likeliest way you are going to find out that they are technically quasiparticles is through the science media. So here it is. 😄

Setting aside their electric charge, protons and neutrons are very similar particles. They have almost the same mass and they’re made up of exactly the same kinds of smaller particles. These smaller particles are called quarks and gluons. Three quarks and three gluons come together to form each proton or neutron. That is, protons and neutrons are technically quasiparticles because they are clumps of smaller particles that are grouped together and behave in a collective and predictable way.

This grainier picture of protons – and neutrons, but we’ll stick to protons because they’re both so similar – is necessary to understand their mass. In classical mechanics, the weight of a bag of oranges is equal to the weight of the bag plus the weight of all the oranges. But in quantum mechanics, and particle physics in particular, the mass of a proton need is not equal to the mass of the quarks that make it up (gluons are massless). This is because there are other energetic phenomena that ‘supply’ mass through the mass-energy equivalence (E = mc2).

Each proton weighs 938.2 MeV/c2 (a unit of mass unique to particle physics). It is made up of two up quarks – 2.4 MeV/c2 ×2 – and one down quark – 5 MeV/c2. That is just 9.8 MeV/c2 together. Where does the remaining 928.4 MeV/c2, or 98.95%, come from?

It comes mostly from the effects of one of the four fundamental forces, the strong nuclear force. A new paper authored by physicists from the US and China claims to show for the first time the precise contributions each of these effects towards the proton’s overall mass.

Since the 20th century, physicists have determined how much protons and neutrons weigh, and how much the quarks weigh, to a large degree of precision using experiments. But this hasn’t helped understand why protons weigh as much as they do because of the ‘bag of oranges’ problem. Additionally, quarks acquire their masses through the Higgs mechanism (involving the Higgs boson) whereas protons don’t because they are not fundamental particles. So there is something else that kicks in between the quarks and protons layers.

To understand what this is, physicists need to perform pen and paper computer calculations using the theory of these particles. There are two theories, as in ways of studying the interactions of particles, they could use here. One is called the Standard Model of particle physics, which strives to predict the properties of all known elementary particles (including the Higgs boson) in a single framework. The other is the framework of quantum chromodynamics, or QCD. It strives specifically to explain the behaviour of the strong nuclear force and the quarks it acts on (the force is mediated by gluons).

While previous studies to determine the proton’s mass using theoretical methods have been attempted, they have focused on using the Standard Model route, which is less difficult (but not significantly so) and involves more assumptions. The US/Chinese study takes the QCD route. This is useful because it will help physicists understand how contributions to the proton’s mass are rooted in concepts specific to QCD.

QCD is a very strange and difficult theory, and its effects show up as weird properties. For example, one effect is called colour confinement: it is impossible to tear apart clumps of quarks and gluons below the Hagedorn temperature (2,000,000,000,000 K, one of two known ‘absolute hot’ temperatures). It arises because of the properties of the energy field – a.k.a. the gluonic field – between two nearby quarks.

Heisenberg’s uncertainty principle states that you can’t know the momentum and position of a particle with the same precision at the same time. But colour confinement actually confines the position of quarks – so the uncertainty principle suggests that its momentum can be quite large. Physicists have previous calculated that this momentum could contribute a mass (through Einstein’s mass-energy-momentum equivalence) of a few hundred MeV/c2. Now we’re getting somewhere, although we still have aways to go.

The US/Chinese scientists used a technique called lattice QCD to take these calculations to the next level. Lattice QCD was developed because QCD is so difficult, and it is so difficult because the strong nuclear force is so strong. In fact, it is the strongest of the four forces, and prevents neutron stars from collapsing into black holes. The studies of other particles, such as quantum electrodynamics of electrons, don’t require specialised techniques because the force between electrons is not so strong.

More importantly, like most areas of modern physics, the real innovation in the present study comes from advancements in computing techniques (see here and here for examples from astronomy and materials science, resp.). The US/Chinese scientists developed new algorithms to solve lattice QCD problems better and also reduce errors. (According to their paper: “We present a simulation strategy to calculate the proton mass decomposition”.) As a result, they have elucidated four distinct contributions to the proton’s mass, from the following sources:

• Quark condensate
• Quark energy
• Gluonic field strength energy
• Anomalous gluonic contribution

The interesting thing here is that the quark condensate is different from the other three sources because it is the only one made up entirely of just quarks. It contributes only ~9% to the proton’s mass. Also, earlier in this post, we saw that just adding up the masses of the constituent quarks yielded 1.05% of the proton’s mass. The new calculation says it is about 9%. The remaining 7.95% appears to come from virtual strange quarks – i.e. strange quarks popping in and out of existence in the vacuum of space – the up and down quarks’ interactions with them. The quark structure of a proton. The wiggly lines represent gluons. Caption: Jacek rybak/Wikimedia Commons, CC BY-SA 4.0

The other three sources involve the dynamics of quark-gluon interactions and the strong nuclear force that keeps them confined inside a proton. Quark energy relates to the kinetic energies of the confined quarks and gluonic field strength, to the kinetic energies of the confined gluons. They contribute 32% and 37% respectively. The anomalous gluonic contribution has to do with complex interactions between the constituent quarks and all virtual quarks (i.e. all charm, strange, bottom and top quarks popping in and out of existence in the vacuum). It pitches in with about 23%.

In sum: 1 proton’s mass = 9% quark condensate + 32% quark energy + 37% gluonic field + 23% anomalous gluonic contribution. (That’s actually 101% but becomes 100% if we use less approximate, more accurate values.)

We could also slice this thus: 1 proton’s mass = 9% quark condensate + 91% quark-gluon dynamics. Imagine there is an alternate universe where all the quarks have zero mass. The quark condensate contributes only ~9% to the proton’s mass, so in this alternate universe, protons and neutrons would still weigh 91% as much as protons and neutrons in our universe. This is possible thanks once again to the effects and strength of the strong nuclear force.

Let us take this just one step further. 1) Each proton and neutron weighs almost 1,900-times as much as an electron. 2) Protons, neutrons and electrons make up all the matter in the universe. 3) Electrons aren’t made up of quarks and gluons (i.e. they are not quasiparticles). All together, the non-quark contribution effectively makes up ~89% of all the mass of all the matter in the universe.