# Why are protons heavier than electrons?

Our teacher told us that protons are nearly 1800 times heavier than electrons. Is there any known reason as to why this is so? Or is this just an empirical value, one we do not know the reason to?

• I don't know what 'why?' means in this context, but almost all of a proton's mass is due to the binding energy between its constituent quarks, if that helps. May 8, 2017 at 12:03
• "Why" is a question best left for philosophy or religion. "How" can be answered by physics, "why" even if it could be answered once will just causes more "why" questions that fall into the other departments. May 8, 2017 at 14:51
• @Adwaenyth but isn't this ‘answering of one “why” to cause more “why” questions in other departements’ precisely what physics is all about? May 8, 2017 at 15:01
• OP's use of "why" seems to be more like "I'd expect the positive and negative charge carriers to be of similar mass, and thus find this difference surprising, is there a reason for this?" Perhaps the answer should be to compare the electron to the positron - the antimatter partner of the electron. The proton is the wrong object of comparison for opposite charge in this case, even though it also is a positive charge carrier. It's like asking why a bike is so much bigger than a roller skate because of their different wheel arrangement, when one should be comparing to an inline rollerblade. May 8, 2017 at 23:11
• @leftaroundabout the problem is, whe cannot find a reason behind the laws of physics other than "They make sense the way they are because our universe works like this". The "why" essentially boils down to that most of the time. "Why does the apple fall down?" - "Because it is attracted by gravity." - "Why is it attracted by gravity?" - "Because space time folds around masses." - "Why does space time fold around masses?" And right at that point you're chain of why question exactly ends there, because we can't go much further. The question is of course interesting but sadly unanwserable that way May 9, 2017 at 5:30

There are multiple reasons why protons are heavier than electrons. As you suggested, there are empirical and theoretical evidence behind this. I'll begin with the empirical, since they have important historical context associated with them. As a preface, this will be a fairly long post as I'll be explaining the context behind the experiments and the theories.

# Empirical Electron Mass

Measuring the mass of an electron historically is a multi-step process. First, the charge is measured with the Millikan oil drop experiment, then the charge-to-mass ratio is measured with a variation of J.J. Thomson's experiment.

## Millikan Oil Drop

In 1909, Robert A. Millikan and Harvey Fletcher measured the mass of an electron by suspending charged droplets of oil in an electric field. By suspending the oil droplets such that the electric field cancelled out the gravitational force, the charge of the oil droplet may be determined. Repeat the experiment many times for smaller and smaller oil droplets, and it may be determined that the charges measured are integer multiples of a singular value: the charge of an electron.

$$e = 1.60217662 \times 10^{-19} \, \mathrm{C}$$

## J.J. Thomson's Experiments

In 1897 J. J. Thomson proved that cathode rays (a beam of electrons) were composed of negatively charged particles with a massive charge-to-mass ratio (as compared to ionized elements). The experiment began with first determining if cathode rays could be deflected by an electric field. The cathode ray was shot into a vacuumed Crookes tube, within which it'd pass between two plates before impacting an electric screen. When the plates were charged, the beam would deflect and hit the electric screen thereby proving that cathode rays contained a charge.

Later he would perform a similar experiment, but exchange the electric field for a magnetic field. This time though, the magnetic field would induce centripetal acceleration upon the cathode ray and produce circles. By measuring the radius of the circle and the strength of the magnetic field produced, the charge-to-mass ratio ($e/m_e$) of the cathode ray would be obtained.

$$e/m_e = 1.7588196 \times 10^{11} \, \mathrm{C} \cdot \mathrm{kg}^{-1}$$

Multiply this by the elementary charge obtained in the Millikan oil experiment, and account for uncertainty, and the mass of the electrons in the cathode ray is obtained.

$$m_e = \frac{e}{\frac{e}{m_e}} = \frac{1.60217662 \times 10^{-19} \, \mathrm{C}}{1.7588196 \times 10^{11} \, \frac{\mathrm{C}}{\mathrm{kg}}} = 9.10938575 \times 10^{-31} \, \mathrm{kg}$$

# Empirical Proton Mass

Ernest Rutherford is credited with the discovery of the proton in 1917 (reported 1919). In that experiment he detected the presence of the hydrogen nucleus in other nuclei. Later he named that hydrogen nucleus the proton, believing it to be the fundamental building block to other elements. Since ionized hydrogen consisted only of a proton, he correctly deduced that protons are fundamental building blocks to the nuclei of elements; however, until the discovery of the neutron, ionized hydrogen and the proton would remain interchangeable. How then, was the proton mass measured? By measuring the mass of ionized hydrogen.

$$m_p = 1.6726219 \times 10^{-27} \mathrm{kg}$$

This is done in one of several ways, only one of which I'll cite here.

## J.J. Thomson Variation

Repeat J.J. Thomson's experiment with magnetic deflection; but, swap out the cathode ray for ionized hydrogen. Then you may measure the charge to mass ratio ($e/m$) of the ions. Since the charge of a proton is equivalent to the charge of an electron:

$$m_p = \frac{e}{\frac{e}{m}} = \frac{1.60217662 \times 10^{-19} \, \mathrm{C}}{9.5788332 \times 10^{7} \, \frac{\mathrm{C}}{\mathrm{kg}}} = 1.67262 \times 10^{-27} \, \mathrm{kg}$$

## Other variations

Other variations may include the various methods used in nuclear chemistry to measure hydrogen or the nucleus. Since I'm not familiar with these experiments, I'm omitting them.

# Empirical Proton to Electron Mass Ratio

So now we've determined: $$m_p = 1.6726219 \times 10^{-27} \, \mathrm{kg}$$ and $$m_e = 9.10938575 \times 10^{-31} \, \mathrm{kg}$$

Using the two values and arithmetic:

$\frac{m_p}{m_e} = \frac{1.6726219 \times 10^{-27} \, \mathrm{kg}}{9.10938575 \times 10^{-31} \, \mathrm{kg}} = 1836$, or $1800$ if you round down.

# Theoretical Proton to Electron Mass Ratio

Theoretically, you first need to understand a basic principal of particle physics. Mass and Energy take on very similar meanings in particle physics. In order to simplify calculations and use a common set of units in particle physics variations of $\mathrm{eV}$ are used. Historically this was developed from the usage of particle accelerators in which the energy of a charged particle was $\mathrm{qV}$. For electrons or groups of electrons, $\mathrm{eV}$ was convenient to use. As this extends into particle physics as a field, the convenience remains, because anything develop theoretically needs to produce experimental values. Using variations of $\mathrm{eV}$ thus removes the need for complex conversions. These "fundamental" units, called the planck units, are:

$$\begin{array}{|c|c|c|} \hline \text{Measurement} & \text{Unit} & \text{SI value of unit}\\ \hline \text{Energy} & \mathrm{eV} & 1.602176565(35) \times 10^{−19} \, \mathrm{J}\\ \hline \text{Mass} & \mathrm{eV}/c^2 & 1.782662 \times 10^{−36} \, \mathrm{kg}\\ \hline \text{Momentum} & \mathrm{eV}/c & 5.344286 \times 10^{−28} \, \mathrm{kg \cdot m/s}\\ \hline \text{Temperature} & \mathrm{eV}/k_B & 1.1604505(20) \times 10^4 \, \mathrm{K}\\ \hline \text{Time} & ħ/\mathrm{eV} & 6.582119 \times 10^{−16} \, \mathrm{s}\\ \hline \text{Distance} & ħc/\mathrm{eV} & 1.97327 \times 10^{−7} \, \mathrm{m}\\ \hline \end{array}$$

Now then, what's the rest energies of a proton and electron?

$$\text{electron} = 0.511 \, \frac{\mathrm{MeV}}{c^2}$$

$$\text{proton} = 938.272 \, \frac{\mathrm{MeV}}{c^2}$$

As we did with the experimentally determined masses,

$$\frac{m_p}{m_e} = \frac{938.272 \, \frac{\mathrm{MeV}}{c^2}}{0.511 \, \frac{\mathrm{MeV}}{c^2}} = 1836$$

which matches the previously determined value.

# Why?

I'll preface this section by pointing out that "why" is a contentious question to ask in any science without being much more specific. In this case, you may be wondering what causes the proton mass to 1800× larger than the electron. I'll attempt an answer here:

Electrons are elementary particles. They cannot (or at least have never been observed to) break down into "constituent" particles. Protons, on the other hand, are composite particles composed of 2 up quarks, 1 down quark, and virtual gluons. Quarks and gluons in turn are also elementary particles. Here are their respective energies:

$$\text{up quark} = 2.4 \, \frac{\mathrm{MeV}}{c^2}$$

$$\text{down quark} = 4.8 \, \frac{\mathrm{MeV}}{c^2}$$

$$\text{gluon} = 0 \, \frac{\mathrm{MeV}}{c^2}$$

If you feel that something is off, you're correct. If you assume

$$m_p = 2m_{\uparrow q} + m_{\downarrow q}$$

you'll find:

$$m_p = 2m_{\uparrow q} + m_{\downarrow q} = 2 \times 2.4 \, \frac{\mathrm{MeV}}{c^2} + 4.8 \, \frac{\mathrm{MeV}}{c^2} = 9.6 \, \frac{\mathrm{MeV}}{c^2}$$

but

$$9.6 \, \frac{\mathrm{MeV}}{c^2} \ne 938.272 \, \frac{\mathrm{MeV}}{c^2}$$

This begs the question: what happened, why is the proton mass 100 times larger than the mass of its constituent elementary particles? Well, the answer lies in quantum chromodynamics, the 'currently' governing theory of the nuclear force. Specifically, this calculation performed above omitted a very important detail: the gluon particle field surrounding the quark that binds the proton together. If you're familiar with the theory of the atom, a similar analogy may be used here. Like atoms, protons are composite particles. Like atoms, those particles need to be held together by a "force".

For atoms, the Electromagnetic Force binds electrons to the atomic nucleus with photons (who mediate the EM force). For protons, the Strong Nuclear Force binds quarks together with gluons (who in turn mediate the SN force). The difference between the two though, is that photons can exist independently of the electron and nucleus. Thus we can detect it and perform a host of measurements with them. For gluons though, they not only mediate the strong force between quarks, but may also interact with each other via the Strong Nuclear Force. As a result, strong nuclear interactions are much more complex than electromagnetic interactions.

## Gluon Color Confinement

This goes further. Gluons carry a property called color. When two quarks share a pair of gluons, the gluon interaction is color constrained. This means that as the quarks are brought apart, the 'color field' between them increases in strength linearly. As a result, they require an ever increasing amount of energy to be pulled apart from each other. Compare this to the EM force. When you try to pull an electron from its atom, it requires enough energy to be plucked from its shell into the vacuum. If you don't, it'll jump up one or more energy levels, then fall back to its original shell and release a photon that carries the difference.

Similarly, if you want to pluck an object from a planet, you need to provide it with enough energy to escape the planet's gravity indefinitely (energy needed to reach escape velocity). Unlike the gravitational force and the electromagnetic force, the force binding gluons to each other grows stronger as they grow apart. As a result, there comes an inevitable point where the it becomes increasingly more energetically favorable for a quark-antiquark pair to be produced than for the gluons to be pulled further. When this occurs, the quark and antiquark bind to the 2 quarks that were being pulled apart, and the gluons that were binding them are now binding the new pair of quarks.

This animation is from Wikipedia, courtesy of user Manishearth under the Creative Commons Attribution-Share Alike 3.0 Unported license.

But wait! Where did those two quarks came from? Recall how pulling the quarks apart requires energy? Well that energy is on the scale of $\mathrm{GeV}$. At these scales, the energy may convert to particles with kinetic energy. In fact in particle accelerators, we typically see jets of color neutral particles (mesons and baryons) clustered together instead of individual quarks. This process is called hadronization but is also referred to as fragmentation or string breaking depending on the context or year. Finally I must point out that this one of the least understood processes in particle physics because we cannot study or observe gluons alone.

## Proton Mass

So, now going back to the original question. Earlier we noticed that the empirical proton mass was $938.272 \, \frac{\mathrm{MeV}}{c^2}$; but, theoretically its mass should be $9.6 \, \frac{\mathrm{MeV}}{c^2}$. The $928.672 \, \frac{\mathrm{MeV}}{c^2}$ difference arises from the color constraints that binds the three quarks together. In simpler terms: the nuclear binding energy of the proton.

• Is the actual mass $938.272 eV/c^2$ or $938.272 MeV/c^2$? May 9, 2017 at 1:13
• @KareemElashmawy, you still need to correct a few more "Mev/c^2" errors, in the "Theoretical" section and the "Why" section of your reply. May 9, 2017 at 19:21
• @KareemElashmawy Is $9.6MeVc^{2}≠938.272eVc^{2}$? likely to be $9.6MeVc^{2}≠938.272MeVc^{2}$
– L.K.
May 9, 2017 at 19:32
• I'm a little puzzled why no one is pointing out that the Muon mass is 105.658MeV/c^2, and the Tau lepton mass is 1776.82 MeV/c^2. It's interesting to point out the proton gains so much energy from its binding energy, but that doesn't really address the question.
– Hod
May 10, 2017 at 1:44
• OP's question only asks about protons and electrons. Reading between the lines, I suggest 1) OP is unaware of the broader range of elementary particles and 2) the real question is whether there's a theoretical connection between the masses. Regardless, that's what I'm interested in. :) You give no reason (and I'm not saying there is one) why proton constituents being 18x more massive than the electron is anything but arbitrary. I point to Muon/Tau as examples of elementary particles essentially identical to the electron except for mass to illustrate this.
– Hod
May 10, 2017 at 21:59

As noted "why" is a tricky question but we may ask what is the most fundamental view known concerning this question.

Electrons and protons are very different beasts. Electrons as far as we can tell are elementary, participating in the electromagnetic and so-called weak interactions. On the other hand protons are known to consist of quarks. Quarks are very similar in many properties to electrons but unlike the latter they also participate in the so-called strong interaction described by the theory called quantum chromodynamics (QCD).

For reasons I won't elucidate here the strong interaction works like a rubber band between quarks permitting them to behave as if they were free on very short distances (which we can see on collider experiments from which we know about their existence) but growing stronger and stronger with distance, so quarks never fly around as free particles, only in the form of the composite particles known as hadrons: protons, neutrons, pions etc.

In addition to the quark masses (which are quite small actually) the proton gets its mass from their interaction energy. Because the strong interaction is (surprise) very strong, this energy is huge, constituting almost 99% of its mass. Now can we calculate it using QCD? This is an extremely hard problem - QCD is easy in the regime when quarks are almost free and the strong interaction can be treated as a perturbation. But to compute protons' mass we need to work in a completely different regime for which most computational methods are useless. However it was successfully done using lattice QCD with an error of less than 2%.

It's just an empirical value. According to our current knowledge, the masses actually come from some more fundamental quantities - the electron yukawa coupling and the Higgs field vev, in the case of the electron mass; and the QCD confinement scale (which in turn comes from the strong coupling constant), in the case of the proton mass. But where those numbers come from, we don't know.

• It's a shame that this answer doesn't have as many upvotes. Though the others are filled with good info, this is the only one that's actually correct. May 10, 2017 at 8:45
• Perhaps this is where I should move my comment to the question: The τ lepton, a sibling of the electron, is almost twice as heavy as the proton. Why would that be? It's mass comes from exactly the same mechanism as the electron's. So, somehow "explaining" why the e is lighter than the p automatically collapses for the τ. Cherry-picking facts to concoct a dubious "paradox" is dangerous. May 10, 2017 at 22:27

You are comparing 2 different items. While the charge of a proton is equal and opposite of the electron, any comparison ends there. An electron is a fundamental particle which can't be broken down further whereas a proton can be broken down further into more fundamental particles. The anti-matter twin of the electron is the positron which is equal in mass but opposite (positively) charge.

In addition to the other excellent answers, Note that with antimatter it is perfectly possible to have a positron or an antiproton (negatively charged equivalent of the proton). You just don't find many on Earth, because they annihilate spectacularly with matter.

As explained in wikipedia, the reason the observable universe is composed almost entirely of matter and very little antimatter is not well understood.

Distant galaxies might be composed of antimatter and would be predicted to have atoms made of positrons and antiprotons with identical chemical properties to the materials we know. But when brought in contact with "normal" matter they would annihilate each other, liberating enormous quantities of energy in the form of gamma rays.

• True, but off topic. May 8, 2017 at 20:41
• @CamilleGoudeseune I disagree - I think this is one point that is missing from the other excellent answers. It seems to me the OP is wondering about the about the apparent asymmetry of matter: light negative charges and heavy positive charges. So it's worth noting that the converse particles do exist. The reason we don't (often) observe the light positive and heavy negative charged particles (which are perfectly stable alone) is because they quickly get annihilated by "normal" matter. In other parts of the universe "antimatter" might be the norm, but nowhere will we find both types of matter May 8, 2017 at 21:09

The $\Delta^+$ particle has the same quark content as the proton has ($uud$), but nevertheless, its mass is 1232 $\frac{MeV}{c^2}$. The mass difference between this particle and the proton is about 575 times the mass of the electron. This surely shows that something is going on between the constituent quarks (the $u$-quark has a "bare" mass of about 4,2 $\frac{MeV}{c^2}$, while the $d$-quark has a "bare" mass of about 7,5 $\frac{MeV}{c^2}$, which of course doesn't mean quarks can really exist without clothes) which imparts a great deal of mass to both the proton and the $\Delta ^+$ particle.

It's remarkable that the $\Delta ^+$ decays in about $0,6$x$10^{-23}$(sec) into a neutron and positive pion. This short time is a sign that the "something" that's going between the constituent quarks is the strong color force playing around to cause the transition. The lifetime of the proton, on the other hand, is infinite; there isn't a lower energy state it can change into [at least in the standard model; in the rishon model, which conjectures that quarks and leptons are composite structures, the change from a proton to positron and a pion, is for example easily explained, just as is the mass difference between an electron and a muon, the last of which can be seen in rishon light as an excited state of the electron; and regarding the distribution of matter and anti-matter, the rishon model solves this conundrum by claiming that there is as much matter as anti-matter! But this is not the place to discuss this model (theory) and I just mention it here as an aside].

Nobody really knows the reason! Even taking into account that the proton is made of quarks, the masses of electrons and quarks and their interaction strengths are numbers that we don't know where they come from.

One day there might be a theory which tells us why these numbers must be the numbers they are. Or it might turn out that these numbers are random. Nobody knows..... yet!