# Electron Charge is 150%?

Is there a theory for why the charge of an electron is precisely 50% larger (magnitude) than a quark's? I have usually thought of this the other way around: the charge of a quark being 2/3 (or -2/3) that of an electron. Is there something fundamental in space/time/mass/etc that makes it a nice proportion? (I understand that this question is not worded perfectly to cover the entire list of standard model components. I tried to word it to cover proton charges and positron charges etc but it became too complicated. I opted for simple.)

• Charge conservation? A proton is made of 3 quarks. – Rob Jeffries Sep 2 '14 at 23:42
• – Emilio Pisanty Sep 2 '14 at 23:46
• Note that some quarks (down, strange and bottom) have charges of -1/3, and their antiparticles +1/3. You might want to change the title to something that specifically refers to the relationship of the electron charge to the quark charges. – Emilio Pisanty Sep 2 '14 at 23:48

The electron charge is called $-e$; let me pick the convention where $e$ is positive.

The atoms (e.g. the hydrogen-1 atom) contain the same number of protons as electrons and they are neutral. They must be neutral because ordinary macroscopic matter is composed of atoms and it has to be neutral because the atoms would otherwise attract the oppositely charged objects and create neutral composites, anyway.

It follows that the protons have to have the charge opposite to the electrons, $+e$.

We also know that there are electrically neutral particles inside the nuclei, neutrons, that only impact the nuclear physics, not the atomic physics (chemical properties). They have the charge $0$.

In the late 1960s and early 1970s, people learned that each proton is composed ot three quarks and so is each neutron. One needs two quark types ("flavors"), $u$ and $d$. Proton has $uud$ and neutron has $udd$. If $u,d$ denote the charges of the quarks for a while, we have the equations for the total charges of these blocks $$2u+d = +e, \quad u+2d = 0$$ The second implies $u=-2d$. Substitute it to the first one and you get $$-4d+d = +e, \quad -3d=+e, \quad d= -\frac e3$$ and $u=+2e/3$ for the charges of the quarks. The electron's $-e$ is therefore 50% larger than $-2e/3$, the charge of the up antiquark ("anti": note that "up" and "electron" have opposite signs of the charge).

If one wants to choose a "fundamental charge" based on quarks, the "elementary charge" should be the minimum one, $e/3$, and not $2e/3$. Then the electron has 200% higher charge than the sub-electron elementary charge (of the down-quark).

But quarks are confined and there are other reasons, although partly flavored with conventions, why people use the term "elementary charge" for $e$ and not $e/3$.

• "It follows that the protons have to have the charge opposite to the electrons." That strikes me as a grand non sequitur. It could be $+2 e$ and attract double electrons. Or $5/7 e$ or maybe even $\sqrt\pi e$. That they balance at exactly 1-1, and not some rational ratio, hardly follows from what you say. Perhaps you can logically exclude irrational ratios, but by what logic is 2-1 necessarily excluded? – zibadawa timmy Sep 3 '14 at 14:31
• Well, indeed, helium's nucleus has charge $+2e$ and it attracts double electrons. But because the minimum atom with 1 electron exists, the minimum nucleus - the proton - has $+e$. There exist high-brow proofs that the electric charge ratios can't be irrational - linking things like magnetic monopoles and quantum gravity - but I am afraid that you don't have the background for those. Sometimes we just use the observations - nuclei with $e/2$ don't exist - for example. – Luboš Motl Sep 3 '14 at 15:27
• But it is true that many of these empirical observations may be deduced by purely theoretical arguments, at least when we assume some of the empirical observations. Also, many worlds with wrong values of charges etc. wouldn't admit life so even without direct observations, one could prove many of the statements by the pure "anthropic" reasoning. – Luboš Motl Sep 3 '14 at 15:31