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I always hear people saying symmetry is beautiful, nature is symmetric intrinsically, physics and math show the inherent symmetry in nature et cetera, et cetera.

Today I learned that half of the quarks have +2/3 electric charge and other half have -1/3 magnitude electric charge. Is there any explanation for this? Why their net charge isn't zero?

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People will often say that there are a great many symmetries in nature, but that is not to say that nature is maximally symmetric. If that were so, the universe would be one big isotropic homogeneous blob. You can tell that there is a great deal of asymmetry in the universe just by looking at the world around you. Indeed there is also much asymmetry and irregularity and almost-symmetry on a fundamental level (consider the seemingly arbitrary mass ratios of the fundamental particles). This is motivation to find more fundamental, symmetric theories, but they won't be 'perfectly' symmetric! –  gj255 Feb 15 at 9:50

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When it comes to fundamental charges, the (left-handed) up-type quarks actually have either the same values of the charge as the down-type quarks, or exactly the opposite ones.

It just happens that the electric charge isn't a fundamental charge in this sense.

Let me be more specific. All the quarks carry a color – red, green, or blue – the charge of the strong nuclear force associated with the $SU(3)$ gauge group. There is a perfect uniformity among all quarks of all types.

The quarks also carry the hypercharge, the charge under the $U(1)$ gauge group of the electroweak force. Both the up-quarks and the down-quarks (well, their left-handed components) carry $Y=+1/6$ charge under this group. The right-handed components carry different values of $Y$ but I won't discuss those because that would make the story less pretty. ;-)

Finally, there is the $SU(2)$ group of the electroweak force. The left-handed parts of the up-quarks and down-quarks carry $T_3=+1/2$ and $T_3=-1/2$, exactly opposite values, and there is a perfect symmetry between them. (The right-handed components carry $T_3=0$.)

It just happens that neither $Y$ nor $T_3$ but only their sum, $$ Q = Y+ T_3$$ known as the electric charge, is conserved. The individual symmetries generated by $Y$ and $T_3$ are "spontaneously broken" due to the Higgs mechanism for which the latest physics Nobel prize was given.

The symmetry is broken because a field, the Higgs field $h$, prefers – in order to lower its energy – a nonzero value of the field. More precisely, one component is nonzero, and this component has $Y\neq 0$, $T_3\neq 0$ but $Q=0$. So the first two symmetries are broken but the last one, the $U(1)$ of electromagnetism generated by the electric charge, is preserved.

It is often the case that every symmetry that is "imaginable" and "pretty" is indeed a symmetry of the fundamental laws but at low energies, due to various dynamical mechanisms, some of these symmetries are broken. The laws of physics may still be seen to respect the symmetry at some level but the vacuum state isn't invariant under it, and the effective low-energy laws therefore break the symmetry, too.

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If you are looking for symmetry, I think one should point out that there IS a particle with a -2e/3 charge and a particle with a +e/3 charge. They are the up antiquark and down antiquark respectively. Now, following that, you would very reasonably ask the question why we observe more up quarks than up antiquarks, and other follow-up questions like Matter-Antimatter Asymmetry in Experiments?

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