# Why are there no composite particles with fractional charges?

1. Mesons are composed of a quark and an anti-quark, so no fractional charge is possible mathematically.
2. Baryons are composed of three quarks, no anti-quarks mixed in with quarks, so no fractional charge is possible.
3. Pentaquarks are composed of four quarks and one antiquark, charge-wise sum of mesons and baryons so no fractional charge is possible.

Is there some theoretical reason for the lack of composite particles that would result in fractional charges?

For example 2 quarks and one anti-quark could result in a fractional charge, and even it seems that all such combinations would be a fractional charge. Four quark composites would allow fractional charge too if there is not an equal balance of quarks and anti-quarks. It has to be possible to detect, so if no actual fractional charges been observed then there must be some theory to explain this. Is this the reason for the theory behind color charge and color confinement?

It is very interesting if there is no fractional charge observed at all across all known particles - composite particle or not.

• The overall normalization of $U(1)$ charges is arbitrary. You can assign, if you want, $q=1/42$ to quarks, and get fractional for composites. Conversely, if your composites have fractional charges, you can GCD'd the quarks and get integers. That being said, see physics.stackexchange.com/q/353346/84967 Jan 6, 2021 at 1:46
• @AccidentalFourierTransform I am not using the right word for charge here. I mean one electron's worth of charge, so not coulombs or other arbitrary scale of charge. Should I rephrase somehow? Jan 6, 2021 at 1:52
• So the question is why hadrons have charges that are integer multiples of the electron charge? See physics.stackexchange.com/q/21753/84967 and links therein. Jan 6, 2021 at 1:55
• @AccidentalFourierTransform, that question and its answers are limited to just the proton and why the two particles have the same charge. Jan 6, 2021 at 2:08
• QCD dictates vanishing triality color singlets. Jan 6, 2021 at 3:17

To keep things manageable, I'll interpret the question like this: Given that quarks have their special pattern of electric charges with magnitudes $$2/3$$ and $$1/3$$ in units of the electron's charge, why do all hadrons (particles made of quarks) have integer electric charges in units of the electron's charge?

I'll use these inputs: Quarks are bound together by the strong force. Each quark species comes in three "colors" (this is what we call the strong-force charge, to distinguish it from electric charge), and the strong force ensures that only color-neutral combinations can occur as isolated particles. I'll explain what this means below.

Let $$q_c$$ denote a quark with color $$c$$, and let $$\bar q_c$$ denote an antiquark with color $$c$$. Isolated hadrons must be color-neutral, meaning that they must be invariant under the transformations \begin{align} q_c &\to \sum_{c'} U_{cc'}q_{c'} \\ \bar q_c &\to \sum_n \bar q_{c'}U^*_{c'c} \tag{1} \end{align} where $$U$$ is a $$3\times 3$$ unitary matrix with determinant $$1$$. (The group of all such matrices is called $$SU(3)$$.) The two basic color-neutral combinations are the meson-like combination $$\sum_c \bar q_c q'_c \tag{2}$$ where $$c$$ is the color index, and the baryon-like combination $$\sum_\pi (-1)^\pi q_{\pi(1)} q'_{\pi{2}} q''_{\pi{3}}. \tag{3}$$ The sum in (3) is over all permutations of the three color index-values, and the signs make the result completely antisymmetric. The fact that $$U$$ is unitary ensures that (2) is invariant, and the fact that $$U$$ has determinant $$1$$ ensures that (3) is invariant. The conjugate of (3) is also invariant, of course. Other invariants are sums and products of these basic invariants.

In units where an electron has charge $$-1$$, quarks $$q$$ have charge $$+2/3$$ modulo an integer, and antiquarks $$\bar q$$ have charge $$-2/3$$ modulo an integer. Since a meson-like combination involves the same number of quarks and antiquarks, we immediately conclude that it must have integer charge. And since a baryon-like combination involves three quarks or three anti-quarks, we immediately conclude that it must also have integer charge. All other color-neutral combinations are built from these, so all hadrons must have integer electric charge.

• This answer didn't try to explain why the strong force ensures that only color-neutral combinations can occur as isolated particles. If you want to learn more about that, the keywords include quantum chromodynamics and confinement.

• This answer also didn't try to explain why quarks have their special pattern of electric charges. If you want to learn more about that, the keywords include electroweak symmetry breaking and chiral anomalies.

• If you want to learn more about general conditions under which all electric charges must be integer multiples of some basic charge (which in the real world is the electric charge of a down-quark), the keywords are charge quantization and compact gauge group.

• For an experimental perspective, which is what makes all of this mathematical stuff relevant, see anna v's answer.

Is there some theoretical reason for the lack of composite particles that would result in fractional charges?

It is an experimental fact that there are no fraction of the electron's charge particles in the data of the large number of experiments in hight energy physics.

So, a theory had to be developed that would fit mathematically this experimental observation.

This theory is the standard model of particle physics, given by the groups of $$SU(3) \times SU(2) \times U(1)$$ which give the allowed representations. It is chosen because it has no fractional charges in order to agree with the experimental data.. So the theory by construction cannot have a particle or particle-antiparticle combination of fractional charge for on mass shell observable particles.

The concept of color charges for the quarks and color neutrality for on mass shell particles can be seen here, which also contributes to the pairings. An interesting observation , though not directly relevant:

The rationale for the concept of color can be highlighted with the case of the omega-minus, a baryon composed of three strange quarks. Since quarks are fermions with spin 1/2, they must obey the Pauli exclusion principle and cannot exist in identical states. So with three strange quarks, the property which distinguishes them must be capable of at least three distinct values.

The standard model mathematics describes data we have at present as far as charges go completely, by construction. If in the future such a particle were detected, the standard model would have to be expanded or changed.

Quark jets have been experimentally studied to see whether their fractional charge is seen in the distribution of the jet particles, and the article says that their study verifies the fractional charges of the top-untitop pairs produced in the experiment.

• Im torn between these two answers but have to choose the other because its addressing the why by explaining the theoretical whys. Without observation and experiment the theory is just hogwash. I appreciate several of the points you've made here and wish they were in the other answer too. Jan 7, 2021 at 0:36
• @Jason Not to worry. There is a tendency to platonism in the physics and general community that visits this site, i.e. the axiom that "mathematics molds reality". As an experimental physicist I often get -1 's without a comment , and I attribute it to that. If the theory of everything is found, I will be happy to join the crowd. "God always geometrizes", the pythagorians stated . Jan 7, 2021 at 6:06
• actually my introducing color in the charges question, is a parallel path but I thought that the omega minus argument for color should get an audience, when charges are discussed. So I clarified a bit Jan 7, 2021 at 6:13
• I am not 100% satisfied how this first suggests that a theory can only reflect already observed experimental data. Special+General Relativity was originally setup to match things like Michelson-Morley, but it allowed surprising solutions such as black holes that could be observed much later. -- Similarly, a theory built to match how charges behave in experiments might allow surprising solutions under weird situations, thereby suggesting new experiments. In a way, the non-integral charge of quarks is already the black-hole-like surprise result Feb 23, 2021 at 3:15
• @HagenvonEitzen when a theory predicts new phenomena that are found experimentally, that is called validation of the theory. If a theory does not predict new observations or new observations do not invalidate it, then it is a mathematical map of reality and can be thought as the theory of everything. Except, from euclidean times on, we find that theories have to be modified and new ones proposed, when observations become finer. The standard model has such disagreements with experiments, example CP violation cannot be completely accounted. Feb 23, 2021 at 4:55