Why do quarks have fractional charges?

tybeedave
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i can't seem to get a handle on this...help :)
 
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The definition of the elementary charge is arbitrary, we could assign the charges +-1 and -+2 to the quarks, then electrons would have +-3.
It is unknown why those charges can be written as integers at all (they could have some arbitrary factors like 2.464235...). If magnetic monopoles exist, they would lead to an explanation why we see multiples of a common charge value only.
There is also a connection between quark and lepton charges, see this post on anomaly cancellation (probably beyond the scope of the original question).
 
Toponium is a hadron which is the bound state of a valance top quark and a valance antitop quark. Oversimplified presentations often state that top quarks don't form hadrons, because they decay to bottom quarks extremely rapidly after they are created, leaving no time to form a hadron. And, the vast majority of the time, this is true. But, the lifetime of a top quark is only an average lifetime. Sometimes it decays faster and sometimes it decays slower. In the highly improbable case that...
I'm following this paper by Kitaev on SL(2,R) representations and I'm having a problem in the normalization of the continuous eigenfunctions (eqs. (67)-(70)), which satisfy \langle f_s | f_{s'} \rangle = \int_{0}^{1} \frac{2}{(1-u)^2} f_s(u)^* f_{s'}(u) \, du. \tag{67} The singular contribution of the integral arises at the endpoint u=1 of the integral, and in the limit u \to 1, the function f_s(u) takes on the form f_s(u) \approx a_s (1-u)^{1/2 + i s} + a_s^* (1-u)^{1/2 - i s}. \tag{70}...

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