# What is the most natural definition of the weak hypercharge coupling constant if grand unification is wrong?

A tricky question. Here is the famous graph of the running of the three coupling constants in the standard model: http://www-ekp.physik.uni-karlsruhe.de/~deboer/html/Forschung/unification_eng.eps .

The graph shows, in its top curve, the running of the coupling constant alpha_1. This is the coupling of the weak hypercharge coupling constant for the weak hypercharge group U(1)_Y, which is one of the three gauge groups of the standard model of particle physics.

But there is a tricky detail. In that curve, alpha_1 is multiplied with 5/3. This factor 5/3 comes from the assumption that GUTs are valid. The factor ensures that the various group traces of U(1)_Y, SU(2) and SU(3) are normalized in the correct way when they form the SU(5), SO(10) or any other grand unification gauge group.

In the case that grand unification is wrong, the factor 5/3 cannot be deduced. Which factor would be natural in this case?

Clarification added, after remarks by Lubos Motl: it is assumed in the question that the usual definition of the weak hypercharge is used, Y_W=2(Q-T_3), in which left-handed quarks have hypercharge 1/3.

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Of course, if you assume a normalization with no relationship between any groups where the hypercharge $U(1)$ is normalized just like any other $U(1)$ would be, and your goal is to make the formulae as simple as possible on the paper (which is not really a physical criterion), then $5/3$ is replaced by $1$. You just omit the $5/3$ factor. But this is a kind of vacuous statement because one may only compare the fine-structure constants of the different group factors if there is some relationship between them which is either grand unification or plays the same role.
One more preemptive comment: at low energies, it is not true that the hypercharge fine-structure constant renormalized by another simple factor such as $5/3$ yields the electromagnetic fine-structure constant. At the GUT scale, similar relations exist but the electromagnetic fine-structure constant is not well-defined there.