Why are there two signs for electric charge, whereas there is only one sign for mass?

A possible answer is the following: the analog of electric charge is not mass, but square-root of mass. Given a mass m, its square-root has two signs: + sqrt{m} and - sqrt{m}. The + sign is for matter and the - sign is for anti-matter. Mass has only one sign because our universe is made only of matter. It also follows that whereas matter-matter gravitational interaction is attractive, the matter-antimatter gravitational interaction is repulsive. To our knowledge, this last claim is not ruled out by experiments to date.

These conclusions follow from trying to understand the standard model in the language of the octonions.

Defining elementary particle states using the octonions shows that electric charge is quantised, as observed. The Clifford algebra Cl(6) can be used to deduce the quarks and leptons of the SM:

Particles and electric charge Anti-particles

Neutrino 0 Anti-neutrino 0

Anti-down quark 1/3 Down quark -1/3

Up quark 2/.3 Anti-up quark -2/3

Positron 1 Electron -1

Quantisation of electric charge is deduced from the eigenvalues of a U(1) number operator interpreted as U(1)_em. Anti-particle states are obtained by complex conjugation of particle states and are shown to have an electric charge value opposite to that of the particles.

It turns out that exactly the same construction can alternatively be used to obtain quantisation of square-root mass [a gauge-gravity duality]. There is no reason to confine oneself to eigenstates of electric charge. There is a dual construction of eigenstates of square-root mass, which gives:

Matter and sq. root mass Anti-matter

Neutrino 0 Anti-neutrino 0

Electron 1/3 Positron -1/3

Up quark 2/3 Anti-up -2/3

Down 1 Anti-down -1

This time the complex conjugation maps matter to anti-matter and changes the sign of sqrt m.

Whatever be the reason for the excess of matter over antimatter in our universe, it is obvious that our universe has only particles with plus sign for sqrt m, even though both signs for electric charge are there. The minus sqrt m is with the antimatter.

From the viewpoint of eigenstates of electric charge, the charge is same across generations, for a given family; but mass changes because charge eigenstates are not mass eigenstates.

Equivalently, from the viewpoint of eigenstates of sqrt mass, the sqrt mass is same across generations, for a given family; but electric charge changes because mass eigenstates are not charge eigenstates.

In summary, the gravitational force is only attractive, and not repulsive, because our universe is made only of matter. if we could observe the gravitational interaction of matter and antimatter (which is repulsive) we will see the two signs of sqrt mass in action.

The above also explains how in our approach pre-gravitation is a spin one vector interaction with two signs for sqrt mass, and yet the emergent gravitational interaction is only attractive. It is because matter and antimatter have been separated from each other!

Pre-gravitation is on the same footing as the SM gauge fields: all are spin one vector interactions. gravity appears different because of the matter-antimatter asymmetry in our universe. Whereas SM is SU(3)_c X SU(2)_L X U(1)_Y, pre-gravitation is brought on by a left-right symmetric extension of the SM, and is SU(3)_grav X SU(2)_R X U(1)_grav. Of these, SU(2)_R gets related to GR, whereas SU(3)_grav and U(1)_grav remain to be understood: they might relate to dark matter / MOND and to dark energy.