# In the Orthogonal series, would the weak nuclear force have infinite range?

I read about the universe described in the Orthogonal series. It has 4 fundamentally similar dimensions, rather than 3 dimensions of space and one dimension of time as is the case with our universe. In it, photons have mass, and so whether the force between two electric charges would be attractive or repulsive depends on the distance between the charges

The equations Greg Egan gives for the Coulomb potential and Coulomb field imply that in the universe described in the Orthogonal series, even with massive photons the electric force between two electric charges would have an infinite range.

I understand that the weak force only being significant at close range is related to the way the W and Z bosons, the carriers of the weak force, are massive. However, I couldn't find anything about how the weak force in the Orthogonal series behaves.

I was wondering if a universe similar to that one had something similar to the weak force, would the W boson have an infinite range, or would two particles still need to be at close range to exchange a W Boson?

## A "classical" view

Although it's not really meaningful to take the "classical" limit of the weak force, we can make an attempt to view it classically by studying something physicists call the Klein-Gordon equation, which describes the propagation of a massive particle. In our universe, for the case of a time-independent potential, it takes the form $$\nabla^2\phi=\frac{m^2c^2}{\hbar^2}\phi\tag{1}$$ where $$m$$ is the mass of the particle. This yields the solution $$\phi_Y(r)\sim\frac{1}{r}e^{-\alpha mr}$$ where $$\alpha=c/\hbar$$; we refer to $$\phi_Y$$ as a Yukawa potential (see these notes for a derivation of how it can be derived from the Klein-Gordon equation).

If we try to formulate the Klein-Gordon equation in Egan's universe, we get $$\nabla^2\phi=-\frac{m^2c^2}{\hbar^2}\phi\tag{2}$$ which is a special case of what Egan refers to as the "Riemann scalar wave equation" if we set $$\omega_m\equiv\frac{mc}{\hbar}$$, remove the time dependence and set the current to $$\mathbf{j}=0$$. Solving this gives us $$\phi_E(r)\sim\frac{\cos(\omega_mr)}{r}$$ which is Egan's expression for his version of the Coulomb potential. (Note that in both cases, if we set $$m=0$$, we recover the Coulomb potential for a massless photon.)

Effectively, if you try to view the weak force classically, you find that it ends up acting essentially the same as the electromagnetic force. This makes sense; in both cases, we're dealing with forces mediated by massive spin-1 gauge bosons. From this perspective, it at first looks like the weak force could indeed have infinite range.

## The Lagrangian

Egan apparently goes into some detail on the quantum mechanical Lagrangian describing the fields in his universe. If we take out the electromagnetic part, we get $$\mathcal{L}_{\text{EM}}=\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{1}{2}m_{\text{ph}}^2A_{\mu}A^{\mu}\tag{3}$$ with $$m_{\text{ph}}$$ the mass of the photon. To me, this looks exactly like what we get if we try to naïvely add a massive photon to a theory of electromagnetism in our universe - the Proca action - multiplied by $$-1$$. Now, you can show by picking a certain gauge transformation$$^{\dagger}$$ that this Lagrangian doesn't have the $$\text{U}(1)$$ gauge symmetry associated with the electromagnetic force in our universe; rather, it has the symmetry associated with the group $$\mathbf{R}$$ under addition (perhaps it's not surprising, as the two are in a sense quite similar).

I bring up $$\mathcal{L}_{\text{EM}}$$ to make the point that the Lagrangian describing the weak force may not look much like the weak force Lagrangian folks are used to in our universe insofar as it would not have to obey the same symmetry as ours does ($$\text{SU}(2)$$), which is why our universe requires the Higgs. Equation $$(3)$$ is a perfectly valid Lagrangian depending on whether you care about certain properties of your theory. This makes it hard to talk about what Egan's weak force Lagrangian would look like. In other words, looking at the Lagrangian tells us very little, or, to be more honest, nothing. We leave with less insight than we started with.

## A quantum view

Egan doesn't seem to elaborate on interactions between particles in his universe, nor the masses of particular elementary particles (aside from photons). This is unfortunate, because those are the key things we would need to know to get the full picture of the weak force - the classical picture I began with is incomplete at best, and misleading at worst. What we can say is that it seems unlikely that there is anything in his universe that changes how particles can interact and decay, and these are what truly limit the range of the weak force.

Assuming that the decay widths $$\Gamma$$ are the same for the W and Z bosons as they are in our universe, the particles would still have lifetimes of $$\tau\sim\hbar/\Gamma$$. They would still be quite massive and would still decay quickly, traveling finite - and small - distances before decaying. Nothing in any of Egan's pages indicates otherwise, meaning that the weak force is, we can assume, still an extremely short-range force, no matter what any version of the Klein-Gordon equation says. Classical assumptions and results can only get you so far.

$$^{\dagger}$$We can pick the transformation $$A_{\mu}\to A_{\mu}-\partial_{\mu}\eta$$ for any $$\eta$$, and expand out $$F_{\mu\nu}$$ as $$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$$ and see that $$\mathcal{L}_{\text{EM}}$$ does not retain its original form if we have the mass term - it's algebra after this.

• Can we still be sure that electromagnetism would have an infinite range, and can we be sure that the electric potential would be as Egan describes it? – Anders Gustafson Jun 14 '20 at 2:02
• @AndersGustafson I don't know Egan's equations well enough to say; the classical potential certainly indicates that the force has an infinite range, but the fact that there's a massive boson mediating it - with the possibility of decays - makes it possible that it's in fact quite finite. – HDE 226868 Jun 14 '20 at 3:17
• Assuming that the weak force would indeed have an infinite range, would the W Boson having an electric charge and weak isospin mean that the force carriers for the weak force would interact with each other, and if so could this also effect the relationship between distance and the weak force as well? – Anders Gustafson Jun 14 '20 at 18:31
• @AndersGustafson Those interactions are certainly possible, though I don't believe they would have much of an effect on the weak force's range. – HDE 226868 Jun 14 '20 at 19:13