Magnetic Monopoles and Generalized Charge Conservation

Sherman Frankel

Dept. of Physics, University of Pennsylvania

Abstract

In a decades-old proof [#!www!#] that both the force between a magnetic charge(g) and an electric charge(e) ( $F = g e (\vec r \times \vec v/c)/r^3$) and the well-known angular momentum in the field ( $l= ge \hat r$)[#!Wilson!#] could be derived from P and T conservation alone without any recourse whatsoever to electromagnetic theory, we found that a ``generalized'' charge conservation would allow monopole charged particles to decay to electric charged particles, accounting for their non-observance. We now return with two other theoretical arguments that support this view stimulated by the possibility that astrophysical data and high energy accelerators might now show evidence for monopole decay. Five possible experimental searches are described and some qualitative theoretical implications are discussed.

We first examine the question of generalized charge conservation more directly than in the analysis in ref [#!www!#]:

Rather than re-examining the deivation of the force between a monopole and an electric charge and the field angular momentum from P conservation and T invariance, in this note we accept them as derivable from the conventional electric fields, $\vec E$ and $\vec B$ and simply use the fact that the total angular momentum, $\vec L$, must be conserved, that is we require that the sum of the motional angular momentum plus the field angular momentum be conserved.

$$d\vec L/dt = d(\vec r \times m\vec v)/dt + d(ge \hat r)/dt = 0 = \vec r \times \vec F + (ge) d\hat r /dt + \hat r d(ge)/dt$$ (1)

Taking the time derivative of the unit vector $\hat r$ one observes that this equation is only satisfied if the last term vanishes so that d(ge)/dt = 0. This is the same conclusion as was obtained in [#!www!#] in a more general way.

Recognizing that charges are quantized and setting dt = t_final - t_initial, and d(ge) = (ge)_final - (ge)_initial we can expand out the equation to read:

 

d(ge) = gde + edg = (ge)_final - (ge)_initial = 0 (2)

(The charge sign convention we use is that a g(+)e(-) pair can be created from the vacuum just as in ordinary charge conservation an e(+)e(-) pair can be created.)

There are two solutions to this equation.

The conventional solution is the case where dg and de are both zero.

The second solution is the case where gde = - edg so the terms cancel, allowing monopole decay.

Before addressing the meaning and consequences of this equation we shall turn to a third way of understanding why the Dirac charge quantization[#!Dirac!#] requires generalized charge conservation and links angular momentum and charge conservation in a world that contains monopoles. The calculation of the Dirac quantization for a ge pair requires that there be no other unpaired monopoles or charges in the universe since, if there were, and the charges were not exactly spatially isotropic, the angular momentum would be $\sum g_i e_j \hat r_{ij}$ which would not be a unit vector and could not be set equal to a multiple of $\hbar$ as Dirac postulated. [#!Dirac!#] This problem is resolved by assuming an isotropic universe, so that $\sum \hat r_{ij}$ is in fact zero and then creating a single (g+) - (e-) pair from the vacuum. Then one can correctly calculate the angular momentum of an isolated (ge) pair and quantize the product. Creating the pair from vacuum requires the generalized charge conservation.

It will now be useful to examine some specific cases:.

I. Stable charges: electron has e = e and g = 0, so (ge) = 0 and remains so, since de = 0 by ordinary charge conservation. Also since ordianry electric charges always have g = 0 dg = 0.

II. Monopole decay: ( $g \rightarrow e$) dg = g_final - g_initial = -g and de = e_final - e_initial = +e Therefore the two terms gde + edg = g(+e) + e(-g) cancel.

III. Dyons: For particles with both e and g, ge = 1.[#!Schwing!#] Therefore a dyon cannot be made from vacuum and cannot decay to ordinary particles. Neither can a dyon plus antidyon be produced since ge would be 2 and not zero in the final state.¹

We now examine five possible ways of looking for the effects of monopole decays taking place at any time in our universe:

A) Monopole leptons, l_g, can decay to electrons by an electro-weak decay with the usual requirement that each lepton have its own neutrino.

$$l_g \rightarrow e + \nu_e + \nu_g$$ (3)

where $\nu_g$ is the neutrino associated with l_g. (Similarly it can decay to a $\tau$ or $\mu$ and their respective neutrinos.)

If the monopole neutrino is not massless this could account for some of the missing mass of the universe.

B) Monopole neutrinos entering the SNO deuterium detector will produce breakup of deuterium and thereby increase the neutral current signal above that expected from conventional estimates.

C) The production and decay of monopoles would always be accompanied by large photon jets because the monopole fine structure constant is (137)² times larger than the electric fine structure constant. Thus one might find very large photon jets coming in from outer space. Also, if the monopole decays were leptonic, the jets would be accompanied by a monopole neutrino and an ordinary neutrino so photon jet-neutrino coincidences might be observed.

D) At Fermilab or LHC the production of a boson monopole near threshold would have in the final state an extremely heavy monopole boson that would decay into a very high momentum pion (say) opposite to a very high equal momentum photon jet. The initial state would contain soft hadrons and a soft photon jet emitted when the monopole was produced. The present Fermilab data base could be searched for candidates and searches could be planned for the LHC.

E) If the universe were formed from zero total angular momentum and monopole pairs were formed such that $\sum g_i e_j\hat r_i$ were not exactly zero, the universe would have rotational angular momentum until the decays of all the monopoles took place. This might have implications for early universe cosmology.

Discussion:

We are unable to predict masses of magnetically charged particles just as present theory is unable to predict the masses of electrically charged particles. Perhaps the simplest approach would be to consider a world of monopole charged particles, as in the Standard Model, simply replacing e by g. Since the W mass is proportional to e, we might estimate the monopole vector boson, W_g to be very crudely g/e times heavier, although we recognize that the large value of g does not allow the same perturbative calculation.

We also cannot calculate the lifetimes for decay of monopole leptons to ordinary leptons without a theory that couples the W_g to the W_e.

Unfortunately without knowing the masses and lifetimes we can only make qualitative predictions for the experiments we have outlined. Further, the coupling of the two sectors might allow for very small violations of ordinary charge conservation that might be inversely proportional to the monopole mass parameters.

Conclusion: The facts that P and T conservation allow us to derive the laws of electromagnetism providing monopoles exist, that stable monopoles have never been observed, and that monopoles introduce a new connection between charge conservation and angular momentum conservation, suggests that attempts should be made to search for the consequences of monopole decay both in the early and the present universe.

I wish to thank many of my colleagues around the world, and especially Gino Segre, Burt Ovrut and William Frati, for being willing to examine and attempt to detect flaws in this work.