# Dirac and non-Dirac conditions in the two-potential theory of magnetic charge

## Abstract

We investigate the Cabbibo–Ferrari, two-potential approach to magnetic charge coupled to two different complex scalar fields, \(\Phi _1\) and \(\Phi _2\), each having different electric and magnetic charges. The scalar field, \(\Phi _1\), is assumed to have a spontaneous symmetry breaking self-interaction potential which gives a mass to the “magnetic” gauge potential and “magnetic” photon, while the other “electric” gauge potential and “electric” photon remain massless. The magnetic photon is hidden until one reaches energies of the order of the magnetic photon rest mass. The second scalar field, \(\Phi _2\), is required in order to make the theory non-trivial. With only one field one can always use a duality rotation to rotate away either the electric or magnetic charge, and thus decouple either the associated electric or magnetic photon. In analyzing this system of two scalar fields in the Cabbibo–Ferrari approach we perform several duality and gauge transformations, which require introducing non-Dirac conditions on the initial electric and magnetic charges. We also find that due to the symmetry breaking the usual Dirac condition is altered to include the mass of the magnetic photon. We discuss the implications of these various conditions on the charges.

## 1 Introduction

From Eq. (3) one can work out the behavior of \(\phi _m\) and \(\mathbf{C}\) under the discrete *P* and *T* symmetries (i.e. parity or space inversion and time reversal). Given that \(\mathbf{E}\) is odd under *P* and even under *T*, while \(\mathbf{B}\) is the opposite, one finds from Eq. (3) that \(\phi _m\) is a pseudoscalar (odd under *P* and odd under *T*) and \(\mathbf{C}\) is a pseudovector (even under *P* and even under *T*).

While the Cabibbo–Ferrari approach avoids the need for a singular Dirac string this comes at the expense of introducing extra degrees of freedom via \(\phi _m\) and \(\mathbf{C}\). Because of the opposite behavior of \(\phi _m\) and \(\mathbf{C}\) under parity and time reversal as compared to \(\phi _e\) and \(\mathbf{A}\) these “magnetic” potentials would give rise to a pseudovector, massless “magnetic” photon in the quantized version of the theory. There are various ways of dealing with these extra degrees of freedom: (i) One can impose extra conditions on the theory so that despite having two 4-vector potentials the final number of degrees of freedom correspond to only a single “electric” photon; (ii) accept the second 4-potential as a real “magnetic” photon [7] but then apply the Higgs mechanism so that this additional photon becomes massive and is “hidden” until one reaches some appropriately, high energy scale [11, 12, 13]. It is the latter choice that we have in mind here. This then expands the usual \(SU(2) \times U(1)\) Standard Model to \(SU(2) \times U(1) \times U(1)\). Introducing an extra, massive *U*(1) gauge boson may be phenomenologically interesting in the light of dark/hidden photon models [16] as well as light-shining-through-wall experiments [17], all of which require an extra *U*(1) gauge boson. The magnetic photon might provide an electromagnetic origin for these dark/hidden photon models. In addition given the opposite behavior of \(A_\mu \) and \(C_\mu \) under *P* and *T*, as discussed above, the “electric” photon associated with \(A_\mu \) and the “magnetic” photon associated with \(C_\mu \) will have different properties under *P* and *T* symmetry and will thus be distinguishable on this basis.

## 2 Scalar fields coupled to electric and magnetic charge

In this section we couple the two 4-vector potentials, \(A_\mu \) and \(C_\mu \), to two different complex scalar fields, \(\Phi _1 , \Phi _2\). We do this in steps – first coupling \(A_\mu , C_\mu \) to \(\Phi _1\) (in Sect. 2.1) and then coupling \(A_\mu , C_\mu \) to \(\Phi _2\) (in Sect. 2.2). The scalar field \(\Phi _1\) has a \(\lambda \Phi _1 ^4\) self-interaction which spontaneously breaks the magnetic gauge symmetry of the theory, making the magnetic photon massive. This makes the magnetic photon un-observable until one probes an energy scale of the order of the rest mass energy of the magnetic photon. However, with only a single scalar field \(\Phi _1\) we find after symmetry breaking that the electric 4-vector potential, \(A_\mu \), completely decouples from \(\Phi _1\) and the theory becomes trivial in regard to \(A_\mu \). This triviality goes back to the well-known result that if all fields/particles have the same ratio of electric charge to magnetic charge (which is true with only one scalar field), one can always perform a duality rotation to rotate away either the electric or magnetic charge. To make the theory non-trivial we introduce a second complex scalar field \(\Phi _2\) with initial charges different from \(\Phi _1\).

### 2.1 Scalar field \(\Phi _1\)

*U*(1). We repeat the details here since it will serve as necessary background for Sect. 2.2. The Lagrangian density is

*v*gives

*both*\(A_\mu \) and \(C_\mu \). The mixed term \(2q_e q_m A_{\mu }C^{\mu }\) is problematic. We can get rid of this term by performing a duality rotation on \(A_\mu \) and \(C_\mu \). This is exactly analogous to the rotation by the Weinberg angle that one performs in the Standard Model in order to get the physical Z-boson and photon fields from the initial \(W^0_\mu \) and \(X_\mu \) bosons. Writing the second line in (10) as

*g*is a combination of the initial charges \(q_e\) and \(q_m\). The mass term for \(C_\mu \) is given by \(\frac{1}{2} m_C ^2 C_\mu C^\mu \rightarrow \frac{1}{2} v^2(q_e^2+q_m^2)C_{\mu }C^{\mu }\). Thus the mass associated with \(C_\mu \) is \(m_C = v \sqrt{q_e^2+q_m^2} = g v\).

### 2.2 Scalar field \(\Phi _2\)

*g*and

*e*, in terms of the initial Lagrangian parameters \(q_e, q_m, q_e ' , q_m '\). These are

## 3 Physical consequences of additional quantization conditions

The first condition can be re-written as \(\frac{q_e'}{q_m '}=-\frac{q_e}{q_m}\) so that the ratio of initial electric to magnetic charges of the two scalar fields are not equal *i.e.* \(\frac{q_e'}{q_m '} \ne \frac{q_e}{q_m}\). This result can be connected with the well-known statement “If particles/fields all have the same ratio of magnetic to electric charge then we can make a duality rotation by choosing \(\theta \) so that \(\rho _m=0\) and \(\mathbf{J}_m =0\)” [1]. Therefore, the first condition in (31), which results from the gauge transformation, ensures that one has a non-trivial theory with the final Lagrangian in (29) having fields with both electric and magnetic charges as given in (30).

If one looks at the last equation in (30) one sees that it can be re-written as \(e g = 2 q_e q_e '\), so that the physical electric and magnetic charges, *e* and *g*, are given in terms of only the starting electric charges, \(q_e\) and \(q_e '\), of the two scalar fields. The general conclusion is that the new, non-Dirac conditions are a re-parametrization of the physical charges in terms of the original Lagrangian charges – \(q_e, q_e ', q_m, q_m '\). Directly from (30) one can see that the physical electric and magnetic charges are given by some combination of the initial “electric” and “magnetic” parameters.

Next we look into what happens with the usual Dirac quantization condition \(eg = n \frac{\hbar }{2}\) in the above Cabbibo–Ferrari approach with symmetry breaking. In the two-potential approach there is no Dirac string, no need for the Dirac veto and thus one might expect there will be no Dirac quantization condition. However, the Dirac quantization condition can be obtained independently of the Dirac string argument by requiring that the field angular momentum of the electric–magnetic charge system be quantization in half-integer units of \(\hbar \) [18, 19, 20]. Moreover, since we have hidden the magnetic gauge symmetry associated with \(C_\mu \) via symmetry breaking, the magnetic field produced by the magnetic charge will have a Yukawa character and this will change the quantization condition.

*i.e.*\(\mathbf{R} = R {\hat{\mathbf{z}}}\). The field angular momentum in (34) for this set up only has a non-vanishing component in the

*z*-direction. After performing the \(d \varphi \) integration one finds

*eg*to find

*x*-dependent factor in parentheses represents the deviations, due to symmetry breaking, from the usual Dirac condition. The plot of (37) is shown in Fig. 1. From this one can see that when we turn off symmetry breaking (i.e. \(m_C\) and \(x \rightarrow 0\)) we recover the expected result of \(eg =\frac{1}{2}\) for the \(n=1\) case given in the graph. When we turn on symmetry breaking and \(m_C >0\) we see that \(eg > \frac{1}{2}\). This implies that the strength of the magnetic coupling will be stronger in the presence of symmetry breaking as compared to when there is no symmetry breaking. This makes sense in the following way: due to the presence of a mass for \(C_\mu \) the magnetic field will fall off more rapidly than in the Coulomb case as can be seen from (33). To compensate for this weaker magnetic field, the magnetic coupling

*g*must be larger in order to still give an angular momentum of \(L_z =\frac{n \hbar }{2} \rightarrow \frac{1}{2}\).

*U*(1) symmetry connected with the magnetic charge will undergo a phase transition so that magnetic charge will be confined [22, 23, 24]. In Refs. [25, 26] lattice gauge calculations indicated that the critical value of the coupling for a

*U*(1) theory to become confining is \(\alpha _c \sim \mathcal{O} (1)\). For the present case \(\alpha _g \approx 137/4\), and this is clearly well above unity so that the magnetic charge in this model should be confined. Furthermore, one could propose that it is this non-pertubative value of \(\alpha _g\) which drives the mass term for the magnetic photon, as in the old techni-color models (for a recent review see [27]). In this picture a monopole–antimonopole pair would form a condensate due to the strong coupling implied by \(\alpha _g\), and this condensate would mix with the magnetic gauge boson to generate a mass, exactly as strongly interacting techni-quarks were supposed to form a condensate which would mix with the \(W^\pm \) and

*Z*bosons to give them their masses.

## 4 Conclusions

We have examined the Cabibbo–Ferrari, two-potential approach to magnetic charge with two complex scalar fields, each carrying different initial “electric” and “magnetic” couplings – \(q_e, q_e ', q_m, q_m '\). In this formulation the Dirac string is replaced by an additional gauge potential, \(C_\mu \). In the usual Dirac string approach the singular string has to be “hidden” or made non-observable. This is accomplished by imposing the Dirac condition on the electric and magnetic charges, namely \(eg = n\frac{\hbar }{2}\). Here we need to “hide” the extra gauge potential. This is accomplished by assuming that \(\Phi _1\) has a Higgs-like self-interaction which breaks the magnetic, *U*(1) symmetry and makes \(C_\mu \) massive so that the magnetic vector potential is un-observable, at least up to an energy scale of \(m_C c^2\). With only a single scalar field, \(\Phi _1\), we find that we have a trivial system in the sense that the massless potential, \(A_\mu \), decouples from everything else, and there is only one matter field, \(\eta \), which carries only a magnetic charge. We next introduced a second scalar field, \(\Phi _2\), with different initial “electric” and “magnetic” charges. With this additional matter field we found that one needed to impose non-Dirac conditions on the Lagrangian charges, \(q_e, q_m, q_e', q_m'\) in order for the Lagrangian to be properly gauge invariant and in order for the duality transformation to give interaction terms that avoided problematic mixed terms proportional to \(A_\mu C^\mu \). These additional, non-Dirac conditions were given in Eqs. (21) and (25).

We also found that the magnetic photon gaining a mass modified the usual Dirac condition since the magnetic field now had a Yukakwa behavior which changed the field angular momentum of the electric charge–magnetic charge system to (36). This modified Dirac quantization condition is given in (37). We found that this modified condition implied a larger magnetic charge as compared to the usual Dirac case. In addition, this new Dirac-like quantization condition involved not only the charges, *e* and *g*, but also the parameter \(x = m_C R\). This is expected, since with symmetry breaking one introduces a mass/distance scale (*i.e.* \(m_C\) or \(d=1/m_C\)), which should then show up in the quantization condition. An interesting conjecture is that the Dirac-like quantization condition in the presence of symmetry breaking might lead to a mass quantization condition. This idea of mass quantization coming from Dirac electric–magnetic charge quantization in the presence of symmetry breaking will be investigated in future work.

## Notes

### Acknowledgements

This work was supported by Grant \(\Phi .0755\) in fundamental research in natural sciences by the Ministry of Education and Science of Republic of Kazakhstan. T.J. Evans would like to acknowledge his father, Charles Evans, for lifelong encouragement of practicing math and science.

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