# Dirac and non-Dirac conditions in the 2-potential theory of magnetic charge

###### Abstract

We investigate the Cabbibo-Ferrari, two potential approach to magnetic charge coupled to two different complex scalar fields, and , each having different electric and magnetic charges. The scalar field, , 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, , 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.

## I Introduction

Maxwell’s equations with magnetic sources in Gaussian units are

(1) |

Eqns. (1) are more symmetric than Maxwell equations with only electric sources. The mathematical expression of this extra symmetry is given by the dual transformation of electric and magnetic quantities jackson

(2) |

where () are the original quantities: , , (, , ) and () are the final quantities: , , (, , ), and is the duality angle by which these quantities are rotated in “dual” space. This dual symmetry does not usually extend down the level of the potentials, since the electric and magnetic fields in terms of the scalar potential, , and vector potential, , take the asymmetric form . In order to extend the dual symmetry of (2) to the level of the potentials one needs to introduce a second set of potentials – a strategy first proposed by Cabibbo and Ferrari cabibbo who added a second scalar and vector potential, and . In terms of these two potentials the electric and magnetic fields can be written as and

(3) |

With the definitions in (3) the dual symmetry of (2) can be extended to the level of the potentials with in (2) being . The 2-potential approach avoids the Dirac string dirac which is the usual way magnetic charge is introduced with only a single 4-vector potential (see for example ryder pgs. 402-405 for a short pedagogical exposition of the Dirac string approach). A short, non-exhaustive list of work on the Cabibbo-Ferrari approach to magnetic charge can be found in references hagen ; salam ; zwang ; barker ; singleton1 ; singleton2 ; singleton3 ; chatterjee ; chatterjee2 .

From equation (3) one can work out the behavior of and under the discrete and symmetries (i.e. parity or space inversion and time reversal). Given that is odd under and even under , while is the opposite, one finds from equation (3) that is a pseudoscalar (odd under and odd under ) and is a pseudovector (even under and even under ).

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 and . Because of the opposite behavior of and under parity and time reversal as compared to and these “magnetic” potentials would give rise to a pseudo-vector, 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 salam but then apply the Higgs mechanism so that this additional photon becomes massive and is “hidden” until one reaches some appropriately, high energy scale singleton1 ; singleton2 ; singleton3 . It is this latter choice that we have in mind here. This then expands the usual Standard Model to . Introducing an extra, massive gauge boson may be phenomenologically interesting in light of dark/hidden photon models darkphoton as well as light-shining-through-wall experiments wall , all of which require an extra gauge boson. The magnetic photon might provide an electromagnetic origin for these dark/hidden photon models. In addition given the opposite behavior of and under and , as discussed above, the “electric” photon associated with and the “magnetic” photon associated with will different properties under and symmetry and will thus be distinguishable on this basis.

The above results can be put in 4-vector notation. The electric and magnetic 4-vector potentials are and , and the electric and magnetic 4-currents are and . From the electric and magnetic 4-vector potentials one can construct the electric and magnetic field strength tensors and their dual field strength tensors

(4) |

In terms of these the electric and magnetic fields from (3) can be written as and . Using all this one can construct a pure electromagnetic field Lagrangian density

(5) |

where for generality we have included two additional mixing terms between the two different field strength tensors – and – with being undetermined constants. The term does not effect the equations of motions. The term would change the equations of motion and is a kinetic mixing term of the kind found in many dark/hidden photon models darkphoton and light-shining-through-wall wall proposals. Neither term will play a role in the considerations in the rest of this paper and so we will drop both terms in the following sections. With the dropping of the term the equations of motion resulting from (5) are and which are the 4-vector form of the equations in (1). (Here and in the rest of the article we set ). In the next section we replace the arbitrary sources by a dynamical source coming from a complex scalar field initially carrying both electric and magnetic charge.

## Ii Scalar fields coupled to electric and magnetic charge

In this section we couple the two, 4-vector potentials, and , to two different complex scalar field, . We do this in steps – first coupling to (in section IIA) and then coupling to (in section IIB). The scalar field has a self-interaction which spontaneously breaks the magnetic gauge symmetry of the theory, making the magnetic photon massive. This makes the magnetic photon unobservable until one probes an energy scale on the order of the rest mass energy of the magnetic photon. However with only a single scalar field we find after symmetry breaking that the electric 4-vector potential, , completely decouples from and the theory becomes trivial in regard to . 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 with different initial charges from .

### ii.1 Scalar field

We begin with one complex, scalar field, , with a symmetry breaking self-interacting potential. initially couples to both and with strength and respectively. provides a dynamical source for the 4-currents and . The following manipulations are the standard symmetry breaking of a gauge symmetry down to . We repeat the details here since it will serve as necessary background for the following section IIB. The Lagrangian density is

(6) |

where the gauge-covariant derivative is . There are two gauge field kinetic terms, and , for and respectively. The self interaction term for is of the form and the mass term is tachyonic (i.e. ). Under these conditions develops a vacuum expectation value

(7) |

where . Expanding around gives

(8) |

where are real fields. Following the standard prescription we perform a gauge transformation to get rid of the phase, , in

(9) | |||||

The Lagrangian in (6) now becomes

In (II.1) we have dropped a constant term, , which has no effect on the equations of motion. Also we have relabeled the gauge fields, by dropped the primes. The second line in (II.1) represents a mass term for the scalar field as well as interaction terms between and both and . The mixed term is problematic. We can get rid of this term by performing a duality rotation on and . 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 and bosons. Writing the second line in (II.1) as

(11) |

we perform the following duality rotation of the potentials

(12) |

This transforms the gauge field part of (11) into

(13) |

To diagonalize the gauge field interaction matrix of (13) one can choose so that and . In this way the interaction matrix of (13) diagonalizes to become

(14) |

This can be written out as

(15) |

Piecing together all of these parts we find that the Lagrangian from (II.1) becomes,

(16) | |||||

In the above we have again dropped the primes. Note that the electric gauge potential, , is completely decoupled from the scalar matter field, , and from ; has become a sterile/decoupled field in since it interacts with nothing else. In the Lagrangian in (16) there is only one interacting gauge potential, , and only one magnetically charged scalar field, . This is related to the fact that in a world with only one type of field/particle one can always rotate away one of the charges via a duality rotation. In summary the final Lagrangian in (16) has a completely decoupled, massless electric gauge potential, , and a massive “magnetic” gauge potential, , which is coupled to a scalar field, . This scalar field carries only a magnetic charge of . This physical magnetic charge is a combination of the initial charges and . The mass term for is given by is . . Thus the mass associated with

### ii.2 Scalar field

In order to move beyond the trivial results of section IIA, and to obtain the advertised non-Dirac conditions, we add a second scalar field, , with different initial electric and magnetic charges, and . The Lagrangian from (6) now expands to become

(17) |

The covariant derivatives for the two fields, and are respectively and . Using these covariant derivatives to expand (17) gives

(18) | |||||

where the currents, and , from and respectively, are defined as

(19) |

As in the previous section, we perform the gauge transformation on given in (II.1). The part of the reduces as in (II.1) and the whole Lagrangian from (18) reduces to

Note that is missing. By the gauge transformation (II.1) (i.e. the scalar field is purely real) and so the associated vanishes. The last two terms in (II.2) depend on the arbitrary gauge function , so the Lagrangian in (II.2) is not invariant under the gauge transformation (II.1). One could try another gauge transformation on in order to restore the invariance of the Lagrangian. However this would then require making an additional gauge transformation of , which then would require another gauge transformation on etc. There is another option: require the charges on and satisfy the relationship

(21) |

This is the first non-Dirac condition on the electric and magnetic charges of the fields. This condition comes from the requirement that the gauge transformation applied to in (II.1) also leave the expanded Lagrangian in (II.2) invariant, especially with respect to the added field which also couples to the same as . This mutual coupling of to is the source of the proposed extra condition above. Applying (21) to in (II.2) leads to

Now in addition to the gauge transformation which we had to perform in the section IIA, we also performed a duality rotation as given in equations (12) (13) which gave the Lagrangian in (16). In the final form of the Lagrangian , the remaining scalar field, , only coupled to the magnetic gauge potential with a physical magnetic charge . Since the in this section are the same as in section IIA we must also perform the duality rotation from (12) to the new terms in (II.2) which can be written out as.

(23) |

From section IIA is defined by and . Performing the matrix multiplication gives

(24) |

In order to diagonalize matrix and get rid of the problematic cross terms, , we require the following conditions

(25) |

These are the second, non-Dirac conditions. As before, the reason for requiring these conditions is that both couple to both . Unlike the condition in (21) which comes from the gauge transformation, these conditions arise from the duality transformation. Actually once the condition, , is imposed the second condition (i.e. ) follows from (21). Using the conditions (25) in (24) leads to

(26) |

Next we examine what happens to the term in (II.2) under the duality rotation. We find

(27) |

where we have replaced the and by their values in terms of and . Finally, applying the conditions from equation (25) we arrive at the dual rotated interaction between and

(28) |

Collecting all of these results together from equations (15) (26) (28), and using these in (II.2) yields

(29) | |||||

In this last step we have dropped the primes. From the last line of (29) one can read off the physical magnetic and electric charges, and , in terms of the initial Lagrangian parameters . These are

(30) |

The point to notice about the final form of the Lagrangian is that now is no longer decoupled, but couples directly to via the last two terms in (29), and couples directly to – both and interact non-trivially.

## Iii Physical consequences of additional quantization conditions

In the previous section we have shown how additional, non-Dirac quantization conditions arise between the initial electric/magnetic charges – – of the fields in the Cabbibo-Ferrari two-potential approach to magnetic charge. These non-Dirac conditions come from requiring that some combination of gauge transformations and duality rotations leads to a consistent, gauge invariant theory. We repeat these additional condition here for convenience

(31) |

In (21) there is an additional condition, , which we leave off here since it is redundant – it comes from combining the two conditions in (31).

The first condition can be re-written as i.e. so that and ” jackson . 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). . 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 so that the ratio of initial electric to magnetic charges of the two scalar fields are not equal

If one looks at the last equation in (30) one sees that it can be re-written as so that the physical electric and magnetic charges, and , are given in terms of only the starting electric charges, and , 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 – . 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 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 saha . Moreover, since we have hidden the magnetic gauge symmetry associated with via symmetry breaking, the magnetic field produced by the magnetic charge will have a Yukawa character and this will change the quantization condition.

We begin by placing the magnetic charge associated with at the origin and the electric charge associated with a distance from . The respective four-vector potentials produced by this configuration are

(32) |

where . The field/particle produces a Coulomb potential, while the particle/field produces a Yukawa potential. Using the definitions of the electric and magnetic fields from (3) gives

(33) |

The field angular momentum associated with the electric and magnetic fields in (33) is given by

(34) |

Since the fields are radial we will, without loss of generality, place the electrically charged particle, , along the z-axis i.e. . The field angular momentum in (34) for this set up only has a non-vanishing component in the z-direction. After performing the integration one finds

(35) |

Doing the remaining integrations singleton2 chatterjee chatterjee2 gives

(36) |

In the limit we recover the results found in saha (also see cart for a more accessible and modern derivation of this result for the case). Now making the assumption that and defining we can solve (36) for the product of charges to find

(37) |

In the last limit we have taken the lowest option of and set . The -dependent factor in parenthesis represents the deviations, due to symmetry breaking, from the usual Dirac condition. The plot of (37) is shown in figure (1). From this one can see that when we turn off symmetry break (i.e. and ) we recover the expected result of for the case given in the graph. When we turn on symmetry break and