All nite-mass Dirac monopoles Filip Blaschke1 2and Petr Bene s2y 1Research Centre for Theoretical Physics and Astrophysics Institute of Physics

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All ﬁnite-mass Dirac monopoles

Filip Blaschke1, 2, ∗and Petr Beneˇs2, †

1Research Centre for Theoretical Physics and Astrophysics, Institute of Physics,

Silesian University in Opava, Bezruˇcovo n´amˇest´ı 1150/13, 746 01 Opava, Czech Republic

2Institute of Experimental and Applied Physics,

Czech Technical University in Prague, Husova 240/5, 110 00 Prague 1, Czech Republic

We present a “primitive” way of realizing ﬁnite-mass Dirac monopoles in U(1) gauge theories

involving a single non-minimally interacting scalar ﬁeld. Typically, the energy density of this type

of monopole is not concentrated at its core, but it is distributed in a spherical shell, as we illustrate

on several exact solutions in the Bogomol’nyi–Prasad–Sommerﬁeld (BPS) limit. We show that our

construction can be interpreted as a limit of inﬁnitely massive Wbosons coupled to electromagnetic

ﬁeld-strength via a dipole moment. Combining our approach with ideas of Weinberg and Lee, we

present a general landscape of U(1) gauge models that support a ﬁnite-mass Dirac monopole. In

fact, all classical monopoles, i.e., Wu–Yang, ’t Hooft–Polyakov, Cho–Maison, etc., are special points

on this landscape.

Keywords: Magnetic monopole; exact solutions; BPS limit

I. BARE MONOPOLES & DRESSED

MONOPOLES

In the classical Maxwell theory (edenotes the electric

charge)

L=−1

2e2Fµν Fµν ,(1)

a magnetic monopole is a singular conﬁguration of U(1)

gauge ﬁelds, i.e., the solution of ~

∇ × ~

A=q ~r/r3, with

qbeing a dimensionless constant. As is well known, the

vector potential can be described everywhere except on

a line going from the monopole to inﬁnity, the so-called

Dirac string [1]. For instance, a static monopole at the

origin with a Dirac string lying on negative z-axis is given

i=qεi3jxj

r(r+z).(2)

The shape of the string can be changed via gauge trans-

formation.

The above description, which we dub a bare monopole,

is singular in three logically distinct ways. Namely, 1)

there is an unphysical singularity along the Dirac string,

2) there is a physical singularity at the origin in both

gauge ﬁelds and in the magnetic ﬁeld ~

B=q ~r/r3(we

deﬁne Bi≡1

2εijkFjk) and 3) the energy density E=

q2/(2e2r4) diverges at the origin in such a way that the

total energy (i.e., the classical mass of the monopole) is

inﬁnite.

These singularities are invariably clues that point to

the incompleteness of the theoretical model. However, in

so far as physics is concerned, they are not equivalently

serious. The presence of Dirac string is simply a failure

∗ﬁlip.blaschke@fpf.slu.cz

†petr.benes@utef.cvut.cz

of global description of the U(1) ﬁbre bundle. The sin-

gularity in ~

Bis a result of taking the bird’s eye point

of view and describing the monopole as a point particle.

By itself, this is not a problem as we can reasonably ex-

pect that in a more fundamental theory this singularity

will be smoothed out by monopole’s microscopic degrees

of freedom. However, the real red ﬂag is the absence of

ﬁnite energy solutions.

All in all, we are compelled to depart from the pure

Maxwell theory. Fortunately, unlike the case of elec-

tric monopoles, we do not have to abandon classical

ﬁeld theory to establish the existence of ﬁnite-mass Dirac

monopoles.

Let us now follow the (somewhat chronological) path

that leads to ﬁnite-mass ﬁeld-theoretical descriptions.

First, there was an observation of Wu and Yang [2] that

one can get rid of Dirac string by embedding (2) into an

SU (2) gauge ﬁeld (q= 1):

AWY

i=εiaj σaxj

2r2=AD

iUσ3U†−iU∂iU†,(3)

where σaare the Pauli matrices and where

U=cos θ/2−sin θ/2e−iϕ

sin θ/2eiϕcos θ/2(4)

is a (singular) SU (2) gauge transformation that relates

the Wu–Yang monopole AWY

iand the Dirac monopole

i. By itself, however, this enhancement of symmetry is

not suﬃcient to get rid of other singularities, neither in

the SU (2) “magnetic” ﬁelds, nor in the energy density.

But, as was realized independently by ’t Hooft [3]

and Polyakov [4], the Wu–Yang monopole can be made

completely regular via spontaneous symmetry break-

ing instigated by adjoint scalars. Loosely speaking,

these additional ﬁelds condense at monopole’s core and

smooth out the r= 0 singularity – the monopole be-

comes dressed. Due to this ﬁeld-dressing, the ’t Hooft–

Polyakov monopole has none of the three singularities

arXiv:2210.11854v2 [hep-th] 3 Jan 2023

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摘要：

Allnite-massDiracmonopolesFilipBlaschke1,2,andPetrBenes2,y1ResearchCentreforTheoreticalPhysicsandAstrophysics,InstituteofPhysics,SilesianUniversityinOpava,Bezrucovonamest1150/13,74601Opava,CzechRepublic2InstituteofExperimentalandAppliedPhysics,CzechTechnicalUniversityinPrague,Husova240/5,110...

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All nite-mass Dirac monopoles Filip Blaschke1 2and Petr Bene s2y 1Research Centre for Theoretical Physics and Astrophysics Institute of Physics

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