Dirac plus Nambu Monopoles in the Standard Model

We show how in the standard electroweak model three $SU(2)_L$ Nambu monopoles, each carrying electromagnetic (EM) and Z- magnetic fluxes, can merge (through Z-strings) with a single $U(1)_Y$ Dirac monopole to yield a composite monopole that only carries EM magnetic flux. Compatibility with the Dirac quantization condition requires this composite monopole to carry six quanta ($12 \pi /e$) of magnetic charge, independent of the electroweak mixing angle $\theta_w$. The Dirac monopole is not regular at the origin and the energy of the composite monopole is therefore divergent. We discuss how this problem is cured by embedding $U(1)_Y$ in a grand unified group such as $SU(5)$. A second composite configuration with only one Nambu monopole and a colored $U(1)_Y$ Dirac monopole that has minimal EM charge of $4\pi/e$ is also described. Finally, there exists a configuration with an EM charge of $8\pi/e$ as well as screened color magnetic charge.


I. INTRODUCTION
It is widely recognized that the Standard Model (SM) based on the gauge group SU (3) c ×SU (2) L ×U (1) Y /Z 3 × Z 2 does not contain topologically stable finite energy monopoles. This stems from the fact that the vacuum manifold in the SM is a three-sphere and does not have a non-trivial second homotopy group that is required for the existence of topological finite energy monopoles. Note that this result is unaffected if additional scalar and/or matter fields are introduced in the SM. Nevertheless, as we show, interesting composite topological structures are present in the SM and its extensions such as grand unified theories (GUTs).
It was recognized sometime ago [1][2][3] that a 't Hooft-Polyakov type [4,5] monopole configuration can be realized within the SU (2) L sector of the SM. This is achieved by utilizing a Higgs doublet field configuration which effectively behaves like a scalar SU (2) L triplet with hypercharge Y = 0. This yields a monopole configuration that carries one unit of T 3 L magnetic charge, where T 3 L = diag(1, −1) denotes the third generator of SU (2) L . In other words, the monopole emits both electromagnetic (EM) and Z magnetic flux, where the latter appears in a flux tube (or Z string). The EM and Z magnetic flux carried by the monopole are found to be (4π/e) sin 2 θ w and (4π/e) sin θ w cos θ w respectively where θ w is the electroweak mixing angle with sin 2 θ w = 0.23. This monopole cannot be purely electromagnetic in nature with the Z flux screened within its core (size ∼ M −1 Z ) because it would violate Gauss's law pertaining to the magnetic component of U (1) Y .
A recent paper [6] shows that the Nambu monopole configurations routinely arise in GUTs, with the simplest example provided by SU (5). A new feature here is the presence of an elementary scalar SU (2) L triplet with Y = 0 that acquires an induced vacuum expectation value (VEV) from the electroweak breaking. In addition, the quantized value of the Z flux is also predicted by exploiting the well-known prediction of sin 2 θ w = 3/8.
Motivated by these considerations, it seems appropriate to enquire whether a purely electromagnetic monopole configuration can be realized in the SM. Clearly, this cannot happen without involving the U (1) Y sector, in contrast to the claim in Refs. [7,8]. Ignoring the singular nature of a Dirac monopole [9] for a moment, it turns out that a purely electromagnetic monopole can be realized by combining via Z strings a single Y monopole of appropriate magnetic charge with three Nambu monopoles. This composite monopole structure carries a magnetic charge of (12π/e) in order to satisfy the Dirac quantization conditions in the presence of quarks and charged leptons. All dependence on sin 2 θ w disappears in arriving at this conclusion. We show how the embedding of U (1) Y in a grand unified framework such as SU(5) resolves the singular nature of the above composite monopole. As expected, the monopole turns out to be superheavy with mass comparable to the GUT scale. Two additional examples are provided of configurations that carry EM charges of 4π/e and 8π/e as well as screened color magnetic charge.

II. COMBINING NAMBU AND DIRAC MONOPOLES IN THE STANDARD MODEL
We are admitting the possibility of a U (1) Y Dirac monopole in the SM. From the Dirac quantization condition such a monopole carries magnetic hypercharge where g is the hypercharge gauge coupling constant, and the left-handed quark doublet (u, d) L has hypercharge 1/6. Next consider a Nambu monopole in the symmetry breaking SU (2) L → U (1) L . Such a monopole has weak magnetic charge given by the condition where g is the SU (2) L coupling constant. We can work in the unitary gauge for the SU (2) L monopole such that the magnetic flux from the monopole consists only of W 3 µ gauge field, where W a µ denote the usual SU (2) L gauge fields.
We now wish to put the Nambu SU (2) L and Dirac U (1) Y monopoles together and re-express the charges in terms of the post-electroweak symmetry breaking gauge fields: Z µ and A µ . The relation between (W 3 µ , Y µ ) and (Z µ , A µ ) is given by Therefore, the Z µ and A µ magnetic charges on n SU (2) L monopoles are and on n U (1) Y monopoles are The net Z µ and A µ magnetic charges on a conglomerate of n U (1) Y and n SU (2) L monopoles are This configuration should not have any net Z magnetic charge because the Z magnetic fields are confined once the electroweak symmetry is broken. Any net Z flux would form a string that would confine the monopole conglomerate to an anti-conglomerate. Thus, we require Together with the relations cos θ w = g the constraint in (9) gives and so the conglomerate should contain three times as many Nambu monopoles as the Dirac Y-monopoles. Next let us consider the EM magnetic charge on the conglomerate. From (8), taking into account (10)  where e = gg / g 2 + g 2 is the electromagnetic coupling constant. Therefore the minimal EM charge of the conglomerate is 6 × 2π/e and this result is independent of sin 2 θ w .
In Fig. 1 we sketch the minimal allowed monopole conglomerate. From (5), we see that the minimal (n = 1) magnetic flux along the Z-string is 4π/g z , where g z = g/c w is the gauge coupling associated with the Z gauge boson with c w = cos θ w . This string corresponds to a 4π rotation about the corresponding generator T 3 L /2 − s 2 w Q and therefore it carries magnetic flux T 3 L − 2s 2 w Q, where s w = sin θ w . The minimal Nambu monopole also carries EM magnetic charge (4π/g)s w = (4π/e)s 2 w corresponding to magnetic flux 2Qs 2 w . In Sec. IV we show how a monopole configuration with EM charge 4π/e is realized by merging a single Nambu monopole with a colored U (1) Y monopole.
The embedding of the SM in a grand unified framework has important ramifications, especially as far as our current discussion is concerned. For instance, in minimal SU(5) the Nambu monopole contains an inner core due to the presence of a heavy SU (2) L scalar triplet field that acquires an induced VEV through mixing with the SM Higgs doublet. The Nambu monopole carries (3/4)(2π/e) unit of Dirac magnetic charge, and the Z magnetic flux is also predicted [6]. The composite electromagnetic monopole discussed above is now realized by considering a U (1) Y monopole with winding number six, which appears from the symmetry breaking At this stage the monopole has a core size of order M −1 GU T and mass around 10 M GU T . Clearly, the singularity encountered with such a monopole in the SM setting has now disappeared.
Following electroweak breaking the monopole carries an appropriate combination of EM and Z magnetic fluxes, with the latter in the form of Z-strings. In order to retain a finite energy configuration this U (1) Y monopole in SU (5) must merge with three Nambu monopoles to yield a composite structure that carries six quanta of Dirac magnetic charge (Fig. 1). Note that in the absence of quarks just two quanta (4π/e) would have been compatible with the Dirac quantization condition. Indeed, in the SU (5) breaking to SU (3) c × SU (2) L × U (1) Y /Z 3 × Z 2 the fundamental topologically stable monopole carries SU (3) c , SU (2) L and U (1) Y magnetic charges. The subsequent breaking of the SM symmetry to SU (3) c × U (1) em /Z 3 yields a monopole carrying a single quantum (2π/e) of Dirac magnetic charge as well as screened color magnetic charge. Quark confinement in this case is required in order for the Dirac quantization condition to be satisfied beyond the screening radius ∼ Λ −1 QCD . Finally, let us note that a purely electromagnetic monopole carrying three quanta of Dirac magnetic charge and accompanied by no other field exists in SU(5), and a classical solution has been shown in Ref. [10].

IV. COLORED DIRAC MONOPOLES
The smallest homotopically non-trivial loop in the SM gauge group SU (3) c × SU (2) L × U (1) Y /Z 3 × Z 2 consists of a 2π rotation about Y -for the left-handed quark doublet Y = 1/6 -accompanied by a π rotation about T 3 L = diag(1, −1) and a 2π/3 rotation about T 8 c = diag(1, 1, −2). The minimal singular Dirac monopole then has magnetic flux Y + T 3 L /2 + T 8 c /3. The magnetic flux of the doubly charged Dirac monopole can be written as 2Y + 2T 8 c /3 since the loop in SU (2) L becomes homotopically trivial and can be removed. After electroweak symmetry breaking, the colored Dirac monopole can absorb the Z-flux emerging from the Nambu monopole (Fig. 2) (see also Ref. [11]). The total EM plus color flux of the conglomerate is 2Q + 2T 8 c /3 with the color part screened for distances larger than the QCD scale.

V. STRUCTURE AND STABILITY
Figs. 1 and 2 show the purely EM monopoles in terms of their constituent Dirac and Nambu monopoles. However, we do not expect these diagrams to realistically show classical solutions of the field theory. For example, the colored monopole of Fig. 2 is more likely to be a structureless, point-like SU (3) × U (1) Y /Z 3 Dirac monopole surrounded by a cloud of electroweak fields that make up . 2: A doubly charged colored Dirac monopole (pink) and a Nambu monopole (blue) connected by a Z-string with magnetic flux T 3 L − 2s 2 w Q. The total flux of this configuration is 2Q + 2T 8 c /3 of which the colored part will be screened due to QCD effects. the Nambu monopole. Other solutions have been proposed in Ref. [12] though these solutions do not correspond to the correctly quantized monopoles depicted in Figs. 1 and 2 -the conglomerate in Fig. 1 has three (not one) Nambu monopoles, and the conglomerate in Fig. 2 has net color charge.
If the Dirac monopoles are regularized as SU (5) monopoles, Fig. 2 corresponds to the winding 2 monopole resulting from the symmetry breaking to SU Fig. 1 corresponds to the winding 6 grand unified monopole. The odd winding SU (5) monopoles carry SU (2) L magnetic charge and once electroweak symmetry breaks, two odd winding monopoles combine to result in an SU (2) L neutral monopole [13]. Further, it has been shown that the SU (5) monopole with winding n = 5 is unstable to fragmentation into n = 2 + 3 monopoles, and monopoles with winding n ≥ 7 are unstable to fragmentation into n = 6 + (n − 6) monopoles [14,15]. Then the only other relevant winding to consider in the SU (5) regularization is n = 4. This monopole will connect with two Nambu monopoles to form a conglomerate that carries color and EM magnetic charges but the color magnetic field will get screened to give a pure EM monopole with flux 4Q (EM charge 8π/e) at large distances.

VI. SUMMARY
We have explored how purely EM monopole configurations may arise in the SM. We do not expect such configurations to have well defined energy because of the singular nature of the Dirac monopole associated with the U (1) Y sector of the electroweak model. Barring this, we identified a configuration consisting of a single U (1) Y Dirac monopole which is linked via Z strings to three Nambu monopoles. This composite monopole carries six units (12π/e) of EM magnetic charge in order to be compatible with the Dirac quantization condition in the presence of quarks. Although the Y and Nambu monopoles both carry θ w dependent EM and Z magnetic charges, the end result leaves θ w undetermined in the SU (2) L ×U (1) Y /Z 2 framework. We also show how this singular composite monopole gets regularized and obtains a finite, albeit superheavy mass, from embedding the SM in a GUT. In the second example a colored U (1) Y monopole merges with a single Nambu monopole to yield a composite topological structure carrying an EM magnetic charge of 4π/e as well as screened color magnetic charge. A third example consists of two Nambu monopoles and a single colored Dirac U (1) Y monopole and has EM magnetic charge 8π/e. Let us recall here that models such as SU (4) c × SU (2) L × SU (2) R [16] and SU (3) c × SU (3) L × SU (3) R predict electric charge quantization [17,18], but gauge coupling unification in these models is not automatic. In this case the masses of the composite monopoles can lie in the multi-TeV range. It is worth stressing that these two models also predict the existence of topologically stable magnetic monopoles that respectively carry two and three Dirac quanta of magnetic charge, and which may be accessible at high energy colliders.
Before concluding we offer some remarks regarding the observability of these composite monopoles at the LHC and future high energy colliders. The production cross section of a composite coherent quantum state including these composite monopole states is expected to be exponentially suppressed in Drell-Yan processes involving particle collisions [19]. This is reminiscent of the exponential suppression in quantum mechanical tunneling. It has been argued that the magnetic analogue of the Schwinger mechanism could be exploited to significantly enhance the production cross sections of these extended structures and propel them in the observable range. With adequately strong magnetic fields the (dual) Schwinger mechanism may lead to an observable cross section for monopole pair production in heavy ion collisions [20]. The MoEDAL experiment has recently presented the first results [21] based on this idea.
For the composite monopoles to be accessible at high energy colliders their mass should lie in the TeV range or so. This means that the U (1) Y component of the electroweak sector of the SM should be embedded in a unifying gauge symmetry such as SU (3) L × SU (3) R , which may be broken at the TeV scale if we do not require gauge coupling unification of the left-right SU (3) symmetries [18,22]. For a discussion of monopoles in cosmic rays, see Ref. [22] and other references listed there.