Electroweak monopoles and magnetic dumbbells in grand unified theories
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Electroweak Monopole and Magnetic Dumbbell in SU (5) George Lazarides, Qaisar Shafi School of Electrical and Computer Engineering, Faculty of Engineering,Aristotle University of Thessaloniki, Thessaloniki 54124, Greece Bartol Research Institute, Department of Physics and Astronomy,University of Delaware, Newark, DE 19716, USA
Abstract
We display in SU (5) the presence of an electroweak monopole and a magnetic dumb-bell (“meson”) made up of a monopole-antimonopole pair connected by a Z-flux tube. Themonopole is associated with the spontaneous breaking of the weak SU (2) L gauge symme-try by the induced vacuum expectation value of a heavy scalar SU (2) L triplet with zeroweak hypercharge contained in the adjoint Higgs 24-plet. This monopole carries a Coulombmagnetic charge of (3 / π/e ) as well as Z-magnetic charge, where 2 π/e denotes the unitDirac magnetic charge. Its total magnetic charge is p / π/e ), which is in agreement withthe Dirac quantization condition. The monopole weighs about 700 GeV, but because of theattached Z-tube it exists in a magnetic dumbbell configuration whose mass is expected to liein the TeV range. The presence of this dumbbell provides an exciting new avenue for testinggrand unified theories such as SU (5) and SO (10) at high energy colliders. rand unification based on gauge groups SU (5) [1], SO (10) [2] and E [3] all predict theexistence of a superheavy and topologically stable magnetic monopole, which carries a singlequantum (2 π/e ) of Dirac magnetic charge [4] as well as color magnetic charge [5, 6]. In SU (5)this is the lightest monopole, but SO (10) and E may also give rise to topologically stableintermediate scale monopoles depending on their symmetry breaking patterns. For instance, SO (10) breaking via SU (4) c × SU (2) L × SU (2) R [7] yields an intermediate scale topologicallystable monopole carrying two units of the Dirac charge as well as color magnetic charge [8].Depending on its symmetry breaking scale, the trinification symmetry group SU (3) c × SU (3) L × SU (3) R can yield a topologically stable triply charged monopole with mass in the TeV region,thus making it potentially accessible at high energy colliders [9].In this paper we display the presence of another class of topological structures that canappear in grand unified theories. These structures are not topologically stable and in SU (5)in particular, they are magnetic dumbbells (or “mesons”) made up of electroweak monopole-antimonopole pairs connected by magnetic Z-flux tubes. The magnetic dumbbell, with massestimated to lie in the TeV range, is closely related, it appears, to objects of essentially the samename discussed sometime ago by Nambu [10] in the standard electroweak model. However,there is an important difference between these two cases, which is related to the fact thatthe electroweak monopole in SU (5) is associated with the induced vacuum expectation value(VEV) of a heavy SU (2) L triplet field with zero weak hypercharge that resides in the adjointHiggs 24-plet. There is, of course, no corresponding elementary SU (2) L triplet scalar in thestandard SU (2) L × U (1) Y model. The electroweak monopole in SU (5) carries a Dirac chargeof (3 / π/e ) as well as Z-magnetic charge, and it is expected to exist in a confined statetogether with its antimonopole. [Monopoles carrying Coulomb and Z-flux with the aim ofconfining primordial monopoles were previously discussed in Ref. [11]. For a recent discussionof monopoles in an extended SU (2) L × U (1) Y model with real and complex scalar triplets seeRef. [12].]It is important to clarify that the SU (5) model does not predict topologically stable elec-troweak monopoles or strings associated with electroweak breaking. To this end, we concen-trate on the electroweak sector of the model, where the generator of U (1) Y is taken as 2 Y sothat it has integral eigenvalues and thus periodicity 2 π . We note that inside SU (5), the elec-troweak gauge symmetry is G = SU (2) L × U (1) Y /Z , where Z is generated by the element( − , −
1) = ( e iπT L , e iπ Y ) of SU (2) L × U (1) Y . Indeed, this element acts as the identity elementon the SU (2) L doublet with Y = − / Y = 0, as well as any other complex triplets with integralhypercharge. The second homotopy group of the vacuum manifold is π (cid:18) GU (1) em (cid:19) = π ( U (1) em ) G , (1)where the right hand side consists of all the homotopically non-trivial loops in U (1) em whichare trivial in G . The smallest non-trivial loop in U (1) em corresponds to a 2 π rotation along Q = ( T L + 2 Y ) /
2, which is equivalent to a π rotation along T L accompanied by a π rotation1long 2 Y interpolating between (1 ,
1) and ( − , −
1) in G . This is the smallest closed loop in G and is homotopically non-trivial. The fundamental (first homotopy) group in the right handside of Eq. (1) is therefore trivial, and there are no topologically stable electroweak monopoles.Moreover, the fundamental group of the vacuum manifold π (cid:18) GU (1) em (cid:19) = π ( U (1) em ) G , (2)is also trivial since both G and U (1) em are connected, and no stable strings appear either.In the electroweak model, we introduce a real Higgs triplet field T = T i σ i / Y , where σ i ( i = 1 , ,
3) are the three Pauli matrices. This triplet, which residesin the SU (5) adjoint Higgs 24-plet, couples to the electroweak doublet H , as displayed in thefollowing additional contribution to the potential energy density: V T = 12 M T (cid:18) T i − λ T M T H † σ i H (cid:19) . (3)Here, M T ≫ M Z is the triplet mass which may be as large as the grand unification scale, and λ T is a dimensionless coupling constant of order unity or less. The cross term in the aboveexpression originates from the Higgs couplings 5 † × × † × × SU (5) Higgs 24-plet and the Higgs 5-plet which contains the electroweak doublet H . It yields a T VEV suppressed relative to the VEV h H i of H by a factor h H i /M T .The spontaneous breaking of the electroweak symmetry is achieved, as usual, via the potential V H = λ (cid:18) H † H − v D (cid:19) , (4)which yields the following VEV for the electroweak doublet H : h H i = v D √ , (5)where v D ≃
246 GeV. From Eq. (3), this induces the triplet VEV given by h T i = λ T v D M T ≡ v T , (6)which breaks SU (2) L to U (1) L with generator T L = diag(1 , − h H i , the breaking of SU (2) L by the Higgs triplet T yields a ’t Hooft-Polyakov type monopole [14] with magnetic flux cor-responding to a 2 π rotation around T L or, equivalently, a 4 π rotation around the customarilynormalized generator T L /
2. Reintroducing h H i , this monopole ceases to be topologically stableand becomes attached to a magnetic flux tube. Indeed, the electroweak symmetry breakingleaves unbroken the electric charge generator Q = T L / Y , where the weak hypercharge oper-ator is given by Y = diag( − / , − / , − / , / , /
2) in SU (5). The corresponding orthogonalbroken generator is B = T L / − Y /
5. 2t this stage it is convenient to consider 5 B , which has the smallest possible integer elementsand thus periodicity 2 π . A rotation by 2 π/ B leaves invariant the VEV of H , andtherefore the associated tube carries Z-magnetic flux corresponding to a 2 π rotation around5 B /
4. From the relation T L = 3 Q/ B /
4, we see that the monopole with one unit of fluxalong T L (i.e. corresponding to a 2 π rotation around this generator) is attached to a Z-fluxtube with one unit of flux along 5 B /
4, and also has Coulomb flux 3 Q/
4. The magnetic chargecorresponding to the Coulomb flux of the monopole is g M = (3 / π/e ). A monopole and anantimonopole are expected to pair up and form a dumbbell connected by this flux tube.A few remarks about the Z-magnetic flux emerging from the electroweak monopole are inorder here. The normalized generator orthogonal to Q is p / T L / − Y / π rotation around (5 / T L / − Y / π/g ) p /
2, where g is the SU (5) gauge coupling which,in the SU (5) limit, coincides with the SU (2) L gauge coupling g = e/ p /
8. The magneticflux along the Z-tube therefore takes the form (4 π/e ) p / p /
8. For completeness, we shouldnote that the expressions above for the Coulomb and Z-magnetic fluxes of the electroweakmonopole coincide, respectively, with the values 4 π sin θ W /e and 4 π sin θ W cos θ W /e found byNambu [10] by recalling the SU (5) prediction sin θ W = 3 /
8, where θ W in the Weinberg angle.Combining appropriately these two fluxes, one obtains the total SU (2) L magnetic charge 4 π/g of the electroweak monopole.It is important to realize that the VEV of H is not affected by the presence of the Higgstriplet T . Indeed, minimization of the combined potential V = V H + V T is achieved at M T (cid:18) T i − λ T M T H † σ i H (cid:19) = 0 , (7)and − M T (cid:18) T i − λ T M T H † σ i H (cid:19) λ T M T H † σ i + λ (cid:16) H † H − v D (cid:17) H † = 0 . (8)In view of Eq. (7), Eq. (8) reduces to the standard equation for the electroweak symmetrybreaking λ (cid:18) H † H − v D (cid:19) = 0 , (9)and so the presence of the triplet T i does not affect the VEV of the doublet H .The ρ parameter [15] in our case is given by ρ = 1+ 4 R , with R = v T /v D (see e.g. Ref. [16]).From the 2- σ upper bound ρ . . v T . . M T & . (cid:18) λ T . (cid:19) TeV . (10)A triplet with mass in the TeV range could provide a new source for Higgs production at highenergy colliders.With the ansatz T i = v T x i /r , where x i ( i = 1 , ,
3) are the spatial coordinates and r is theradial distance, the potential in Eq. (3) is minimized for H † σ i H = v D x i r . (11)3his is achieved by taking [10] H = v D √ cos θ sin θ e iϕ , (12)where 0 ≤ θ < π and 0 ≤ ϕ < π are the polar angles. It is important to note that theformula in Eq. (12) has an ill-defined phase ϕ on the negative x -axis where θ = π . This reflectsthe fact that the monopole is not topologically stable and thus cannot exist on its own. It isaccompanied by a string (Z-flux tube) attached to it. [For a discussion of the stability of thisstring, see Ref. [18] and papers listed therein.]To obtain a rough estimate of the monopole mass, following Ref. [10], we ignore for themoment the attached Z-tube and approximate the monopole by a sphere of radius r withinwhich the gauge fields, H and T are zero. [Being a heavy scalar field we expect T to approachits VEV inside an inner core of radius M − T . The energy stored in this inner core can be ignored.]Outside the sphere all the Higgs fields lie in the vacuum, and we have a Coulomb magnetic fieldcorresponding to the magnetic charge [10, 19] g M = 4 πe sin θ W , (13)where e is the absolute value of the electron charge. The energy of the monopole configurationis then E M = g M πr + 4 π r V , (14)where V , the potential energy density within the sphere, is given by V = λv D
16 = m H v D , (15)and m H = p λ/ v D is the Higgs boson mass. The energy E M is minimized at r min = (cid:18) g M π V (cid:19) = r e sin θ W ( m H v D ) − ≃ × − GeV − , (16)giving the monopole mass m M ≈ / g / M V / π / = 2 / π sin θ W ( m H v D ) / e / ≃
688 GeV . (17)One can calculate the Z-tube radius ρ str and tension µ str following Ref. [10]. We find ρ str ≃ . × − GeV − and µ str ≃ . × GeV . (18)The string radius exceeds the monopole radius by a factor 2.5 or so. So it makes sense to considera string segment at least as long as its radius. The energy of the “minimal” string segment isabout 4.8 TeV, which yields a minimal dumbbell of 5.8 TeV, after including the potential energyfrom the Coulomb attraction between the monopole-antimonopole pair.4ambu has argued [10] that a rotating relativistic dumbbell with energy E and angularmomentum L may yield a Regge trajectory L ∼ α ′ E , with α ′ = 1 / πµ str . Using the relevantformulas in Ref. [10], we find that for string lengths bigger than the minimal length, L &
35 and E & . E/ . × − GeV − & . × − sec [10]. Thestring can also decay by monopole pair creation with decay rate per unit length [20]Γ str = µ str π e − π m Mµ str ≃
127 GeV . (19)The corresponding lifetime for a dumbbell with energy E is ( E/ . × − GeV − & . × − sec, which is shorter than its radiative lifetime by about two orders of magnitude.With the electroweak monopole mass estimated to be around 700 GeV, it is plausible thathighly unstable configurations, consisting of (overlapping) monopole-antimonopole pairs in amass range of 1-2 TeV, may be produced in high energy collisions.Although we have discussed the presence of TeV scale magnetic dumbbells in minimal SU (5),one could reasonably expect that similar structures also appear in larger grand unified theories.Consider SO (10), for instance, in which a non-zero VEV for the SU (2) L triplet T will arisethrough the mixed quartic coupling 45 × ×
10. Here, the 10-plet contains the standard Higgsdoublet, and the 45-plet is the adjoint Higgs field. In a more elaborate SO (10) × U (1) P Q model[21], where U (1) P Q denotes the axion symmetry [22], the trilinear mixed coupling 10 × × SU (2) L triplet in the 45-plet. The presence of TeV scalemagnetic dumbbells, made up of monopole-antimonopole pairs, thus appears to be a rathergeneric feature of grand unified theories. Furthermore, the SO (10) breaking to the standardmodel often proceeds through one or more intermediate steps. Suppose that the low energygroup, excluding QCD, is SU (2) L × SU (2) R × U (1) B − L . In this case we could proceed to break SU (2) R to U (1) R with an SU (2) R triplet scalar, which yields a monopole carrying U (1) R charge.With the next breaking of U (1) R × U (1) B − L to U (1) Y , the monopole gets connected to a string(flux tube). We therefore expect the appearance of new dumbbells, and if the monopole massscale for this case is suitably large compared to the string scale, the flux tubes can be relativelystable and less likely to break via monopole-antimonopole pair creation. The dumbbell mass maylie in the TeV range depending on the symmetry breaking scale of SU (2) L × SU (2) R × U (1) B − L .To summarize, we have identified in SU (5) the presence of an electroweak monopole thatcarries a Dirac magnetic charge of (3 / π/e ) and a Z-magnetic flux. Under plausible assump-tions the monopole mass is estimated to be around 700 GeV, and the associated Z-flux tubewidth and tension are of order M − Z and 30 M Z respectively. The monopole-antimonopole pairsform dumbbells (“mesons”) discovered sometime ago by Nambu. A search for these extendedstructures at the LHC and its upgrades seems worthy of further consideration. Finally, we havenoted that analogous TeV scale extended structures can also appear in larger gauge symmetriessuch SO (10). Acknowledgments.
This work is supported by the Hellenic Foundation for Research and5nnovation (H.F.R.I.) under the “First Call for H.F.R.I. Research Projects to support FacultyMembers and Researchers and the procurement of high-cost research equipment grant” (ProjectNumber: 2251). Q.S. is supported in part by the DOE Grant DE-SC-001380.
References [1] H. Georgi and S.L. Glashow, Phys. Rev. Lett. , 438 (1974).[2] H. Georgi, AIP Conf. Proc. , 575 (1975); H. Fritzsch and P. Minkowski, Annals Phys. ,193 (1975).[3] F. G¨ursey, P. Ramond, and P. Sikivie, Phys. Lett. , 177 (1976); Y. Achiman and B. Stech,Phys. Lett. , 389 (1978); Q. Shafi, Phys. Lett. , 301 (1978).[4] P.A.M. Dirac, Proc. Roy. Soc. Lond. A , no. 821, 60 (1931).[5] C.P. Dokos and T.N. Tomaras, Phys. Rev. D , 2940 (1980).[6] M. Daniel, G. Lazarides, and Q. Shafi, Nucl. Phys. B170 , 156 (1980); G. Lazarides, Q. Shafi,and W.P. Trower, Phys. Rev. Lett. , 1756 (1982).[7] J.C. Pati and A. Salam, Phys. Rev. D , 275 (1974), Erratum: Phys. Rev. D , 703 (1975).[8] G. Lazarides, M. Magg, and Q. Shafi, Phys. Lett. , 87 (1980).[9] G. Lazarides and Q. Shafi, e-Print:2101.01412 [hep-ph].[10] Y. Nambu, Phys. Rev. D , 4262 (1974); Y. Nambu, Nucl. Phys. B130 , 505 (1977).[11] G. Lazarides and Q. Shafi, Phys. Lett. , 149 (1980).[12] P.Q. Hung, Nucl. Phys.
B962 , 11527 (2021); J. Ellis, P.Q. Hung, and N. Mavromatos,e-Print:2008.00464 [hep-ph].[13] A.J. Buras, J.R. Ellis, M.K. Gaillard, and D.V. Nanopoulos, Nucl. Phys.
B135 , 66 (1978).[14] G. ’t Hooft, Nucl. Phys.
B79 , 276 (1974); A.M. Polyakov, J. Exp. Theor. Phys. Lett. ,194 (1974).[15] M.J.G. Veltman, Phys. Lett. B91 , 95 (1980).[16] J.L. D´ıaz-Cruz and D.A. L´opez-Falc´on, Phys. Lett. B , 245 (2003).[17] M. Tanabashi et al., Phys. Rev. D , 030001 (2018).[18] M. James, L. Perivolaropoulos, and T. Vachaspati, Phys. Rev. D , R5232 (1992).[19] A. Achucarro and T. Vachaspati, Phys. Rept. , 347 (2000).620] A. Monin and M.B. Voloshin, Phys. Rev. D , 065048 (2008); L. Leblond, B. Shlaer,and X. Siemens, Phys. Rev. D , 123519 (2009); A. Monin and M.B. Voloshin, Phys. Atom.Nucl. , 703 (2010); W. Buchmuller, V. Domcke, and K. Schmitz, Phys. Lett. B , 135914(2020).[21] R. Holman, G. Lazarides, and Q. Shafi, Phys. Rev. D , 995 (1983); G. Lazarides andQ. Shafi, Phys. Lett. B , 135603 (2020).[22] R.D. Peccei and H.R. Quinn, Phys. Rev. Lett. , 1440 (1977); S. Weinberg, Phys. Rev.Lett. , 223 (1978); F. Wilczek, Phys. Rev. Lett.40