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Stable SU(5) monopoles with higher magnetic charge
Abstract
Taking into account the electroweak breaking effects, some multiply charged monopoles were shown to be stable by Gardner and Harvey. We give the explicit Ansa$uml: tze for finite-energy, nonsingular solutions of these stable higher-strength monopoles with eg = 1,(3/2),3. We also give the general stability conditions and the detailed behavior of the interaction potentials between two monopoles which produce the stable higher-strength monopoles.
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We construct the monopole spectrum in an SO(10) grand unified theory which has an arbitrary pattern of symmetry breakdown to SU(3)×U(1)EM. We show that if the fermion fields are in a 16 of SO(10), then it is impossible to find an SO(10) theory where monopoles do not catalyze proton decay. Furthermore, the branching ratios for monopole-catalyzed proton decay are identical in SO(10) and SU(5) grand unified theories. We also present a criterion for constructing grand unified theories which do not have monopole catalysis of proton decay.
In addition to being bounded from below by the Prasad-Sommerfiled limit, monopole masses are also bounded from above by the limit in which the scalar field variables are frozen to their vacuum values. This upper bound is close to the lower one: The single-, double-, and triple-strength SU(5) monopoles are found to have their masses bound in m<~M<~m×1.7867, 2m<~M<~2m×2.0741, and 3m<~M<~3m×2.3155, respectively, where m=3Mx/8α.
We analyze the behavior of the magnetic-monopole equations as a function of the ratio of Higgs-scalar-to-vector-meson mass MH/MW. For very small MH/MW, we find that the mass of the monopole deviates from the Prasad-Sommerfield limit MW/α by MH/2α, independently of the particular details of the symmetry-breaking potential. In the opposite limit MH/MW→∞, the scalar field contributes negligibly to the mass, which is found to be (MW/α)(1.787-2.228MW/MH+⋯).
We propose a construction of static magnetic Yang-Mills-Higgs monopole solutions of arbitrary topological charge. They are axially symmetric and contain no free parameters except for their position. The regularity of the solutions has yet be proved; doing so would complete the constructive proof of existence.
An extensive and detailed analysis of self-dual gauge fields, in particular with axial symmetry, is presented, culminating in a purely algebraic procedure to generate solutions. The method which is particularly suited for the construction of multimonopole solutions for a theory with arbitrary G, is also applicable to a wide class of non-linear sigma models.The relevant symmetries as well as the associated linear problems which underly the exact solubility of the problem, are constructed and discussed in detail.
The point monopole solution found by Corrigan, Olive, Fairlie and Nuyts for SU(3) broken down to U(2) by an octet Higgs field is shown to survive when the symmetry is further broken to U(1), and therefore is a magnetic monopole in the traditional sense. The demonstration exploits a singular transformation from the Abelian gauge with a Dirac string, to a non-singular gauge in which the vector field has a manifest rotational symmetry. The procedure is generalized to SU(N) broken to U(1), always yielding the smallest strength monopole consistent with the Dirac quantization condition, and in some cases higher-strength monopoles as well. The manifest symmetry of the vector field corresponds to an angular momentum J0, different in general from the physical charge-pole angular momentum J. The latter generates a symmetry of the whole system of Higgs and vector fields. Even that symmetry fails for the interior of a finite-energy solution unless J and J0 coincide, which happens only for the minimal monopole coupled to a charge in an SU(2) subgroup of SU(N). If SU(N) is not broken all the way to U(1), there can be solutions with exact J0 symmetry whose possible significance is discussed. A theorem of Georgi and Glashow is used to show that SU(3) --> U(1) could occur in a natural way if there were two Higgs fields.
Magnetic monopoles with multiple Dirac charge are found to be stable in grand unified theories with symmetry breaking by an adjoint Higgs field under certain conditions. In the SU(5) model, the double, triple, quadruple, and sextuple monopoles are stable for a range of parameters in the Higgs potential. The effects of electroweak symmetry breaking on multiply charged monopoles are discussed. Evidence is also presented for the existence of a stable nonspherically symmetric quadruple monopole.
Gauge fields admitting spherical symmetry are listed. Spherically symmetric solutions of Yang-Mills equations and spherically symmetric magnetic monopoles are studied. A simple exact solution of the Yang-Mills and Einstein equations is found.
We present an exact solution to the nonlinear field equations which describe a classical excitation possessing magnetic and electric charge. This solution has finite energy and exhibits explicitly those properties which have previously been found by numerical analysis.
In this paper we develop a method for constructing point-singular spherically symmetric monopoles in theories with an arbitrary compact simple gauge group. A.P. Sloan Foundation Reserach Fellow.
We extend our analysis of the fundamental SU(5) monopole to include the 5-plet of scalars responsible for the weak breaking. We solve numerically the corresponding Yang-Mills equations and calculate the classical monopole mass. The dependence on the weak scale is found to be small but non-vanishing.
We solve numerically the Yang-Mills equations corresponding to the fundamental monopole in SU(5) broken spontaneously by the usual 24-plet of scalars. We also calculate the classical monopole mass as a function of the scalar potential.
The static spherically symmetric self-dual SU(5) monopole solutions which correspond to stable monopoles outside of the Prasad-Sommerfield limit are constructed. These are the monopole solutions appropriate for perturbative calculations (of monopole masses, stability, proton decay and weak interaction catalysis, fermion-monopole bound states, etc.) about the PS limit. The symmetries of the monopoles as r-->0 are also discussed. My thanks are due to Alan Guth, Jeffrey Harvey, Nick Manton, and Henry Tye for valuable discussions. I am also grateful for the opportunity of attending the Seattle Institute on Phase Transitions in Gauge Theories at the University of Washington, where part of this work was completed.
I give a general, gauge-invariant, definition of spherical symmetry in a gauge theory. I show that if the fields are required to be non-singular at the origin, such symmetry can occur only for certain values of the magnetic charge. One consequence is the existence of stable monopoles which are not spherically symmetric.
The task of finding all spherically symmetric three-dimensional point monopoles in a gauge theory with arbitrary compact semisimple group is completely formulated in an Abelian gauge, where the problem is purely group-theoretical. The form of the gauge transformation to the spherically symmetric gauge is explicitly given in the general case. For SU(N) groups this result is reduced to a simple diagrammatic method which yields all such point monopoles by inspection. Also for SU(N) groups, a technique is given for efficient construction of the radial differential equations satisfied by the corresponding finite-energy solutions.
A solution-generating method for the self-duality and the Bogomolny equations is given. We point out the existence of an infinite-parameter invariance group of these equations.
The spherically symmetric monopoles and dyons of the SU(5) model of grand unification (without quarks and leptons) are discussed. It is shown that such monopoles and dyons can exist only in the sectors corresponding to magnetic charges m=±1/2e, ±1/e, ±3/2e, and ±2/e, where e is the charge of the positron. We investigate in detail the properties of the dyons with the smallest possible magnetic charge (|m|=1/2e). By semiclassical reasoning we show that apart from the magnetic charge the properties of the dyons are described by two quantum numbers n and k. The dyons come in families, denoted by n=0,1,2,…, with electric charge qn=n(-4e/3), baryon minus lepton number =n(-2/3), and the kth member of the nth family (k=0,1,2,…) transforms according to the (n+k,k) for n≥0 or the (k,|n|+k) for n<0 representation of SU(3)C. We argue that all the members of a given family are degenerate at the level we are working. This degeneracy is expected to be lifted in the full quantum theory, in which case each family collapses to one stable dyon, characterized by one integer n and whose quantum numbers are as follows: It has electric charge =n(-4e/3) and baryon number minus lepton number =n(-2/3), and it transforms under SU(3)C like the symmetric combination of n 3's for n≥0, or |n| 3̅ 's for n<0. Interesting processes involving monopoles and dyons are discussed; we show, for example, that the presence of a dyon strongly enhances baryon-number-violating processes. Finally, a less detailed discussion of poles with the other possible magnetic charges is included.
Properties of spherically symmetric monopoles are discussed. Inversion symmetry is also considered.
The Dirac magnetic monopole potential is a stable solution to the Abelian Maxwell equations. The simple generalization is a solution to the classical non-Abelian generalized Yang-Mills equations, where is a matrix in the Lie algebra of the gauge group. In this paper, the stability problem for these non-Abelian monopoles is posed and solved. Although is essentially Abelian in that , the stability analysis is non-trivial because it involves the full non-Abelian structure of the theory. It is first shown that the potential leads to a rotationally invariant classical theory only if the quantization condition is satisfied. (In contrast, the Abelian Dirac quantization condition is necessary only in quantum mechanics.) The stability analysis is performed by solving the linearized equations for the perturned potentials . Thus the existence of a solution which grows exponentially in time t is equivalent to the instability of . Using a convenient choice for the basis of , and the background gauge condition, the equation for the Fourier transform ψ(r,ω) of a is seen to be equivalent to the Schrödinger equation for a particle of unit spin, unit charge and unit anamolous magnetic moment moving in the potential 4πgβA. This equation is separated using the Wu-Yang monopole harmonics, generalized to include unit spin. The radial equation is then solved in terms of the eigenvalues of an operator related to the spin and orbital angular momentum operators. The result is that is stable if and only if each integer nβ is either 0 or ±1. The monopoles with are thus unstable and therefore have no quantum-mechanical significance. This conclusion is used to speculate about the empirical absence of monopoles, the stability of the non-singular ('t Hooft-Polyakov) monopoles, and the existence of magnetic confinement.
A new potential is presented which leads to the symmetry-breaking scheme of Georgi and Glashow. Using this potential the classical masses of the particles of the theory are calculated. The model contains monopoles. The vector boson masses and the masses of the monopoles satisfy , where v is the length of the Higgs field transforming according to the adjoint representation when the field is in the vacuum, y is the hypercharge of the particle, and g its magnetic charge. θw is the weak mixing angle. The symmetric monopoles of the theory are explicitly shown to obey a generalised Dirac quantisation condition.
The monopoles of the unified SU(5) gauge theory broken down to HE = SU(3)c ⊗ U(1)EM [or to KE = SU(3)c ⊗ SU(2) ⊗ U(1)Y], are classified. They belong to representations of a magnetic group HM(KM), which is found to be isomorphic to HE(KE). For SU(5) broken down to HE, there exists a regular and stable monopole which is a colour magnetic triplet, and carries a non-zero abelian magnetic charge. It is suggested that composite operators made out of this monopole and its antiparticle fields develop a non-zero vacuum expectation value, and so lead to a squeezing of the colour electric flux. Finally, we comment on the cosmological production of SU(5) monopoles.
Grand unified theories, in which the strong and electroweak interactions are embedded into an underlying theory with a single gauge coupling constant, are reviewed. A detailed description is given of many of the necessary background topics, including gauge theories, spontaneous symmetry breaking, the standard SU2 x U1 electroweak model and its modifications and extensions, Majorana and Dirac neutrino masses, the induced cosmological term, CP violation, quantum chromodynamics and its symmetries, and dynamical symmetry breaking. The Georgi-Glashow SU5 model is examined in detail. Models based on unitary, orthogonal, exceptional, and semi-simple groups and general constraints on model building are surveyed. Phenomenological aspects of grand unified theories are described, including the determination of the unification mass, the prediction of sin2 θw in various models, existing and planned nucleon decay experiments, the predictions for the proton lifetime and branching ratios, general baryon number violating interactions, and the possible explanation of the matter-antimatter asymmetry of the universe. Other aspects of grand unified theories are discussed, including horizontal symmetries, neutrino and fermion masses, topless models, asymptotic freedom, implications for the neutral current, CP violation, superheavy magnetic monopoles, dynamical symmetry breaking, and the hierarchy problem.
With the aid of Bäcklund transformations we construct exact multimonopole solutions of the axially and mirror symmetric Bogomolny equations. The explicit form of the length of the Higgs field is given and is studied both analytically and numerically. The energy density for monopoles with charges 2, 3, 5 is also calculated.
A new static, purely magnetic Yang-Mills-Higgs monopole solution is presented. It is axisymmetric and has a topological charge of 2; the charge is located at a single point.