Supersymmetric Cosmic Strings

Supersymmetric Cosmic Strings Marcos Lima March 11, 2005 Abstract We study the formation of cosmic strings in the context of spontaneous symmetry breaking of gauge symmetry groups. We discuss the conditions for formation and stability of defects in nonsupersymmetric theories and also F-strings in the supersymmetry context. Finally, we discuss how cosmological observations can constrain supersymmetric grand unified theories. 1 Introduction In the modern view of cosmology, the universe underwent a series of phase transitions as it expanded and cooled down after the big bang. As a result of these transitions, most models of grand unified theories (GUT) predict the existence of topological defects. These defects have a large energy density and would be expected to have impact on the structure formation of the universe, gravitational lensing, CMB anisotropies, gravitational waves and cosmic rays, if they are formed after or at the late stages of inflation. On the other hand, inflationary models solve many problems of the standard model of cosmology and explain structure formation adn CMB anisotropies. If we believe that inflationary models are true then there is not much room left for topological defects. Since most GUT models predict the existence of these defects, we should be able to constrain these GUT models by allowing only those that produce defects that are consistent with the inflationary picture. For instance, domain walls, because they are heavy, and monopoles, because they are too abun- dant, would have a great gravitational effect, which is inconsistent with current observations. Therefore, we have to either rule out GUT models that predict the existence of monopoles and domain walls or ensure that these defects are formed before or in the early stages of inflation. In the latter case, the exponential expansion of inflation dilutes the defects’ energy density and they are not observed today. Cosmic strings are also predicted by most GUT models and, differently from domain walls and monopoles, are not inconsistent with the inflationary picture. They can actually help us 1 understand aspects of cosmology for which we have no explanation in view of the standard inflationary scenario. It is believed that supersymmetry (SUSY) is a general symmetry of nature, connecting fermions and bosons. But if this is true, SUSY must be broken since we do not observe super- partners for all the particles in the standard model of particle physics. Studying SUSY GUT theories in the context of inflation and the defects produced by these theories may tell us then about constraints in the physics at high energies, by constraining GUT models, and also about cosmology. This paper is organized in the following way. In section (2), we give an overview of topological defects and their formation. In section (3), we provide examples of formation of domain walls, monopoles and strings and discuss the conditions for the formation and topological stability of the defects in the context of quantum field theories. In section (4), we discuss cosmic strings in light of SUSY models, specializing to the F-theory. In section (5) the cosmological constraints of those models are analyzed and we conclude in section (6). 2 Topological defects overview During its first seconds, the universe had a series of phase transitions, which might have left important observational signatures, including topological defects [9]: • GUT transition : E ∼ 1015 GeV. Bellow this energy, the gauge group of particle physics degenerates from the grand unified G to the usual standard model SU(3) ⊗ SU(2) ⊗ U(1) by the Higgs mechanism. • Electroweak transition: E ' 300 GeV. Bellow this energy, the Higgs mechanism breaks the SU(2) ⊗ U(1) part of the standard model to yield distinct electromagnetic and weak interactions. • Quark-hadron transition: E ' 0.2 GeV. This transition is related to quark confine- ment. During these phase transitions, it is thought that defects are formed by the Kibble mechanism [7]. Suppose we have a N-component real scalar field φ, with a Lagrangian invariant under O(N) and coupled to N(N − 1)/2 vector fields Aµ 1 1 L = (D φ)2 − F F µν − V (φ) (1) 2 µ 4 µν eff where 2 Dµφ = ∂µφ − eAµφ (2) µν F = ∂µAν − ∂νAµ + e[Aµ,Aν] (3) At zero temperature, the effective potential is simply 1 V (φ) = V (φ) = g2(φ2 − η2)2 (4) eff 2 In this case the O(N) symmetry is spontaneously broken to O(N − 1), and φ acquires a vacuum expectation value η, i.e. in tree approximation < φ >2= η2 (5) Notice that there is a manifold M of degenerate vacua solutions, in this case a circle of radius η. At a finite temperature T , the leading temperature dependence of the effective potential at high T and small coupling constant is calculated from one-loop diagrams [14] 1 1 V (φ) = g2(φ2 − η2)2 + [(N + 2)g2 + 6(N − 1)e2]T 2φ2 (6) eff 2 48 In this case, the minimum occurs at φ = 0, and the symmetry is unbroken for T larger than the critical temperature N + 2 N − 1 e2 !−1/2 T = η + (7) c 12 2 g2 Thus, the Higgs field φ starts at high temperatures with a zero expectation value and when the symmetry breaks bellow Tc, it acquires an expectation value rolling down to one of the possible vacua. Since there are no correlations on distances greater than the horizon scale, different points in space will choose different vacua in the manifold M. As these regions of different vacua come into causal contact, topological defects might be formed. The conditions for topological defects formation are topological constraints of the manifold M (see section (3)). Defect formation depends then on the structure of the manifold M, and ultimately on the symmetry that is being broken. In our O(N) case in 3 spatial dimensions, different values of N will lead to different defects, namely: 3 • (N = 1) Domain walls: Two-dimensional defects dividing space into connected regions which are in one of the two possible phases. • (N = 2) Strings: Linear defects, where the phase of φ changes by 2π in making one loop around the string. • (N = 3) Monopoles: Point defects where the field φ points radially away from the defect. • (N = 4) Textures: Here it is easier to think of the case of only 2 spatial dimensions. Then textures correspond to field configurations in which φ makes a rotation of π in its 3-dimensional internal space in going from r = 0 to r = ∞. 3 Defect formation and stability The precise condition on defect formation rests on the topological structure of the manifold of degenerate vacua M. Let us see some examples before we define these conditions. Example 1 - Domain Walls Suppose we have a real scalar field φ with a Lagrangian invariant under the reflection φ → −φ 1 L = (∇φ)2 − V (φ) (8) 2 1 2 2 2 2 Here the zero temperature potential is again V (φ) = 2 g (φ − η ) . The equilibrium states are φ = φ± = ±η. Therefore the manifold of degenerate vacua M is a disconnected set of two points in this case. When different regions come into contact, the set of points in which the field vanishes form a two-dimensional surface that separates the two phases and constitute the domain wall. Example 2 - Monopoles Suppose we have a scalar field with internal space φi, i = 1, 2, 3. The gauge group to be i broken is SO(3) and there are three gauge fields Aµ, i = 1, 2, 3. The Lagrangian is 1 1 L = DµφiD φi − F i F iµν − V (φ) (9) 2 µ 4 µν where the covariant derivatives and tensors are i i ijk j k Dµφ = ∂µφ − e Aµφ (10) i i i ijk j k Fµν = ∂µAν − ∂νAµ − e AµAν (11) 4 Suppose the potential V (φ) is arranged to give symmetry breaking with |φ| = v, along some arbitrary direction, which at a given point we can choose to be the 3-axis of the scalar field’s internal space. After some time, the scalar field will align itself in such a way that at all points of space the fields will all be parallel in their internal space. There will exist then an internal U(1) symmetry of rotations about the Higgs axis that remained unbroken, 3 and the corresponding gauge field Aµ would be the electromagnetic 4-potential. This model is not realistic, but it provides an example of a model that yields the U(1) of electromagnetism after symmetry breaking. We shall see now that we actually have a monopole solution. It is important the definition of the covariant derivative here, because the extra piece in it provides a cancellation of terms that otherwise would produce a divergent energy. Consider now the time-independent solution r φi = v i (12) r r Aj = k (13) i ijk er2 where r denotes real radial position and ri the internal coordinate of the scalar field. This solution is clearly a topological defect in the sense that there is a winding of the Higgs as one performs circles about the origin. Now, at large radii, the Higgs field does become uniform , and we suppose it becomes uniform in the direction of third axis. The magnetic field associated 1 3 j i ri is then given by Bi = 2 ijkFjk. Inserting the solution for Ai we obtain B = er3 [9]. Example 3 - Strings a) Global Suppose we have a complex scalar field φ(x) and a Lagrangian with a global U(1) symmetry φ → φeiα, with α constant ∗ µ L = ∂µφ ∂ φ − V (φ) (14) 1 1 2 V (φ) = λ |φ|2 − η2 (15) 2 2 The equation of motion can be obtained from the Lagrangian Eq.(14) 1 ∂2φ + λφ(|φ|2 − η2) = 0 (16) 2 The vacuum solution (ground state) is the one that solves√ the equation of motion and also minimizes the scalar potential V (φ).

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