Magnetic monopole solutions with a massive dilaton

Static, spherically symmetric monopole solutions of a spontaneously broken SU(2) gauge theory coupled to a massive dilaton field are studied in detail in function of the dilaton coupling strength and of the dilaton mass. corresponding author E-mail: [email protected] 1 In this paper we present some results on finite energy solutions in an SU(2) Yang-Mills (YM) and YM-Higgs (YMH) theory (with the Higgs field in the adjoint representation) coupled to a massive dilaton field. The present work is an extension of previous investigations with a massless dilaton field [1, 2, 4]. In Refs. [1, 2] it has been found that the SU(2) dilaton-YM (DYM) theory admits static, finite energy (‘particle-like’) solutions (absent in the pure YM case). They are in close analogy to the particle-like solutions found by Bartnik and McKinnon in an Einstein-YM (EYM) system [3]. In Ref. [4] it has been shown that in the SU(2) dilaton-YMH (DYMH) theory in addition to the analogue of the (nonabelian) ‘t Hooft-Polyakov monopole [5] there is a discrete family of finite energy solutions, which can be interpreted as radial excitations of the monopole. The mass scale of the radial excitations is inversely proportional to the dilaton coupling. In Ref. [4] it has also been found that there is a maximal dilaton coupling, αmax, above which only an abelian solution exists. Although the abelian solution (which has a simple analytical form) is singular at the origin, its total energy is finite. The numerical results of Ref. [4] indicate that the nonabelian monopole merges with the abelian one for a critical value of the dilaton coupling, α. The critical dilaton coupling, αcrit, depends on the value of the Higgs self-coupling strength, β. As found in Ref. [4] when β is sufficiently small the largest possible value of the dilaton coupling, αmax, for which a nonabelian solution still exists is different from the critical value, αcrit, i.e. we have αmax > αcrit. This implies that if α ∈ [αcrit, αmax] a bifurcation takes place and there are two different monopole solutions for the same value of α. All these findings are again in close analogy with the results of Refs. [6, 7] found in the EYMH case. A massless dilaton, which necessarily appears in string theories [8, 9, 10], violates the equivalence principle, and therefore its (dimensionful) coupling strength is expected to be extremely weak, of the order of 1/MPl where MPl is the Planck mass. It is very natural to assume, however, that the dilaton gets a mass (possibly related to supersymmetry breaking) then, however, there is no strong experimental constraint on the dilaton coupling. Therefore it might be of some importance to study the effect of mass of the dilaton in DYMH theories. Regular and black hole solutions in EYM theories coupled to a massive dilaton and axion have investigated in Ref. [11]. We have carried out a rather thorough numerical investigation and we have found that even if the dilaton is massive, most phenoma associated with the presence of the dilaton found in the massless case persist (the existence of radial excitations, αmax 6= αcrit). Also we have good numerical evidence that αcrit (where the solution become abelian) is independent of the dilaton mass. We have also investigated in detail the limit m → ∞ of the mass of the dilaton and we were able to show its expected decoupling. Our numerical results show that αmax grows as m increases, consistently with the expectation that for m → ∞ αmax(m) → ∞. We have also found that not just a single maximal dilaton coupling, αmax exists, but there are several local extrema of α too, e.g. αmax1 > αmax2 > αcrit > αmin1 . These bifurcation points are more clearly distinguishable as the dilaton mass becomes larger. This implies that there are certain values of the dilaton coupling, α, where three or even more different monopole solutions exists for the same value of α with different masses. By using the minimal spherically symmetric and static ansatz A0 = 0 , A a i = ǫaik 1−W (r) r xk r , Φ = H(r) xa r , φ = φ(r) , (1) where Aμ is the gauge potential, Φ a is the Higgs triplet and φ is the dilaton field, the reduced energy functional reads: