Magnetic Monopoles As a New Solution to Strong CP Problem

A non-perturbative solution to strong CP problem is proposed. It is shown that the gauge orbit space with gauge potentials and gauge tranformations restricted on the space boundary in non-abelian gauge theories with a θ term has a magnetic monopole structure if there is a magnetic monopole in the ordinary space. The Dirac’s quantization condition in the corresponding quantum theories ensures that the vacuum angle θ in the gauge theories must be quantized. The quantization rule is derived as θ = 2π/n (n 6= 0) with n being the topological charge of the magnetic monopole. Therefore, we conclude that the strong CP problem is automatically solved non-perturbatively with the existence of a magnetic monopole of charge ±1 with θ = ±2π. This is also true when the total magnetic charge of monopoles are very large (|n| ≥ 102π) if it is consistent with the abundance of magnetic monopoles. This implies that the fact that the strong CP violation can be only so small or vanishing may be a signal for the existence of magnetic monopoles. ∗This work was supported by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of High Energy Physics of the U.S. Department of Energy under Contract DE-AC03-76SF00098. Since the discovery of Yang-Mills theories, the non-perturbative effects of gauge theories have played one of the most important roles in particle physics. It is known that, in non-abelian gauge theories a Pontryagin or θ term, Lθ = θ 32π ǫF a μνF a λσ, (1) can be added to the Lagrangian density of the system due to instanton effects in gauge theories. The θ term can induce CP violations. An interesting fact is that the θ angle in QCD can be only very small (θ ≤10) or vanishing. Where in our discussions of QCD, θ is used to denote θ + arg(detM) effectively with M being the quark mass matrix, when the effects of electroweak interactions are included. One of the most interesting understanding of the strong CP problem has been the assumption of an additional Peccei-Quinn U(1)PQ symmetry , but the observation has not given evidence for the axions needed in this approach. Thus the other possible solutions to this problem are of fundamental interest. In this paper, we will extend the method of Wu and Zee for the discussions of the effects of the Pontryagin term in pure Yang-Mills theories in the gauge orbit spaces in the Schrodinger formulation. This formalism has also been used with different methods to derive the mass parameter quantization in three-dimensional Yang-Mills theory with Chern-Simons term. Wu and Zee showed that the Pontryagin term induces an abelian background field or an abelian structure in the gauge configuration space of the Yang-Mills theory. In our discussions, we will consider the case with the existence of a magnetic monopole. We will show that magnetic monopoles in space will induce an abelian gauge field with non-vanishing field strength in gauge configuration space, and magnetic flux through a two-dimensional sphere in the induced gauge orbit space is non-vanishing. Then, Dirac condition in the corresponding quantum theories leads to the result that the relevant vacuum angle θ must be quantized as θ = 2π/n with n being the topological charge of the monopole 1 to be generally defined. Therefore, the strong CP problem can be solved with the existence of magnetic monopoles. To the knowledge of the author, such an interesting result has never been given before in the literature. We will now consider the Yang-Mills theory with the existence of a magnetic monopole at the origin. As we will see that an interesting feature in our derivation is that we will use the Dirac quantization condition both in the ordinary space and restricted gauge orbit space to be defined. The Lagrangian of the system is given by L = ∫ dx{− 1 4 F a μνF aμν + θ 32π ǫF a μνF }. (2) We will use the Schrodinger formulation and the Weyl gauge A = 0. The conjugate momentum corresponding to Ai is given by π i = δL δȦi = Ȧai + θ 8π ǫijkF a jk. (3) In the Schrodinger formulation, the system is similar to the quantum system of a particle with the coordinate qi moving in a gauge field Ai(q) with the correspondence qi(t) → A a i (x, t), (4) Ai(q) → A a i (A(x)), (5) where Aai (A(x)) = θ 8π ǫijkF a jk. (6) Thus there is a gauge structure with gauge potential A in this formalism within a gauge theory with the θ term included. Note that in our discussion with the presence of a magnetic monopole, the gauge potential A outside the monopole generally need to be understood as well defined in each local coordinate region. 2 In the overlapping regions, the separate gauge potentials can only differ by a welldefined gauge transformation. In fact, single-valuedness of the gauge function corresponds to the Dirac quantization condition. For a given r, we can choose two extended semi-spheres around the monopole, with θ ∈ [π/2−δ, π/2+δ](0 < δ < π/2) in the overlapping region, where the θ denotes the θ angle in the spherical polar coordinates. For convenience, we will use differential forms in our discussions, where A = Aidx , F = 1 2 Fjkdx dx, with F = dA + A locally. For our purpose to discuss about the effects of the abelian gauge structure on the quantization of the vacuum angle, we will now briefly clarify the relevant topological results needed, then we will realize the topological results explicitly. With the constraint of Gauss’ law, the quantum theory in this formalism is described in the gauge orbit space U/G which is a quotient space of the gauge configuration space U with gauge potentials connected by gauge transformations in each local coordinate region regarded as equivalent. Where G denotes the space of continuous gauge transformations, and each gauge potential as an element in U may be defined up to a gauge transformation in the overlapping regions. Now consider the following exact homotopy sequence: ΠN (U) P∗ −→ ΠN(U/G) ∆∗ −→ ΠN−1(G) i∗ −→ ΠN−1(U) (N ≥ 1). (7) Note that homotopy theory has also been used to study the global gauge anomalies , especially by using extensively the exact homotopy sequences and in terms of James numbers of Stiefel manifolds14−19. One can easily see that U is topologically trivial, thus ΠN (U) = 0 for any N. Since the interpolation between any two gauge potentials A1 and A2 At = tA1 + (1− t)A2 (8) for any real t is in U (Theorem 7 in Ref.9, and Ref.6). since At is transformed as a gauge potential in each local coordinate region, and in an overlapping 3 region, both A1 and A2 are gauge potentials may be defined up to a gauge transformation, then At is a gauge potential which may be defined up to a gauge transformation, namely, At ∈ U . Thus, we have 0 P∗ −→ ΠN(U/G) ∆∗ −→ ΠN−1(G) i∗ −→ 0 (N ≥ 1). (9) This implies that ΠN (U/G) ∼= ΠN−1(G) (N ≥ 1). (10) As we will show that in the presence of a magnetic monopole, the topological properties of the system are drastically different. This will give important consequences in the quantum theory. Actually, it is interesting to note more generally that the topological results in Eq.(9-10) are true if U and G are the corresponding induced spaces with A and g restricted to certain region of the ordinary space, especially the 2-sphere S as the space boundary since the restricted gauge configuration space U is topologically trivial. This is in fact the the relevant case in our discussion, since only the integrals on the space boundary S are relevant in the quantization equation as we will see. We will call the induced spaces of U , G and U/G when A and g are restricted on the space boundary S as restricted gauge configuration space, restricted gauge orbit space and restricted gauge transformation space respectively, and restricted spaces collectively. Now for the restricted spaces, the main topological result we will use is given by Π2(U/G) ∼= Π1(G), (11) The condition Π2(U/G) 6= 0 corresponds to the existence of a magnetic monopole in the restricted gauge orbit space. In the usual unrestricted case based on the whole compactified space M as that for pure Yang-Mills theory, there can not be monopole structure constructed. We will first show that in this case F 6 = 0, and then demonstrate explicitly that the magnetic flux ∫ S F̂ 6 = 0 can be 4 nonvanishing in the restricted gauge orbit space, where F̂ denotes the projection of F into the restricted gauge orbit space. Denote the differentiation with respect to space variable x by d, and the differentiation with respect to parameters {ti | i = 1, 2...} which A(x) may depend on in the gauge configuration space by δ, and assume dδ+ δd=0. Then, similar to A = Aμdx μ with μ replaced by a, i, x, A = AiL dx, F = 1 2 F a jkL dxdx and tr(LL) = − 2 δ for a basis {L | a = 1, 2, ..., rank(G)} of the Lie algebra of the gauge group G, the gauge potential in the gauge configuration space is given by A = ∫ d3xAai (A(x))δA a i (x). (12) Using Eq.(6), this gives A = θ 8π ∫ d3xǫijkF a jk(x)δA a i (x) = − θ 2π ∫