Standard Model and Beyond · Problem 4.1

The Strong Interactions and QCD

Physics — Particle Physics · cosmic-strings-higgs-mechanism

Problem

The spontaneous breaking of a global or local U(1)U(1) symmetry allows one-dimensional cosmic string classical solutions, analogous to the two-dimensional domain walls considered in Section 3.3.2. Consider the complex scalar ϕ\phi in a U(1)U(1) gauge theory, with L=−14FμνFμν+(Dμϕ)†Dμϕ−V(ϕ),V(ϕ)=μ2ϕ†ϕ+λ(ϕ†ϕ)2,\mathcal{L} = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu} + (D_\mu\phi)^\dagger D^\mu\phi - V(\phi), \quad V(\phi) = \mu^2\phi^\dagger\phi + \lambda\left(\phi^\dagger\phi\right)^2, where Dμϕ=(∂μ+igAμ)ϕD^\mu\phi = (\partial^\mu + igA^\mu)\phi, FμνF_{\mu\nu} is the field strength tensor, λ>0\lambda > 0, and μ2<0\mu^2 < 0. It is convenient to rewrite V(ϕ)=λ(ϕ†ϕ−12ν2)2,ν=−μ2λ,V(\phi) = \lambda\left(\phi^\dagger\phi - \frac{1}{2}\nu^2\right)^2, \quad \nu = \sqrt{\frac{-\mu^2}{\lambda}}, where we have dropped the irrelevant constant −λν4/4-\lambda\nu^4/4. It was shown (Nielsen and Olesen, 1973) that there is a classical solution for ϕ(x)\phi(x) and Aμ(x)A^\mu(x) corresponding to a vortex or string running along the zz axis from −∞-\infty to +∞+\infty. In cylindrical coordinates (r,θ,z)(r, \theta, z) the solution exhibits ϕ→r→∞ν2e−inθ,\phi \xrightarrow{r\to\infty} \frac{\nu}{\sqrt{2}}e^{-in\theta}, so that V→0V \to 0 for r→∞r \to \infty. Single-valuedness requires that nn is an integer. Find the asymptotic expression for the vector potential AμA^\mu for the string solution, for which Fμν→0F^{\mu\nu} \to 0, Dμϕ→0D^\mu\phi \to 0, and calculate the magnetic flux ∫B⃗⋅dS⃗=∫(∇⃗×A⃗)⋅dS⃗\int \vec{B} \cdot d\vec{S} = \int\left(\vec{\nabla} \times \vec{A}\right) \cdot d\vec{S} for the solution.

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