Musings

It has been taken for granted that, whatever their microscopic origins, cosmic strings could be characterized by their string tension, $G\mu$, and their reconnection probability, $P$. Observationally, we need $G\mu\lessapprox 10^{-7}$.

In the Abelian Higgs model, $P=1$. The same is true is a surprisingly wide variety of weakly-coupled field theory generalizations. By contrast, $P\sim g_{st}^2$ for weakly-coupled fundamental strings and $P\lt 1$ for D-strings. For QCD flux strings, $P\sim 1/N_c^2$.

This difference in the reconnection probability is taken to be one of the hallmarks of string-theoretic cosmic strings. (The other being the fact that $(p,q)$-strings form 3-string junctions.) You might ask to what extent do we know that’s true and, more generally, to what extent do we understand the range of behaviours available in field-theoretic cosmic strings?

One can rather cheaply get $P\ll 1$, even in weakly-coupled field theory. Take $N$ identical decoupled Abelian Higgs models. There are then $N$ superficially indistiguishable types of strings. Strings of the same type reconnect with probability $1$; strings of different types pass right through each other. So, overall, the observable reconnection probability is $1/N$.

That’s sorta cheap, but, recently, Hashimoto and Tong considered a more interesting model. Let $\phi_i$ be $N_f$ scalars in the fundamental representation of $U(N_c)$. The Lagrangian $$L = \frac{1}{4e^2} Tr(F_{\mu\nu}^2) + \sum_{i=1}^{N_f} D_\mu \phi_i^\dagger D^\mu \phi_i - \frac{\lambda e^2}{2} Tr\left(\sum_{i=1}^{N_f} \phi_i\phi_i^\dagger -v^2 𝟙 \right)^2 + L_{\text{soft}}$$ where $L_{\text{soft}}$ explicitly breaks the $SU(N_f)$ global symmetry of the model. Neglecting $L_{\text{soft}}$, the theory is invariant under $U(N_c)\times SU(N_f)$. For $e^2 v^2 \gg \Lambda_{QCD}$, the theory is weakly coupled, and the symmetry is broken spontaneously. For $N_c=N_f\equiv N$, $$U(N_c)\times SU(N_f)\to SU(N)_{\text{diag}}$$ Since $\pi_1$ of the vacuum manifold is $\mathbb{Z}$, the low-energy theory has strings. But, unlike the Abelian Higgs model, the holonomy of the gauge connection around the string (or vortex in 2+1 dimensions) picks out a direction in the gauge group. Given a pair of strings, the relative orientation in the gauge group is physically significant.

According to Hashimoto and Tong, the strings will reconnect (with probability $P=1$), unless the relative orientation in the gauge group is such that the strings lie in commuting, mutually-orthogonal $U(1)$ subgroups of the gauge group.

That requires a fine-tuning. So, for generic initial conditions, the strings will reconnect.

Things change when we include the explicit symmetry-breaking term $$L_{\text{soft}}= \frac{1}{2}Tr(D_\mu \Phi D^\mu\Phi) - \sum_i \phi_i^\dagger (\Phi-m_i)^2 \phi_i - \frac{1}{2}m^2 Tr(\Phi^2)$$ where $\Phi$ is an adjoint-valued scalar field. Now, instead of a continuous family of strings, there are $N$ distinct ones, spanning the Cartan subalgebra. They all have the same tension, and so are indistinguishable. Indeed, at sufficiently low energies, it’s just like $N$ decoupled Abelian Higgs models, and $P=1/N$. But, at energies above the scale of explicit symmetry-breaking (but still low enough so that the moduli space approximation is valid), the behaviour returns to that of the model without $L_{\text{soft}}$, and $P=1$.

So we have an example of a velocity-dependent reconnection probability.

That’s very interesting. It ought to have a rather dramatic impact on the evolution of string networks. All the simulations assume that $P$ is just a constant, rather than being velocity-dependent. Now we have an effect which makes fast-moving strings reconnect more readily than slow-moving ones.