Open effective field theories from deeply inelastic reactions

Effective field theories have often been applied to systems with deeply inelastic reactions that produce particles with large momenta outside the domain of validity of the effective theory. The effects of the deeply inelastic reactions have been taken into account in previous work by adding local anti-Hermitian terms to the effective Hamiltonian. Here, we show that when multiparticle systems are considered, an additional modification is required in equations governing the density matrix. We define an effective density matrix by tracing over the states containing high-momentum particles and show that it satisfies a Lindblad equation, with local Lindblad operators determined by the anti-Hermitian terms in the effective Hamiltonian density.

Figure 1

In the amplitude for muon decay, the $W$ can propagate only over short distances. Its exchange can therefore be approximated by a contact interaction.

Figure 2

In the amplitude for ${\mu }^{-}\to {\nu }_{\mu }{e}^{-}{\overline{\nu }}_e\to {\mu }^{-}$, the high-momentum intermediate-state particles are created and annihilated in a localized region. Their effects can therefore be reproduced by local operators.

Figure 3

In the amplitude for ${\mu }^{-}{e}^{+}\to {\nu }_{\mu }{\overline{\nu }}_e\to {\mu }^{-}{e}^{+}$, the high-momentum intermediate-state neutrinos are created and annihilated in a localized region. Their effects can therefore be reproduced by local operators.

Figure 4

Self-energy diagram of order $g^2$ for a $\psi$ particle. The propagators for $\psi$ and $\varphi$ are represented by solid lines and dashed lines, respectively.

Figure 5

The operator $\frac{1}{2}g{\tilde{\varphi }}^2(\mathit{r},t)$ can be replaced by $\tilde{\mathrm{\Pi }}(M,0)\psi (\mathit{r},t)$ in a nonrelativistic correlator.

Figure 6

Diagram of order $g^2$ for $\psi \psi \to \psi \psi$ through an intermediate state with two $\varphi$ particles.