Basis light-front quantization approach to positronium

We present the first application of the recently developed basis light-front quantization (BLFQ) method to self-bound systems in quantum field theory, using the positronium system as a test case. Within the BLFQ framework, we develop a two-body effective interaction, operating only in the lowest Fock sector, that implements photon exchange, neglecting fermion self-energy effects. We then solve for the mass spectrum of this interaction at the unphysical coupling $\alpha =0.3$. The resulting spectrum is in good agreement with the expected Bohr spectrum of nonrelativistic quantum mechanics. We examine in detail the dependence of the results on the regulators of the theory.

Figure 1

Representative spectrum of positronium ($\alpha =0.3$) calculated in BLFQ at $K=N_{\max }=19$ and $\mu =0.1m_f$, using the unregulated (top panel) and regulated (bottom panel) effective interactions. The exact energies shown should not be interpreted as the final converged results. Using the unregulated interaction (top panel), the approximate rotational invariance allows for the clear identification of the $1{}^1{S}_0$, $1{}^3{S}_1$, $2{}^1{S}_0$, $2{}^3{S}_1$, $2{}^1{P}_1$, $2{}^3{P}_0$, $2{}^3{P}_1$ and $2{}^3{P}_2$ states of the positronium system (see text for details). Using the regulated interaction (bottom panel), the approximate rotational invariance is more strongly broken.

Figure 2

Singlet and triplet ground states of positronium ($\alpha =0.3$) as a function of $1/N_{\max }$ for indicated values of $K$ and $\mu$. The curves are second-order polynomial fits of the $K=55$ results used to extrapolate the curves to the limit $N_{\max }\to \infty$ ($1/N_{\max }\to 0$). The basis energy scale $b=0.4m_f$ is chosen to optimize the $N_{\max }$ convergence rate for these states.

Figure 3

The $2{}^3{P}_2$ state of positronium ($\alpha =0.3$) as a function of $1/N_{\max }$ for indicated values of $K$ and $\mu$ (values of $K$ increase in increments of 10). The discrete points represent the lowest mass eigenstate obtained in calculations with fixed total magnetic projection $M_J=2$. The curves are second-order polynomial fits used to extrapolate the curves to the limit $N_{\max }\to \infty$ ($1/N_{\max }\to 0$). The basis energy scale $b=0.1m_f$ is chosen to optimize the $N_{\max }$ convergence rate of the $2{}^3{P}_2$ state binding energy. We include higher values of $K$ as $\mu$ is decreased because convergence with $K$ becomes slower. For $\mu \ge 0.04m_f$, the energy converges to $2m_f$ (the threshold for binding) in the continuum limit, indicating that the state is not bound at these high values of $\mu$.

Figure 4

Dependence of the $1{}^1{S}_0$, $1{}^3{S}_1$ and $2{}^3{P}_2$ states on the fictitious photon mass $\mu$. The calculation is performed at a large, unphysical coupling of $\alpha =0.3$. The plotted blue crosses are the results of the $N_{\max }\to \infty$ extrapolations (at fixed $K$) performed in Figs. 2 and 3. The fixed $K$ values chosen for the $1/N_{\max }$ extrapolations are $K=55$ for $\mu =0.06–0.10m_f$, $K=65$ for $\mu =0.04–0.05m_f$, $K=75$ for $\mu =0.03m_f$, $K=85$ for $\mu =0.02m_f$ and $K=95$ for $\mu =0.01m_f$. These values of $K$ are chosen to be sufficiently high such that the results can be considered converged with respect to $K$. For comparison, the results found using a simple $K\to \infty$ extrapolation (see text) are shown as solid red circles. The curves are second-order polynomial fits, used to extrapolate to the physical limit $\mu \to 0$. The crosses on the energy axis represent the predictions of nonrelativistic quantum mechanics, including first-order perturbative relativistic corrections, as discussed in the text.

Figure 5

Basic Hamiltonian interaction vertices $P_{e\to e\gamma }^{-}$ (left panel) and $P_{\overline{e}\gamma \to \overline{e}}^{-}$ (right panel).

Figure 6

Iterated interactions generated in the two-body effective interaction. The photon-exchange diagrams correspond to $(H_{\mathrm{eff}}^{\prime }{)}_{e\to e\gamma }$ (left panel) and $(H_{\mathrm{eff}}^{\prime }{)}_{\overline{e}\to \overline{e}\gamma }$ (right panel). The fermion-self-energy contributions are neglected in $H_{\mathrm{eff}}^{\prime }$.

Figure 7

Instantaneous photon-exchange interaction, $P_{\text{inst}}^{-}$. The tick mark on the photon denotes instantaneous exchange (in light-front time).