Matrix algorithms for solving (in)homogeneous bound state equations

Abstract

In the functional approach to quantum chromodynamics, the properties of hadronic bound states are accessible via covariant integral equations, e.g. the Bethe–Salpeter equation for mesons. In particular, one has to deal with linear, homogeneous integral equations which, in sophisticated model setups, use numerical representations of the solutions of other integral equations as part of their input. Analogously, inhomogeneous equations can be constructed to obtain off-shell information in addition to bound-state masses and other properties obtained from the covariant analogue to a wave function of the bound state. These can be solved very efficiently using well-known matrix algorithms for eigenvalues (in the homogeneous case) and the solution of linear systems (in the inhomogeneous case). We demonstrate this by solving the homogeneous and inhomogeneous Bethe–Salpeter equations and find, e.g. that for the calculation of the mass spectrum it is as efficient or even advantageous to use the inhomogeneous equation as compared to the homogeneous. This is valuable insight, in particular for the study of baryons in a three-quark setup and more involved systems.

Charming quasi-exotic open-flavor mesons

“…On the other hand, since the RL truncated BSE does not include hadronic decay channels, which one would expect when finding a decay width of the state, it is not clear how to interpret such a scenario. Note that an iterative procedure in finding the eigenvalues [83] cannot produce complex conjugate eigenvalue pairs correctly; more sophisticated methods are needed to obtain correct results in this case [32,52].…”

“…Within this approach, the nonperturbatively dressed quark propagator is obtained as solution to the quark Dyson-Schwinger equation (DSE) and serves as an input for the quark-antiquark meson Bethe-Salpeter equation (BSE) [31,32]. The RL truncation provides a reliable meson phenomenology for heavy-quark mesons [33][34][35][36].…”

Exciting flavored bound states

“…(25) by means of Arnoldi factorization implemented in the ARPACK library [65,66] which computes the eigenvalue spectrum for a given N × N matrix. Practical implementation implies a transcription of the BSE kernel K αβ (p, k, P ) in Eq.…”

Light-quarkonium spectra and orbital-angular-momentum decomposition in a Bethe–Salpeter-equation approach

“…For the cases where our calculations did not produce a numerically convincing result, "empty grid points" may appear in the plot. This is the result of some of the numerical techniques used in order to arrive at our computed results; details of the numerical setup and difficulties as well as solution strategies can be found in [72,[75][76][77]85,86]. Error bars corresponding to our numerical uncertainties for our results are generated by our calculations automatically.…”