Abstract
We propose a method to reconstruct the Bethe–Salpeter amplitude in Minkowski space given the Euclidean Bethe–Salpeter amplitude – or alternatively the light-front wave function – as input. The method is based on the numerical inversion of the Nakanishi integral representation and computing the corresponding weight function. This inversion procedure is, in general, rather unstable, and we propose several ways to considerably reduce the instabilities. In terms of the Nakanishi weight function, one can easily compute the BS amplitude, the LF wave function and the electromagnetic form factor. The latter ones are very stable in spite of residual instabilities in the weight function. This procedure allows both, to continue the Euclidean BS solution in the Minkowski space and to obtain a BS amplitude from a LF wave function.
Highlights
Among the methods in quantum field theory and quantum mechanical approaches to relativistic few-body systems, the Bethe–Salpeter (BS) equation [1], based on first principles and existing already for more than 60 years, remains rather popular.Other methods, based on the appropriate treatment of the singularities in the BS equation, provide the Minkowski solution directly, both for bound and scattering states [12].The straightforward numerical extrapolation of the solution from Euclidean to Minkowski space is very unstable and has not achieved any significant progress of practical interest
Since the LF wave function can be found independently either by solving the corresponding equation [13] or by means of quantum field theory (QFT) inspired approaches like the discrete light cone quantization [15] or the basis LF quantization approach [16], one can pose the problem of finding the Nakanishi weight function from Eq (3)
We find the Minkowski BS amplitude M and using it, we calculate observables, namely, EM form factor and momentum distribution, represented by the LF wave function
Summary
Among the methods in quantum field theory and quantum mechanical approaches to relativistic few-body systems, the Bethe–Salpeter (BS) equation [1], based on first principles and existing already for more than 60 years, remains rather popular. The above mentioned Nakanishi integral representation provides a more reliable method It is valid for the Euclidean E as well as for the Minkowski M solutions and both solutions are expressed via one and the same Nakanishi weight function g. Solving the integral equation (1) relative to g and substituting the result into Eq (2), one can in principle determine the Minkowski amplitude M starting with the Euclidean one E. This strategy seems applicable since, in contrast to the direct extrapolation E → M , it uses the analytical
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
