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Matrix algorithms for solving (in)homogeneous bound state equations.

Blank M, Krassnigg A - Comput Phys Commun (2011)

Bottom Line: 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.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).This is valuable insight, in particular for the study of baryons in a three-quark setup and more involved systems.

View Article: PubMed Central - PubMed

Affiliation: Institut für Physik, Universität Graz, Universitätsplatz 5, 8010 Graz, Austria.

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.

No MeSH data available.


Related in: MedlinePlus

The homogeneous equation for a three-body bound state (covariant Faddeev equation), Eq. (5).
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fg0040: The homogeneous equation for a three-body bound state (covariant Faddeev equation), Eq. (5).

Mentions: For Baryons, an analogous construction can be made, where the homogeneous equation for the on-shell amplitude is a covariant three-quark equation often referred to as a covariant Faddeev equation [42,70,71], which may be written as(5)Γ[3h](k1,k2,P)=∫q1,q2Sa(p1)Sb(p2)Sc(p3)K[3](k1,k2,q1,q2,P)Γ[3h](q1,q2,P), and a pictorial representation is given in Fig. 4. Here, the kernel subsumes all interactions of the three quarks with the individual momenta , , and the bound state is described by the covariant three-quark on-shell amplitude , which depends on the total momentum P as well as two relative (Jacobi) momenta and . Note that this equation contains an integral over two momenta, namely and , thus inflating the size of the problem in terms of a numerical setup.


Matrix algorithms for solving (in)homogeneous bound state equations.

Blank M, Krassnigg A - Comput Phys Commun (2011)

The homogeneous equation for a three-body bound state (covariant Faddeev equation), Eq. (5).
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC3095096&req=5

fg0040: The homogeneous equation for a three-body bound state (covariant Faddeev equation), Eq. (5).
Mentions: For Baryons, an analogous construction can be made, where the homogeneous equation for the on-shell amplitude is a covariant three-quark equation often referred to as a covariant Faddeev equation [42,70,71], which may be written as(5)Γ[3h](k1,k2,P)=∫q1,q2Sa(p1)Sb(p2)Sc(p3)K[3](k1,k2,q1,q2,P)Γ[3h](q1,q2,P), and a pictorial representation is given in Fig. 4. Here, the kernel subsumes all interactions of the three quarks with the individual momenta , , and the bound state is described by the covariant three-quark on-shell amplitude , which depends on the total momentum P as well as two relative (Jacobi) momenta and . Note that this equation contains an integral over two momenta, namely and , thus inflating the size of the problem in terms of a numerical setup.

Bottom Line: 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.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).This is valuable insight, in particular for the study of baryons in a three-quark setup and more involved systems.

View Article: PubMed Central - PubMed

Affiliation: Institut für Physik, Universität Graz, Universitätsplatz 5, 8010 Graz, Austria.

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.

No MeSH data available.


Related in: MedlinePlus