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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

Momentum flow of the loop-diagram present in Eqs. (1) and (2).
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fg0020: Momentum flow of the loop-diagram present in Eqs. (1) and (2).

Mentions: In order to study mesons as bound states of quarks, anti-quarks and gluons in the DSE approach to QCD, one can consider general vertices connecting (anti-)quarks to objects carrying the appropriate quantum numbers as demanded by the respective superselection rules. These vertices are the so-called (inhomogeneous) Bethe–Salpeter amplitudes (BSAs), denoted by , which describe a two-particle system, denoted by the subscript [2], with total momentum P and relative momentum k of the constituents. The inhomogeneous BSA satisfies the inhomogeneous (vertex) BSE,(1)Γ[2](k,P)=Γ0(k,P)+∫qK[2](k,q,P)Sa(q+)Γ[2](q,P)Sb(q−). Here is a renormalized current (cf. [68]) with the quantum numbers of the system which acts as a driving term, and denote the renormalized dressed (anti-)quark propagators. As opposed to bare propagators, they are obtained as solutions of the quark DSE, Eq. (6). Together with the renormalized quark–anti-quark scattering kernel the propagators provide the input necessary to solve the BSE. This equation is formulated in Euclidean space, and the four-dimensional momentum integration (including a translationally invariant regularization) is given by . A graphical representation of Eq. (1) is given in Fig. 1, where the arrows denote dressed-quark propagators (analogously in Figs. 3 and 4). The momentum flow (detailed in Fig. 2) is defined such that the total momentum P is given by the difference of the (anti-)quark momenta . represents the momentum partitioning parameters, which satisfy . The relative momentum q of the BSA is therefore given by . On the left-hand side of Eqs. (1) and (2) the relative momentum is denoted by k.


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

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

Momentum flow of the loop-diagram present in Eqs. (1) and (2).
© Copyright Policy
Related In: Results  -  Collection

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

fg0020: Momentum flow of the loop-diagram present in Eqs. (1) and (2).
Mentions: In order to study mesons as bound states of quarks, anti-quarks and gluons in the DSE approach to QCD, one can consider general vertices connecting (anti-)quarks to objects carrying the appropriate quantum numbers as demanded by the respective superselection rules. These vertices are the so-called (inhomogeneous) Bethe–Salpeter amplitudes (BSAs), denoted by , which describe a two-particle system, denoted by the subscript [2], with total momentum P and relative momentum k of the constituents. The inhomogeneous BSA satisfies the inhomogeneous (vertex) BSE,(1)Γ[2](k,P)=Γ0(k,P)+∫qK[2](k,q,P)Sa(q+)Γ[2](q,P)Sb(q−). Here is a renormalized current (cf. [68]) with the quantum numbers of the system which acts as a driving term, and denote the renormalized dressed (anti-)quark propagators. As opposed to bare propagators, they are obtained as solutions of the quark DSE, Eq. (6). Together with the renormalized quark–anti-quark scattering kernel the propagators provide the input necessary to solve the BSE. This equation is formulated in Euclidean space, and the four-dimensional momentum integration (including a translationally invariant regularization) is given by . A graphical representation of Eq. (1) is given in Fig. 1, where the arrows denote dressed-quark propagators (analogously in Figs. 3 and 4). The momentum flow (detailed in Fig. 2) is defined such that the total momentum P is given by the difference of the (anti-)quark momenta . represents the momentum partitioning parameters, which satisfy . The relative momentum q of the BSA is therefore given by . On the left-hand side of Eqs. (1) and (2) the relative momentum is denoted by k.

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