Derivation of the three-body bound-state equation from the effective action

Komachiya, M.; Ukita, M.; Fukuda, R.

doi:10.1103/physrevd.40.2654

Cited by 7 publications

(8 citation statements)

References 12 publications

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“…The Bethe-Salpeter kernel corresponding to the truncation at hand can readily be derived from the 3PI effective action [38,40,41,[62][63][64]. In the case of the Bethe-Salpeter equation (BSE) for a meson, its quark-antiquark kernel is obtained by twice differentiating with respect to the quark propagator S; in Appendix A 3 we discuss how consistency with the axial-vector Ward-Takahashi identity leads to the appearance of a Goldstone boson in the chiral limit and the formation of the Gell-Mann-Oakes-Renner relation.…”

Section: E Application To Bound-statesmentioning

confidence: 99%

Light mesons in QCD and unquenching effects from the 3PI effective action

2016

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We investigate the impact of unquenching effects on QCD Green's functions, in the form of quark-loop contributions to both the gluon propagator and three-gluon vertex, in a three-loop inspired truncation of the three-particle irreducible (3PI) effective action. The fully coupled system of Dyson-Schwinger equations for the quark-gluon-, ghost-gluon-and three-gluon vertices, together with the quark propagator, are solved self-consistently; our only input are the ghost and gluon propagators themselves that are constrained by calculations within Lattice QCD. We find that the two different unquenching effects have roughly equal, but opposite, impact on the quark-gluon vertex and quark propagator, with an overall negative impact on the latter. By taking further derivatives of the 3PI effective action, we construct the corresponding quark-antiquark kernel of the Bethe-Salpeter equation for mesons. The leading component is gluon exchange between two fullydressed quark-gluon vertices, thus introducing for the first time an obvious scalar-scalar component to the binding. We gain access to time-like properties of bound-states by analytically continuing the coupled system of Dyson-Schwinger equations to the complex plane. We observe that the vector axial-vector splitting is in accord with experiment and that the lightest quark-antiquark scalar meson is above 1 GeV in mass.

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Section: E Application To Bound-statesmentioning

confidence: 99%

Light mesons in QCD and unquenching effects from the 3PI effective action

2016

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“…Such an enhancement is needed in order to account for the effect on the quark-gluon vertex of higher loop terms in the 2PI effective action. An alternative to this is using a 3PI or higher effective action to define the truncation, which means that both vertices and propagators are fully dressed and independent objects [15,18]. Further details on (8) and its solution can be found in [12].…”

Section: Frameworkmentioning

confidence: 99%

“…The analogue of the Bethe-Salpeter equation for baryons can now be formulated. It contains the permuted sum of the two-body quark-quark kernel K (qq) and an irreducible three-body kernel K (qqq) [15,34]…”

Section: Frameworkmentioning

confidence: 99%

“…One reason that makes nPI techniques a powerful tool is that they provide a natural link between bound-state equations described by BSEs and the propagators and vertices provided by DSEs [14][15][16][17][18]. A truncation of the loop expansion at a certain order translates into a unique prescription for the truncation of the DSEs and the BSEs that maintains symmetries.…”

Section: Introductionmentioning

confidence: 99%

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Probing the quark–gluon interaction with hadrons

Sanchis-Alepuz

Williams

2015

Physics Letters B

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We present a unified picture of mesons and baryons in the Dyson-Schwinger/Bethe-Salpeter approach, wherein the quark-gluon and quark-(anti)quark interaction follow from a systematic truncation of the QCD effective action and includes all its tensor structures. The masses of some of the ground state mesons and baryons are found to be in reasonable agreement with the expectations of a 'quark-core calculation', suggesting a partial insensitivity to the details of the quark-gluon interaction. However, discrepancies remain in the meson sector, and for excited baryons, that suggest higher order corrections are relevant and should be investigated following the methods outlined herein.

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“…Then the following equations are easily verified: 4 xi and ("')0 signifies that it is evaluated at the solution rp/O) which can be either the perturbative or the non-perturbative condensed vacuum.…”

Section: Derivation Of the 4-body Bound-state Equation From The Effecmentioning

confidence: 99%

Derivation of the 4-Body Bound-State Equation from the Effective Action: Diagrammatical Legendre Transformation

Okumura¹

1992

Progress of Theoretical Physics

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The 4-body bound-state equation is derived by use of on-shell expansion of the effective action. Derivation is based on the second derivative of the effective action which makes the graphical rule much easier. The essential tool is the sum-up rule of the graphs which is equivalent to the Legendre transformation. This rule can be widely used for any operator that has external point(s). Applied to the 1,2, 3-body channels our method gives another way of deriving the known bound-state equations. The obtained equations hold whether the vacuum state is condensed or not. § 1. IntroductionThe bound state problem in relativistic field theory has long been discussed starting from the bosonic Bethe-Salpeter (BS) equation for the 2-body bound state (Refs_ 1))_ Since then many authors discussed the solution and the formal properties of the bound-state equation (for example, see Refs_ 2)). The 3-body bound-state equation has been obtained (Refs. 3),4)) in the relativistic field theory. It will not be necessary to emphasize here the importance of generalizing BS equation for N-body channel. Indeed the Nth-order Bethe-Salpeter kernel was defined as a generalization of lower-order kernels (Ref. 5)) and its grahical properties have been studied by several authors (Refs. 6), 7)). But the derivation of the bound state equation for N-body channel is another subject.N-body bound sate equation for general N has been recently obtained by use of the on-shell expansion of the effective action. However the arguments of Ref. 8) are limited to the case where the vacuum state is not a cOl)densed one.The purpose of this paper is to extend the on-shell expansion to derive 1, 2, 3 and 4-body bound-state equation in a way that they are valid whether the vacuum state is a non-perturbative one or not. By non-perturbative vacuum we mean the case that the true vacuum is a condensed state. A newly obtained result is the 4-body boundstate equation which can be used to determine the bound state excited above the condensed vacuum. (This will help us to understand what problem has to be. solved in deriving the rules for the graphical expression of the N-body BS kernel (N)4) for the system whose vacuum is a condensed state.) The physical importance of the above problem is clear since we encounter the condensed vacuum in various systems; ferromagnet, superconductivity or quark-gluon system described by quantum chromo dynamics (QeD).In the on-shell expansion the bound-state equation is obtained by the second derivative of the effective action (Ref. 9)). Our result indeed reproduces the second . Their result has some exceptional graphs but ours does not contain them because they disappear when we take the second derivative. Their derivation of the 4th effective action has much combinatorial complexity. Several authors have tried to reduce the complexity and to make the proof simpler by using equations of motion for the Legendre transforms (Ref. 11)) or Spencer's t-derivative (Refs. 12), 6), 7)). Here instead we establish the sum-up rule which plays the cen...

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Derivation of the three-body bound-state equation from the effective action

Abstract: Using a recently developed method, a formal novel derivation of the three-body bound-state equation is presented. It is based on the second derivative of the effective action. The baryonic equation in the case of quantum chromodynamics is also derived.

Cited by 7 publications

References 12 publications

Light mesons in QCD and unquenching effects from the 3PI effective action

Light mesons in QCD and unquenching effects from the 3PI effective action

Probing the quark–gluon interaction with hadrons

Derivation of the 4-Body Bound-State Equation from the Effective Action: Diagrammatical Legendre Transformation

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