Solving the three-body bound-state Bethe-Salpeter equation in Minkowski space

Ydrefors, E.; Nogueira, J. H. Alvarenga; Karmanov, V. A.; Frederico, T.

doi:10.1016/j.physletb.2019.02.046

Cited by 19 publications

(31 citation statements)

References 12 publications

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“…Recent progress has also been made in calculating propagators and BS amplitudes in Minkowski space, see [32][33][34][35][36][37][38][39][40][41] and references therein. Here we want to point out that there is no intrinsic difference between Euclidean and Minkowski space approaches: To obtain scattering amplitudes in the complex plane, contour deformations are necessary both in a Euclidean and Minkowski metric.…”

Section: Introductionmentioning

confidence: 99%

Scattering amplitudes and contour deformations

et al. 2019

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We employ a scalar model to exemplify the use of contour deformations when solving Lorentzinvariant integral equations for scattering amplitudes. In particular, we calculate the onshell 2 → 2 scattering amplitude for the scalar system. The integrals produce branch cuts in the complex plane of the integrand which prohibit a naive Euclidean integration path. By employing contour deformations, we can also access the kinematical regions associated with the scattering amplitude in Minkowski space. We show that in principle a homogeneous Bethe-Salpeter equation, together with analytic continuation methods such as the Resonances-via-Padé method, is sufficient to determine the resonance pole locations on the second Riemann sheet. However, the scalar model investigated here does not produce resonance poles above threshold but instead virtual states on the real axis of the second sheet, which pose difficulties for analytic continuation methods. To address this, we calculate the scattering amplitude on the second sheet directly using the two-body unitarity relation which follows from the scattering equation. arXiv:1907.05402v1 [hep-ph] 11 Jul 2019 ₊ ₋ Bound states Resonances � �1 Re Im FIG. 2: Complex t plane for a typical current correlator. The leading perturbative loop diagram only produces the cut.

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Section: Introductionmentioning

confidence: 99%

Scattering amplitudes and contour deformations

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“…1 it is shown as an example the calculated vertex function, v(q 0 , q v = 0.5m), for the three-body binding energy B 3 /m = 0.395. As discussed in more detail in [17], the vertex function has peaks at the values of q v and q 0 , corresponding to that M 2 12 = 0 or M 2 12 = 4m 2 . In the figure these positions are indicated with dashed lines.…”

Section: Resultsmentioning

confidence: 97%

“…In Ref. [17], we solved Eq. (5) by adopting a bi-cubic spline decomposition of the vertex function v(p, q).…”

Section: Resultsmentioning

confidence: 99%

“…where the propagator singularities have been eliminated by subtractions. The kernel Π has now only weak (logarithmic) singularities which will be treated numerically and is given explicitly in [17]. It is not possible to directly compare the Minkowski vertex function v(q 0 , q v ) with the corresponding Euclidean one.…”

Section: Three-body Bethe-salpeter Equationmentioning

confidence: 99%

“…As already pointed out, it is crucial to acquire the BS amplitude directly in Minkowski space. To this end, we solved in the recent work [17] the BS equation by direct integration in Minkowski space and some of the results are presented in this contribution. The results for the binding energies and transverse amplitudes are also compared with the ones obtained in Euclidean space.…”

Section: Introductionmentioning

confidence: 99%

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Bethe-Salpeter Approach to Three-Body Bound States with Zero-Range Interaction

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The Bethe-Salpeter equation for three scalar bosons, with zero-range interaction, is solved in Minkowski space by direct integration of the four-dimensional integral equation. The singularities occuring in the propagators are treated properly by standard analytical and numerical methods, without relying on any ansatz for the Bethe-Salpeter amplitude. The results for the binding energies and transverse amplitudes are compared with the results computed in Euclidean space. A fair agreement between the calculations is obtained.

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Quark Propagator in Minkowski Space

Solis

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et al. 2019

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The analytic structure of the quark propagator in Minkowski space is more complex than in Euclidean space due to the possible existence of poles and branch cuts at timelike momenta. These singularities impose enormous complications on the numerical treatment of the nonperturbative Dyson-Schwinger equation for the quark propagator. Here we discuss a computational method that avoids most of these complications. The method makes use of the spectral representation of the propagator and of its inverse. The use of spectral functions allows one to handle in exact manner poles and branch cuts in momentum integrals. We obtain model-independent integral equations for the spectral functions and perform their renormalization by employing a momentum-subtraction scheme. We discuss an algorithm for solving numerically the integral equations and present explicit calculations in a schematic model for the quark-gluon scattering kernel.

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Solving the three-body bound-state Bethe-Salpeter equation in Minkowski space

Cited by 19 publications

References 12 publications

Scattering amplitudes and contour deformations

Scattering amplitudes and contour deformations

Bethe-Salpeter Approach to Three-Body Bound States with Zero-Range Interaction

Quark Propagator in Minkowski Space

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