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M. N. Ferreira, A. S. Miramontes, J. M. Morgado, J. Papavassiliou, J. M. Pawlowski

Abstract

We present a novel version of the pion Bethe-Salpeter equation in the chiral limit, solved using as ingredients state-of-the-art QCD correlation functions. The constraints imposed by the axial Ward-Takahashi identities are exactly fulfilled, both formally and numerically.

Pions reloaded

Abstract

We present a novel version of the pion Bethe-Salpeter equation in the chiral limit, solved using as ingredients state-of-the-art QCD correlation functions. The constraints imposed by the axial Ward-Takahashi identities are exactly fulfilled, both formally and numerically.

Paper Structure

This paper contains 5 sections, 4 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Main ingredients
  3. The symmetric-vertex approximation
  4. Numerical analysis
  5. Conclusions

Figures (4)

  • Figure 1: Panel A: The exact SDE for the axial-vector $\Gamma_{\!5}^\mu$ . Panel B: The SDE for the vertex $G_5^{\mu\nu}$, after implementing the SV approximation.
  • Figure 2: Panel A: The SDE for the quark-gluon vertex in the SV approximation. Panel B: The corresponding gap equation. Panel C: The final pion BSE, after renormalization. The index "1" indicates the part of the cyan vertex that contains an odd number of Dirac $\gamma$ matrices, while the square vertex stands for its "quantum" part, namely the sum of the graphs $c_1$ and $c_2$.
  • Figure 3: Left panel: Schematic representation of the SV approximation. Right panel: The classical form factor $\lambda_1$ obtained in Aguilar:2024ciu; the orange line indicates the symmetric configuration $r^2=p^2=q^2$, used for $V(q)$.
  • Figure 4: Left panel: Contributions to the eigenvalue of the pion BSE, from each of the three diagrams shown in \ref{['fig:gapeq']}. Right panel: Numerical confirmation of the WTI-imposed relation $\chi_1(p)=2B(p)$.