Partial wave analysis for the in-hadron condensate

In-hadron condensates, defined as the scalar form factors at zero-momentum transfer, are investigated for flavor-symmetric mesons in pseudoscalar and vector channels under the rainbow-ladder (RL) truncation within the Dyson-Schwinger equations framework. We confirm the efficiency of the in-hadron condensates in describing the effects of dynamical chiral symmetry breaking from both global and structural perspectives by comparing the meson masses, the dimensionless in-hadron condensates, and the partial wave decompositions of in-hadron condensates as functions of current-quark mass. From partial wave analysis, $\pi (1300)$ is inferred as a radial excitation dominated by $s$ waves, while $\rho (1450)$ needs further studies beyond the RL truncation. This work provides a new insight into the studies of hadron properties with partial wave analysis for the in-hadron condensates.

Figure 1

The diagram for the in-hadron condensate in Eq. (9). $P$ and $q_{\pm }$ are the momenta of the mesons and quark propagators. $Q$ is the transfer momentum with $Q^2=0$.

Figure 2

The diagram for the gap equation in Eqs. (10) and (11).

Figure 3

The diagram for the homogeneous BS equation in Eq. (14).

Figure 4

The diagram for the scalar vertex in Eq. (17).

Figure 5

The meson masses $M_P$ as functions of the current-quark mass $m_q$ in different channels: pseudoscalar ground state ($\pi$ channel), pseudoscalar first-excited state (${\pi }^{\prime }$ channel), vector ground state ($\rho$ channel), and vector first-excited state (${\rho }^{\prime }$ channel). The gray dashed line represents $2m_q$. The light quark region [0, 1] GeV (left) and the heavy quark region [3, 4] GeV (right) are shown.

Figure 6

Similar to Fig. 5 but for dimensionless in-hadron condensates $S_P(0)/M_P$.

Figure 7

The relative partial wave contributions to the in-hadron condensates $S_P^{(S,L)}(0)/S_P(0)$ as functions of current-quark mass $m_q$ for pseudoscalar mesons at ground state (left) and first-excited state (right).

Figure 8

Similar to Fig. 7 but for vector mesons.