Equivalence of three-particle scattering formalisms

In recent years, different on-shell $\mathbf{3}\to \mathbf{3}$ scattering formalisms have been proposed to be applied to both lattice QCD and infinite-volume scattering processes. We prove that the formulation in the infinite volume presented by Hansen and Sharpe in [M. T. Hansen and S. R. Sharpe, Phys. Rev. D 92, 114509 (2015).] and subsequently Briceño et al. in [R. A. Briceño, M. T. Hansen, and S. R. Sharpe, Phys. Rev. D 95, 074510 (2017).] can be recovered from the $B$-matrix representation, derived on the basis of $S$-matrix unitarity, presented by Mai et al. in [M. Mai, B. Hu, M. Döring, A. Pilloni, and A. Szczepaniak, Eur. Phys. J. A 53, 177 (2017).] and Jackura et al. in [A. Jackura, C. Fernández-Ramírez, V. Mathieu, M. Mikhasenko, J. Nys, A. Pilloni, K. Saldaña, N. Sherrill, and A. P. Szczepaniak (JPAC Collaboration), Eur. Phys. J. C 79, 56 (2019).] Therefore, both formalisms in the infinite volume are equivalent and the physical content is identical. Additionally, the Faddeev equations are recovered in the nonrelativistic limit of both representations.

Figure 1

A three-particle state in the (a) center of momentum frame (CMF) with fixed total momentum $\mathbf{P}=\mathbf{0}$ and (b) in the helicity frame at fixed ${\mathbf{P}}^{\star }-{\mathbf{p}}^{\star }=\mathbf{0}$. Standard Lorentz transformation with the boost $\mathit{\beta }=-(\mathbf{P}-\mathbf{p})/(E-{\omega }_{\mathbf{p}})$ transforms the system from the total CMF to the pair rest frame.

Figure 2

Diagrammatic representation of (a) the $B$-matrix representation for the on-shell amplitude, Eq. (11), and (b) the $B$ matrix that is composed of the OPE ${\mathcal{G}}_{{\mathbf{p}}^{\prime }\mathbf{p}}$, Eq. (13), and the $R$ matrix.

Figure 3

Diagrammatic representation for the ladder series generated by particle exchanges, Eq. (15), where the black box represents ${\mathcal{D}}_{{\mathbf{p}}^{\prime }\mathbf{p}}$.

Figure 4

Diagrammatic representation of (a) the $\mathbf{3}\to \mathbf{3}$ amplitude Eq. (16) in terms of the ladder series and the $B$-matrix divergence-free amplitude dressed by initial and final state rescatterings, (b) the integral equation for the $B$-matrix divergence-free amplitude Eq. (17), and (c) the $B$-matrix rescattering function Eq. (19).

Figure 5

Diagrammatic representation of (a) the $\mathbf{3}\to \mathbf{3}$ amplitude Eq. (20) in terms of the ladder series and the HS-BHS divergence-free amplitude dressed by initial and final state rescatterings, (b) the integral equation for the HS-BHS divergence-free amplitude Eq. (23), and (c) the HS-BHS rescattering function Eq. (21).

Figure 6

Diagrammatic representation of the leading order relation between ${\mathcal{R}}_{{\mathbf{p}}^{\prime }\mathbf{p}}$ and ${\mathcal{K}}_{\mathrm{df}}({\mathbf{p}}^{\prime },\mathbf{p})$, Eq. (39), where the black diamond represents ${\mathcal{K}}_{\mathrm{df}}({\mathbf{p}}^{\prime },\mathbf{p})$.

Figure 7

Diagrammatic representation for the $\mathbf{3}\to \mathbf{3}$ unitarity relation for the amplitude ${\mathcal{A}}_{{\ell }^{\prime }{m_{\ell }}^{\prime };\ell m_{\ell }}({\mathbf{p}}^{\prime },s,\mathbf{p})$. Closed loops yield three-dimensional integrations over the labeled spectator momentum, and the dashed vertical lines represent placing all three intermediate state particles on their mass shell. Momentum flow is from right to left, as before, and each amplitude on the left of the dashed line is Hermitian conjugated.