Multiple dispersive bounds. II) Sub-threshold branch-cuts

Abstract

We apply the strategy proposed in the companion paper [1] for dealing with multiple dispersive bounds, to the case of sub-threshold branch-cuts, which is a topic addressed extensively in the literature (see, e.g., Refs. [2-8]). We consider the simultaneous application of a double dispersive bound as a proper way to take into account unitarity constraints within phenomenological analyses of hadronic form factors in the presence of sub-threshold branch-cuts. Accordingly, the standard $z$-expansion of hadronic form factors, commonly referred to as the Boyd-Grinstein-Lebed approach [4, 9-11], is modified by including simultaneously the dispersive bounds related to the pair-production and to the sub-threshold regions. For the latter one the effects of above-threshold poles are described through a simple resonance model and the possible choices of the outer function outside the pair-production region are discussed. A detailed numerical analysis of the experimental data or lattice QCD results in the spacelike region for the charged kaon form factor is presented as a direct application of the procedure of double dispersive bound. The comparison with other methodologies present in literature and with the $z$-expansion based on the single, total dispersive bound clearly shows that the $z$-expansion including the double dispersive bound provides the most precise extrapolation at large momentum transfer as well as the most stable results with respect to the choice of the outer function outside the pair-production region.

Figures (10)

• Figure 1: Analytical domains in terms of the conformal variables $z_+$ and $z$, given respectively by Eqs. (\ref{['eq:zplus']}) and (\ref{['eq:z']}) with $t_{th} < t_+ = (m_1 + m_2)^2$. The blue lines correspond to the pair-production branch-cut $t \geq t_+$, while the red ones to the extra branch-cut ranging from $t_{th}$ to $t_+$. The lowest branch-point $t_{th}$ is located at $z_+^{th} = z_+(t_{th}; t_0)$ chosen arbitrarily to be equal to $z_+^{th} = - 0.5$.Correspondingly, the angle $\alpha_{th}$ is given by $\hbox{cos}(\alpha_{th})= 2 (1 + z_+^{th})^2 / (1 - z_+^{th})^2 - 1 \simeq - 0.78$.
• Figure 2: Left panel: the absolute value of the electromagnetic pion form factor $F_\pi^{(em)}(t)$, determined by the dispersive analysis of experimental data made in Ref. Colangelo:2018mtw, labelled as CHS19, compared with the corresponding predictions of the approximation (\ref{['eq:frho']}), obtained using $M_\rho = 775$ MeV and $\Gamma_\rho = 147$ MeV from PDG ParticleDataGroup:2024cfk. Right panel: the experimental data on $\pi-\pi$ scattering phase shifts from Ref. Protopopescu:1973sh (green squares) and Ref. Estabrooks:1974vu (black diamonds) compared with the phase of the form factor (\ref{['eq:frho']}).
• Figure 3: Absolute value of the electromagnetic form factors of charged (left panel) and neutral kaons (right panel) versus the squared momentum transfer $t$ both in the spacelike region ($t \leq 0$) and in the timelike one ($t > 0$). Dotted blue lines represent the results of the dispersive analysis of Ref. Stamen:2022uqh, labelled as SHHKS '22, while the solid red lines correspond to the predictions of the resonance model given by Eqs. (\ref{['eq:Kcharged_model']})-(\ref{['eq:Kneutral_model']}) using the values $M_\rho = 775$ MeV, $\Gamma_\rho = 147$ MeV, $M_\omega = 782.7$ MeV, $\Gamma_\omega = 8.68$ MeV, $M_\phi = 1019.5$ MeV and $\Gamma_\phi = 4.25$ MeV taken from the PDG ParticleDataGroup:2024cfk.
• Figure 4: Absolute values of the kinematical function (\ref{['eq:phi_q1q2']}) versus the squared 4-momentum transfer $t$ corresponding to $q_1 = q_2 = 0$ (solid lines) and to the choice of $q_1$ and $q_2$ made in Ref. Flynn:2023nhi (dashed lines) and labelled with the suffix $RBC$ (see text). The value of the auxiliary variable $t_0$ is chosen to be equal to $t_0 = t_-$. Left panel: vector and scalar kinematical functions for the $B_s \to K$ transition with $t_- = (m_{B_s} - m_K)^2 \simeq 23.7$ GeV$^2$, $t_{th} = t_{B \pi} = (m_B + m_\pi)^2 \simeq 29.3$ GeV$^2$ and $t_+ = t_{B_s K} = (m_{B_s} + m_K)^2 \simeq 34.4$ GeV$^2$. Right panel: vector kinematical function corresponding to the case of the electromagnetic form factor of the charged kaon with $t_- = 0$, $t_{th} = t_{2\pi} = 4 m_\pi^2 \simeq 0.08$ GeV$^2$ and $t_+ = t_{2K} = 4 m_K^2 \simeq 0.98$ GeV$^2$.
• Figure 5: The electromagnetic form factor of the charged kaon $F_{K^\pm}^{(em)}(Q^2)$ versus the 4-momentum transfer $Q^2 = - t$. The red and blue bands, evaluated at $1\sigma$ level, correspond to the results obtained using the BGL expansion (\ref{['eq:BGL_truncated']}) at $M = 8$ and adopting respectively the double bound (\ref{['eq:unitarity_pair_truncated']})-(\ref{['eq:unitarity_extra_truncated']}) and the single, total bound (\ref{['eq:unitarity_truncated']}). The dispersive bounds $\chi_+^U$ and $\chi_{\rm extra}^U$ are given respectively by Eqs. (\ref{['eq:chi+U']}) and (\ref{['eq:chiextraU_q0']}), and the total bound $\chi^U$ is $\chi^U = \chi_+^U + \chi_{\rm extra}^U$. The green band represents the results obtained using the expansion (\ref{['eq:BGL_VD']}) with the unitarity constraint (\ref{['eq:unitarity_VD']}) corresponding to the pair-production arc only. In all cases the choice $q = 0$ for the kinematical function $\phi^{q_1 q_2}(z)$ is considered. The black band (having tiny errors) corresponds to the results of the dispersive approach of Ref. Stamen:2022uqh, while the blue squares and the red circles are respectively the experimental data from the FNAL Dally:1980dj and CERN Amendolia:1986ui experiments. The bottom panel is a zoom in the range of values of $Q^2$ covered by the two experiments.