Springer : Variations on the Maiani-Testa approach and the inverse problem

Bruno, Mattia; Hansen, Maxwell T.

doi:10.1007/JHEP06(2021)043

Article
Report number	arXiv:2012.11488 ; CERN-TH-2020-412
Title	Variations on the Maiani-Testa approach and the inverse problem
Author(s)	Bruno, Mattia (CERN) ; Hansen, Maxwell T. (U. Edinburgh, Higgs Ctr. Theor. Phys. ; Edinburgh U.)
Publication	2021-06-07
Imprint	2020-12-21
Number of pages	32
In:	JHEP 2106 (2021) 043
DOI	10.1007/JHEP06(2021)043 (publication)
Subject category	hep-th ; Particle Physics - Theory ; hep-ph ; Particle Physics - Phenomenology ; hep-lat ; Particle Physics - Lattice
Abstract	We discuss a method to construct hadronic scattering and decay amplitudes from Euclidean correlators, by combining the approach of a regulated inverse Laplace transform with the work of Maiani and Testa. Revisiting the original result, we observe that the key observation, i.e. that only threshold scattering information can be extracted at large separations, can be understood by interpreting the correlator as a spectral function, $\rho(\omega)$, convoluted with the Euclidean kernel, $e^{- \omega t}$, which is sharply peaked at threshold. We therefore consider a modification in which a smooth step function, equal to one above a target energy, is inserted in the spectral decomposition. This can be achieved either through Backus-Gilbert-like methods or more directly using the variational approach. The result is a shifted resolution function, such that the large $t$ limit projects onto scattering or decay amplitudes above threshold. The utility of this method is highlighted through large $t$ expansions of both three- and four-point functions that include leading terms proportional to the real and imaginary parts (separately) of the target observable. This work also presents new results relevant for the un-modified correlator at threshold, including expressions for extracting the $N \pi$ scattering length from four-point functions and a new strategy to organize the large $t$ expansion that exhibits better convergence than the expansion in powers of $1/t$.
Copyright/License	preprint: (License: CC-BY-4.0) publication: © 2021-2025 The Authors (License: CC-BY-4.0), sponsored by SCOAP³

$Sketch of the normalized effective resolution functions, $\widehat \delta(\mathcal E,t_2)$ and $\widehat \delta^\Theta(\mathcal E,t_2)$, defined in eqs.~\eqref{eq:deltahat} and \eqref{eq:deltahatTheta} respectively. The corresponding correlators can be written as convolution integrals of these resolution functions with the same spectral function, $\rho_{\vec q}(\mathcal E)$, which contains the time-like information we are after. Thus, the functional forms plotted here give an indication of the energies that predominantly contribute to the correlation functions. The left panel illustrates that $c_{\vec q,-\vec q}(t_2)$ is dominated by $\mathcal E \approx 2 m_\pi$ and that large $t_2$ sharpens the near-threshold resolution. By contrast, the right panel illustrates the modification involving $\Theta(\mathcal E - 2\omega_{\vec q}, \Delta)$ (also plotted here in gray) for $2\omega_{\vec q}=5 m_\pi$. The three curves correspond to the same $\Delta$ value, and the sharpening around $\mathcal E = 5 m_\pi$ is achieved by increasing $t_2$, with the dashed curves showing the corresponding factors of $e^{- t_2 ( \mathcal E - 2 \omega_{\vec q})}$.$ $Plot of the first few $\mathcal I^{(n)}(b)$ functions (solid lines), defined in \Eq~\eqref{eq:Idef} of the main text. The dashed lines show the large $b$ expansion for each function, scaling as $b^{-n-1/2}$ as shown in \Eq~\eqref{eq:Iexp}.$ $Plot of the first few $\mathcal J^{(n)}(t_2, \omega_q, \Delta)$ functions, defined in \Eq~\eqref{eq:Jdef} of the main text, vs $t_2$ with $\omega_q=3m_\pi$ and $\Delta = m_\pi$.$ Show more plots

Corresponding record in: Inspire

Naspäť k hľadaniu

Záznam vytvorený 2020-12-23, zmenený 2023-12-14

Podobné záznamy

Article from SCOAP3:

PDF

Plný text:

PDF

Pridaj do osobného košíka
Exportuj ako BibTeX, MARC, MARCXML, DC, EndNote, NLM, RefWorks

CERN Document Server

Access articles, reports and multimedia content in HEP

Main menu

CERN Accelerating science