arXiv : Cornering Large-$N_c$ QCD with Positivity Bounds

Fernandez, Clara; Pomarol, Alex; Riva, Francesco; Sciotti, Francesco

doi:10.1007/JHEP06(2023)094

Article
Report number	arXiv:2211.12488
Title	Cornering Large-$N_c$ QCD with Positivity Bounds
Author(s)	Fernandez, Clara (Barcelona, IFAE ; BIST, Barcelona) ; Pomarol, Alex (Barcelona, IFAE ; BIST, Barcelona ; Barcelona, Autonoma U. ; CERN) ; Riva, Francesco (Geneva U., Dept. Theor. Phys.) ; Sciotti, Francesco (Barcelona, IFAE ; BIST, Barcelona)
Publication	2023-06-16
Imprint	2022-11-22
Number of pages	39
Note	38 pages, 7 figures
In:	JHEP 2306 (2023) 094
DOI	10.1007/JHEP06(2023)094
Subject category	hep-ph ; Particle Physics - Phenomenology ; hep-lat ; Particle Physics - Lattice ; hep-th ; Particle Physics - Theory
Abstract	The simple analytic structure of meson scattering amplitudes in the large-$N_c$ limit, combined with positivity of the spectral density, provides precise predictions on low-energy observables. Building upon previous studies, we explore the allowed regions of chiral Lagrangian parameters and meson couplings to pions. We reveal a structure of kinks at all orders in the chiral expansion and develop analytical tools to show that kinks always correspond to amplitudes with a single light pole. We build (scalar- and vector-less) deformations of the Lovelace-Shapiro and Coon UV-complete amplitudes, and show that they lie close to the boundaries. Moreover, constraints from crossing-symmetry imply that meson couplings to pions become smaller as their spin increases, providing an explanation for the success of Vector Meson Dominance and holographic QCD. We study how these conclusions depend on assumptions about the high-energy behavior of amplitudes. Finally, we emphasize the complementarity between our results and Lattice computations in the exploration of large-$N_c$ QCD.
Copyright/License	publication: (License: CC-BY-4.0), sponsored by SCOAP³ preprint: (License: CC BY 4.0)

$\it Null constraints from \eq{nullconst}, as a function of $J$, for fixed $m$.$ $\it Allowed regions in the $\tilde g_{2,1}$--$\, \tilde g_{2,0}$ plane, divided into regions with $J=0$ states only (red line) and $J>0$ states (blue) -- using null constraints with $\nmax=3$ (dashed light-blue line) and $\nmax=11$ (dashed blue line). The dark blue area spans \eq{interpol} for $0\leq m_\rho\leq M$ and $m_\rho\leq m\leq \infty$. Moreover, $J=1$ models lie on the blue line, $J>0$ $su$-models on the black line (\eq{suamp} with \eq{lambda} and $0\leq m\leq M$), and the magenta line corresponds to \eq{sutomrho}. The brown and yellow dot correspond to the Lovelace-Shapiro amplitude with and without scalars (\eq{LS-Samp})) respectively. The dashed brown and yellow line are the Coon amplitude (\eq{coonamp} with $C=1$) with and without $J=0$ states respectively.$ $\it Allowed regions from positivity with $J\geq1$ (as in Fig.~\ref{fig:g2g2pdec}). The green regions have heavier resonances of all spin $J\geq 1$ with masses $M'\geq 1.65\, m_\rho$: the dashed green line uses null constraints up to $\nmax=11$. In orange the allowed region for a fixed $\tilde g_\rho=1/2$ (left) and $\tilde g_\rho=1/3$ (right), with still $M'/m_{\rho}=1.65$. In purple, the allowed region from Lattice simulations \cite{Bali:2013kia,Hernandez:2019qed,Baeza-Ballesteros:2022azb}. In red the QCD experimental value from Ref.~\cite{Bijnens:1994ie}.$ Show more plots

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