Abstract
XEFT is a low-energy effective field theory for charm mesons and pions that provides a systematically improvable description of the $X(3872)$ resonance. To simplify calculations beyond leading order, we introduce a new formulation of XEFT with a dynamical field for a pair of charm mesons in the resonant channel. We simplify the renormalization of XEFT by introducing a new renormalization scheme that involves the subtraction of amplitudes at the complex $D^{*0} \bar D^0$ threshold. The new formulation and the new renormalization scheme are illustrated by calculating the complex pole energy of $X$ and the $D^{*0} \bar D^0$ scattering amplitude to next-to-leading order using Galilean-invariant XEFT.
Highlights
The Xð3872Þ was the first of the dozens of exotic hadrons whose fundamental constituents include a heavy quark and its antiquark that have been discovered since the beginning of the century [1,2,3,4]
The possibilities for the particle structure of X that are compatible with this information include (i) the χc1ð2PÞ charmonium state, whose quark constituents are cc, (ii) a compact isospin-1 tetraquark meson, whose diquark constituents are ðcuÞðcu Þ − ðcdÞðcd Þ, (iii) an isospin-0 charm-meson molecule, whose hadron constituents are ðDÃ0D 0 þ D0D Ã0Þ þ ðDÃþD−þ DþDÃ−Þ, which correspond to quark constituents ðcu ÞðcuÞ þ ðcd ÞðcdÞ
We introduce a new formulation of Galileaninvariant XEFT with a dynamical field for a pair of charm mesons in the resonant channel
Summary
The Xð3872Þ was the first of the dozens of exotic hadrons whose fundamental constituents include a heavy quark and its antiquark that have been discovered since the beginning of the century [1,2,3,4]. If the model has an adjustable parameter that can be used to tune the resonance energy to the DÃ0D 0 threshold, the resonance will in the limit develop the particle structure in Eq (2) with the universal wave function expð−r=aÞ=r This remarkable phenomenon is widely recognized in the case of a charm-meson molecule. [12], the elastic scattering amplitude for DÃ0D 0 was calculated at NLO using dimensional regularization in Galilean-invariant XEFT There are both linear and logarithmic UV divergences, and they were removed by subtractions at the complex pole energy of X. We introduce a simpler renormalization scheme for XEFT in which divergences are removed instead by subtractions at the complex threshold energy of DÃ0D 0 This new renormalization scheme greatly simplifies analytic results at NLO.
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