Molecular partners of the X(3872) from heavy-quark spin symmetry: a fresh look

¹ Institut für Theoretische Physik II, Ruhr-Universität Bochum, D-44780 Bochum, Germany
² Institute for Theoretical and Experimental Physics, B. Cheremushkinskaya 25, Moscow, Russia
³ Forschungszentrum Jülich, Institute for Advanced Simulation, Institut für Kernphysik and Jülich Center for Hadron Physics, D-52425 Jülich, Germany
⁴ National Research Nuclear University MEPhI, 115409, Kashirskoe highway 31, Moscow, Russia
⁵ Moscow Institute of Physics and Technology, 141700 Institutsky lane 9, Dolgoprudny, Moscow Region, Russia

^a e-mail: vadimb@tp2.rub.de

Published online: 22 March 2017

Abstract

The heavy-quark spin symmetry (HQSS) partners of the X(3872) molecule are investigated in a chiral effective field theory (EFT) approach which incorporates contact and one-pion exchange interactions. The integral equations of the Lippmann-Schwinger type are formulated and solved for the coupled-channel problem for the DD̄, DD̄*, and D*D̄* systems with the quantum numbers J^PC = 1⁺⁺, ¹⁺⁻, 0⁺⁺, and 2⁺⁺. We confirm that, if the X(3872) is a 1⁺⁺DD̄* molecular state then, in the strict heavy-quark limit, there exist three partner states, with the quantum numbers 1⁺⁻, 0⁺⁺, and 2⁺⁺, which are degenerate in mass. At first glance, this result looks natural only for the purely contact pionless theory since pions contribute differently to different transition potentials and, therefore, may lift the above degeneracy. Nevertheless, it is shown that, by an appropriate unitary transformation, the Lippmann-Schwinger equation in each channel still can be brought to a block-diagonal form, with the same blocks for all quantum numbers, so that the degeneracy of the bound states in different channels is preserved. We stress that neglecting some of the coupled-channel transitions in an inconsistent manner leads to a severe violation of HQSS and yields regulator-dependent results for the partner states. The effect of HQSS violation in combination with nonperturbative pion dynamics on the pole positions of the partner states is discussed.