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 DD , DD *, and D*D * systems with the quantum numbers J PC = 1 ++ , 1+− , 0 ++ , and 2 ++ . We confirm that, if the X (3872) is a 1 ++ DD * 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.
Highlights
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
In the molecular picture a relation between the poles of the molecular partner states of the isovector resonances Zb+(10610) and Zb+(10650) was derived in Refs. [4,5,6]. It was argued in Refs. [7, 8] that one should expect a shallow molecular partner state for the X(3872) in the D∗D ∗ channel with the quantum numbers JPC = 2++ while, in Ref. [9], the width of this state was estimated to be as small as a few MeV using an effective field theory with perturbative pions (X-EFT)
Assuming the X(3872) with the quantum numbers JPC = 1++ to be an isosinglet molecule state, in this contribution, we discussed the implications of heavy-quark spin symmetry (HQSS) for the molecular partners of the X(3872)
Summary
In a recent study [2], it was emphasized that the implications of HQSS for the X(3872), such as the number of its HQSS partners, their location and decay properties, depend drastically on the nature of this state. In this contribution, we analyze the implications of HQSS under the assumption that the X(3872) is a weakly bound D D∗ molecule.. We demonstrate that non-perturbative pion dynamics together with the leading HQSS breaking corrections result in a significant shift of the 2++ partner binding energy and make it as broad as 50 MeV.
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