Predicting isovector charmonium-like states from X(3872) properties

A bstract Using chiral effective field theory, we predict that there must be isovector charmonium-like $$ D{\overline{D}}^{\ast } $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>D</mml:mi> <mml:msup> <mml:mover> <mml:mi>D</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> hadronic molecules with J PC = 1 ++ denoted as W c 1 . The inputs are the properties of the X (3872), including its mass and the ratio of its branching fractions of decays into J / ψρ 0 and J / ψω . The predicted states are virtual state poles of the scattering matrix, pointing at a molecular nature of the X (3872) as well as its spin partners. They should show up as either a mild cusp or dip at the $$ D{\overline{D}}^{\ast } $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>D</mml:mi> <mml:msup> <mml:mover> <mml:mi>D</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> thresholds, explaining why they are elusive in experiments. The so far negative observation also indicates that the X (3872) is either a bound state with non-vanishing binding energy or a virtual state, only in these cases the X (3872) signal dominates over that from the $$ {W}_{c1}^0 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>W</mml:mi> <mml:mrow> <mml:mi>c</mml:mi> <mml:mn>1</mml:mn> </mml:mrow> <mml:mn>0</mml:mn> </mml:msubsup> </mml:math> . The pole positions are $$ {3881.2}_{-0.0}^{+0.8} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mn>3881.2</mml:mn> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>0.0</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>+</mml:mo> <mml:mn>0.8</mml:mn> </mml:mrow> </mml:msubsup> </mml:math> − $$ i{1.6}_{-0.9}^{+0.7} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>i</mml:mi> <mml:msubsup> <mml:mn>1.6</mml:mn> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>0.9</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>+</mml:mo> <mml:mn>0.7</mml:mn> </mml:mrow> </mml:msubsup> </mml:math> MeV for $$ {W}_{c1}^0 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>W</mml:mi> <mml:mrow> <mml:mi>c</mml:mi> <mml:mn>1</mml:mn> </mml:mrow> <mml:mn>0</mml:mn> </mml:msubsup> </mml:math> on the fourth Riemann sheet of the $$ {D}^0{\overline{D}}^{\ast 0} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>D</mml:mi> <mml:mn>0</mml:mn> </mml:msup> <mml:msup> <mml:mover> <mml:mi>D</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mrow> <mml:mo>∗</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:msup> </mml:math> - D + D ∗− coupled-channel system, and $$ {3866.9}_{-7.7}^{+4.6} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mn>3866.9</mml:mn> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>7.7</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>+</mml:mo> <mml:mn>4.6</mml:mn> </mml:mrow> </mml:msubsup> </mml:math> − i (0 . 07 ± 0 . 01) MeV for $$ {W}_{c1}^{\pm } $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>W</mml:mi> <mml:mrow> <mml:mi>c</mml:mi> <mml:mn>1</mml:mn> </mml:mrow> <mml:mo>±</mml:mo> </mml:msubsup> </mml:math> on the second Riemann sheet of the $$ {\left(D{\overline{D}}^{\ast}\right)}^{\pm } $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mfenced> <mml:mrow> <mml:mi>D</mml:mi> <mml:msup> <mml:mover> <mml:mi>D</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mo>∗</mml:mo> </mml:msup> </mml:mrow> </mml:mfenced> <mml:mo>±</mml:mo> </mml:msup> </mml:math> single-channel system. The findings imply that the peak in the J / ψπ + π − invariant mass distribution is not purely from the X (3872) but contains contributions from $$ {W}_{c1}^0 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>W</mml:mi> <mml:mrow> <mml:mi>c</mml:mi> <mml:mn>1</mml:mn> </mml:mrow> <mml:mn>0</mml:mn> </mml:msubsup> </mml:math> predicted here. The states should have isovector heavy quark spin partners with J PC = 0 ++ , 2 ++ and 1 +− , with the last one corresponding to Z c . We suggest to search for the charged 0 ++ , 1 ++<