In the past decade, due to the experimental observation of many charmonium-like states, there has been a revival of hadron spectroscopy. In particular, the experimental observation of charged charmonium-like, $Z_c$ states, and bottomonium-like, $Z_b$
states, represents a challenge since they can not be accommodated within the naive quark model. These charged states are good candidates of either tetraquark or molecular states and their observation motivated a vigorous theoretical activity. This is a rapidly evolving field with enormous amount of new experimental information. In this work, we review the current experimental progress and investigate various theoretical interpretations of these candidates of the multiquark states. The present review is written from the perspective of the QCD sum rules approach, where we present the main steps of concrete calculations and compare the results with other approaches and with experimental data.
Using the QCD sum rules we test if the charmonium-like structure Y(4260), observed in the $J/psipipi$ invariant mass spectrum, can be described with a $J/psi f_0(980)$ molecular current with $J^{PC}=1^{--}$. We consider the contributions of condensat
es up to dimension six and we work at leading order in $alpha_s$. We keep terms which are linear in the strange quark mass $m_s$. The mass obtained for such state is $m_{Y}=(4.67pm 0.09)$ GeV, when the vector and scalar mesons are in color singlet configurations. We conclude that the proposed current can better describe the Y(4660) state that could be interpreted as a $Psi(2S) f_0(980)$ molecular state. We also use different $J^{PC}=1^{--}$ currents to study the recently observed $Y_b(10890)$ state. Our findings indicate that the $Y_b(10890)$ can be well described by a scalar-vector tetraquark current.
Using the QCD sum rules we test if the new narrow structure, the X(4350) recently observed by the Belle Collaboration, can be described as a $J^{PC}=1^{-+}$ exotic $D_s^*D_{s0}^*$ molecular state. We consider the contributions of condensates up to di
mension eight, we work at leading order in $alpha_s$ and we keep terms which are linear in the strange quark mass $m_s$. The mass obtained for such state is $m_{D_s^*{D}_{s0}^*}=(5.05pm 0.19)$ GeV. We also consider a molecular $1^{-+}, D^{*}{D}_0^{*}$ current and we obtain $m_{D^*{D}_0^*}=(4.92pm 0.08)$ GeV. We conclude that it is not possible to describe the X(4350) structure as a $1^{-+} D_s^*{D}_{s0}^*$ molecular state.
We use the QCD sum rules to evaluate the mass of a possible scalar mesonic state that couples to a molecular $D_{s}^{*}bar{D}_s^{*}$ current. We find a mass $m_{D_s^*D_s^*}=(4.14pm 0.09)$ GeV, which is in a excellent agreement with the recently obser
ved Y(4140) charmonium state. We consider the contributions of condensates up to dimension eight, we work at leading order in $alpha_s$ and we keep terms which are linear in the strange quark mass $m_s$. We also consider a molecular $D^{*}bar{D}^{*}$ current and we obtain $m_{D^*{D}^*}=(4.13pm 0.10)$, around 200 MeV above the mass of the Y(3930) charmonium state. We conclude that it is possible to describe the Y(4140) structure as a $D_s^*bar{D}_s^*$ molecular state.
