Erratum to: Spin-1 diquark contributing to the formation of tetraquarks in light mesons

Abstract

In the paper above, we have proposed a tetraquark picture with the mixing scheme for the I z = 1 members of the isovector (I = 1) resonances, a + 0 (980), a + 0 (1450). In particular, their mass splittings fit relatively well with the hyperfine mass splittings if they are viewed as mixtures of two spin-configurations of diquark-antidiquark constituents, |J, J 12 , J 34 = |000 , |011 , where J is the tetraquark spin, J 12 the diquark spin, J 34 the antidiquark spin. The second configuration involving the spin-1 diquark, |011 , is found to be an important ingredient in explaining the resonances of our concern in this tetraquark picture. However, the existence of the |011 component requires additional tetraquarks to be found in J = 1 and J = 2 resonances with the spin configurations, |J, J 12 , J 34 = |111 and |211 , respectively.In this erratum, we point out that our assignment of a + 1 (1260) as a candidate for the J = 1 tetraquark with the |111 configuration is incorrect because of the C-parity for its corresponding member in I z = 0. Specifically, we would like to demonstrate that the |111 state with I = 1, I z = 0 must have the C-parity odd and, in this regard, a relevant candidate for the |111 state should be b 0 1 (1235). So its charged member (I = 1, I z = 1), which in fact was considered in our paper, must be b To demonstrate that C|111 = −|111 for the isospin member of I = 1, I z = 0, we take the state with J = 1 and the spin projection M = 1 among three spin states in |111 , and we denote this state as |J M = |11 . The same proof can be done for the other spin states, |J M = |10 , |1 − 1 . The flavor structure of the member I = 1, Now it is straightforward to prove that the state above has C = − by applying the charge conjugation [Eq. (2) 123

Testing the tetraquark mixing framework from QCD sum rules for a0(980)

“…To motivate the construction of an interpolating field in our QCD sum rule study, we briefly look at the mathematical structure of the tetraquark mixing framework advocated by Refs. [19][20][21][22]. The mixing framework has been developed as a possible structure for the two nonets in PDG, the light nonet composed of a 0 (980), K * 0 (800), f 0 (500), f 0 (980), and the heavy nonet of a 0 (1450), K * 0 (1430), f 0 (1370), f 0 (1500).…”

“…The second tetraquark type, |011 , which was suggested as another possibility in Refs. [19][20][21][22]32], is constructed by combining the spin-1 diquark in the structure of (6 c ,3 f ) and its antidiquark. By construction, the two tetraquark types differ by color and spin configurations but they have the same flavor structure, namely the nonet.…”

“…Here α, β are the mixing parameters determined in each isospin channel by the diagonalization process. But they are found to be almost independent of isospin [19][20][21][22] and their values are approximately close to α ≈ 2/3, β ≈ 1/3. This basically implies that the wave functions in Eqs.…”

“…Indeed, as recently advocated by Refs. [19][20][21][22], one can construct the second tetraquark by using the spin-1 diquark with the color and flavor structure (6 c ,3 f ). Even though the spin-1 diquark is less compact than the spin-0 diquark, the total binding energy of the second tetraquark, if calculated from the color-spin interaction for all the pairs, is found to be more attractive than the binding from the first tetraquark type involving the spin-0 diquark.…”

“…In practice, the QCD sum rule [24][25][26] or lattice QCD calculation [27] can be used for this purpose. One surprising result to test is the statement that the light nonet has more probability to stay in the second type tetraquark containing the spin-1 diquark rather than in the first type involving the spin-0 diquark [19][20][21][22]. This picture is very different from the common expectation that the light nonet has the structure of the first tetraquark type only [15-18, 28, 29].…”