Comment on "σ-meson: Four-quark versus two-quark components and decay width in a Bethe-Salpeter approach"
aa r X i v : . [ h e p - ph ] F e b Comment on “ σ -meson: Four-quark versus two-quark componentsand decay width in a Bethe-Salpeter approach” B. Blankleider
College of Science and Engineering, Flinders University, Bedford Park, SA 5042, Australia ∗ A. N. Kvinikhidze
Andrea Razmadze Mathematical Institute of Tbilisi State University,6, Tamarashvili Str., 0186 Tbilisi, Georgia † (Dated: March 2, 2021) Abstract
In a recent paper by N. Santowsky et al. [Phys. Rev. D , 056014 (2020)], covariant coupledequations were derived to describe a tetraquark in terms of a mix of four-quark states 2 q q and two-quark states q ¯ q . These equations were expressed in terms of vertices describing the disintegration of atetraquark into identical two-meson states, into a diquark-antidiquark pair, and into a quark-antiquarkpair. We show that these equations are inconsistent as they imply a q ¯ q Bethe-Salpeter kernel that is q ¯ q -reducible. ∗ boris.blankleider@flinders.edu.au † sasha [email protected]
1n 2012, Heupel, Eichmann and Fischer (HEF) [1] developed covariant equations describinga tetraquark using a model where the two-quark plus two-antiquark (2 q q ) system is describedby four-body (4 q ) Faddeev-like equations of Khvedelidze and Kvinikhidze [2], and where thedynamics is dominated by the formation of either two identical mesons or a diquark-antidiquarkpair. These equations are represented graphically in Fig. 1, and relate the form factors φ M and φ D of the tetraquark, describing its disintegration into two identical mesons, and a diquark-antidiquark pair, respectively. As is evident from Fig. 1, the input interactions to these equa-tions consist of vertices for the transitions between a meson ( M ) and a quark-antiquark pair( q ¯ q ↔ M ), a diquark ( D ) and a quark-quark pair ( qq ↔ D ), and between an antidiquark ( ¯ D )and an antiquark-antiquark pair (¯ q ¯ q ↔ ¯ D ). Missing from these equations is the phenomenonof quark-antiquark annihilation which would result in coupling to two-body (2 q ) q ¯ q states.There have since been two attempts to extend the equations of HEF to include coupling to q ¯ q channels. The first of these was our derivation of 2014 [3] where disconnected contributions wereadded to the usual connected part of the q ¯ q interaction. The second was a recent derivation ofSantowsky et al. (SEFWW) [4] where coupling to q ¯ q channels was included phenomenologically.The tetraquark equations of SEFWW are represented graphically in Fig. 2, and include anadditional tetraquark form factor Γ ∗ , describing the disintegration of a tetraquark into a q ¯ q pair.For the tetraquark equations of Fig. 2 to be meaningful, it is essential that the form factorΓ ∗ be identified with the residue of the q ¯ q Green function G (2) describing the formation of thetetraquark in the scattering of a quark from an antiquark; that is, as P → M , where P is thetotal momentum of the q ¯ q system and M is the mass of the tetraquark, G (2) → G (2)0 Γ ∗ ¯Γ ∗ G (2)0 P − M , (1)where G (2)0 is the fully disconnected q ¯ q propagator corresponding to the independent propaga-tion of q and ¯ q in the s channel. This implies that Γ ∗ satisfies the bound state equationΓ ∗ = K ir G (2)0 Γ ∗ (2)where K ir is the q ¯ q -irreducible Bethe-Salpeter kernel for the q ¯ q system.Here we would like to point out that the tetraquark equations of SEFWW cannot be correctas they imply a kernel K ir that is q ¯ q -reducible. To show this, we write the three coupled φ M = φ M + φ D φ D = φ M FIG. 1. Tetraquark equations without coupling to q ¯ q channels, as first developed in Ref. [1].Tetraquark form factors φ M (displayed in red) couple to two mesons (dashed lines), and tetraquarkform factors φ D (displayed in blue) couple to diquark and antidiquark states (double-lines). M = Φ M + Φ D + Γ ∗ Φ D = Φ M + Γ ∗ Γ ∗ = K (2) Γ ∗ + K (2) Φ M + K (2) Φ D FIG. 2. The tetraquark equations of SEFWW [4] which include coupling to q ¯ q channels. In additionto the tetraquark form factors as in Fig. 1, these equations involve the tetraquark form factor Γ ∗ (displayed in yellow) that couples to q ¯ q states (solid lines). The amplitude K (2) (displayed in lightblue) represents the q ¯ q kernel in a theory without q ¯ q annihilation. equations corresponding to Fig. 2 as Φ M = V MM G M Φ M + V MD G D Φ D + N M G (2)0 Γ ∗ , (3a)Φ D = V DM G M Φ M + N D G (2)0 Γ ∗ , (3b)Γ ∗ = K (2) G (2)0 Γ ∗ + K (2) G (2)0 ¯ N M G M Φ M + K (2) G (2)0 ¯ N D G D Φ D , (3c)where V MM , V MD , and V DM , are quark-exchange potentials for the processes M M ← M M , M M ← D ¯ D , and D ¯ D ← M M , respectively, and where N M , N D , ¯ N M , and ¯ N D describe thetransitions between 4 q and 2 q states via the processes M M ← q ¯ q , D ¯ D ← q ¯ q , q ¯ q ← M M ,and q ¯ q ← D ¯ D . Note that these equations also involve a q ¯ q kernel K (2) which should not beconfused with K ir as it does not contain terms that involve 2 q ↔ q transitions.Writing Eqs. (3) in matrix form asΦ = V G Φ +
N G (2)0 Γ ∗ , (4a)Γ ∗ = K (2) G (2)0 (cid:0) Γ ∗ + ¯ N G Φ (cid:1) , (4b)where Φ = (cid:18) Φ M Φ D (cid:19) , G = (cid:18) G M G D (cid:19) , (5) N = (cid:18) N M N D (cid:19) , ¯ N = (cid:0) ¯ N M ¯ N D (cid:1) , (6)and V = (cid:18) V MM V MD V DM (cid:19) , (7) For simplicity, we ignore all symmetry factors in Eqs. (3) as they do not affect our argument. ∗ = h K (2) + K (2) G (2)0 ¯ N G (cid:0) − V G (cid:1) − N i G (2)0 Γ ∗ . (8)Comparison with Eq. (2) shows that K ir = K (2) + K (2) G (2)0 ¯ N G (cid:0) − V G (cid:1) − N, (9)which is in conflict with the very definition of a q ¯ q kernel since this expression for K ir is q ¯ q reducible (notice the presence of the q ¯ q propagator G (2)0 ). For this reason the tetraquarkequations of Ref. [4] are inconsistent. [1] W. Heupel, G. Eichmann, and C. S. Fischer, Tetraquark Bound States in a Bethe-Salpeter Ap-proach, Phys. Lett. B718 , 545 (2012), arXiv:1206.5129 [hep-ph].[2] A. M. Khvedelidze and A. N. Kvinikhidze, Pair interaction approximation in the equations ofquantum field theory for a four-body system, Theor. Math. Phys. , 62 (1992).[3] A. N. Kvinikhidze and B. Blankleider, Covariant equations for the tetraquark and more,Phys. Rev. D , 045042 (2014), arXiv:1406.5599 [hep-ph].[4] N. Santowsky, G. Eichmann, C. S. Fischer, P. C. Wallbott, and R. Williams, σ -meson:Four-quark versus two-quark components and decay width in a Bethe-Salpeter approach,Phys. Rev. D , 056014 (2020), arXiv:2007.06495 [hep-ph]., 056014 (2020), arXiv:2007.06495 [hep-ph].