Covariant tetraquark equations in quantum field theory

We derive general covariant coupled equations of QCD describing the tetraquark in terms of a mix of four-quark states $2q2\bar q$, and two-quark states $q\bar q$. The coupling of $2q2\bar q$ to $q\bar q$ states is achieved by a simple contraction of a four-quark $q\bar q$-irreducible Green function down to a two-quark $q\bar q$ Bethe-Salpeter kernel. The resulting tetraquark equations are expressed in an exact field theoretic form, and are in agreement with those obtained previously by consideration of disconnected interactions; however, despite being more general, they have been derived here in a much simpler and more transparent way.


I. INTRODUCTION
In Quantum Field Theory (QFT) the number of particles is not conserved. This fact necessitates a careful theoretical definition of an exotic particle. In particular, to define a tetraquark, an exotic bound state of two quarks and two antiquarks (2q2q) whose existence has recently been evidenced [1][2][3], requires more subtlety than to simply associate it with a pole in the 4-body 2q2q Green function G (4) . Indeed, in QFT the existence of a tetraquark is signalled by a pole in G (4) ir , the qq-irreducible part of G (4) , even though the physical mass of this tetraquark is determined by the corresponding pole position in the full Green function G (4) (which will generally be shifted, perhaps slightly, with respect to the pole position in its qq-irreducible part). This fact is made clear in Fig. 1 which demonstrates that any pole in the two-body qq Green function G (2) will automatically appear in G (4) , making a pole in G (4) an insufficient criterion for a tetraquark.
Similarly, if a candidate tetraquark pole is found in the qq Green function G (2) , a sign that it is indeed a tetraquark is the existence of a corresponding pole in K (2) 4q−red , the 2q2q (4q)-reducible part of the two-body Bethe-Salpeter (BS) kernel K (2) . These observations are particularly relevant to the recent efforts to describe tetraquarks using covariant few-body equations [4][5][6]. The initial such formulation [4] was based on an analysis of only the qq-irreducible part of the 2q2q Green function, G (4) ir (although this fact was not emphasized at the time), and as coupling to qq channels was neglected, this analysis may not be the most accurate. The present paper therefore addresses the more recent formulations of covariant few-body equations describing the four-body 2q2q system where coupling to two-body qq states is included [5,6]. The ultimate goal of such equations is to describe tetraquarks in terms of identical poles in the full 2q2q and qq Green functions, G (4) and G (2) , with both poles being due to a common pole in G (4) ir . Unfortunately there is currently no consensus on the form such equations take in the approximation where only two-body forces are retained in the equation for G (4) ir and where the underlying dynamics is dominated by meson-meson (MM) and diquark-antidiquark (DD) components.
A serious issue facing all relativistically covariant derivations of equations that couple fourbody to two-body states, is the appearance of overcounted terms. This type of overcounting first came to light in the analogous problem of formulating covariant few-body equations for the pion-two-nucleon (πNN) system where coupling to two-nucleon (NN) states is included [7]; there, it was found that in order to attain overcounting-free equations where all possible twobody interactions are retained, special subtraction terms needed to be introduced and certain three-body forces needed to be retained. Although covariant few-body equations for the coupled qq − 2q2q system can be derived in an analogous manner to that for the NN − πNN system, as in Ref. [7], current interest is in formulating a less detailed approach that applies specifically to the case where the qq − 2q2q system is dominated by MM and DD components. It is in this context that we would like to revisit the formulation of the coupled qq − 2q2q system. In the process, we aim to help resolve the differences between the aforementioned tetraquark equations ir is the qq-irreducible part of G (4) , with M by presenting a derivation that does not involve the introduction of any explicit disconnected contributions to two-body (qq) interactions, viewed as potentially problematic in Ref. [6], but which were pivotal to our previous derivation [5]. Despite the very different approach presented here, the tetraquark equations resulting from the new derivation are in agreement with those of Ref. [5], and moreover, are obtained in a simpler and much more transparent way.

II. TETRAQUARK POLES AND WAVE FUNCTIONS
In the context of QFT, to formulate a few-body approach for a system of particles where some of them can be absorbed by others (e.g. π by N in the πNN system or a qq pair that is annihilated in the 2q2q system) one starts with the general structure of the full few-body Green function, which in the case of the 2q2q system is manifested by the decomposition where G (2) 0 is the disconnected part of the two-body qq Green function G (2) corresponding to the independent propagation of q andq in the s channel, and G ) is the sum of all qq-irreducible diagrams corresponding to the transition qq ← 2q2q (2q2q ← qq). Equation (1) is illustrated in Fig. 1 where G The main problem is then to express G ir can be expressed in terms of four-body scattering equations that are valid in the absence of qq annihilation.
As discussed above, the signature for the formation of a tetraquark in the 2q2q system is the occurrence of a pole in G (4) ir . In the current problem setting, a pole in G (4) ir is similarly the signature for the formation of a tetraquark bound state in the qq system. The contribution of G (4) ir to the qq Green function G (2) takes place through the BS kernel K (2) which, by definition, is qq-irreducible and is related to G (2) through the Dyson equation 1 In particular, the signature tetraquark bound state pole occurs in the 2q2q-reducible part of K (2) , which can be expressed as where A (4−2) (A (2−4) ) is some amplitude (of course qq-irreducible) corresponding to the transition 2q2q ← qq (qq ← 2q2q). The full analysis of these functions would be similar to the one used for the πNN system in the covariant approach of Ref. [7], but this will be investigated elsewhere. In the present paper we use the 4q-reducible part of the two-body potential, Eq. (3), to look for a tetraquark solution in the 2-body equation (of course if the potential itself possesses the corresponding pole at the tetraquark "bare" mass value); however, we follow the existing approaches where MM and DD dominance is assumed. Ultimately, we shall be interested in the case where G (4) and G (2) display simultaneous poles corresponding to a tetraquark of mass M, so that as P 2 → M 2 where P is the total momentum of each system, In Eq. (4), Ψ is the tetraquark 4-body (2q2q) bound state wave function, while Γ * is the form factor for the disintegration of a tetraquark into a qq pair. We note that the definition of Ψ and Γ * via the pole parts of G (4) and G (2) in Eq. (4), together with Eq. (1) relating G (4) and G (2) , leads to the relation between Ψ and Γ * , As is evident from Eq. (2) and the second of the relations in Eq. (4), a tetraquark state will also satisfy the two-body (not 4-body) equation, It is Eq. (6) which will be used in this paper to formulate the tetraquark equations. This will be achieved by first constructing G (4) ir using the four-body equations of Khvedelidze and Kvinikhidze [8], together with a pole approximation Ansatz for all quark pair scattering amplitudes, and then using Eq. (3) to generate the essential part of the qq kernel.

III. TETRAQUARK FEW-BODY EQUATIONS
The approach used here to derive covariant equations for the coupled qq − 2q2q system is different from that employed by us in Ref. [5]. Instead of incorporating coupling to qq states right from the outset, as embodied in the full 4-body Green function G (4) , here we first consider a formulation of 4-body tetraquark equations for the case where there is no coupling to qq states; that is, we first consider a formulation based on G (4) ir , the qq-irreducible part of G (4) . Coupling to qq states is then achieved by generating the qq kernel K (2) through a simple contraction of 4-body to 2-body states as in Eq. (3).
One can express G (4) ir in terms of the qq-irreducible 4-body interaction kernel K (4) ir through the Dyson equation where G (4) 0 is the fully disconnected part of G (4) . For simplicity, we start out by treating the quarks as distinguishable particles; however, the full antisymmetry of quark states will be taken into account shortly. The kernel K (4) ir can be formally expressed as 2F consists of only pair-wise interactions, and K (4) 3F consists of all other contributions, necessarily involving three-and four-body forces. Assigning labels 1,2 to the quarks and 3,4 to the antiquarks, one can write K 2F as a sum of three terms whose structure is illustrated in Fig. 2, and correspondingly expressed as where the index a ∈ {12, 13, 14, 23, 24, 34} enumerates six possible pairs of particles, the double index aa ′ ∈ {(13, 24), (14, 23), (12, 34)} enumerates three possible two pairs of particles, and the Greek index α is used as an abbreviation for aa ′ such that α = 1 denotes aa ′ = (13, 24), α = 2 denotes aa ′ = (14, 23), and α = 3 denotes aa ′ = (12, 34). Thus K (4) aa ′ describes the part of the four-body kernel where all interactions are switched off except those within the pairs a and a ′ . As is well known [4,5,8], K aa ′ can be expressed in terms of the two-body kernels K (2) a and K (2) where G 0 a ( G 0 a ′ ) is the 2-body disconnected Green function for particle pair a (a ′ ). It is also useful to introduce the corresponding 4-body qq-irreducible t matrix T (4) ir defined by equation One can similarly express T ir as a sum of three terms [8] T with components T (4) α satisfying Faddeev-like equations whereδ αβ = 1 − δ αβ and where the Greek subscripts run over the three possible "two pairs" of particles as in Eq. (9). In Eq. (13), T α is the t matrix corresponding to kernel K (4) α , that is with T α being expressed in terms of two-body t matrices T (2) a and T a ′ as A. Tetraquark equations with no coupling to qq states To compare with the existing approaches [4][5][6], our aim is to describe the tetraquark using two-body equations that couple identical meson-meson (MM), and diquark-antidiquark (DD) channels. To this end we consider G (4) ir in the approximation where the two-body t matrices T a and T a ′ are expressed in the bound state pole approximation where D a (P a ) = 1/(P 2 a − m 2 a ) is the propagator for the bound particle (diquark, antidiquark, or meson) in the two-body channel a. Showing explicit dependence on momentum variables, T a , for a = 12, can be expressed as T (2) 12 (p ′ 1 p ′ 2 , p 1 p 2 ) = iΓ(p ′ 1 p ′ 2 )D 12 (P )Γ(p 1 p 2 ), where P = p 1 + p 2 is the total off-mass-shell momentum of the bound particle.
As discussed above, the signature for a tetraquark is the existence of a pole in G ir . In turn, this means the existence of a 4-body tetraquark wave function Ψ ir ≡ G (4) 0 ψ for the case where all coupling to qq states is switched off. We therefore begin by considering the corresponding bound state form factor ψ for the case of 2 indistinguishable quarks and 2 indistinguishable antiquarks, and relate it to the corresponding form factor ψ d for distinguishable quarks as where P ij is the operator exchanging the quantum numbers of particles i and j. The Faddeevlike equations for ψ d are [4,8], whereδ αβ = 1−δ αβ and the Greek subscripts run over the three possible "two pairs" of particles as in Eq. (9). Using the approximations of Eqs. (16), one can write it follows from Eq. (18b) that Noting that Γ 12 = −Γ 21 and Γ 34 = −Γ 43 , it follows that V 12 = V 21 , V 23 = −V 13 and V 32 = −V 31 . We can now use Eq. (17) to derive MM and DD components of the tetraquark form factor ψ in the case of indistinguishable quarks. These are defined by the pole contributions to ψ at p 2 13 = M 2 π , p 2 24 = M 2 π , p 2 14 = M 2 π , p 2 23 = M 2 π , p 2 12 = M 2 D , and p 2 34 = M 2 D , where p ij = p i + p j is the total momentum of particles i and j, M π is the mass of the meson and M D is the mass of the diquark or antidiquark. To this end consider the use of Eq. (21) in Eq. (17): where are symmetric functions under the exchange of the meson quantum numbers, and define the MM and DD components of the tetraquark form factor ψ where quarks are identical.
To derive equations for the tetraquark vertex functions for identical quarks, we first write out Eq. (21) for distinguishable quarks using notation Then, subtracting the second line from the first, we obtain a set of two equations for φ where we used D 1 = D 2 ≡ MM, D 3 ≡ DD. Equations (27) can be written in matrix form as To finally derive the tetraquark equations in the case of indistinguishable quarks, note that according Eqs. (24), where P is permutation operator of the meson state labels. Thus, symmetrizing Eqs. (27) with respect to meson legs gives where we have used the following symmetry properties of V 12 and V 1D : Equations (30) can be written in matrix form as and thereby revealing V of Eq. (34) to be the interaction kernel for the coupled MM − DD system. The elements of V involve the potentials as illustrated in Fig. 3. With the kernel matrix V established, one can determine the t matrix T defined by and thereafter G M 0 + G M 0 T G M 0 , which is the matrix Green function in coupled MM-DD space corresponding to G (4) ir . As indicated by Eq. (3), the 4q-reducible part of the two-body qq kernel, K 4q−red , can be found by sandwiching G (4) ir between amplitudes that contract 2q2q states to qq states. In the present case of coupled MM-DD channels, this contraction can be expressed as where S 23 is the quark propagator connecting quark lines 2 and 3. Equations (38) are illustrated in Fig. 4.
Using the formal solution to Eq. (36), we can write the general expression for the two-body qq kernel as where ∆ is defined to be the sum of all qq-irreducible contributions allowed by QFT that are not accounted for by the last term of of Eq. (40). In particular, ∆ includes correction terms that account for the difference between the approximations used in Eqs. (16), and exact QFT, thus making Eq. (40) an exact expression for K (2) . It is important that none of the of 2q2qreducible diagrams in the last term of Eq. (40) are overcounted, therefore ∆ should not contain counter-terms for eliminating overcounting. 2 As such, ∆ can be used in future studies to take into account effects such as one-gluon exchange, one-meson exchange, etc.
B. Tetraquark equations with coupling to qq states Equation (32) constitutes the matrix form of the tetraquark equations without coupling to qq states. It expresses the column matrix φ of tetraquark form factors φ M and φ D , in terms of potentials contained in matrix V . To derive the corresponding tetraquark equations with coupling to qq states, we simply use the kernel K (2) of Eq. (40) in Eq. (6), the bound state equation for the tetraquark form factor Γ * : Equation (42) is the matrix form of the sought-after tetraquark equations with coupling to qq states. It expresses the column matrix Φ of tetraquark form factors Φ M and Φ D in terms of both the potentials contained in matrix V , and the tetraquark form factor Γ * describing the disintegration of a tetraquark into a qq pair. We can write Eq. (42) explicitly as where Γ M = Γ 13 Γ 24 ,Γ M =Γ 13Γ24 , Γ D = Γ 12 Γ 34 ,Γ D =Γ 12Γ34 , and P ij is the operator exchanging quarks i and j, thereforē Thus the tetraquark equations with coupling to qq included take the form of three coupled equations which are illustrated in Fig. 5. Since ∆ is defined in a way that makes the expression used for K (2) exact, Eqs. (45) represent the most general form of the tetraquark equations in QFT.

IV. CONCLUSIONS
We have derived a set of covariant coupled equations for the tetraquark, Eqs. (45), using a model where the two-body qq, qq, andqq interactions are dominated by the formation of a meson, a diquark, and an antidiquark, respectively. Nevertheless, Eqs. (45) constitute the most general form of the tetraquark equations in QFT since all differences between the model used and exact QFT are accounted for by correction terms formally included in the term ∆. These equations determine the form factors Φ M , Φ D , and Γ * of the tetraquark, describing its disintegration into two identical mesons, a diquark-antidiquark pair, and a quark-antiquark pair. As such, they extend the purely four-body (4q) tetraquark equations of Ref. [4] to include coupling to two-body(2q) qq states.
The motivation for the present work comes from the need to have exact quantum field theoretic equations describing the tetraquark, but formulated for the case where the dynamics is dominated by meson-meson and diquark-antidiquark components. This is especially important in view of the lack of agreement between two previous attempts to calculate tetraquark equations with 4q-2q mixing. The first of these was our derivation of 2014 [5] using a careful but involved incorporation of disconnected qq interactions as a means of incorporating qq annihilation into a 4q description. The second of these was a recent derivation [6] where coupling to 2q channels was included phenomenologically, and where some doubt was expressed regarding the incorporation of disconnected qq interactions. Our present derivation of Eqs. (45) has therefore been based on a method that avoids any explicit introduction of disconnected qq interactions, and which, in the absence of approximations for ∆, provides an exact field-theoretic description. It is therefore gratifying to note that in the absence of the term ∆, Eqs. (45) coincide with the equations derived by us in Ref. [5]. Indeed, settting ∆ = 0 in Eq. (45c) and substituting into Eq. (43) gives Φ in the form presented in Ref. [5] Here NG (2) 0N is the qq reducible part of the kernel which is denoted by V qq in Ref. [5]. It accounts for the qq admixture through the qq propagator G (2) 0 . By contrast, the tetraquark equations of Ref. [6] are not consistent with the general form prescribed by Eqs. (45).
Finally, it is worth noting that in comparison with our previous derivation [5] , the approach taken in the present work allows for the derivation of the tetraquark equations in a much simpler and more clear way.