Covariant tetraquark equations in quantum field theory
aa r X i v : . [ h e p - t h ] F e b Covariant tetraquark equations in quantum field theory
A. N. Kvinikhidze
Andrea Razmadze Mathematical Institute of Tbilisi State University,6, Tamarashvili Str., 0186 Tbilisi, Georgia ∗ B. Blankleider
College of Science and Engineering, Flinders University, Bedford Park, SA 5042, Australia † (Dated: February 19, 2021) Abstract
We derive general covariant coupled equations of QCD describing the tetraquark in terms of a mixof four-quark states 2 q q , and two-quark states q ¯ q . The coupling of 2 q q to q ¯ q states is achieved bya simple contraction of a four-quark q ¯ q -irreducible Green function down to a two-quark q ¯ q Bethe-Salpeter kernel. The resulting tetraquark equations are expressed in an exact field theoretic form, andare 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. ∗ sasha [email protected] † boris.blankleider@flinders.edu.au . INTRODUCTION In Quantum Field Theory (QFT) the number of particles is not conserved. This fact neces-sitates a careful theoretical definition of an exotic particle. In particular, to define a tetraquark,an exotic bound state of two quarks and two antiquarks (2 q q ) whose existence has recentlybeen evidenced [1–3], requires more subtlety than to simply associate it with a pole in the4-body 2 q q Green function G (4) . Indeed, in QFT the existence of a tetraquark is signalled bya pole in G (4) ir , the q ¯ q -irreducible part of G (4) , even though the physical mass of this tetraquarkis determined by the corresponding pole position in the full Green function G (4) (which willgenerally be shifted, perhaps slightly, with respect to the pole position in its q ¯ q -irreduciblepart). This fact is made clear in Fig. 1 which demonstrates that any pole in the two-body q ¯ q Green function G (2) will automatically appear in G (4) , making a pole in G (4) an insufficientcriterion for a tetraquark.Similarly, if a candidate tetraquark pole is found in the q ¯ q Green function G (2) , a sign that it isindeed a tetraquark is the existence of a corresponding pole in K (2)4 q − red , the 2 q q (4 q )-reduciblepart of the two-body Bethe-Salpeter (BS) kernel K (2) . These observations are particularlyrelevant to the recent efforts to describe tetraquarks using covariant few-body equations [4–6]. The initial such formulation [4] was based on an analysis of only the q ¯ q -irreducible partof the 2 q q Green function, G (4) ir (although this fact was not emphasized at the time), andas coupling to q ¯ q channels was neglected, this analysis may not be the most accurate. Thepresent paper therefore addresses the more recent formulations of covariant few-body equationsdescribing the four-body 2 q q system where coupling to two-body q ¯ q states is included [5, 6].The ultimate goal of such equations is to describe tetraquarks in terms of identical poles inthe full 2 q q and q ¯ q Green functions, G (4) and G (2) , with both poles being due to a commonpole in G (4) ir . Unfortunately there is currently no consensus on the form such equations take inthe approximation where only two-body forces are retained in the equation for G (4) ir and wherethe underlying dynamics is dominated by meson-meson ( M M ) and diquark-antidiquark ( D ¯ D )components.A serious issue facing all relativistically covariant derivations of equations that couple four-body to two-body states, is the appearance of overcounted terms. This type of overcountingfirst came to light in the analogous problem of formulating covariant few-body equations for thepion-two-nucleon ( πN N ) system where coupling to two-nucleon ( N N ) states is included [7];there, it was found that in order to attain overcounting-free equations where all possible two-body interactions are retained, special subtraction terms needed to be introduced and certainthree-body forces needed to be retained. Although covariant few-body equations for the coupled q ¯ q − q q system can be derived in an analogous manner to that for the N N − πN N system,as in Ref. [7], current interest is in formulating a less detailed approach that applies specificallyto the case where the q ¯ q − q q system is dominated by M M and D ¯ D components. It is in thiscontext that we would like to revisit the formulation of the coupled q ¯ q − q q system. In theprocess, we aim to help resolve the differences between the aforementioned tetraquark equations G (4) = G (4) ir + M (4 − ir G (2) M (2 − ir FIG. 1. Field theoretic structure of the 2 q q Green function G (4) , where G (4) ir is the q ¯ q -irreducible partof G (4) , with M (4 − ir and M (2 − ir being q ¯ q -irreducible 2 q ← q ¯ q and q ¯ q ← q q transition amplitudes,respectively.
2y presenting a derivation that does not involve the introduction of any explicit disconnectedcontributions to two-body ( q ¯ q ) interactions, viewed as potentially problematic in Ref. [6], butwhich were pivotal to our previous derivation [5]. Despite the very different approach presentedhere, the tetraquark equations resulting from the new derivation are in agreement with thoseof 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 wheresome of them can be absorbed by others (e.g. π by N in the πN N system or a q ¯ q pair thatis annihilated in the 2 q q system) one starts with the general structure of the full few-bodyGreen function, which in the case of the 2 q q system is manifested by the decomposition G (4) = G (4) ir + G (4 − ir G (2) − G (2) G (2) − G (2 − ir , (1)where G (2)0 is the disconnected part of the two-body q ¯ q Green function G (2) corresponding tothe independent propagation of q and ¯ q in the s channel, and G (2 − ir ( G (4 − ir ) is the sum ofall q ¯ q -irreducible diagrams corresponding to the transition q ¯ q ← q q (2 q q ← q ¯ q ). Equation(1) is illustrated in Fig. 1 where G (4 − ir G (2) − ≡ M (4 − ir and G (2) − G (2 − ir ≡ M (2 − ir . The mainproblem is then to express G (2 − ir and G (4 − ir in terms of G (4) ir , while G (4) ir can be expressed interms of four-body scattering equations that are valid in the absence of q ¯ q annihilation.As discussed above, the signature for the formation of a tetraquark in the 2 q q system isthe occurrence of a pole in G (4) ir . In the current problem setting, a pole in G (4) ir is similarly thesignature for the formation of a tetraquark bound state in the q ¯ q system. The contribution of G (4) ir to the q ¯ q Green function G (2) takes place through the BS kernel K (2) which, by definition,is q ¯ q -irreducible and is related to G (2) through the Dyson equation G (2) = G (2)0 + G (2)0 K (2) G (2) . (2)In particular, the signature tetraquark bound state pole occurs in the 2 q q -reducible part of K (2) , which can be expressed as K (2)4 q − red = A (2 − G (4) ir A (4 − (3)where A (4 − ( A (2 − ) is some amplitude (of course q ¯ q -irreducible) corresponding to the tran-sition 2 q q ← q ¯ q ( q ¯ q ← q q ). The full analysis of these functions would be similar to theone used for the πN N system in the covariant approach of Ref. [7], but this will be investi-gated 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 itselfpossesses the corresponding pole at the tetraquark ”bare” mass value); however, we follow theexisting approaches where M M and D ¯ D dominance is assumed.Ultimately, we shall be interested in the case where G (4) and G (2) display simultaneous polescorresponding to a tetraquark of mass M , so that as P → M where P is the total momentumof each system, G (4) → i Ψ ¯Ψ P − M , G (2) → i G (2)0 Γ ∗ ¯Γ ∗ G (2)0 P − M . (4)In Eq. (4), Ψ is the tetraquark 4-body (2 q q ) bound state wave function, while Γ ∗ is the formfactor for the disintegration of a tetraquark into a q ¯ q pair. We note that the definition of Ψ Note that our definition of K (2) is different from the one used in Ref. [6]. ∗ 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 ∗ ,Ψ = G (4 − Γ ∗ . (5)As is evident from Eq. (2) and the second of the relations in Eq. (4), a tetraquark state willalso satisfy the two-body (not 4–body) equation,Γ ∗ = K (2) G (2)0 Γ ∗ . (6)It is Eq. (6) which will be used in this paper to formulate the tetraquark equations. Thiswill be achieved by first constructing G (4) ir using the four-body equations of Khvedelidze andKvinikhidze [8], together with a pole approximation Ansatz for all quark pair scattering am-plitudes, and then using Eq. (3) to generate the essential part of the q ¯ q kernel. III. TETRAQUARK FEW-BODY EQUATIONS
The approach used here to derive covariant equations for the coupled q ¯ q − q q system isdifferent from that employed by us in Ref. [5]. Instead of incorporating coupling to q ¯ q statesright from the outset, as embodied in the full 4-body Green function G (4) , here we first considera formulation of 4-body tetraquark equations for the case where there is no coupling to q ¯ q states;that is, we first consider a formulation based on G (4) ir , the q ¯ q -irreducible part of G (4) . Couplingto q ¯ q states is then achieved by generating the q ¯ q kernel K (2) through a simple contraction of4-body to 2-body states as in Eq. (3).One can express G (4) ir in terms of the q ¯ q -irreducible 4-body interaction kernel K (4) ir throughthe Dyson equation G (4) ir = G (4)0 + G (4)0 K (4) ir G (4) ir (7)where G (4)0 is the fully disconnected part of G (4) . For simplicity, we start out by treating thequarks as distinguishable particles; however, the full antisymmetry of quark states will be takeninto account shortly. The kernel K (4) ir can be formally expressed as K (4) ir = K (4)2 F + K (4)3 F (8)where K (4)2 F consists of only pair-wise interactions, and K (4)3 F consists of all other contributions,necessarily involving three- and four-body forces. Assigning labels 1,2 to the quarks and 3,4to the antiquarks, one can write K (4)2 F as a sum of three terms whose structure is illustrated inFig. 2, and correspondingly expressed as K (4)2 F = X aa ′ K (4) aa ′ = X α K (4) α (9)where the index a ∈ { , , , , , } enumerates six possible pairs of particles, the doubleindex aa ′ ∈ { (13 , , (14 , , (12 , } enumerates three possible two pairs of particles, andthe Greek index α is used as an abbreviation for aa ′ such that α = 1 denotes aa ′ = (13 , α = 2 denotes aa ′ = (14 , α = 3 denotes aa ′ = (12 , K (4) aa ′ describes the partof 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 (4) aa ′ can be expressed in terms of the two-body kernels K (2) a and K (2) a ′ as K (4) aa ′ = K (2) a G a ′ − + K (2) a ′ G a − − K (2) a K (2) a ′ , (10)4 (4)1 = ¯ qq ¯ qq , K (4)2 = ¯ qq ¯ qq , K (4)3 = ¯ q ¯ qqq K (4) α ( α = 1 , ,
3) making up the 4-body kernel K (4)2 F where onlytwo-body forces are included. The three terms are summed as in Eq. (9). where G a ( G a ′ ) is the 2-body disconnected Green function for particle pair a ( a ′ ). It is alsouseful to introduce the corresponding 4-body q ¯ q -irreducible t matrix T (4) ir defined by equation G (4) ir = G (4)0 + G (4)0 T (4) ir G (4)0 . (11)One can similarly express T (4) ir as a sum of three terms [8] T (4) ir = X aa ′ T (4) aa ′ = X α T (4) α (12)with components T (4) α satisfying Faddeev-like equations T (4) α = T (4) α + X β T (4) α ¯ δ αβ G (4)0 T (4) β (13)where ¯ δ αβ = 1 − δ αβ and where the Greek subscripts run over the three possible ”two pairs” ofparticles as in Eq. (9). In Eq. (13), T (4) α is the t matrix corresponding to kernel K (4) α , that is T (4) α = K (4) α + K (4) α G (4)0 T (4) α , (14)with T (4) α being expressed in terms of two-body t matrices T (2) a and T (2) a ′ as T (4) α = T (4) aa ′ = T (2) a G a ′ − + T (2) a ′ G a − + T (2) a T (2) a ′ . (15) A. Tetraquark equations with no coupling to q ¯ q states To compare with the existing approaches [4–6], our aim is to describe the tetraquark usingtwo-body equations that couple identical meson-meson (
M M ), and diquark-antidiquark ( D ¯ D )channels. To this end we consider G (4) ir in the approximation T (4) aa ′ = T (2) a T (2) a ′ (16a)where the two-body t matrices T (2) a and T (2) a ′ are expressed in the bound state pole approximation T (2) a = i Γ a D a ¯Γ a , (16b)where D a ( P a ) = 1 / ( P a − m a ) is the propagator for the bound particle (diquark, antidiquark, ormeson) in the two-body channel a . Showing explicit dependence on momentum variables, T (2) a ,for a = 12, can be expressed as T (2)12 ( p ′ p ′ , p p ) = i Γ( p ′ p ′ ) D ( P )¯Γ( p p ) , where P = p + p isthe 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 (4) ir . In turn,this means the existence of a 4-body tetraquark wave function Ψ ir ≡ G (4)0 ψ for the case where5ll coupling to q ¯ q states is switched off. We therefore begin by considering the correspondingbound state form factor ψ for the case of 2 indistinguishable quarks and 2 indistinguishableantiquarks, and relate it to the corresponding form factor ψ d for distinguishable quarks as ψ = 14 (1 − P )(1 − P ) ψ d (17)where P ij is the operator exchanging the quantum numbers of particles i and j . The Faddeev-like equations for ψ d are [4, 8], ψ d = X α ψ dα (18a) ψ dα = X β T (4) α ¯ δ αβ G (4)0 ψ dβ (18b)where ¯ δ αβ = 1 − δ αβ and the Greek subscripts run over the three possible ”two pairs” of particlesas in Eq. (9). Using the approximations of Eqs. (16), one can write T (4) α = T (2) a T (2) a ′ = i Γ a Γ a ′ D a D a ′ ¯Γ a ¯Γ a ′ ≡ − Γ α D α ¯Γ α (19)where Γ α ≡ Γ a Γ a ′ , ¯Γ α ≡ ¯Γ a ¯Γ a ′ , and D α ≡ D a D a ′ . Further, defining the vertex functions φ dα bythe relation ψ dα = Γ α D α φ dα , (20)it follows from Eq. (18b) that φ dα = X β V αβ D β φ dβ (21)where V αβ = − ¯ δ αβ ¯Γ α G (4)0 Γ β . (22)Noting that Γ = − Γ and Γ = − Γ , it follows that V = V , V = − V and V = − V .We can now use Eq. (17) to derive M M and D ¯ D components of the tetraquark form factor ψ in the case of indistinguishable quarks. These are defined by the pole contributions to ψ at p = M π , p = M π , p = M π , p = M π , p = M D , and p = M D , where p ij = p i + p j is thetotal momentum of particles i and j , M π is the mass of the meson and M D is the mass of thediquark or antidiquark. To this end consider the use of Eq. (21) in Eq. (17): ψ = 14 (1 − P )(1 − P ) (cid:2) Γ D φ d ( p , p ) + Γ D φ d ( p , p ) + Γ D φ d ( p , p ) (cid:3) = 14 Γ Γ D (cid:2) φ d ( p , p ) + φ d ( p , p ) (cid:3) −
14 Γ Γ D (cid:2) φ d ( p , p ) + φ d ( p , p ) (cid:3) + 14 Γ Γ D (cid:2) φ d ( p , p ) + φ d ( p , p ) (cid:3) −
14 Γ Γ D (cid:2) φ d ( p , p ) + φ d ( p , p ) (cid:3) + Γ Γ D φ d ( p , p )= 12 Γ Γ D (cid:2) φ S ( p , p ) − φ S ( p , p ) (cid:3) + 12 Γ Γ D (cid:2) φ d ( p , p ) − φ d ( p , p ) (cid:3) + Γ Γ D φ d ( p , p )= 12 Γ Γ D φ M ( p , p ) −
12 Γ Γ D φ M ( p , p ) + Γ Γ D φ D ( p , p ) , (23)6here φ S ( p, q ) = 12 (cid:2) φ d ( p, q ) + φ d ( q, p ) (cid:3) , (24a) φ S ( p, q ) = 12 (cid:2) φ d ( p, q ) + φ d ( q, p ) (cid:3) . (24b)are symmetric functions under the exchange of the meson quantum numbers, and φ M ( p, q ) = φ S ( p, q ) − φ S ( p, q ) , (25a) φ D ( p, q ) = φ d ( p, q ) (25b)define the M M and D ¯ D 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 outEq. (21) for distinguishable quarks using notation V = V D , V = V D , V = V D , V = V D ,and φ D = φ d , : φ d = V D φ d + V D D φ D (26a) φ d = V D φ d + V D D φ D = V D φ d − V D D φ D (26b) φ D = V D D φ d + V D D φ d = V D ( D φ d − D φ d ) . (26c)Then, subtracting the second line from the first, we obtain a set of two equations for φ − = φ d − φ d and φ D , φ − = − V D φ − + 2 V D D φ D = − V (cid:18) M M (cid:19) φ − + 2 V D D ¯ Dφ D , (27a) φ D = V D D φ − = 2 V D (cid:18) M M (cid:19) φ − (27b)where we used D = D ≡ M M , D ≡ D ¯ D . Equations (27) can be written in matrix form as (cid:18) φ − φ D (cid:19) = 2 (cid:18) − V V D V D (cid:19) (cid:18) M M D ¯ D (cid:19) (cid:18) φ − φ D (cid:19) . (28)To finally derive the tetraquark equations in the case of indistinguishable quarks, note thataccording Eqs. (24), φ M = Φ S − Φ S = Φ S − = 12 [ φ − ( p, q ) + φ − ( q, p )]= 12 (1 + P ) φ − (29)where P is permutation operator of the meson state labels. Thus, symmetrizing Eqs. (27) withrespect to meson legs gives φ M = − P ) V (cid:18) M M (cid:19) φ − + 2 12 (1 + P ) V D D ¯ Dφ D , = − P ) V (cid:18) M M (cid:19) φ M + 2 V D D ¯ Dφ D , (30a) φ D = 2 V D (cid:18) M M (cid:19) φ − = 2 V D (cid:18) M M (cid:19) φ M , (30b)7 a ) ( b ) ( c )FIG. 3. The potentials making up the elements of the coupled channel M M − D ¯ D kernel matrix V of Eq. (34): (a) V , (b) V D , and (c) V D . Solid lines represent quarks or antiquarks, dashed linesrepresent mesons, and double-lines represent diquarks and antidiquarks. where we have used the following symmetry properties of V and V D :(1 + P ) V = (1 + P ) V
12 (1 + P ) , (31a)12 (1 + P ) V D = V D . (31b)Equations (30) can be written in matrix form as φ = V G M φ (32)where φ = (cid:18) φ M φ D (cid:19) , G M = (cid:18) M M D ¯ D (cid:19) , (33)and V = (cid:18) − (1 + P ) V V D V D (cid:19) , (34)thereby revealing V of Eq. (34) to be the interaction kernel for the coupled M M − D ¯ D system.The elements of V involve the potentials V = − ¯Γ G (4)0 Γ = − ¯Γ ¯Γ G (4)0 Γ Γ , (35a) V D = − ¯Γ G (4)0 Γ = − ¯Γ ¯Γ G (4)0 Γ Γ , (35b) V D = − ¯Γ G (4)0 Γ = − ¯Γ ¯Γ G (4)0 Γ Γ , (35c)as illustrated in Fig. 3. With the kernel matrix V established, one can determine the t matrix T defined by T = V + V G M T, (36)and thereafter G M + G M T G M , which is the matrix Green function in coupled M M - D ¯ D spacecorresponding to G (4) ir . As indicated by Eq. (3), the 4 q -reducible part of the two-body q ¯ q kernel, K (2)4 q − red , can be found by sandwiching G (4) ir between amplitudes that contract 2 q q states to q ¯ q states. In the present case of coupled M M - D ¯ D channels, this contraction can be expressed as K (2)4 q − red = ¯ N ( G M + G M T G M ) N (37)where ¯ N = ( ¯ N M , ¯ N D ) is the two-component amplitude whose elements ¯ N M and ¯ N D describetransitions of two-meson and diquark-antidiquark states to the quark-antiquark state, q ¯ q ← M ( p ) M ( k ) and q ¯ q ← D ( p ) ¯ D ( k ), respectively. Similarly, N = ( N M , N D ) describes transitions M ( p ) M ( k ) ← q ¯ q and D ( p ) ¯ D ( k ) ← q ¯ q . Explicitly, these transition amplitudes are given by¯ N M = S (Γ p Γ k + Γ k Γ p ) , ¯ N D = S Γ p Γ k , (38a) N M = (¯Γ p ¯Γ k + ¯Γ k ¯Γ p ) S , N D = ¯Γ p ¯Γ k S , (38b)8 N M = kp + pk , ¯ N D = kpN M = kp + pk , N D = kp FIG. 4. Illustration of Eqs. (38). Lines have the same meaning as in Fig. 3. where S is the quark propagator connecting quark lines 2 and 3. Equations (38) are illustratedin Fig. 4.Using the formal solution to Eq. (36), T = (cid:0) − V G M (cid:1) − G M − − G M − , (39)we can write the general expression for the two-body q ¯ q kernel as K (2) = ∆ + K (2)4 q − red = ∆ + ¯ N G M (cid:0) − V G M (cid:1) − N (40)where ∆ is defined to be the sum of all q ¯ q -irreducible contributions allowed by QFT that arenot accounted for by the last term of of Eq. (40). In particular, ∆ includes correction termsthat 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 2 q q -reducible diagrams in the last term of Eq. (40) are overcounted, therefore ∆ should not containcounter-terms for eliminating overcounting. As such, ∆ can be used in future studies to takeinto account effects such as one-gluon exchange, one-meson exchange, etc.
B. Tetraquark equations with coupling to q ¯ q states Equation (32) constitutes the matrix form of the tetraquark equations without coupling to q ¯ q states. It expresses the column matrix φ of tetraquark form factors φ M and φ D , in termsof potentials contained in matrix V . To derive the corresponding tetraquark equations withcoupling to q ¯ q states, we simply use the kernel K (2) of Eq. (40) in Eq. (6), the bound stateequation for the tetraquark form factor Γ ∗ :Γ ∗ = K (2) G (2)0 Γ ∗ = h ∆ + ¯ N G M (cid:0) − V G M (cid:1) − N i G (2)0 Γ ∗ = ∆ G (2)0 Γ ∗ + ¯ N G M Φ (41)where Φ =
V G M Φ +
N G (2)0 Γ ∗ . (42)Equation (42) is the matrix form of the sought-after tetraquark equations with coupling to q ¯ q In fact our choice of the last term of Eq. (40) in this note is motivated by physics arguments and the possibilityof close comparison with existing studies. M = 1 + P M − D + Γ ∗ Φ D = − Φ M + Γ ∗ Γ ∗ = ∆ Γ ∗ + 12 Φ M + Φ D FIG. 5. Illustration of the tetraquark equations, Eqs. (45), with coupling to q ¯ q states included.Tetraquark form factors Φ M (displayed in red) couple to two mesons (dashed lines), tetraquarkform factors Φ D (displayed in blue) couple to diquark and antidiquark states (double-lines), andthe tetraquark form factors Γ ∗ (displayed in yellow) couple to q ¯ q states (solid lines). The amplitude∆ (displayed in green) represents all contributions to the q ¯ q kernel K (2) that are not included in thelast term of Eq. (40). states. It expresses the column matrix Φ of tetraquark form factors Φ M and Φ D in terms ofboth the potentials contained in matrix V , and the tetraquark form factor Γ ∗ describing thedisintegration of a tetraquark into a q ¯ q pair. We can write Eq. (42) explicitly as (cid:18) Φ M Φ D (cid:19) = (1 + P )¯Γ M G (4)0 P Γ M − M G (4)0 Γ D − D G (4)0 Γ M ! (cid:18) M M D ¯ D (cid:19) (cid:18) Φ M Φ D (cid:19) + (cid:18) N M N D (cid:19) G (2)0 Γ ∗ (43)where Γ M = Γ Γ , ¯Γ M = ¯Γ ¯Γ , Γ D = Γ Γ , ¯Γ D = ¯Γ ¯Γ , and P ij is the operatorexchanging quarks i and j , therefore¯Γ M G (4)0 P Γ M = ¯Γ ¯Γ G (4)0 Γ Γ , ¯Γ M G (4)0 Γ D = ¯Γ ¯Γ G (4)0 Γ Γ . (44)Thus the tetraquark equations with coupling to q ¯ q included take the form of three coupledequations Φ M = (1 + P )¯Γ M G (4)0 P Γ M M M M − M G (4)0 Γ D D ¯ D Φ D + N M G (2)0 Γ ∗ , (45a)Φ D = − D G (4)0 Γ M M M M + N D G (2)0 Γ ∗ , (45b)Γ ∗ = ∆ G (2)0 Γ ∗ + ¯ N M M M M + ¯ N D D ¯ D Φ D , (45c)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), usinga model where the two-body q ¯ q , qq , and ¯ q ¯ q interactions are dominated by the formation of10 meson, a diquark, and an antidiquark, respectively. Nevertheless, Eqs. (45) constitute themost general form of the tetraquark equations in QFT since all differences between the modelused 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 itsdisintegration into two identical mesons, a diquark-antidiquark pair, and a quark-antiquarkpair. As such, they extend the purely four-body (4 q ) tetraquark equations of Ref. [4] to includecoupling to two-body(2 q ) q ¯ q states.The motivation for the present work comes from the need to have exact quantum field the-oretic equations describing the tetraquark, but formulated for the case where the dynamics isdominated by meson-meson and diquark-antidiquark components. This is especially importantin view of the lack of agreement between two previous attempts to calculate tetraquark equa-tions with 4 q -2 q mixing. The first of these was our derivation of 2014 [5] using a careful butinvolved incorporation of disconnected q ¯ q interactions as a means of incorporating q ¯ q annihila-tion into a 4 q description. The second of these was a recent derivation [6] where coupling to 2 q channels was included phenomenologically, and where some doubt was expressed regarding theincorporation of disconnected q ¯ q interactions. Our present derivation of Eqs. (45) has thereforebeen based on a method that avoids any explicit introduction of disconnected q ¯ q 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 theequations derived by us in Ref. [5]. Indeed, settting ∆ = 0 in Eq. (45c) and substituting intoEq. (43) gives Φ in the form presented in Ref. [5]: Φ = " (1 + P )¯Γ M G (4)0 P Γ M − M G (4)0 Γ D − D G (4)0 Γ M ! + N G (2)0 ¯ N M M D ¯ D (cid:19) Φ . (46)Here N G (2)0 ¯ N is the q ¯ q reducible part of the kernel which is denoted by V q ¯ q in Ref. [5]. Itaccounts for the q ¯ q admixture through the q ¯ q propagator G (2)0 . By contrast, the tetraquarkequations 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 approachtaken in the present work allows for the derivation of the tetraquark equations in a much simplerand more clear way. ACKNOWLEDGMENTS
A.N.K. was supported by the Shota Rustaveli National Science Foundation (Grant No.FR17-354). The expression in the square bracket in Eq. (46) may appear to come with an opposite sign in Ref. [5], butthis is not the case as the definitions of ¯Γ M , Γ M , ¯Γ D , Γ D , ¯ N , and N used in Ref. [5] differ from the ones usedhere.
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