OPE and quark-hadron duality for two-point functions of tetraquark currents in $1/N_c$ expansion

We discuss the Operator Product Expansion (OPE) and quark-hadron duality for two-point Green functions of tetraquark currents. We emphasize that the factorizable part of the OPE series for such Green functions, including nonperturbative contributions described by QCD condensates, is saturated by the full system of ordinary hadrons and therefore cannot have any relationship to the possible tetraquark bound states. Possible tetraquark bound states may be contained in nonfactorizable parts of these Green functions. In the framework of the $1/N_c$ expansion in QCD$(N_c)$, nonfactorizable parts of the two-point Green functions of tetraquark currents provide $N_c$-suppressed contributions compared to the $N_c$-leading factorizable parts. A possible exotic tetraquark state may appear only in $N_c$-subleading contributions to the QCD Green functions, in full accord with the well-known rigorous properties of large-$N_c$ QCD.


INTRODUCTION
The correlation functions of two local colorless currents are the simplest gauge-invariant Green functions that have a unique decomposition in terms of the physical hadron states. These correlation functions are defined as vacuum expectation values of the time-ordered-products of two local gauge-invariant quark currents taken at different locations: (1.1) We shall discuss and compare two cases: j being a local bilinear quark current and j being a tetraquark current. The Dirac structure of the currents will be of no relevance for our arguments and will not be specified; we therefore do not explicitly write the appropriate combinations of Dirac matrices between the quark fields. The Wilson Operator Product Expansion (OPE) [1] provides the following expansion for the T -product: T {j(x)j † (0)} = C 0 (x 2 , µ)1 + n C n (x 2 , µ) :Ô n (x = 0, µ) : , (1.2) where µ is a renormalization scale. The two-point function (for those cases where only light quarks are involved) is then expanded in the form Π(p 2 ) = Π pert (p 2 , µ) + n C n (p 2 ) n 0| :Ô n (x = 0, µ) : |0 . (1. 3) The QCD vacuum is nonperturbative and its properties are characterized by the condensates -nonzero vacuum expectation values of gauge-invariant operators, depending, in general, on the renormalization scale µ, 0| :Ô(0, µ) : |0 = 0 [2][3][4][5]. Hereafter, we denote 0|...|0 ≡ ... . In the OPE context, perturbative diagrams describe the contribution of the unit operator. The Wilson coefficients of the local operators :Ô(0, µ) : are obtained from perturbative diagrams according to known rules [3]. The appropriate diagrams will be displayed below; we show only diagrams containing quark and gluon lines but do not show those diagrams that contain Faddeev-Popov ghosts. It proved efficient to generalize QCD, based on the color gauge group SU (3), to the case of the color gauge group SU(N c ) and to consider the 1/N c expansion of the Green functions in QCD(N c ) in the so-called 't Hooft limit, the strong coupling constant scaling as α s ∼ 1/N c [6]. For the discussion of exotic states of any structure [in QCD(N c ) one may have a very rich structure of colorless hadron states] the following features of large-N c QCD are of special significance: as shown by Witten [7], large-N c Green functions are saturated by noninteracting ordinary mesons. This means, in particular, that any possible exotic states may appear only in N c -subleading contributions to the QCD(N c ) Green functions [7,8]. This is a distinguishing property of exotic tetraquark mesons compared to ordinary quark-antiquark mesons: the latter appear already in the N c -leading parts of the QCD(N c ) Green functions.
The possibility of the existence of narrow exotic states in QCD at large N c has been studied in recent years in a number of publications [9][10][11][12][13][14][15][16][17][18][19][20]. The present paper makes use of large-N c arguments in a slightly different context: We study two-point functions of bilinear quark currents and of tetraquark currents and discuss the appearance of the tetraquark bound states in the latter, giving further theoretical arguments in the derivation of the tetraquark-adequate (T -adequate) sum rules, formulated in our recent publications [21,22]. Here, we make the following new steps: (i) We show that the T -adequate sum rules in SU(N c ) should be based on appropriate nonfactorizable parts of the OPE for two-point functions of the tetraquark currents. In this way, the tetraquark contributions are compatible with the well-known rigorous property of QCD at large N c : The N c -leading Green functions are saturated by the ordinary mesons; any exotic states may appear only in N c -subleading contributions.
(ii) We discuss nonperturbative effects in two-point functions of tetraquark currents [our analyses [21,22] considered perturbative diagrams and did not address nonperturbative effects] and identify those condensates contributions that appear in the T -adequate duality relations and T -adequate sum rules.
The paper is organized as follows: Section 2 recalls the large-N c behavior of the two-point Green functions of bilinear quark currents and compares the OPE with the hadron saturation of these Green functions. Section 3 studies the OPE for the two-point function of tetraquark currents including nonperturbative condensate contributions and discusses the quark-hadron duality relations that may involve possible tetraquark states. Section 4 presents our conclusions and outlook.

TWO-POINT FUNCTION OF BILINEAR QUARK CURRENTS
Let us start with some uselful algebraic relations for the group SU(N c ) [23]. The generators T A , A = 1, . . . , N 2 c − 1, considered in the fundamental representation, satisfy the color Fierz rearrangement where the generators are normalized according to The relation (2.1) suggests that, with respect to counting an overall N c -leading color factor of a Feynman diagram, any gluon line may be replaced by aqq double line. To calculate the N c -subleading terms in the expansion of a Green function, one has to take into account, in the gluon lines, the second term of the right-hand side of Eq. (2.1). We now briefly recall the properties of the 1/N c -expansion of the OPE series for bilinear quark currents, j =qq.
A. Perturbative diagrams Figure 1 shows diagrams according to their behavior in the framework of the 1/N c expansion, assuming that the strong coupling constant scales as α s ∼ 1/N c . When calculating the 1/N c behavior of a diagram, we replace the gluon line by a doubleqq line. Doing so, we pick up the leading behavior at large N c , but omit corrections of the order 1/N 2 c . [The second term in Eq. (2.1) contains a factor 1/N c and, in addition, the number of color loops generated by the second term is reduced by one compared to the first term, thus yielding an overall suppression factor 1/N 2 c ]. So, all diagrams with a specific large-N c behavior generate also contributions to lower orders of the 1/N c expansion. For instance, the O(N c ) diagrams in Fig. 1  Power corrections scale with N c as follows: Obviously, nonperturbative effects described by the condensates contribute on the same footing as perturbative effects at each order in 1/N c . So, even at N c -leading order QCD is not fully perturbative: although the strong coupling may be made arbitrary small, nonperturbative effects described by the condensates do not disappear and survive the large-N c limit.

C. Hadron saturation of two-point function and sum rules
As is well known [7,8], the spectrum of states of QCD(N c ) in the limit N c → ∞ contains towers of an infinite number of free, stable and noninteracting mesons. Meson-meson elastic scattering amplitudes are of order 1/N c and the decay amplitudes of mesons into two mesons are of order 1/N 1/2 c . The notion of valence quarks takes in the above limit a precise meaning. Mesons are made of pureqq states, rather than ofqqqq states, which only appear at subleading orders of N c . Conversely, states, whose N c -leading element is composed ofqqqq states, correspond to two-meson states. These properties allow us to make a systematic correspondence between the OPE and the hadron saturation of two-point correlation functions.
Let us consider the two-point function of the elastic vector current, V µ =qγ µ q, and denote it as Obviously, we have light pseudoscalar mesons (hereafter referred to as pions). Figures 3(a,b) show the scaling of the hadron diagrams and vertices at large N c and Fig. 3(c) shows the N c -leading sum rule: the sum over stable vector mesons is dual to the N c -leading OPE. Obviously, one can include some N c -subleading effects on the hadron and/or on the OPE side of this sum rule. However, for the consistency of the full approach, it is mandatory that the N c -leading contribution on the hadron side matches the N c -leading contribution on the OPE side. In fact, the vector sum rule in real QCD [2] looks very similar to the relation shown in Fig. 3(c) and thus perfectly satisfies the large-N c consistency. Closing this discussion, we notice that the correlation function of bilinear quark currents describes the "minimal" colorless cluster and thus does not contain inside it any other factorizable colorless clusters. As a result, the hadron saturation of the N c -leading part of the correlation function contains all intermediate hadron states with the appropriate quantum numbers, starting from the one-meson state.

OPE FOR CORRELATORS OF TETRAQUARK CURRENTS
Let us now turn to two-point functions of tetraquark currents. We consider four-quark currents consisting of two antiquarks of generic flavors b and c and two quarks of generic flavors u and d. For the sake of argument, we make two assumptions: First, we assume that all quark flavors are different -this simplifies the topology of the appropriate QCD diagrams, avoiding, in particular, the discussion of quark annihilation diagrams. Our second assumption is that the two antiquarks,b andc, are heavy and therefore do not produce quark condensates; the two quarks, u and d, are light and therefore develop local vacuum condensates (quark condensates ūu , d d , mixed quark-gluon condensates four-quark condensates ūudd , etc.). These assumptions simplify the discussion but do not change any essential qualitative feature of our analysis.
As follows from the property of cluster reducibility of multiquark operators [24], any gauge-invariant multiquark operator can be reduced to a combination of products of colorless clusters. In our case of thebcud flavor content of the tetraquark current, colorless clusters of two different flavor structures emerge in QCD (see [25,26]): with jā f =q a q f . We therefore should distinguish between the diagrams where quark flavors in the initial and final states are combined in the same way (direct diagrams) and in a different way (quark-exchange or recombination diagrams). The Feynman diagrams for the corresponding four-point functions have different topologies and structures of their four-quark singularities and therefore require separate analyses [21,22]. Here, we discuss the direct Green function

2)
A. Perturbative diagrams Perturbative diagrams without gluon exchanges between the quark loops have a factorizable structure (i.e., factorize, in coordinate space, into the product of two-point functions of bilinear quark currents, Πb u (x)Πc d (x), where One can refer to these diagrams as to "disconnected" diagrams, bearing in mind, however, that the vertices corresponding to the initial (final) bilinear currents are connected to each other. 1 So the term "disconnected" should apply to the internal parts of the diagrams. We will prefer to term these diagrams "factorizable". Since any of the quark loops behaves as ∼ N c , the factorizable diagrams are of order N 2 c . Factorizable diagrams can be isolated in a unique way and provide the N c -leading behavior of Π dir (x).
In the diagrams of Fig. 4(b) two quark loops talk to each other via gluon exchanges. Since both quark loops represent colorless clusters, one needs at least two gluons to be exchanged between the loops. Diagrams with two gluon exchanges between the loops and an arbitrary number of planar gluon exchanges inside each of the loops bu and cd have cylinder topology, see Figs. 1 and 2 of Ref. [20]. Their behavior at large N c is N 0 c . One can also have three or more gluon exchanges between the quark loops. All these diagrams have a topology of a cylinder with a number of handles. Each handle reduces the large-N c behavior by two powers of N c .

B. Diagrams containing condensates
Diagrams containing condensates may be obtained from the perturbative diagrams by breaking the internal quark and gluon lines and sending the corresponding particles to vacuum condensates.
Let us start with the factorizable perturbative diagrams of Fig. 4(a). By breaking a light-quark or a gluon line in these diagrams, one obtains the contributions of condensates of lowest dimensions, the quark condensate qq , the gluon condensate α s GG , or the mixed quark-gluon condensate, shown in Fig. 5(a). These diagrams have the same large-N c behavior, O(N 2 c ), as the original perturbative diagrams of Fig. 4(a). Important for us is that diagrams containing the quark, the gluon, and the mixed quark-gluon condensates are of the factorizable type, same as the original perturbative diagrams.  Fig. 4(a). (b) Power corrections of the mixed type: they are proportional to condensates of dimension-6 or higher ( ūudd , etc.) and obtained by sending to quark or gluon fields from different quark loops of factorizable perturbative diagrams of Fig. 4(a) to vacuum condensates. In the four-quark (and higher) condensates, factorizable and nonfactorizable parts may be isolated, see Fig. 6. (c) Power corrections of nonfactorizable type: they are obtained by sending to the condensate quarks and gluons in nonfactorizable diagrams of Fig. 4(b).
A new feature emerges when one calculates the contributions (i.e., the Wilson coefficients) of higher-dimension four-quark and four-quark-gluon condensates, see Fig. 5 Fig. 6: Splitting of the condensates of higher dimensions into factorizable and nonfactorizable (NF) parts. The factorizable parts provide the Nc-leading contribution, whereas the nonfactorizable pieces have an Nc-subleading behavior. For instance, Here, both light quarks, from the upper and the lower loops, can be sent to the condensate simultaneously. For further analysis, it is convenient to isolate factorizable contributions from higher-dimension condensates. For instance, the four-quark condensate may be split into factorizable and nonfactorizable (NF) parts in a unique way, see Fig. 6: The relevance of isolating factorizable parts out of the condensates of higher dimensions will become clear shortly.

C. Isolating the factorizable part from the OPE for two-point functions of tetraquark currents Π dir
We are now fully prepared to isolate the factorizable part from the OPE for the direct two-point function of the tetraquark currents Π dir , including both perturbative and nonperturbative condensate contributions. At the level of diagrams, one may naively suspect the following decomposition of the OPE for Π dir shown in Fig. 7: As we shall see, this formula contains double counting of some of the nonperturbative contributions, and the correct decomposition is a bit more tricky. The factorizable part, Πb u (x)Πc d (x), including the appropriate nonperturbative condensate contributions is obvious and is shown in Fig. 7(a).
More cumbersome are nonfactorizable (NF) contributions. Here, we encounter two types of such NF contributions: • Π dir NF,1 (x), shown in Fig. 7(b), describes nonfactorizable parts of the condensates of higher dimension corresponding to Fig. 5(b); somewhat tricky, these NF nonperturbative power corrections are generated by factorizable perturbative diagrams of Fig. 4(a). Radiative corrections due to gluon exchanges inside the loops bu and cd (and not between the loops) are included in this class of NF contributions.
To understand the proper way to take into account such contributions, let us recall that nonzero vacuum condensates emerge due to interactions with the nonperturbative soft gluon fields. If we look into the anatomy of ūudd NF , this quantity is nonzero due to the interactions with the soft gluons of the type shown in Fig. 8. Taking into account both effects described by Π dir NF,1 (x) and by nonperturbative corrections in those parts of Π dir NF,2 (x) which correspond to two-gluon exchanges between u and d quarks from the different quark loops would be a double counting. We therefore take the appropriate nonperturbative contributions into account as a part of Π dir NF,2 (x) and omit Π dir NF,1 (x).
• Π dir NF,2 (x), Fig. 7(c), describes the "genuinely" nonfactorizable perturbative diagrams of Fig. 4(b) and power corrections of Fig. 5(c); the latter are obtained via the conventional rules by breaking the propagating lines of light quarks and gluons in the perturbative diagrams of Fig. 4(b).
In the end, the proper decomposition of Π dir that avoids the double counting of the nonperturbative corrections has the form Π dir (x) = Πb u (x)Πc d (x) + Π dir NF,2 (x). (3.5) Let us insert the full system of hadron states in the factorizable part. We then obtain where Rb u (x) and Rc d (x) are the quantities coming from hadron saturation, the explicit form of which is irrelevant. Important for us is the fact that the sum runs over the full system of hadron states with flavorsbu (hb u ) andcd (hc d ), respectively. Consequently, the system of the intermediate hadron states that emerges in the factorizable part of Π dir is just the direct product of these two systems, hb u ⊗ hc d . No other hadron state, in particular, no exotic state, may contribute here. So, we conclude that an exotic state, if it exists in the hadron spectrum ofbucd states, contributes only to the nonfactorizable part of an exotic correlation function.
D. T -adequate sum rule and the couplings of tetraquark bound states to tetraquark currents The T -adequate sum rule emerges after Eqs. (3.6) and (3.7) have been taken into account, leading to exact cancellations between the factorizable O(N 2 c ) contributions on the OPE side and the hadron side of the duality relation for the direct two-point function Π dir . Figure 9 shows the corresponding T -adequate sum rule: its OPE side contains and is expressed via the tetraquark coupling to the interpolating tetraquark current: Obviously, the T -adequate sum rule implies f T ∼ N 0 c . This feature is in full agreement with the known property of large-N c QCD that only noninteracting ordinary mesons saturate the N c -leading QCD diagrams. We emphasize once more that for the consideration of exotic states, the factorizable part of the OPE is irrelevant.
Moreover, we would like to point out the following qualitative difference between the correlation functions of tetraquark versus those of bilinear quark currents: As we have discussed, the existence of stable vector mesons in the hadron spectrum at large N c is required by matching the large-N c behavior of the OPE side and of the hadron side of the vector two-point function Π V ; without vector mesons populating the O(N c ) part of the hadronic side no matching may be obtained. For the two-point functions of the tetraquark currents, the situation is qualitatively different: the factorizable OPE and hadronic sides match each other just due to the duality relations for the two-point functions of the bilinear currents. The nonfactorizable OPE side and its nonfactorizable hadronic side have the same large-N c behavior with or without the tetraquark bound state. So, the existence of narrow tetraquark hadrons cannot be established merely on the basis of the large-N c behavior of the exotic Green functions; large-N c QCD does not exclude narrow exotic states in N c -subleading parts of the Green functions, but remains consistent also if such exotic states do not exist in the hadron spectrum. 2 To be more precise, there are two tetraquark currents of globalbcud flavor content: θb ucd and θb dcu [Eq. (3.1)]. Respectively, there might be two tetraquark states of thebcud flavor content, T 1 and T 2 : T 1 couples stronger to the θb ucd current (the corresponding coupling scales as N 0 c ) and weaker to the θb dcu current (the corresponding coupling scales as 1/Nc), and vice versa for T 2 . So, the contribution of T 1 to the correlation function (3.2) scales like N 0 c , whereas the contribution of T 2 is suppressed and scales like 1/N 2 c . These subtleties are, however, a bit outside the main discussion of this paper, so we refer for details to Sect. 2 of [20].

CONCLUSIONS AND OUTLOOK
We discussed in great detail the OPE for two-point Green functions of the bilinear and quadrilinear colorless quark currents at large N c and emphasized the qualitative differences between these two objects: (i) In the case of two-point functions of bilinear quark currents, the contributions of single-meson states with appropriate quantum numbers emerge in the N c -leading part of the Green function. Matching the large-N c behavior of the OPE series and of the hadron saturation series requires the existence of stable mesons with large couplings, f V ∼ √ N c , in the limit N c → ∞. The typical QCD sum rule then relates the N c -leading O(N c ) part of the OPE to the O(N c ) part of the sum over hadron states, and is therefore fully consistent at large N c .
(ii) In the case of two-point functions of tetraquark currents, the N c -leading part of the OPE factorizes into a product of two colorless clusters. Each of them is saturated by the ordinary hadrons that may emerge in the quarkantiquark correlation functions with appropriate quark-flavor content. As a result, tetraquark states (whatever generalization of the N c = 3 tetraquark to N c = 3 is considered) cannot contribute to the N c -leading parts of the Green functions. This property fully agrees with the well-known rigorous property of large-N c QCD: N c -leading Green functions are fully saturated by noninteracting ordinary mesons. The contribution of any colorless state with a more complicated quark structure, for instance, of an exotic meson, may only appear in N c -subleading nonfactorizable parts of the Green functions of tetraquark currents. Moreover, this property is perfectly satisfied by the T -adequate QCD sum rules formulated in [21,22]: one of the outcomes of the T -adequate sum rules is the scaling of the tetraquark coupling to the tetraquark current f T ∼ N 0 c . In the present paper, we have complemented our previous analysis with a detailed discussion of the vacuum condensate contributions.
Existing typical applications of QCD sum rules to the analysis of the tetraquark candidates (see, e.g., recent publications [27][28][29]) relate the tetraquark properties to the low-energy part of the factorizable two-point Green functions of the tetraquark currents. Such an approach copies the route of the sum-rule analysis of the ordinary mesons and does not take into account the fundamental differences between the correlation functions of the bilinear quark currents and of the tetraquark currents. The tetraquark properties are then extracted exclusively from those parts of QCD Green functions which do not have tetraquarks as intermediate states [27][28][29], a feature which does not seem to us physically meaningful. Also, one can easily take the large-N c limit of the corresponding sum-rule analytic expressions for the couplings to immediately find that f T ∼ N c . This would mean that, in contradiction to the rigorous property of QCD at large N c , tetraquark poles appear in the N c -leading QCD diagrams. T -adequate sum rules of [21,22], now complemented with the appropriate account of condensate contributions, are free from these shortcomings and lead to fully consistent relations.