Tetraquark-adequate formulation of QCD sum rules

We study details of QCD sum rules \`a la Shifman-Vainshtein-Zakharov for exotic tetraquark states. We point out that duality relations for correlators involving exotic currents have fundamental differences compared with the duality relations for the correlators of bilinear quark currents: namely, the $O(1)$ and $O(\alpha_s)$ terms in the OPE for the exotic correlators exactly cancel against the contributions of the two-meson states on the phenomenological side of QCD sum rules. As a result, the tetraquark properties turn out to be related to the specific non-factorizable parts of the OPE for the exotic Green functions; the relevant non-factorizable diagrams start at order $O(\alpha_s^2)$. Moreover, we show that all appropriate diagrams may be easily obtained from those Feynman diagrams for the four-point function of bilinear quark currents which contain four-quark $s$-channel singularities.

Motivated by the increasing experimental evidence for narrow near-threshold hadron resonances with a favourable interpretation as tetraquark and pentaquark hadrons (i.e., hadrons with minimal parton configurations consisting of four and five quarks, respectively) [1][2][3], extensive theoretical studies of such objects have been carried out. This letter focuses on the subtleties of the description of tetraquark mesons within the method of Shifman-Vainshtein-Zakharov (SVZ) sum rules in QCD [4]; we demonstrate that some essential criteria for selecting QCD diagrams appropriate for the tetraquark properties in the method of QCD sum rules have not been properly taken into account.
For a proper analysis of possible tetraquarks in QCD and for the selection of the appropriate Feynman diagrams, the understanding of the four-quark singularities of Feynman diagrams plays a crucial role. In a number of publications [5][6][7][8][9][10][11][12][13][14], the four-quark singularities of Feynman diagrams describing the four-point functions of bilinear quark-antiquark interpolating currents have been carefully studied. In recent papers [13,14], the notion of tetraquark-phile diagrams (T -phile diagrams) has been introduced: by definition, the T -phile diagrams are those Feynman QCD diagrams that have four-quark singularities in the appropriate kinematic variable. For the four-point function, those diagrams that contain at least two gluon exchanges with a special topology have been shown to belong to the set of T -phile diagrams.
Independently of this line of research, numerous works deal with the analysis of tetraquark states within the method of QCD sum rules (see the review papers [15,16] and references therein). The emphasis has been laid on two-point functions Π θθ (x) = T {θ(x)θ(0)} of the tetraquark interpolating currents θ(x) =q(x)q(x)q(x)q(x), and three-point functions, Γ θjj (0|x, y) = T {θ(0)j(x)j(y)} , involving one tetraquark current θ and two ordinary bilinear quark currents j(x) =q(x)q(x). (The quark flavour content of the currents will be specified in the forthcoming sections.) All these applications of SVZ sum rules (SR) to exotic states share one common feature: they calculate the leading-order O(1) diagrams (and in some cases also radiative corrections) and power corrections induced by these leading-order diagrams, and borrow exactly the same criteria for continuum subtraction as prescribed by the SVZ sum rules for the ordinary mesons [4]. As a result, the tetraquark contribution is obtained to be dual to the low-energy spectral integral of the QCD diagrams for the corresponding Green function. In particular, the tetraquark receives substantial contributions of the O(1) and O(α s ) QCD diagrams.
One may easily see that the procedures adopted in the SR analyses of exotic states [15,16] in fact do not properly take into account the properties of four-quark singularities of Feynman diagrams. To show this, we make two almost self-evident observations: (i) It is sufficient to consider tetraquark interpolating currents composed as the product of two colorless bilinear quark currents [17]. All other structures of the tetraquark currents are reduced to the product of colorless bilinears by performing Fierz transformations. Next, a tetraquark interpolating current may be consistently defined as the product of two point-splitted bilinear quark currents by sending the displacement parameter δ to zero: θ(x) = lim δ→0 j(x)j(x + δ). Then any diagram describing Π θθ and Γ θjj may be obtained from the diagrams of the 4-point function Γ 4j ≡ T {jjjj} by merging two pairs of vertices (in the case of Π θθ ) or one pair of vertices (in the case of Γ θjj ).
(ii) The procedures adopted in the method of QCD sum rules relate the tetraquark properties to those contributions to Π θθ and Γ θjj that are obtained by merging vertices in non-T -phile diagrams of Γ 4j . Recall that such contributions to Γ 4j have no four-quark cuts [18] and therefore may not be related to tetraquark properties [10][11][12].
In this paper, we show that no contradiction emerges as soon as one makes use of quark-hadron duality relations for correlation functions involving exotic tetraquark currents in a proper way: one observes an exact cancellation of the O(1) and O(α s ) contributions on the OPE side against the two-meson contribution on the phenomenological side of the SVZ sum rule [4], due to quark-hadron duality relations for correlation functions of ordinary bilinear quark currents describing properties of ordinary mesons. Consequently, a consistent QCD sum rule for any exotic state should be formulated in the following way: on the OPE side for Π θθ and Γ θjj , one takes into account only T -phile diagrams, i.e., those diagrams which are obtained from T -phile diagrams for Γ 4j ; on the phenomenological side, one has the suspected tetraquark pole and the interacting mesons. One may then assume, similarly to the conventional procedures of QCD sum rules for ordinary correlators, that the tetraquark contribution is dual to the low-energy part of the T -phile contributions to Π θθ and Γ θjj .

DIRECT GREEN FUNCTIONS INVOLVING TETRAQUARK CURRENTS
Let us discuss tetraquarks involving two quarks of flavours a and c and two antiquarks of flavours b and d. We therefore consider interpolating currents with two different flavour structures, θā bcd = jā b jc d and θā dcb = jā d jc b , with jā b =q a q b . We do not specify here the Dirac structure of the currents, so we do not explicitly write the appropriate combinations of γ matrices between the quark fields. This technical complication does not change the argument developed in this paper.
As mentioned above, an appropriate definition of the four-quark operator may be given by point-splitting in the product of bilinear quark current operators. From this perspective, any diagram involving a tetraquark interpolating current may be obtained from the four-point function of bilinear quark currents studied in detail in [11]. One should distinguish between the diagrams where quark flavours in the initial state and the final state are combined in the same way (direct diagrams, this section) and in a different way (recombination diagrams, the next section). Feynman diagrams for the corresponding four-point functions have different topologies and different structures of four-quark singularities. Accordingly, the duality relations for these correlators should be discussed separately.
A. Two-point function Π dir θθ Figure 1 shows the direct four-point function Γ dir 4j and the corresponding two-point function of the tetraquark currents. In Fig. 1, only diagrams (c) are T -phile diagrams, so we expect that the r.h.s. diagrams (a,b) should drop out from the tetraquark sum rule. Diagrams with one-gluon exchanges between the disconnected loops are null.  To show that this is indeed the case, let us start by writing the OPE (theoretical side of the sum rule) and the hadron representation (phenomenological side of the sum rule) for the two-point direct correlation function Π θθ , Fig. 2. There is an infinite subset of diagrams in the OPE for Π dir θθ that factorize in coordinate space into two parts separated by the red dash-dotted line. The O(1) and O(α s ) diagrams belong to this subset. On the phenomenological side, there is also an infinite subset of meson contributions that factorize in coordinate space. It is straightforward to check that the factorizable subset of diagrams on the OPE side is exactly equal to the factorizable subset on the phenomenological side as soon as the QCD sum rule for the two-point function of ordinary bilinear of quark currents Π jj (Fig. 3) is used. After cancelling out the equal factorizable parts on both sides of the sum rule of Fig. 2, we arrive at the tetraquark sum rule given by Fig. 4. Now, similarly to the case of ordinary mesons, we consider a single spectral representation in the variable p 2 of the QCD diagrams in the l.h.s. of Fig. 4 and introduce the effective threshold s eff [19][20][21] such that the contribution of the QCD diagrams in the l.h.s. of Fig. 4 above s eff cancels the non-factorizable meson-meson interaction diagrams on the r.h.s. of Fig. 4. Then, after Borel transformation, we obtain the tetraquark sum rule T is the spectral density in the variable s of the r.h.s. diagram of Fig. 1(c) with two-gluon exchanges of order O(α 2 s ). Power corrections in the r.h.s. of Eq. (2.1) correspond to condensate insertions in the diagram of Fig. 1(c). Let us emphasize that power corrections generated by the r.h.s. diagrams in Fig. 1(a,b) do not contribute to the tetraquark sum rule (2.1), as they have cancelled against the factorizable meson-meson contributions. Here, M T is the tetraquark mass and fā bcd T = T |θā bcd |0 . (2.2) As argued above, only the T -phile diagram of Fig. 1(c) and the corresponding power corrections contribute to the tetraquark sum rule (2.1). Similarly to Π dir θθ , the direct Green functions Γ dir θjj may be obtained from the direct Green functions Γ dir 4j by merging in the latter only one (say, the left) pair of coordinates. The corresponding procedure is shown in Fig. 5; here, only diagram (c) is the T -phile diagram, so the r.h.s. diagrams (a) and (b) should not contribute to the tetraquark coupling to two mesons. For the direct three-point function this is very easy to show: the tetraquark would lead to the pole 1/(p 2 − M 2 T ) in the full Green function Γ dir θjj (p is the total momentum of the currents), and the residue in this pole would be related to the coupling T → This completes, for the direct Green function, the proof of the statement that only diagrams obtained from the set of T -phile diagrams of Γ 4j contribute to the tetraquark sum rule.

RECOMBINATION GREEN FUNCTIONS INVOLVING TETRAQUARK CURRENTS
We now briefly discuss diagrams with a different -"recombination" -topology (Figs. 6 and 7), where the quark color singlets in the initial and final currents have different flavour structures. Similarly to the case of the direct diagrams, only the diagrams of Figs. 6(c) and 7(c) are T -phile diagrams and contribute to the sum rules for the tetraquark properties. The proof of this statement is technically not as simple as for the direct diagrams, where the contribution of O(1) and O(α s ) diagrams just dropped out from the tetraquark sum rule after the duality relations for the correlators of bilinear currents have been taken into account. Nevertheless, in [11] its was shown that the recombination Green functions (a) and (b) on the l.h.s. of Fig. 6 and Fig. 7 do not contribute to the tetraquark poles and are related to specific meson amplitudes without s-channel four-quark singularities. This property also holds after merging the initial and/or final vertices. Thus, the r.h.s. diagrams (a) and (b) of Fig. 6 and Fig. 7 still do not contribute to the T -pole properties. In the end, only the diagrams in Fig. 6(c) and Fig. 7(c) are T -phile diagrams and appear in the tetraquark sum rules, which take the forms In summary, a proper application of QCD sum rules to tetraquarks requires the calculation of the presently unknown non-factorizable O(α 2 s ) radiative corrections and calls for further efforts in order to obtain reliable conclusions about the specific tetraquark candidates.