OPE and Quark-Hadron Duality for Two-Point Functions of Tetraquark Currents in the 1=Nc Expansion

PHYSICAL REVIEW D 103, 014012 (2021)

OPE and quark-hadron duality for two-point functions of tetraquark currents in the 1=Nc expansion

Wolfgang Lucha ,1 Dmitri Melikhov ,2,3,4 and Hagop Sazdjian 5 1Institute for High Energy Physics, Austrian Academy of Sciences, Nikolsdorfergasse 18, A-1050 Vienna, Austria 2D. V. Skobeltsyn Institute of Nuclear Physics, M. V. Lomonosov Moscow State University, 119991 Moscow, Russia 3Joint Institute for Nuclear Research, 141980 Dubna, Russia 4Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria 5Universit´e Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France

(Received 7 December 2020; accepted 24 December 2020; published 12 January 2021)

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=Nc expansion in QCDðNcÞ, nonfactorizable parts of the two-point Green functions of tetraquark currents provide Nc-suppressed contributions compared to the Nc-leading factorizable parts. A possible exotic tetraquark state may appear only in Nc-subleading contributions to the QCD Green functions, in full accord with the well-known rigorous properties of large-Nc QCD.

DOI: 10.1103/PhysRevD.103.014012

X † 2 ˆ 2 ˆ I. INTRODUCTION TfjðxÞj ð0Þg ¼ C0ðx ;μÞ1 þ Cnðx ;μÞ∶Onðx ¼ 0;μÞ∶; n The correlation functions of two local colorless currents are the simplest gauge-invariant Green functions that have a ð1:2Þ unique decomposition in terms of the physical hadron states. These correlation functions are defined as vacuum where μ is a renormalization scale. The two-point function expectation values of the time-ordered products of two (for those cases where only light quarks are involved) is local gauge-invariant quark currents taken at different then expanded in the form locations: X C Π p2 Π p2; μ n 0 ∶Oˆ x 0; μ ∶ 0 : Z ð Þ¼ pertð Þþ 2 n h j nð ¼ Þ j i n ðp Þ Πðp2Þ¼i d4xeipxh0jTfjðxÞj†ð0Þgj0i: ð1:1Þ ð1:3Þ

We shall discuss and compare two cases: j being a local The QCD vacuum is nonperturbative, and its properties are bilinear quark current and j being a tetraquark current. The characterized by the condensates—nonzero vacuum expect- Dirac structure of the currents will be of no relevance for ation values of gauge-invariant operators, depending, in our arguments and will not be specified; we therefore do general, on the renormalization scale μ, h0j∶Oˆ ð0; μÞ∶j0i ≠ 0 not explicitly write the appropriate combinations of Dirac [2–5]. Hereafter, we denote h0j…j0i ≡ h…i. In the OPE matrices between the quark fields. The Wilson operator context, perturbative diagrams describe the contribution of product expansion (OPE) [1] provides the following the unit operator. The Wilson coefficients of the local expansion for the T-product: operators ∶Oˆ ð0; μÞ∶ are obtained from perturbative diagrams according to known rules [3]. The appropriate diagrams will be displayed below; we show only diagrams Published by the American Physical Society under the terms of containing quark and gluon lines but do not show those the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to diagrams that contain Faddeev-Popov ghosts. the author(s) and the published article’s title, journal citation, It proved efficient to generalize QCD, based on the color and DOI. Funded by SCOAP3. gauge group SU(3), to the case of the color gauge group

2470-0010=2021=103(1)=014012(10) 014012-1 Published by the American Physical Society LUCHA, MELIKHOV, and SAZDJIAN PHYS. REV. D 103, 014012 (2021)

SUðNcÞ and to consider the 1=Nc expansion of the Green considered in the fundamental representation, satisfy the functions in QCDðNcÞ in the so-called ’t Hooft limit, the color Fierz rearrangement strong coupling constant scaling as αs ∼ 1=Nc [6]. For the 1 1 discussion of exotic states of any structure [in QCDðNcÞ A A ðT Þii0 ðT Þjj0 ¼ δij0 δi0j − δii0 δjj0 ; ð2:1Þ one may have a very rich structure of colorless hadron 2 2Nc states] the following features of large-Nc QCD are of special significance: as shown by Witten [7], large-Nc where the generators are normalized according to Green functions are saturated by noninteracting ordinary 1 mesons. This means, in particular, that any possible exotic TrðTATBÞ¼ δAB: ð2:2Þ states may appear only in Nc-subleading contributions to 2 the QCDðNcÞ Green functions [7,8]. This is a distinguish- The relation (2.1) suggests that, with respect to counting an ing property of exotic tetraquark mesons compared to N ordinary quark-antiquark mesons: the latter appear already overall c-leading color factor of a Feynman diagram, any gluon line may be replaced by a qq¯ double line. To in the Nc-leading parts of the QCD Nc Green functions. ð Þ N The possibility of the existence of narrow exotic states in calculate the c-subleading terms in the expansion of a Green function, one has to take into account, in the gluon QCD at large Nc has been studied in recent years in a number of publications [9–20]. The present paper makes lines, the second term of the right-hand side of Eq. (2.1). We now briefly recall the properties of the 1=Nc- use of large-Nc arguments in a slightly different context: we study two-point functions of bilinear quark currents and of expansion of the OPE series for bilinear quark currents, j qq¯ tetraquark currents and discuss the appearance of the ¼ . tetraquark bound states in the latter, giving further theoretical arguments in the derivation of the tetraquark- A. Perturbative diagrams T adequate ( -adequate) sum rules, formulated in our recent Figure 1 shows diagrams according to their behavior publications [21,22]. Here, we make the following new in the framework of the 1=Nc expansion, assuming that the steps: strong coupling constant scales as αs ∼ 1=Nc. When (i) We show that the T-adequate sum rules in SUðNcÞ calculating the 1=Nc behavior of a diagram, we replace should be based on appropriate nonfactorizable parts the gluon line by a double qq¯ line. Doing so, we pick up the of the OPE for two-point functions of the tetraquark leading behavior at large Nc, but omit corrections of the currents. In this way, the tetraquark contributions 2 order 1=Nc. [The second term in Eq. (2.1) contains a factor are compatible with the well-known rigorous prop- 1=Nc and, in addition, the number of color loops generated erty of QCD at large Nc: the Nc-leading Green by the second term is reduced by one compared to the first functions are saturated by the ordinary mesons; 2 term, thus yielding an overall suppression factor 1=Nc]. So, any exotic states may appear only in Nc-subleading N contributions. all diagrams with a specific large- c behavior also generate 1=N (ii) We discuss nonperturbative effects in two-point contributions to lower orders of the c expansion. For O N functions of tetraquark currents [our analyses instance, the ð cÞ diagrams in Fig. 1(a) also generate O N−1 N [21,22] considered perturbative diagrams and did contributions to diagrams of the order ð c Þ. The c- not address nonperturbative effects] and identify leading perturbative diagrams of Fig. 1(a) are planar those condensate contributions that appear in diagrams without sea-quark loops. Using the language of the T-adequate duality relations and T-adequate sum intermediate states, these diagrams can be identified as rules. those diagrams that have intermediate valence qq¯ states The paper is organized as follows: section II recalls the plus an arbitrary number of gluons; the Nc-leading pertur- large-Nc behavior of the two-point Green functions of bative QCD diagrams do not have cuts corresponding to bilinear quark currents and compares the OPE with the four quarks and an arbitrary number of gluons, six quarks hadron saturation of these Green functions. Section III and an arbitrary number of gluons, etc. Diagrams with studies the OPE for the two-point function of tetraquark multiquark intermediate states have an Nc-subleading currents including nonperturbative condensate contribu- behavior. tions and discusses the quark-hadron duality relations that may involve possible tetraquark states. Section IV presents B. Power corrections our conclusions and outlook. Power corrections are shown in Fig. 2: the Wilson coefficients describing the contribution of the appropriate II. TWO-POINT FUNCTION OF BILINEAR operators may be obtained from the perturbative diagrams QUARK CURRENTS of Fig. 1 by breaking one or more quark and gluon lines. Let us start with some useful algebraic relations for the For instance, the diagram of Fig. 2(a) provides the Wilson A 2 group SUðNcÞ [23]. The generators T , A ¼ 1; …;Nc − 1, coefficient of the operator qq¯ ; the diagram of Fig. 2(b)

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O(Nc ):

O(N ) c 2 2 3 4 α 3 O( α N c ) O( α Nc ) O( N c ) (a)

0 O(Nc ): α2 2 α3 3 α4 4 α2 2 O( N c ) O( Nc ) O( N c ) O( Nc ) (b)

−1 O(Nc ):

α2 α3 2 4 3 O( Nc ) O( N c ) O( α N c ) (c)

FIG. 1. Perturbative series for the two-point function of bilinear quark currents and its classification in powers of 1=Nc. (a) OðNcÞ. Nc- leading diagrams are diagrams of planar topology without sea-quark loops (i.e., containing only the loop of valence quarks that enter the interpolating current) and with an arbitrary number of planar gluon exchanges. No annihilation-type diagrams appear at this Nc-order. 0 Taking into account that αs ∼ 1=Nc, they have the overall dependence OðNcÞ. (b) OðNcÞ. Diagrams with the Nc-leading contribution of this order are shown; these diagrams are (i) planar diagrams with one sea-quark loop and an arbitrary number of gluon exchanges or −1 (ii) quark-annihilation diagrams. (c) OðNc Þ. Diagrams with Nc-leading behavior of this order are of two different classes: (i) nonplanar diagrams with one gluonic handle and an arbitrary number of planar gluon exchanges but no sea-quark loops and (ii) planar diagrams containing two sea-quark loops and an arbitrary number of planar gluon exchanges.

u u u u u O(N ): q c q q q q

(a) (b) (c) (d)

FIG. 2. Nonperturbative power corrections in hjj†i of bilinear quark currents: (a) contribution of dimension-3 quark condensate huu¯ i; μν (b) contribution of dimension-4 gluon condensate hαsGGi; (c) contribution of dimension-5 mixed condensate hu¯σμνG ui, with 1 σμν ¼ 2i ½γμ; γν, the γs being the Dirac matrices; (d) contributions emerging if q is the light quark: that of dimension-3 quark condensate hqq¯ i and that of dimension-6 four-quark condensate huu¯ qq¯ i. gives the Wilson coefficient of the operator GG, where G is infinite number of free, stable and noninteracting mesons. the gluon field strength, etc. Meson-meson elastic scattering amplitudes are of order Power corrections scale with Nc as follows: 1=Nc, and the decay amplitudes of mesons into two mesons 1=2 are of order 1=Nc . The notion of valence quarks takes in ¯ ¯ ¯ 2 hqqi ∼ Nc; hαsGGi ∼ Nc; hqqqqi ∼ Nc: ð2:3Þ the above limit a precise meaning. Mesons are made of pure qq¯ states, rather than of q¯qqq¯ states, which only appear at Obviously, nonperturbative effects described by the con- subleading orders of Nc. Conversely, states whose Nc- densates contribute on the same footing as perturbative leading element is composed of q¯qqq¯ states correspond to 1=N N effects at each order in c. So, even at c-leading order two-meson states. These properties allow us to make a QCD is not fully perturbative: although the strong coupling systematic correspondence between the OPE and the may be made arbitrarily small, nonperturbative effects hadron saturation of two-point correlation functions. described by the condensates do not disappear and survive Let us consider the two-point function of the elastic the large-Nc limit. V vector current, Vμ ¼ q¯γμq, and denote it as ΠμνðxÞ ≡ hTfVμðxÞVνð0Þgi. Obviously, we have light pseudoscalar C. Hadron saturation of two-point function mesons (hereafter referred to as pions). Figures 3(a) and sum rules and 3(b) show the scaling of the hadron diagrams and As is well known [7,8], the spectrum of states of vertices at large Nc, and Fig. 3(c) shows the Nc-leading QCDðNcÞ in the limit Nc → ∞ contains towers of an sum rule: the sum over stable vector mesons is dual to the

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π π fv fv V V π g4 V π fv gv gv fv V g4 0 −2 O(Nc ) O(N c ) O(N c ) (a) π π π π π 1/2 1/2 V ~Nc V ~1/N ~1/N ~1/Nc fv g π c π π c π gv gv π v g4 (b)

fv f O(N ): v =~ ++ c V V +++ +...

(c)

FIG. 3. Duality relation for the elastic vector two-point function at large Nc. (a) Typical hadron diagrams emerging in the hadron representation of ΠV . (b) Scaling of the hadron couplings at large Nc. (c) Nc-leading sum rule [of order OðNcÞ].

Nc-leading OPE. Obviously, one can include some Nc- quark annihilation diagrams. Our second assumption is that subleading effects on the hadron and/or on the OPE side of the two antiquarks, b¯ and c¯, are heavy and therefore do not this sum rule. However, for the consistency of the full produce quark condensates; the two quarks, u and d, are approach, it is mandatory that the Nc-leading contribution light and therefore develop local vacuum condensates ¯ on the hadron side matches the Nc-leading contribution on (quark condensates huu¯ i, hddi, mixed quark-gluon con- A A ¯ A A the OPE side. In fact, the vector sum rule in real QCD [2] densates hu¯σμνT Gμνui, hdσμνT Gμνdi, four-quark con- looks very similar to the relation shown in Fig. 3(c) and uu¯ dd¯ N densates h i, etc.). These assumptions simplify the thus perfectly satisfies the large- c consistency. discussion but do not change any essential qualitative Closing this discussion, we notice that the correlation “ ” feature of our analysis. function of bilinear quark currents describes the minimal As follows from the property of cluster reducibility of colorless cluster and thus does not contain any other multiquark operators [24], any gauge-invariant multiquark factorizable colorless clusters. As a result, the hadron operator can be reduced to a combination of products of N saturation of the c-leading part of the correlation function colorless clusters. In our case of the b¯cud¯ flavor content of contains all intermediate hadron states with the appropriate the tetraquark current, colorless clusters of two different quantum numbers, starting from the one-meson state. flavor structures emerge in QCD (see [25,26]):

III. OPE FOR CORRELATORS θbu¯ cd¯ ¼ jbu¯ jcd¯ ; θbd¯ cu¯ ¼ jbd¯ jcu¯ ; ð3:1Þ OF TETRAQUARK CURRENTS

Let us now turn to two-point functions of tetraquark with jaf¯ ¼ q¯ aqf. We therefore should distinguish between currents. We consider four-quark currents consisting of two the diagrams where quark flavors in the initial and final antiquarks of generic flavors b and c and two quarks of states are combined in the same way (direct diagrams) and generic flavors u and d. For the sake of argument, we make in a different way (quark-exchange or recombination two assumptions: first, we assume that all quark flavors are diagrams). The Feynman diagrams for the corresponding different—this simplifies the topology of the appropriate four-point functions have different topologies and struc- QCD diagrams, avoiding, in particular, the discussion of tures of their four-quark singularities and therefore require

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b b b b b u u u u O(N2 ): u c d d d d d c c c c c 2 α 3 2 4 α3 5 O(Nc ) O( Nc ) O( α Nc ) O( N c ) (a) b b b b u u u u d ddd c ccc

2 2 O(α N c) 0 O(N c ): b b b b u u u u d d d d c c c c

3 3 O( α Nc ) (b)

FIG. 4. Typical perturbative diagrams emerging in the OPE for Πdir. (a) Planar diagrams with an arbitrary number of planar gluon exchanges inside the bu or cd quark loops, whereas there are no gluon exchanges between the quark loops. These diagrams have a factorizable structure, i.e., may be represented, in coordinate space, as a product of two expressions—one corresponding to the bu loop 2 and the other corresponding to the cd loop. They have the behavior ∼Nc. (b) Diagrams with two-gluon exchanges between the loops bu and cd; the two gluons may be attached to quarks/antiquarks in different quark loops in any combination. Such diagrams have a cylinder 0 topology and behave as Nc at large Nc. Adding an arbitrary number of planar gluon exchanges inside the loops bu or cd does not change the large-Nc scaling behavior. separate analyses [21,22]. Here, we discuss the direct Factorizable diagrams can be isolated in a unique way and dir Green function provide the Nc-leading behavior of Π ðxÞ. In the diagrams of Fig. 4(b) two quark loops talk to each Πdir x ≡ T θ x θ† 0 : : ð Þ h f bu¯ cd¯ ð Þ bu¯ cd¯ ð Þgi ð3 2Þ other via gluon exchanges. Since both quark loops re- present colorless clusters, one needs at least two gluons to be exchanged between the loops. Diagrams with two gluon A. Perturbative diagrams exchanges between the loops and an arbitrary number of Figure 4 shows perturbative diagrams in the OPE for Πdir planar gluon exchanges inside each of the loops bu and cd with different types of gluon exchanges. have cylinder topology; see Figs. 1 and 2 of Ref. [20]. Their 0 Perturbative diagrams without gluon exchanges between behavior at large Nc is Nc. the quark loops have a factorizable structure, i.e., factorize, One can also have three or more gluon exchanges in coordinate space, into the product of two-point functions between the quark loops. All these diagrams have a of bilinear quark currents, Πbu¯ ðxÞΠcd¯ ðxÞ, where topology of a cylinder with a number of handles. Each handle reduces the large-Nc behavior by two powers of Nc. Π x ≡ T j x j† 0 ; bu¯ ð Þ h f bu¯ ð Þ bu¯ ð Þgi † Πcd¯ ðxÞ ≡ hTfjcd¯ ðxÞjcd¯ ð0Þgi: ð3:3Þ B. Diagrams containing condensates Diagrams containing condensates may be obtained from One can refer to these diagrams as “disconnected” dia- the perturbative diagrams by breaking the internal quark grams, bearing in mind, however, that the vertices corre- and gluon lines and sending the corresponding particles to sponding to the initial (final) bilinear currents are connected vacuum condensates. 1 to each other. So the term “disconnected” should apply to Let us start with the factorizable perturbative diagrams of the internal parts of the diagrams. We will prefer to term Fig. 4(a). By breaking a light-quark or a gluon line in these these diagrams “factorizable.” Since any of the quark loops diagrams, one obtains the contributions of condensates of 2 behave as ∼Nc, the factorizable diagrams are of order Nc. lowest dimensions, the quark condensate hqq¯ i, the gluon condensate hαsGGi, or the mixed quark-gluon condensate, 1 N Notice that the terms “connected” and “disconnected” in shown in Fig. 5(a). These diagrams have the same large- c 2 lattice calculations have quite different meanings; see, e.g., [11]. behavior, OðNcÞ, as the original perturbative diagrams of

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b b bb u u 2 u u u u u O(Nc ): d dd d d c c c c (a) b b u u u u 2 O(Nc ): d c d d c d (b) b bb b u u uuu uu O(N0 ): c dd d c d c c c (c)

FIG. 5. Typical diagrams belonging to different classes containing condensate contributions in the OPE for Πdir. (a) Factorizable diagrams obtained by inserting condensate contributions in perturbative diagrams of Fig. 4(a). (b) Power corrections of the mixed type: they are proportional to condensates of dimension-6 or higher (huu¯ dd¯ i, etc.) and obtained by sending 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).

Fig. 4(a). Important for us is that diagrams containing the C. Isolating the factorizable part from the OPE for quark, the gluon, and the mixed quark-gluon condensates two-point functions of tetraquark currents Πdir are of the factorizable type, which is the same as that of the We are now fully prepared to isolate the factorizable part original perturbative diagrams. from the OPE for the direct two-point function of the A new feature emerges when one calculates the tetraquark currents Πdir, including both perturbative and contributions (i.e., the Wilson coefficients) of higher- nonperturbative condensate contributions. At the level of – dimension four-quark and four-quark gluon condensates; diagrams, one may naively suspect the following decom- see Fig. 5(b). position of the OPE for Πdir shown in Fig. 7: Here, both light quarks, from the upper and the lower loops, can be sent to the condensate simultaneously. For Πdir x Π¯ x Π x Πdir x Πdir x : further analysis, it is convenient to isolate factorizable ð Þ¼ buð Þ cd¯ ð Þþ NF;1ð Þþ NF;2ð Þ contributions from higher-dimension condensates. For instance, the four-quark condensate may be split into As we shall see, this formula contains double counting of factorizable and nonfactorizable (NF) parts in a unique some of the nonperturbative contributions, and the correct way, see Fig. 6: decomposition is a bit more tricky. The factorizable part, Πbu¯ ðxÞΠcd¯ ðxÞ, including the appropriate nonperturbative condensate contributions, is huu¯ dd¯ i ≡ huu¯ ihdd¯ iþhuu¯ dd¯ i : ð3:4Þ NF obvious and is shown in Fig. 7(a). More cumbersome are NF contributions. Here, we The relevance of isolating factorizable parts out of the encounter two types of such NF contributions: Πdir x condensates of higher dimensions will become clear (i) NF;1ð Þ, shown in Fig. 7(b), describes nonfactor- shortly. izable parts of the condensates of higher dimension corresponding to Fig. 5(b); somewhat tricky, these NF nonperturbative power corrections are generated b bb by factorizable perturbative diagrams of Fig. 4(a). u u u u u u = + NF Radiative corrections due to gluon exchanges inside d d d d d d bu cd c cc the loops and (and not between the loops) are included in this class of NF contributions. FIG. 6. Splitting of the condensates of higher dimensions into To understand the proper way to take into account factorizable and NF parts. The factorizable parts provide the Nc- such contributions, let us recall that nonzero vacuum leading contribution, whereas the nonfactorizable pieces have an condensates emerge due to interactions with the ¯ Nc-subleading behavior. For instance, huu¯ i ∼ Nc, hddi ∼ Nc, nonperturbative soft gluon fields. If we look into the uu¯ dd¯ ∼ N2 uu¯ dd¯ ∼ N0 uu¯ dd¯ h i c, whereas h iNF c. anatomy of h iNF, this quantity is nonzero due

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b b d d + + u u c c

b b d d ++ ++ u u c c

bbb d ++++... + d ++d +... u u u u u c c c (a) b b b b u u u u NF + NF + NF + +... d d d d NF c c cc (b) b b b b b b u u u u u u u u + + + + + +... d d d d d d d d c c c c c c (c)

FIG. 7. Different types of contributions to the OPE for Πdir corresponding to the decomposition (3.5). (a) Factorizable part of the OPE given by the product of Πbu¯ ðxÞ and Πcd¯ ðxÞ; Πbu¯ ðxÞ and Πcd¯ ðxÞ here contain both perturbative and nonperturbative condensate Πdir x Πdir x contributions. (b) Typical diagrams for the nonfactorizable contribution NF;1ð Þ. (c) Nonfactorizable contribution NF;2ð Þ. Only those diagrams where both gluons are exchanged between the quark u of the bu loop and the quark d of the dc loop are displayed. Diagrams corresponding to other gluon exchanges between the bu and cd quark loops and the appropriate power corrections can be easily drawn.

to the interactions with the soft gluons of the type power corrections of Fig. 5(c); the latter are obtained shown in Fig. 8. Precisely the same nonperturbative via the conventional rules by breaking the propa- corrections emerge in the nonperturbative contribu- gating lines of light quarks and gluons in the tions related to the diagrams of Fig. 7(c) where perturbative diagrams of Fig. 4(b). gluons are exchanged between the u quark of the bu In the end, the proper decomposition of Πdir that avoids quark loop and the d quark of the cd quark loop. the double counting of the nonperturbative corrections (This actually explains the fact that the large-Nc has the form qq¯ 2 qq¯ qq¯ behaviors of h i and h iNF differ from each Πdir x Π¯ x Π x Πdir x : : other.) ð Þ¼ buð Þ cd¯ ð Þþ NF;2ð Þ ð3 5Þ Taking into account both effects described by Πdir x NF;1ð Þ and by nonperturbative corrections in those Πdir x Let us insert the full system of hadron states in the parts of NF;2ð Þ which correspond to two-gluon exchanges between u and d quarks from the different factorizable part. We then obtain quark loops would be a double counting. We therefore X Π x ≡ T j x j† 0 R x ; : take the appropriate nonperturbative contributions bu¯ ð Þ h f bu¯ ð Þ bu¯ ð Þgi ¼ bu¯ ð Þ ð3 6Þ Πdir x Πdir x hbu¯ into account as a part of NF;2ð Þ and omit NF;1ð Þ. Πdir x “ ” X (ii) NF;2ð Þ, Fig. 7(c), describes the genuinely non- † factorizable perturbative diagrams of Fig. 4(b) and Πcd¯ ðxÞ ≡ hTfjcd¯ ðxÞjcd¯ ð0Þgi ¼ Rcd¯ ðxÞ; ð3:7Þ hcd¯ u where Rbu¯ ðxÞ and Rcd¯ ðxÞ are the quantities coming from u u ... hadron saturation, the explicit form of which is irrelevant. NF <>== Important for us is the fact that the sum runs over the full ¯ d d system of hadron states with flavors bu (hbu¯ ) and cd¯ (hcd¯ ), d respectively. Consequently, the system of the intermediate dir FIG. 8. Emergence of the nonfactorizable part of the four-quark hadron states that emerges in the factorizable part of Π is ¯ condensate huu¯ ddi through interactions with the nonperturbative just the direct product of these two systems, hbu¯ ⊗ hcd¯ .No gluon background. other hadron state, in particular, no exotic state, may

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b bb u u u + + d d d c c c = + +... b b T u u + + +... d d c c Hadronic side OPE side

FIG. 9. T-adequate sum rule that emerges after the exact cancellations of factorizable contributions on the OPE side vs the hadronic side have been taken into account. The OPE side contains nonfactorizable diagrams with two gluon exchanges between the quark loops bu and cd. A typical contribution with gluon exchanges between the loops, joining bc and bd quark pairs, and the corresponding condensate corrections are explicitly shown. [It is noteworthy that all appropriate condensate contributions are obtained according to the known rules [3] (i.e., by breaking the lines of light quarks and gluons) from the perturbative nonfactorizable Feynman diagrams with two or more gluon exchanges between the loops; no other condensate contributions appear.] The dots stand for other two-gluon exchanges 0 (i.e., joining uc and ud, bc and ud, uc and bd pairs from the different loops). All nonfactorizable diagrams scale as Nc. The hadronic side contains the assumed tetraquark contribution and nonfactorizable meson interaction diagrams.

contribute here. So, we conclude that an exotic state, if it fT ¼h0jθbu¯ cd¯ jTi: ð3:9Þ exists in the hadron spectrum of bu¯ cd¯ states, contributes only to the nonfactorizable part of an exotic correlation 0 function. Obviously, the T-adequate sum rule implies fT ∼ Nc. This feature is in full agreement with the known property of N D. T-adequate sum rule and the couplings of large- c QCD that only noninteracting ordinary mesons N tetraquark bound states to tetraquark currents saturate the c-leading QCD diagrams. We emphasize once more that for the consideration of exotic states, the T The -adequate sum rule emerges after Eqs. (3.6) and factorizable part of the OPE is irrelevant. (3.7) have been taken into account, leading to exact Moreover, we would like to point out the following O N2 cancellations between the factorizable ð cÞ contributions qualitative difference between the correlation functions of on the OPE side and the hadron side of the duality relation tetraquarks versus those of bilinear quark currents: as we dir for the direct two-point function Π . Figure 9 shows the have discussed, the existence of stable vector mesons in the T corresponding -adequate sum rule: its OPE side contains hadron spectrum at large Nc is required by matching the O N0 nonfactorizable diagrams of order ð cÞ (both perturba- large-Nc behavior of the OPE side and of the hadron side of tive and condensate contributions); its hadron side contains the vector two-point function ΠV; without vector mesons the assumed tetraquark contribution and nonfactorizable populating the OðNcÞ part of the hadronic side no matching meson diagrams. The contribution of the tetraquark T of may be obtained. For the two-point functions of the ¯ flavor content bcud¯ and mass M to the hadronic side in tetraquark currents, the situation is qualitatively different: 2 momentum space has the form the factorizable OPE and hadronic sides match each other just due to the duality relations for the two-point functions 1 f2 : of the bilinear currents. The nonfactorizable OPE side and T 2 2 ð3 8Þ M − p its nonfactorizable hadronic side have the same large-Nc behavior with or without the tetraquark bound state. So, the and is expressed via the tetraquark coupling to the existence of narrow tetraquark hadrons cannot be estab- interpolating tetraquark current: lished merely on the basis of the large-Nc behavior of the exotic Green functions; large-Nc QCD does not exclude 2 narrow exotic states in Nc-subleading parts of the Green To be more precise, there are two tetraquark currents of global ¯ functions, but also remains consistent if such exotic states bcud¯ flavor content: θbu¯ cd¯ and θbd¯ cu¯ [Eq. (3.1)]. Respectively, ¯ there might be two tetraquark states of the bcud¯ flavor content, T1 do not exist in the hadron spectrum. and T2: T1 couples more strongly to the θbu¯ cd¯ current (the 0 corresponding coupling scales as Nc) and more weakly to the θbd¯ cu¯ current (the corresponding coupling scales as 1=Nc), and IV. CONCLUSIONS AND OUTLOOK vice versa for T2. So, the contribution of T1 to the correlation 0 We discussed in great detail the OPE for two-point Green function (3.2) scales like Nc, whereas the contribution of T2 is 2 functions of the bilinear and quadrilinear colorless quark suppressed and scales like 1=Nc. These subtleties are, however, a bit outside the main discussion of this paper, so we refer, for currents at large Nc and emphasized the qualitative details, to Sec. 2 of [20]. differences between these two objects:

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(i) In the case of two-point functions of bilinear quark Existing typical applications of QCD sum rules to the currents, the contributions of single-meson states analysis of the tetraquark candidates (see, e.g., recent with appropriate quantum numbers emerge in the publications [27–29]) relate the tetraquark properties to Nc-leading part of the Green function. Matching the low-energy part of the factorizable two-point Green the large-Nc behavior of the OPE series and of the functions of the tetraquark currents. Such an approach hadron saturation series requires the existence of pffiffiffiffiffiffi copies the route of the sum-rule analysis of the ordinary stable mesons with large couplings, fV ∼ Nc,in mesons and does not take into account the fundamental the limit Nc → ∞. The typical QCD sum rule then differences between the correlation functions of the bilinear relates the Nc-leading OðNcÞ part of the OPE to the quark currents and of the tetraquark currents. The tetra- OðNcÞ part of the sum over hadron states, and is quark properties are then extracted exclusively from those therefore fully consistent at large Nc. parts of QCD Green functions which do not have tetra- (ii) In the case of two-point functions of tetraquark quarks as intermediate states [27–29], a feature which does currents, the Nc-leading part of the OPE factorizes not seem physically meaningful to us. Also, one can easily into a product of two colorless clusters. Each of them take the large-Nc limit of the corresponding sum-rule is saturated by the ordinary hadrons that may emerge analytic expressions for the couplings to immediately find in the quark-antiquark correlation functions with that fT ∼ Nc. This would mean that, in contradiction to the appropriate quark-flavor content. As a result, tetra- rigorous property of QCD at large Nc, tetraquark poles quark states (whatever generalization of the Nc ¼ 3 appear in the Nc-leading QCD diagrams. T-adequate sum tetraquark to Nc ≠ 3 is considered) cannot contrib- rules of [21,22], now complemented with the appropriate ute to the Nc-leading parts of the Green functions. account of condensate contributions, are free from these This property fully agrees with the well-known shortcomings and lead to fully consistent relations. rigorous property of large-Nc QCD: Nc-leading Green functions are fully saturated by noninteracting ACKNOWLEDGMENTS ordinary mesons. The contribution of any colorless state with a more complicated quark structure, for D. M. and H. S. are grateful for support under joint instance, of an exotic meson, may only appear in CNRS/RFBR Grant No. PRC Russia/19-52-15022. Nc-subleading nonfactorizable parts of the Green H. S. acknowledges support from the EU research and functions of tetraquark currents. Moreover, this innovation program Horizon 2020, under Grant Agreement property is perfectly satisfied by the T-adequate No. 824093. D. M. would like to thank the Organizers of QCD sum rules formulated in [21,22]: one of the the MIAPP program “Deciphering Strong-Interaction outcomes of the T-adequate sum rules is the scaling Phenomenology through Precision Hadron-Spectroscopy” of the tetraquark coupling to the tetraquark current held in October 2019 at the Excellence Cluster “Universe” 0 fT ∼ Nc. In the present paper, we have comple- in Garching, Germany, for financial support of his partici- mented our previous analysis with a detailed dis- pation at this workshop, where a part of this work was cussion of the vacuum condensate contributions. presented.

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