Landau equation and QCD sum rules for the tetraquark molecular states

In"Landau equation and QCD sum rules for the tetraquark molecular states", I list out six reasons to refute the assertion of Lucha, Melikhov and Sazdjian. In"Comment on"Landau equation and QCD sum rules for the tetraquark molecular states"", they refute my viewpoint. If they want to the refute my viewpoint robustly, they should refute the six reasons one by one, and provide at least one example to illustrate that they have obtained reliable QCD sum rules for the tetraquark states and free two-meson states, respectively.


Introduction
In 2003, the Belle collaboration observed a narrow charmonium-like structure X(3872) in the π + π − J/ψ invariant mass spectrum in the exclusive B-decays [1], which cannot be accommodated in the traditional or normal quark-antiquark model. Thereafter, more than twenty charmonium-like exotic states were observed by the BaBar, Belle, BESIII, CDF, CMS, D0, LHCb collaborations [2], some exotic states are still needed confirmation and their quantum numbers have not been established yet. There have seen several possible interpretations for those X, Y and Z states, such as the tetraquark states, tetraquark (or hadronic) molecular states, dynamically generated resonances, hadroquarkonium, kinematical effects, cusp effects, etc [3,4].
Among those possible interpretations, the tetraquark states and tetraquark molecular states are outstanding and attract much attention as the exotic X, Y and Z states lie near the thresholds of two charmed mesons. In 2006, R. D. Matheus et al assigned the X(3872) to be the J P C = 1 ++ diquark-antidiquark type tetraquark state, and studied its mass with the QCD sum rules [5]. It is the first time to apply the QCD sum rules to study the exotic X, Y and Z states. Thereafter the QCD sum rules become a powerful theoretical approach in studying the masses and widths of the exotic X, Y and Z states, irrespective of assigning them as the hidden-charm (or hidden-bottom) tetraquark states or tetraquark (or hadronic) molecular states, and has given many successful descriptions of the hadron properties [4,5,6,7,8,9,10,11,12]. In the QCD sum rules for the tetraquark states and tetraquark molecular states, we choose the diquark-antidiquark type currents or meson-meson type (more precisely, the color-singlet-color-singlet type currents), respectively, they can be reformed into each other via Fierz rearrangements, for example, where the i, j, k, m, n are color indices.
In the correlation functions for the color-singlet-color-singlet type currents, Lucha, Melikhov and Sazdjian assert that the Feynman diagrams can be divided into or separated into factorizable diagrams and nonfactorizable diagrams in the color space, the factorizable diagrams contribute to the meson-meson scattering states, while the nonfactorizable diagrams, if have a Landau pole, begin to make contribution to the tetraquark (molecular) states, the tetraquark (molecular) states begin to receive contributions at the order O(α 2 s ) (according to the Fierz rearrangements, see Eq. (1)) [13].
In this article, we will examine the assertion in details and use two examples to illustrate the Landau equation is of no use in the QCD sum rule for the tetraquark molecular states.
The article is arranged as follows: in Sect.2, we discuss the usefulness of the Landau equation in the QCD sum rules for the tetraquark molecular states; in Sect.3, we obtain the QCD sum rules for the meson-meson scattering states and tetraquark molecular states as an example; in Sect.4, we present the numerical results and discussions; Sect.5 is reserved for our conclusion.
2 Is Landau equation useful in the QCD sum rules for the tetraquark molecular states?
In the following, we write down the two-point correlation function Π µν (p) in the QCD sum rules as an example, where The color-singlet-color-singlet type current J µ (x) has the quantum numbers J P C = 1 +− , at the hadron side, the quantum field theory allows non-vanishing couplings to the DD * +D * D scattering states or tetraquark molecular states with the J P C = 1 +− . At the QCD side, when we carry out the operator product expansion, Lucha, Melikhov and Sazdjian assert that the Feynman diagrams can be divided into or separated into factorizable diagrams and nonfactorizable diagrams, the factorizable diagrams contribute to the meson-meson scattering states, while the nonfactorizable diagrams, if have a Landau pole, begin to contribute to the tetraquark (molecular) states, the tetraquark (molecular) states begin to receive contributions at the order O(α 2 s ), see Fig.1 [13]. Such an assertion is questionable. Firstly, we cannot assert that the factorizable Feynman diagrams contribute to the mesonmeson scattering states, because the meson-meson scattering state and tetraquark molecular state both have four valence quarks, which can be divided into or separated into two color-neutral clusters. We cannot distinguish which Feynman diagrams contribute to the meson-meson scattering state or tetraquark molecular state based on the two color-neutral clusters. Secondly, the quarks and gluons are confined objects, they cannot be put on the mass-shell, it is questionable to assert that the Landau equation is applicable in the nonperturbative calculations dealing with the quark-gluon bound states [14].
If we insist on applying the Landau equation to study the Feynman diagrams in the QCD sum rules, we should choose the pole masses rather than the M S masses to warrant that there exists a mass pole which corresponds to the mass-shell in pure perturbative calculations, just like in the quantum electrodynamics, where the electron, muon and tau can be put on the mass-shell.
If the Landau equation is applicable in the QCD sum rules for the tetraquark states and tetraquark molecular states, it is certainly applicable in the QCD sum rules for the traditional or normal charmonium and bottomonium states. In the case of the c-quark, the pole massm c = 1.67 ± 0.07 GeV from the Particle Data Group [2], the Landau singularity appears at the s-channel √ s = p 2 = 2m c = 3.34 ± 0.14 GeV > m ηc and m J/ψ . While in the case of the b-quark, the pole massm b = 4.78 ± 0.06 GeV from the Particle Data Group [2], the Landau singularity appears at the s-channel √ s = p 2 = 2m b = 9.56 ± 0.12 GeV > m η b and m Υ . It is odd or unreliable that the masses of the charmonium (bottomonium) states lie below the threshold 2m c (2m b ) in the QCD sum rules for the η c and J/ψ (η b and Υ), as the integrals of the forms at the hadron side are meaningless, where the T 2 is the Borel parameter. The tiny widths of the η c , J/ψ, η b and Υ valuate the zero-width approximation, the hadronic spectral densities are of the form δ s − m 2 ηc/J/ψ/η b /Υ . Thirdly, the nonfactorizable Feynman diagrams begin to appear at the order O(α 0 s /α 1 s ) rather than at the order O(α 2 s ), and make contribution to the tetraquark molecular states, if the assertion of Lucha, Melikhov and Sazdjian is right.
The nonperturbative contributions play an important role and serve as a hallmark for the nonperturbative nature of the QCD sum rules, the nonfactorizable contributions appear at the order O(α s ) due to the operatorsqg s Gqqg s Gq, which come from the Feynman diagrams shown in Fig.2. Such Feynman diagrams can be taken as annihilation diagrams, which play an important role in the tetraquark molecular states [15]. If we insist on applying the landau equation to study the Feynman diagrams shown in Fig.2 and choose the pole mass of the c-quark, we obtain a sub-leading Landau singularity at the s-channel s = p 2 = (m c +m c ) 2 , which indicates that it contributes to the tetraquark molecular states. From the operatorsqg s Gqqg s Gq, we can obtain The nonfactorizable Feynman diagrams contribute to the vacuum condensates qg s σGq 2 for the color-singlet-color-singlet type currents, where the solid lines and dashed lines denote the light quarks and heavy quarks, respectively. the vacuum condensate qg s σGq 2 , where the g 2 s = 4πα s is absorbed into the vacuum condensate, so the Feynman diagrams in Fig.2 can be counted as of the order O(α 0 s ). The nonfactorizable Feynman diagrams appear at the order O(α 0 s ) or O(α 1 s ) (based on how to account for the g 2 s in the vacuum condensates), not at the order O(α 2 s ) asserted in Ref. [13]. Fourthly, the Landau equation servers as a kinematical equation in the momentum space, and is independent on the factorizable and nonfactorizable properties of the Feynman diagrams in color space.
In the leading order, the factorizable Feynman diagrams shown in Fig.3 can be divided into or separated into two color-neutral clusters, each cluster corresponds to a trace both in the color space and in the Dirac spinor space. However, in the momentum space, they are nonfactorizable diagrams, the basic integrals are of the form If we choose the pole masses, there exists a Landau singularity or an s-channel singularity at s = p 2 = (m u +m d +m c +m c ) 2 , which is just a signal of a four-quark intermediate state. We cannot assert that it is a signal of a meson-meson scattering state or a tetraquark molecular state, because the meson-meson scattering state and tetraquark molecular state both have four valence quarks, q, q, c andc, which form two color-neutral clusters. The Landau pole is just a kinematical singularity, not a dynamical singularity [16], it is useless in distinguishing the contributions to meson-meson scattering state and tetraquark molecular state. If we switch off the assertion that the factorizable Feynman diagrams shown in Fig.3 make contribution to the meson-meson scattering states alone, the s-channel singularity at s = p 2 = (m u +m d +m c +m c ) 2 supports that they contribute to the tetraquark molecular states. Fifthly, only formal QCD sum rules for the tetraquark states or tetraquark molecular states are obtained in Ref. [13], no feasible QCD sum rules with predictions can be confronted to the experimental data are obtained up to now.
Sixthly, in the QCD sum rules for the traditional or normal heavy mesons, the lowest Feynman diagram contributes to the vacuum condensate qq is shown in Fig.4, which has no loop-integral in the momentum space to apply the Landau equation and the c-quark is not on the mass-shell. Should this Feynman diagram be discarded? In Ref. [17], Lucha, Melikhov and Simula take into account the Feynman diagram and choose the M S mass m c (µ), why?
In fact, when we carry out the operator product expansion in the QCD sum rules for the hidden-charm tetraquark (molecular) states, we will encounter the terms of the forms m c qq , m c qg s σGq , m 2 c qq 2 , m 2 c qq qg s σGq , m 2 c qg s σGq 2 , the M S mass of the c-quark is preferred.

QCD sum rules with color-singlet-color-singlet type currents
Now let us assume that the assertion of Lucha, Melikhov and Sazdjian is right, the tetraquark molecular states begin to receive contributions at the order O(α 2 s ). Then the contributions at the order O(α k s ) with k ≤ 1 contribute to the meson-meson scattering states, we saturate the QCD sum rules with the meson-meson scattering states and examine whether or not we can obtain feasible QCD sum rules.
In the following, we write down the two-point correlation functions Π µν (p) and Π µναβ (p) in the QCD sum rules, where The current J µ (x) has the quantum numbers J P C = 1 +− , while the current J µν (x) has definite charge conjugation, the components J 0i (x) and J ij (x) have positive-parity and negative-parity, respectively, where the space indexes i, j = 1, 2, 3. The charged current J µ (x) couples potentially DD * + D * D scattering state or tetraquark molecular state with the J P C = 1 +− , while the neutral current J µν couples potentially to the D * sDs1 − D s1D * s meson-meson scattering state or tetraquark molecular state with the J P C = 1 ++ and 1 −+ .
In the following, we write down the possible current-hadron couplings, the ε µ are the polarization vectors of the vector and axialvector mesons or tetraquark molecular , f Ds0 and f Ds1 are the decay constants of the traditional or normal heavy mesons, the λ Z and λ X ± are the pole residues of the tetraquark molecular states, the superscripts ± denote the positive-parity and negative-parity, respectively. The charged DD * + D * D tetraquark molecular state with the J P C = 1 +− and the neutral D * sD s1 − D s1D * s tetraquark molecular state with the J P C = 1 −+ differ from the traditional mesons significantly, and are good subjects to study the exotic states.
At the hadron side, we insert a complete set of intermediate hadronic states with the same quantum numbers as the current operators J µ (x) and J µν (x) into the correlation functions Π µν (p) and Π µναβ (p) to obtain the hadronic representation [18,19]. We isolate the contributions of the meson-meson scattering states and the lowest axialvector and vector tetraquark states and get the results, where λ(a, b, c) = a 2 + b 2 + c 2 − 2ab − 2bc − 2ca, we discard the component Π + (p 2 ) which receives contributions from the positive-parity states, thereafter, we will neglect the superscript − in the X − for simplicity. In this article, we carry out the operator product expansion to the vacuum condensates up to dimension-10, and take into account the vacuum condensates which are vacuum expectations of the quark-gluon operators of the order O(α k s ) with k ≤ 1. In calculations, we assume vacuum saturation for the higher dimensional vacuum condensates. For the current J µ (x), we take into account the vacuum condensates qq , αs π GG , qg s σGq , qq 2 , g 2 s qq 2 , qq αs π GG , qq qg s σGq , qq 2 αs π GG , qg s σGq 2 . The four-quark condensate g 2 s qq 2 comes from the terms qγ µ t a qg s D η G a λτ , q j D † µ D † ν D † α q i and q j D µ D ν D α q i , rather than comes from the perturbative corrections of the qq 2 . The four-quark condensate g 2 s qq 2 plays an important role in choosing the input parameters due to the relation g 2 s = 4πα s (µ), which introduces explicit energy scale dependence, on the other hand, it plays a minor important role in numerical calculations. For the current J µν (x), we take into account the vacuum condensates ss , αs π GG , sg s σGs , ss 2 , ss αs π GG , ss sg s σGs , ss 2 αs π GG , sg s σGs 2 , and neglect the condensate g 2 s ss 2 . According to the arguments of Lucha, Melikhov and Sazdjian [13], all the contributions of the order O(α k s ) with k < 2 only contribute to the meson-meson scattering states. Now take the quarkhadron duality below the continuum threshold s 0 and saturate the hadron side of the correlation functions with the meson-meson scattering states, then perform Borel transform with respect to the variable P 2 = −p 2 to obtain the QCD sum rules: Here we introduce the parameter κ to measure the deviations from 1, if κ ≈ 1, we can get the conclusion tentatively that the meson-meson scattering states can saturate the QCD sum rules. The interested readers can acquire the explicit expressions of the QCD spectral densities ρ A (s) and ρ V (s) via contacting me with E-mail. Then we derive Eqs.(21)- (22) with respect to 1 T 2 , and obtain the two additional QCD sum rules, Thereafter, we will denote the QCD sum rules in Eqs.(23)-(24) as the QCDSR I, and the QCD sum rules in Eqs. (21)- (22) as the QCDSR II.
On the other hand, if the meson-meson scattering states cannot saturate the QCD sum rules, we have to introduce the tetraquark molecular states to saturate the QCD sum rules, We derive Eqs.(25)-(26) with respect to 1 T 2 , and obtain the two QCD sum rules for the masses of the tetraquark molecular states, (28)
The DD * +D * D and D * sD s1 −D s1D * s thresholds are m D +m D * = 3.88 GeV and m D * s +m Ds1 = 4.65 GeV, respectively. For the conventional heavy mesons, the mass-gaps between the ground states and the first radial excited states are about 0.4 − 0.6 GeV, so the continuum threshold parameters can be chosen as √ s 0 = 4.40 ± 0.10 GeV and 5.15 ± 0.10 GeV, respectively.
We search for the acceptable Borel parameters T 2 to warrant convergence of the operator product expansion and pole dominance via try and error. Firstly, let us define the pole contributions PC, and the contributions of the vacuum condensates D(n), where the subscript n in the QCD spectral densities ρ A/V ;n (s) represents the vacuum condensates of dimension n. In Fig.5, we plot the pole contributions with variations of the energy scales of the QCD spectral densities with the parameters T 2 A = 2.9 GeV 2 , s 0 A = 4.40 GeV and T 2 V = 3.9 GeV 2 , s 0 V = 5.15 GeV for the QCD spectral densities ρ A (s) and ρ V (s), respectively. We choose those typical values because the continuum threshold parameters s 0 and Borel parameters T 2 have the relation , the weight functions exp − s T 2 have the same values. From Fig.5, we can see that the pole contributions increase monotonically and considerably with the increase of the energy scales at the region µ < 3.0 GeV, then the pole contributions increase monotonically but slowly with the increase of the energy scales. The pole contributions exceed 50% at the energy scales µ = 1.3 GeV and 2.7 GeV for the QCD spectral densities ρ A (s) and ρ V (s), respectively.
In Fig.6, we plot the absolute values of the D(6) with variations of the energy scales µ of the QCD spectral densities with the parameters T 2 A = 2.9 GeV 2 , s 0 A = 4.40 GeV and T 2 V = 3.9 GeV 2 , s 0 V = 5.15 GeV for the QCD spectral densities ρ A (s) and ρ V (s), respectively. The contributions of the vacuum condensates of dimension 6 play a very important role in the QCD sum rules for the hidden-charm or hidden-bottom tetraquark (molecular) states. From Fig.6, we can see that the contributions |D(6)| decrease monotonically with the increase of the energy scales. A larger energy scale µ leads to a larger pole contribution, but a smaller contribution of the vacuum condensate D (6). Too small contributions of the vacuum condensates will impair the stability of the QCD sum rules.

Meson-meson scattering states alone
We saturate the hadron side of the QCD sum rules with the meson-meson scattering states alone, and study the QCD sum rues shown in Eqs. (21)-(24). In this article, we choose the pole contributions as (40 − 60)%, the pole dominance criterion is satisfied. The Borel windows, continuum threshold parameters, energy scales of the QCD spectral densities and pole contributions are shown explicitly in Table 1. In the Borel windows, the contributions of the higher dimensional vacuum condensates are |D (8) Table 1: The Borel parameters, continuum threshold parameters, energy scales of the QCD spectral densities, pole contributions and κ for the QCDSR I and II, where we show the quark constituents of the meson-meson scattering states in the brackets.  spectral densities ρ A (s) and ρ V (s), respectively. The operator product expansion converges very very good. We take into account all uncertainties of the input parameters at the QCD side, and obtain the values of the κ, which are shown in Table 1. In calculations, we add an uncertainty δµ = ±0.1 GeV to the energy scales µ. From Table 1, we can see that the values κ I = 1.55±0.40 and κ II = 1.37±0.40 overestimate the contributions of theūccd meson-meson scattering states with J P C = 1 +− , while the values κ I = 0.50 ± 0.09 and κ II = 0.46 ± 0.09 underestimate the contributions of thesccs meson-meson scattering states with J P C = 1 −+ .
In Fig.7, we plot the values of the κ with variations of the Borel parameters T 2 with the continuum threshold parameters √ s 0 = 4.40 GeV and 5.15 GeV for theūccd andsccs mesonmeson scattering states, respectively. From Fig.7, we can see that the values of the κ increase monotonically and quickly with the increase of the Borel parameters, no platform appears. Now we can obtain the conclusion tentatively that the meson-meson scattering states cannot saturate the QCD sum rules at the hadron side.

Tetraquark molecular states alone
We saturate the hadron side of the QCD sum rules with the tetraquark molecular states alone, and study the QCD sum rues shown in Eqs.(25)-(28).
In Fig.8, we plot the masses with variations of the energy scales of the QCD spectral densities with the parameters T 2 A = 2.9 GeV 2 , s 0 A = 4.40 GeV and T 2 V = 3.9 GeV 2 , s 0 V = 5.15 GeV for theūccd andsccs tetraquark molecular states, respectively. From Fig.8, we can see that the values  Table 2: The Borel windows, continuum threshold parameters, energy scales of the QCD spectral densities, pole contributions, masses and pole residues of theūccd andsccs tetraquark molecular states.
of the masses decrease monotonically and slowly with the increase of the energy scales µ. Now we encounter the problem how to choose the pertinent energy scales of the QCD spectral densities. We describe the heavy tetraquark system QQqq (or the exotic X, Y , Z states) by a doublewell potential with the two light quarks q andq lying in the two potential wells, respectively. In the heavy quark limit, the Q-quark serves as an static well potential, and attracts the light quark q to form a diquark in the color antitriplet channel or attracts the light antiquarkq to form a meson in the color singlet channel. Then the heavy tetraquark (molecular) states are characterized by the effective heavy quark mass M Q (or constituent quark mass) and the virtuality It is natural to choose the energy scales of the QCD spectral densities as,  [11,12]. We can set the value of the effective c-quark mass as M c = 1.84 ± 0.01 GeV. In this article, we use the energy scale formula µ = M 2 X/Z − 4 × (1.84 GeV) 2 as the constraints to choose the best energy scales of the QCD spectral densities.
Again, we choose the pole contributions as (40−60)%. The Borel windows, continuum threshold parameters, energy scales of the QCD spectral densities and pole contributions are shown explicitly in Table 2, just like in Table 1. Again, in the Borel windows, |D(8)| = (3 − 5)%, |D(10)| ≪ 1% and D(8) = (1 − 2)%, D(10) ≪ 1% for the QCD spectral densities ρ A (s) and ρ V (s), respectively. Now let us take into account all uncertainties of the input parameters, and obtain the values of the masses and pole residues of the tetraquark molecular states, which are shown in Table 2 and Figs.9-10. In calculations, we add an uncertainty δµ = ±0.1 GeV to the energy scales µ according the uncertainty in the effective c-quark mass M c = 1.84 ± 0.01 GeV. From Figs.9-10, we can see that there appear Borel platforms in the Borel windows indeed. The tetraquark molecular states alone can satisfy the QCD sum rules.

How to take into account the meson-meson scattering states at the hadron side
The quantum field does not forbid the couplings between the currents and meson-meson scattering states if they have the same quantum numbers.   Figure 9: The masses with variations of the Borel parameters T 2 , where the A and B correspond to theūccd andsccs tetraquark molecular states, respectively, the regions between the two vertical lines are the Borel windows.  Error bounds Figure 10: The pole residues with variations of the Borel parameters T 2 , where the A and B correspond to theūccd andsccs tetraquark molecular states, respectively, the regions between the two vertical lines are the Borel windows.

Conclusion
The quarks and gluons are confined objects, they cannot be put on the mass-shell, it is questionable to use the Landau equation to study the quark-gluon bound states. If we insist on applying the Landau equation to study the Feynman diagrams in the QCD sum rules, we should choose the pole masses rather than the M S masses, which lead to obvious problems in the QCD sum rules for the traditional or normal charmonium and bottomonium states. The meson-meson scattering state and tetraquark molecular state both have four valence quarks, which form two color-neutral clusters, we cannot distinguish which Feynman diagrams contribute to the meson-meson scattering state or tetraquark molecular state based on the two color-neutral clusters in the factorizable Feynman diagrams. The Landau equation servers as a kinematical equation in the momentum space, and is independent on the factorizable and nonfactorizable properties of the Feynman diagrams in color space.
Furthermore, the nonfactorizable Feynman diagrams begin to appear at the order O(α 0 s /α 1 s ) rather than at the order O(α 2 s ), and make contribution to the tetraquark molecular states. If the assertion of Lucha, Melikhov and Sazdjian is right, the tetraquark molecular states begin to receive contributions at the order O(α 2 s ). Then the contributions at the order O(α k s ) with k ≤ 1 contribute to the meson-meson scattering states. We choose the axialvector current J µ (x) and tensor current J µν (x) to examine the outcome if the assertion is right. The axialvector current J µ (x) couples potentially to the charged DD * + D * D meson-meson scattering states or tetraquark molecular states with the J P C = 1 +− , while the tensor current J µν (x) couples potentially to the neutral D * sDs1 − D s1D * s meson-meson scattering states or tetraquark molecular states with the J P C = 1 −+ . The quantum numbers of the DD * + D * D and D * sDs1 − D s1D * s differ from the traditional or normal mesons significantly, and are good subjects to study the exotic states. After detailed analysis, we observe that the meson-meson scattering states cannot saturate the QCD sum rules, while the tetraquark molecular states can saturate the QCD sum rules. We can take into account the meson-meson scattering states reasonably by adding a finite width to the tetraquark molecular states.
The Landau equation is useless to study the Feynman diagrams in the QCD sum rules for the tetraquark molecular states, the tetraquark molecular states begin to receive contributions at the order O(α 0 s /α 1 s ) rather than at the order O(α 2 s ).