Comment on "OPE and quark-hadron duality for two-point functions of tetraquark currents in 1/N_c expansion"
aa r X i v : . [ h e p - ph ] F e b Comment on ”OPE and quark-hadron duality for two-pointfunctions of tetraquark currents in /N c expansion” Zhi-Gang Wang Department of Physics, North China Electric Power University, Baoding 071003, P. R. China
Abstract
Without excluding the contributions of factorizable Feynman diagrams in the color spaceto the QCD sum rules by hand, we cannot obtain the conclusion that the factorizable parts ofthe operator product expansion series cannot have any relationship to the possible tetraquarkbound states. The tetraquark couplings f T are of the order O ( N c ) rather than of the order O ( N c ) in the large N c limit, the conclusion ”a possible exotic tetraquark state may appearonly in N c -subleading contributions to the QCD Green functions” is a paradox. A hadron has many Fock states, a tetraquark state, which has four valence quarks, maybehave color-singlet-color-singlet (11) type, color-antitriplet-color-triplet (¯33) type or color-sextet-color-antisextet (6¯6) type Fock states. If a hidden-charm tetraquark state has the 11-type Fockstates, we can construct the 11-type four-quark currents, which should have the same quantumnumbers, to interpolate this hidden-charm tetraquark state, because the quantum field theory doesnot forbid such current-tetraquark couplings. The argument applies to other Fock states.Now let us construct the four-quark currents to interpolate the hidden-charm tetraquark states, J ( x, ǫ ) = ¯ c ( x + ǫ )Γ q ( x + ǫ ) ¯ q ′ ( x )Γ ′ c ( x ) ,J ¯33 ( x, ǫ ) = ε ijk ε imn q Tj ( x ) C Γ c k ( x )¯ q ′ m ( x + ε )Γ ′ C ¯ c Tn ( x + ε ) ,J ( x, ǫ ) = q Tj ( x ) C Γ c k ( x )¯ q ′ j ( x + ε )Γ ′ C ¯ c Tk ( x + ε ) + q Tj ( x ) C Γ c k ( x )¯ q ′ k ( x + ε )Γ ′ C ¯ c Tj ( x + ε ) , (1)where the i , j , k , m , n are color indexes, the Γ and Γ ′ are Dirac γ -matrixes, the ǫ is the spatialseparation between the two clusters in the color space.The tetraquark states are spatial extended objects, if the mean spatial sizes h r i ≥ ǫ , we canchoose the currents J ( x, ǫ ), J ¯33 ( x, ǫ ) and J ( x, ǫ ) to interpolate the 11-type, ¯33-type and 6¯6-type tetraquark states, respectively, although the diquark states in color-sextet is not favored byrepulsive interaction originates from the one-gluon exchange.In fact, it is difficult to take into account the nonlocal effects in the QCD sum rules due to theappearance of the ǫ , as we have to deal with a bound state problem, so we usually take the locallimit ǫ →
0. In the local limit ǫ →
0, the currents J ( x,
0) couple potentially to the 11-typetetraquark states rather than two-meson pairs, because in such a small spatial separation, the ¯ cq and ¯ q ′ c mesons lose themselves and merge into tetraquark states. Direct calculations based on theQCD sum rules support such arguments [1, 2].If the mean spatial sizes h r i < ǫ , the currents J ( x, ǫ ) couple potentially to the two-mesonpairs, because in such large spatial separations, the ¯ cq and ¯ q ′ c mesons retain themselves.Generally speaking, the ¯33-type and 6¯6-type tetraquark states can have larger spatial exten-sions, as the confinement forbids the appearance of the free diquark states [1].In the local limit, the currents J ¯33 ( x,
0) and J ( x,
0) can be transformed into the currents J ( x,
0) freely through Fierz rearrangements in the color and Dirac-spinor spaces [1], we canstudy the current J ( x,
0) as an example. The J (0 , J ¯33 (0 ,
0) and J (0 ,
0) couple potentiallyto the tetraquark states ( T ), h | J / ¯33 / (0 , | T ( p ) i = f T , (2)where the f T are the pole residues or decay constants or tetraquark couplings.Now we write down the two-point correlation functions Π( p ) in the QCD sum rules,Π( p ) = i Z d xe ip · x h | T n J ( x, J † (0 , o | i , (3) E-mail: [email protected]. N c in the large N c limit. In the momentum space,they are nonfactorizable diagrams, the basic integrals are of the form, Z d qd kd l p + q − k + l ) − m c + iǫ q − m ′ q + iǫ k − m q + iǫ l − m c + iǫ . (4)Such integrals certainly have an imaginary part, and we can obtain imaginary parts throughdispersion relation after carrying out the integrals over q , k , l , and they make contributions to theQCD sum rules. From the Fig.1, we can obtain the relation, f T ∝ N c , (5)rather than the relation f T ∝ N c [3] in the large N c limit. In Ref.[4], Lucha, Melikhov andSazdjian discard the factorizable Feynman diagrams in the color space via putting the confinedquark and gluons on the mass-shell and applying Landau equation to study them by hand. InRef.[2], we present detailed discussions to show that the Landau equation is of no use to study theFeynman diagrams in the QCD sum rules for the tetraquark states, the tetraquark states beginto receive contributions at the order O ( α s /α s ) rather than at the order O ( α s ) claimed in Ref.[4].In Ref.[3], Lucha, Melikhov and Sazdjian re-express their viewpoint as the tetraquark states beginto receive contributions at the order O ( α s N c ) in the large N c limit and obtain the conclusion”a possible exotic tetraquark state may appear only in N c -subleading contributions to the QCDGreen functions”, which is a paradox. Acknowledgements
This work is supported by National Natural Science Foundation, Grant Number 11775079.
References [1] Z. G. Wang, Int. J. Mod. Phys.
A35 (2020) 2050138.[2] Z. G. Wang, Phys. Rev.
D101 (2020) 074011.23] W. Lucha, D. Melikhov and H. Sazdjian, Phys. Rev.
D103 (2021) 014012.[4] W. Lucha, D. Melikhov and H. Sazdjian, Phys. Rev.