Tetraquarks composed of 4 heavy quarks
aa r X i v : . [ h e p - ph ] N ov Tetraquarks composed of 4 heavy quarks
A.V. Berezhnoy, ∗ A.V. Luchinsky, † and A.A. Novoselov ‡ SINP of Moscow State University, Russia Institute for High Energy Physics, Protvino, Russia
In the current work spectroscopy and possibility of observation at the LHC of tetraquarks com-posed of heavy quarks is discussed. Tetraquarks concerned are T c = [ cc ][¯ c ¯ c ] , T b = [ bb ][¯ b ¯ b ] and T bc ] = [ bc ][¯ b ¯ c ] . By solving nonrelativistic Schroedinger equation masses of these states are foundwith the hyperfine splitting accounted for. It is shown that masses of tensor tetraquarks T c (2 ++ ) and T bc ] (2 ++ ) are high enough to observe these states as peaks in the invariant mass distribu-tions of heavy quarkonia pairs in pp → T c + X → J/ψ + X , pp → T bc ] + X → B c + X and pp → T bc ] + X → J/ψ
Υ(1 S ) + X channels while T b is under the threshold of decay into a vectorbottomonia pair. I. INTRODUCTION
Recent observation of
J/ψ -meson pairs production in proton-proton collisions at energy at LHC renewedinterest to the heavy quarks final states. In the low invariant mass region these quarks can form bound states(called tetraquarks) which can be produced in hadronic experiments. Thereby we would like to discuss physics ofthese states, elaborate their mass spectrum and possibility of experimental observation.Conception of tetraquarks, i.e. mesons composed of valent quarks ( qq ¯ q ¯ q ), was first introduced in works [1, 2] in1976. For example, a -meson and σ -mesons was treated as possible tetraquark candidate [3–6]. However, it is hardto determine quark composition of a particle in the light meson domain so these ideas were not developed further.Observation of new unexpected states such as X (3872) [7, 8] gave this idea a new impetus [9, 10]. Eccentricity ofthese particles consists in fact that according to the modes of their production and decay they contain a c ¯ c pair butthey can not be included in the well known systematics of charmonia. Later similar particles were also found in thebottomonia sector [11–14]. It is natural to ascribe these mesons to tetraquarks ( Qq ¯ Q ¯ q ) , where q are light and Q —heavy ( b or c ) quarks.However, situation when all quarks composing a tetraquark are heavy was not treated in details yet. This possi-bility seems to be quite interesting as in this case determination of meson‘s quark composition becomes simpler andits parameters can be determined by solving nonrelativistic Schroedinger equation. Our work is devoted to theseparticular questions.In our recent paper [15] we consider tetraquark T c = [ cc ][¯ c ¯ c ] in the framework of diquark model.The hyperfinesplitting in that paper was described through interaction of total diquark spins. Now we would like to study alsotetraquarks T b = [ bb ][¯ b ¯ b ] and T bc ] = [ bc ][¯ b ¯ c ] . The last state is especially interesting since, in contrast to tetraquarksbuild from four identical quarks, both singlet and triplet spin states of the diquark are possible. It is clear, thathyperfine interaction of spin-singlet diquark cannot be described with the method used in our previous paper, so someother approach should be applied.In the following section spectroscopy of ( cc ¯ c ¯ c ) , ( bb ¯ b ¯ b ) and ( bc ¯ b ¯ c ) tetraquarks is concerned. Possibility of observationof these particles in hadronic experiments is discussed in the third section. We summarize our results in the shortconclusion. II. SPECTROSCOPYA. General preliminaries
In the current study diquark model of tetraquark is used. According to this approach tetraquark T = Q Q ¯ Q ¯ Q ∗ [email protected] † [email protected] ‡ [email protected] consists of 2 almost point-like diquarks ¯ D = [ Q Q ] and D = [ ¯ Q ¯ Q ] with certain quantum numbers (such asangular momentum, spin, color) and mass. What concerns color configuration, two quarks in diquark can be intriplet or sextet color state. According to the manuscript [16] in the sextet configuration diquarks experience mutualrepulsion so we restrict ourselves to the (anti)triplet color configurations. Angular momentum of the diquark systemequals in the ground state, so its spin is equal to the sum of quark spins which is or . What concerns diquarkmass, it can be determined by solving Schroedinger equation with a correctly selected potential. According to work[17] it is possible to use quark-antiquark interaction potential used in heavy quarkonia calculations with additionalfactor / due to the different color structures.In the diquark model tetraquark mass can be determined by solving -particle Schroedinger equation with point-like diquarks. As [ Q Q ] and [ ¯ Q ¯ Q ] diquarks are in (anti)triplet color configuration, potential of their interactioncoincides with that of quark and antiquark in heavy quarkonia. Hyperfine splitting in this system can be describedby the hamiltonian [3] H = M + 2 X i 19 GeV and h T s i = 0 . 38 GeV for the triplet and singlet states respectively. Values of diquark wave functions in origin are presented in manuscript[17] while for mesons they can be calculated from the leptonic width Γ ee or leptonic constant f of meson in question: | R (0) | = 1 α e q Γ ee M M f . Up to the hyperfine splitting tetraquark can be described by its total spin J , spins of diquarks S and S constituting it and its spatial and charge parities P and C : (cid:12)(cid:12) ++ (cid:11) = | 0; 0 , i , (cid:12)(cid:12)(cid:12) ++ ′ E = | 0; 1 , i , (cid:12)(cid:12) + ± (cid:11) = 1 √ | 1; 0 , i ± | 1; 0 , i ) , (cid:12)(cid:12)(cid:12) + − ′ E = | 1; 1 , i , (cid:12)(cid:12) ++ (cid:11) = | 2; 1 , i In this treatment all states are confluent with mass M . If spin-spin interaction is accounted for, masses of | ++ i and | ++ i states shift: M (cid:0) ++ (cid:1) = (cid:10) ++ | H | ++ (cid:11) = M − κ − κ − ,M (cid:0) ++ (cid:1) = 2 m [12] + κ + κ + , where following designations are introduced: κ ± = 2 κ ± κ ± κ , and | ++ i , | ++ ′ i and | + − i , | + − ′ i states mix with each other. In the scalar tetraquarks case mixing matrix has thefollowing form: H " | ++ i (cid:12)(cid:12)(cid:12) ++ ′ E = (cid:20) M − κ −√ κ − −√ κ − M + κ − κ + (cid:21) " | ++ i (cid:12)(cid:12)(cid:12) ++ ′ E , and for | + − i tetraquarks: H (cid:20) | + − i| + − ′ i (cid:21) = (cid:20) M − κ + κ − κ − κ κ − κ M + κ − κ + (cid:21) (cid:20) | + − i| + − ′ i (cid:21) . B. [ QQ ][ ¯ Q ¯ Q ] In the case when quarks of the same flavor are involved, Fermi-Dirac statistics leads to the additional restrictionson the diquark quantum numbers. Indeed, permutation of quark indices should change sign of the total diquark wavefunction. As quarks are in the antitriplet color state, color part of this function is antisymmetric. Radial wave functionis symmetric as quarks are in the S -wave. So spin part of the wave function is to be symmetric too. Consequentlythe total spin of the S -wave diquark can only be equal to . As a result only | ++ ′ i , | ++ ′ i and | ++ i diquark statesremain. They do not mix with each other after the spin-spin interaction is accounted for. Masses of these states areequal M (cid:16) ++ ′ (cid:17) = M + κ − κ + ,M (cid:16) + − ′ (cid:17) = M + κ − κ + ,M (cid:0) ++ (cid:1) = M + κ + κ + . It worth mentioning that this splitting scheme agrees with the result of the work [15] in which interaction of the totaldiquark spins was concerned.To obtain numerical values of tetraquark masses one needs to know the unsplitted mass M and coefficients κ ij inthe Hamiltonian (1). These coefficients can be calculated using expressions (2) and (3). The following values of quarkmasses were used: m c = 1 . 468 GeV , m b = 4 . 873 GeV . Diquark masses without hyperfine splitting are given in [17] while M mass of the tetraquark was calculated using aprocedure similar to that described in [17].Let us begin with the tetraquark composed of c -quarks, T c = [ cc ][¯ c ¯ c ] . Mass of a ground state and value of radialwave function at origin for a [ cc ] diquark given in [17] are m [ cc ] = 3 . 13 GeV , R [ cc ] (0) = 0 . 523 GeV / . Value of radial wave function at origin of the ( c ¯ c ) state determined from the leptonic width of J/ψ -meson equals R ( c ¯ c ) (0) = 0 . 75 GeV / . Spin-spin interaction coefficients calculated using expressions (2) and (3) are equal κ = κ = κ [ cc ] = 12 . ,κ = κ = κ = κ = κ ( c ¯ c ) = 42 . . (4)Without hyperfine splitting T c tetraquark mass equals M = 6 . 124 GeV , and with it this state splits into scalar, axial and tensor mesons with masses ++ ′ : M = 5 . 966 GeV , M − M th = − . MeV , + − ′ : M = 6 . 051 GeV , M − M th = − . MeV , ++ : M = 6 . 223 GeV , M − M th = 29 . . In expressions above differences between the tetraquark masses and a J/ψ -meson pair formation threshold are alsonoted. It can be seen that only tensor state lies above this threshold and can be observed in the T c (2 ++ ) → J/ψ mode. It worth mentioning that scalar tetraquark which is slightly under the J/ψ -pair threshold can however decayby the T c (0 ++ ′ ) → ( J/ψ ) ∗ J/ψ → µ + µ − J/ψ channel i.e. with one J/ψ -meson being virtual. So it can be observedas a peak in the µ + µ − J/ψ invariant mass distribution.For a tetraquark built from b -quarks, i.e. T b = [ bb ][¯ b ¯ b ] , situation is entirely similar to the previous case. Mass ofthe [ bb ] diquark and values of radial wave function at origin for it and for the ( b ¯ b ) ground state are m [ bb ] = 9 . 72 GeV , R [ bb ] (0) = 1 . 35 GeV / , R ( b ¯ b ) (0) = 2 . 27 GeV / . Mass of the T b tetraquark without hyperfine splitting is equal to M = 18 . 857 GeV , and spin-spin interaction coefficients are κ = κ = κ [ bb ] = 5 . 52 MeV ,κ = κ = κ = κ ( b ¯ b ) = 27 . , (5)With hyperfine splitting one obtains the following masses of the T b states: ++ ′ : M = 18 . 754 GeV , M − M th = − . MeV , + − ′ : M = 18 . 808 GeV , M − M th = − . MeV , ++ : M = 18 . 916 GeV , M − M th = − . MeV . It can be seen that in this case all the states are under the Υ(1 S ) pair production threshold M th = 2 m Υ(1 S ) . C. [ bc ][¯ b ¯ c ] The situation is more interesting in the T bc ] = [ bc ][¯ b ¯ c ] tetraquark case. In this case [ bc ] diquark spin can be or and all states mentioned in Section II-A exist. Diquark mass and value of its radial wave function at origin are [17] m [ bc ] = 6 . 45 GeV R [ bc ] (0) = 0 . 722 GeV / . Radial wave function at origin for the color singlet ( b ¯ c ) state can be determined by its leptonic constant f B c = 500 MeV [17]: R ( b ¯ c ) = 1 . 29 GeV / . So spin-spin interaction coefficients are equal κ = κ = κ [ bc ] = 6 . 43 MeV ,κ = κ = κ ( b ¯ c ) = 27 . . Values of κ = κ ( b ¯ b ) and κ = κ ( c ¯ c ) constants were given in expressions (4) and (5). Without hyperfine splitting T bc ] tetraquark mass equals M = 12 . 491 GeV , and with spin-spin interaction accounted for this state splits (see Fig.1) into• Two scalar states with masses ++ a : M = 12 . 359 GeV , M − M th = − . MeV0 ++ b : M = 12 . 471 GeV , M − M th = − . , ++ a0 ++ b 1 + - a1 + - b1 ++ ++ - - - - - M th . MeV Figure 1. [ bc ][¯ b ¯ c ] tetraquark mass spectrum • Two + − states with masses + − a : M = 12 . 424 GeV , M − M th = − . MeV1 + − b : M = 12 . 488 GeV , M − M th = − . , • One ++ meson with mass ++ : M = 12 . 485 GeV , M − M th = − . , • One tensor meson with mass ++ : M = 12 . 566 GeV , M − M th = 16 . . Mass of the two B c mesons is selected for the threshold value in these expressions, M th = 2 m B c = 12 . 55 GeV . It canbe seen that only tensor tetraquark T bc ] (2 ++ ) lies above this threshold and thus can be observed as a peak in the B c -meson pair invariant mass distribution.In paper [18] tetraquark states were also considered in the framework of diquark model. Picture of hyperfinesplittings of T bc ] -tetraquark, presented in this paper is in good agreement with our results. Predictions for masses,on the other hand, are about 700 MeV higher, than our values. As a result, according to this paper all tetraqurkstates should lie above B ∗ c and J/ψ Υ thresholds. We think, that the main reason for difference between these twoworks is the neglection of binding energy in tetraquark tetraquark and diquark spectra, that is negative. For example,for tetraquark state before hyperfine splitting we have δE = M − m [ bc ] ≈ 410 MeV . III. PRODUCTIONA. Duality relations Duality relations can be used to estimate production cross sections of the particles in question. Let us considerformation of two diquarks in gluon interaction gg → [ Q Q ][ ¯ Q ¯ Q ] . Above the doubly-heavy baryon production m gg , GeV510152025 Σ` @ gg ® (cid:144) Ψ D , pb Figure 2. Invariant mass distribution of J/ψ -meson pairs with expected T c tetraquark contribution. Dashed vertical linecorresponds to the two doubly-heavy baryons formation threshold M Ξ cc . threshold these diquarks can hadronize into open heavy flavor mesons, heavy quarkonia or form a bound state,i.e. tetraquark. According to our estimations this tetraquark would preferably decay into a vector quarkonia pair.Indeed, decay into light mesons is suppressed by the Zweig rule, production of open heavy flavor mesons is prohibitedkinematically and formation of pseudoscalar quarkonia requires flip of the heavy quark spin. That is why the followingduality relation can be written: S T = M Ξ QQ ˆ M Q dm gg ˆ σ [ gg → T → Q ] = ǫ M Ξ QQ ˆ m [ QQ ] dm gg ˆ σ ( gg → [ Q Q ] + [ ¯ Q ¯ Q ]) , (6)where ǫ factor stands for the other possible decay modes. This value is to be compared with the integrated non-resonant cross section of quarkonia pairs production in the same duality window: S Q = M Ξ QQ ˆ M Q dm gg ˆ σ [ gg → Q ] . (7)As tetraquark states are typically narrow, these mesons can be observed as peaks in the quarkonia pairs invariantmass distributions despite that S T ≪ S Q relation holds. As tetraquark width is small compared to the detectorresolution ∆ ≈ 50 MeV , its peak can be modeled by a Gaussian form with corresponding width. Therefore crosssection of the gg → T → Q Breit-Wigner process is replaced with the following expression: ˆ σ ( gg → T → Q ) = S T √ π ∆ exp ( − ( m gg − M T ) ∆ ) , where preexponential factor is selected according to the duality relation (6). B. T Q Let us begin with the tetraquarks composed of the identical quarks. In the T c case only tensor state lies abovethe vector charmonia production threshold. Integrated cross sections calculated using expressions (6) and (7) areequal S T c = 0 . S J/ψ = 20 pb GeV , where suppression factor is selected to be ǫ = 0 . . Invariant mass distribution for the J/ψ -meson pairs with expected T c tetraquark contribution is shown in Fig. 2.As already mentioned, in the T b tetraquark case even tensor state is under the two vector bottomonia formationthreshold so its observation in their invariant mass distribution is doubtful. m gg , GeV1234567 Σ` @ gg ® B c D , fb H a L m gg , GeV1020304050 Σ` @ gg ® J (cid:144) ΨU H LD , fb H b L Figure 3. Invariant mass distribution of B c -meson pairs (left plot) and J/ψ Υ(1 S ) (right plot) with expected contribution ofthe T bc ] tetraquark. Dashed vertical line corresponds to the two doubly-heavy baryons formation threshold M Ξ bc . C. T bc ] Let us turn to the T bc ] = [ bc ][¯ b ¯ c ] tetraquark. In this case pseudoscalar B c -meson and vector B ∗ c -meson decayinginto B c γ can be experimentally observed. Thus tensor tetraquark T bc ] (2 ++ ) can be observed as a peak in the B c -meson pairs invariant mass distribution. However T bc ] (2 ++ ) → B c decay requires flip of the heavy quark spin andis suppressed by the factor ǫ ∼ M T − m [ bc ] m [ bc ] ≈ . × − . Integrated cross sections (6) and (7) are equal S T bc ] = 0 . 13 fb GeV S B c = 6 fb GeV , where M Ξ bc = 6 . 82 GeV [19] is used. Invariant mass distribution of the B c -meson pairs with expected contribution ofthe T bc ] tetraquark is shown in Fig. 3a. All T bc ] mesons lie under the B ∗ c pair production threshold.Decay of the T bc ] tetraquark into the J/ψ Υ(1 S ) vector quarkonia pair is also possible. In the color singlet modelreaction gg → J/ψ Υ(1 S ) is prohibited so accounting for octet components of vector quarkonia is needed. Thisprocess was elaborated in the work [20] results of which are used as the background for the tetraquark contribution.Suppression factor ǫ ∼ × − was used to obtain integrated cross sections (6) and (7): S T bc ] = 1 . , S ψ Υ = 33 fb GeV . Invariant mass distribution of the J/ψ Υ(1 S ) pairs with expected contribution of the T bc ] tetraquark is shown in Fig.3b. IV. CONCLUSION In our work tetraquarks composed of heavy quarks are concerned. Spectroscopy of T c = [ cc ][¯ c ¯ c ] , T b = [ bb ][¯ b ¯ b ] and T bc = [ bc ][¯ b ¯ c ] states is elaborated and possibility of observation at the LHC is studied.Model used implies diquark structure of tetraquarks i.e. Q Q ¯ Q ¯ Q tetraquark is assumed to consist of almostpoint-like diquarks [ Q Q ] and [ ¯ Q ¯ Q ] being in triplet color configuration. In such a treatment tetraquark is entirelyanalogous to a doubly-heavy meson. So non-relativistic Schroedinger equation which gives reliable results for thecharmonia and bottomonia description can be used to obtain its parameters.In the case of T c and T b tetraquarks Fermi principle imposes constraints on the possible quantum numbers ofdiquarks: in the antitriplet color configuration total spin of the S -wave diquark can be only equal to . So with thehyperfine splitting accounted for states with the following quantum numbers appear: ++ , + − and ++ . In thecharmed tetraquark case only tensor meson lies above the vector meson pair formation threshold and its peak can beobserved at LHC in their invariant mass distribution. For the T b tetraquark all the states lie under the Υ(1 S ) pairformation threshold. Both and spins are possible for the diquark in the T bc tetraquark. So states arise afterthe hyperfine splitting of the unaffected state: two of ++ , one ++ , two of + − and one ++ .In the last section we estimated the possibility to observe tetraquarks concerned in the inclusive reactions gg → T → Q . According to our calculations T c (2 ++ ) tensor state can be observed as a peak in the J/ψ -meson pairsinvariant mass distribution. T bc ] (2 ++ ) tetraquark can be observed in both B c and J/ψ Υ(1 S ) modes.Authors would like to thank A.K. Likhoded and V.V. Kiselev for the fruitful discussions. The work was financiallysupported by Russian Foundation for Basic Research (grant [1] R. L. Jaffe, Phys.Rev. D15 , 267 (1977).[2] R. L. Jaffe, Phys.Rev. D15 , 281 (1977).[3] L. Maiani, F. Piccinini, A. Polosa, and V. Riquer, Phys.Rev. D71 , 014028 (2005), hep-ph/0412098.[4] S. S. Gershtein, A. K. Likhoded, and A. V. Luchinsky, Phys. Rev. D74 , 016002 (2006), hep-ph/0602048.[5] N. Mathur et al., Phys. Rev. D76 , 114505 (2007), hep-ph/0607110.[6] S. Prelovsek et al., PoS LAT2009 , 103 (2009), arXiv:0910.2749.[7] V. M. Abazov et al. (D0), Phys. Rev. Lett. , 162002 (2004), hep-ex/0405004.[8] B. Aubert et al. (BABAR), Phys. Rev. D71 , 071103 (2005), hep-ex/0406022.[9] E. S. Swanson, Phys. Rept. , 243 (2006), hep-ph/0601110.[10] D. Ebert, R. N. Faustov, and V. O. Galkin, Phys. Lett. B634 , 214 (2006), hep-ph/0512230.[11] K. F. Chen et al. (Belle), Phys. Rev. Lett. , 112001 (2008), arXiv:0710.2577.[12] I. Adachi et al. (Belle), Phys. Rev. D82 , 091106(R) (2010), arXiv:0808.2445.[13] A. Ali, C. Hambrock, and M. J. Aslam, Phys. Rev. Lett. , 162001 (2010), arXiv:0912.5016.[14] A. Ali, C. Hambrock, and W. Wang (2011), arXiv:1110.1333.[15] A. Berezhnoy, A. Likhoded, A. Luchinsky, and A. Novoselov (2011), arXiv:1101.5881.[16] K. Terasaki, Prog.Theor.Phys. , 199 (2011), arXiv:1008.2992.[17] V. Kiselev, A. Likhoded, O. Pakhomova, and V. Saleev, Phys.Rev. D66 , 034030 (2002), hep-ph/0206140.[18] A. Ali, C. Hambrock, I. Ahmed, and M. J. Aslam, Phys. Lett. B684 , 28 (2010), arXiv:0911.2787.[19] V. Kiselev and A. Likhoded, Phys.Usp. , 455 (2002), hep-ph/0103169.[20] P. Ko, C. Yu, and J. Lee, JHEP1101