Heavy light tetraquarks from Lattice QCD
HHeavy light tetraquarks from Lattice QCD
Parikshit
Junnarkar ,(cid:63) , M Padmanath , and Nilmani
Mathur Tata Institute of Fundamental Research, Mumbai 400005 Institüt für Theoretische Physik, Universität Regensburg, D-93040 Regensburg, Germany.
Abstract.
We present preliminary results from a lattice calculation of tetraquark states inthe charm and bottom sector of the type ud ¯ b ¯ b , us ¯ b ¯ b , ud ¯ c ¯ c and sc ¯ b ¯ b . These calculationsare performed on N f = + + a = .
12 fmand a = .
06 fm. A relativistic action with overlap fermions is employed for the lightand charm quarks while a non-relativistic action with non-perturbatively improved co-e ffi cients is used in the bottom sector. Preliminary results provide a clear indicationof presence of energy levels below the relevant thresholds of di ff erent tetraquark states.While in double charm sector we find shallow bound levels, our results suggest deeplybound levels with double bottom tetraquarks. The discovery of the resonances Z b (10607) & Z b (10650) by BELLE [1] in 2012 has shown the exis-tence of multiquark exotic states in the bottom sector. Eventually the existence of a tetraquark state Z c (4430) was firmly established by the LHCb collaboration [2]. These new discoveries on the exis-tence of a new bound state of matter have generated a lot of interest in exploring its hadronic structurewith the leading candidate being that of a tetraquark state. A tetraquark state, first employed by Ja ff e[3] in the context of light scalar mesons and later for exotic spectroscopy [4], is a colour neutral stateformed as a bound system of diquarks and antidiquarks. The tetraquark structure has been recentlyemployed to identify favourable flavour, spin channels in the bottom sector. However, the tools em-ployed in such searches are typically sum rule type calculations. A first principles approach of latticeQCD is more desirable for such searches and in the past year, two lattice studies [5–7] have identifieda promising channel with the flavor content ud ¯ b ¯ b . The calculation in Ref [5] computed a potential oftwo heavy static antiquarks in presence of two light quarks using lattice QCD. This was then used tosolve a coupled non-relativistic Schrödinger equation to find a binding energy of ∆ E = + − MeVin the ud ¯ b ¯ b channel with I ( J P ) = + ). In a later extension to this work in Ref [7], a resonanceprediction was made by searching for the poles in the S and T matrices decaying in two B mesons.The work in Ref [6] also confirmed a presence of a deeply bound level in the flavour channels of ud ¯ b ¯ b and us ¯ b ¯ b with the binding of 189(10) MeV and 98(7) MeV respectively. In the work presented at theconference, we explore the tetraquarks of type ud ¯ b ¯ b , as in Ref [6], confirming a presence of a levelwell below the threshold state and explore other flavour channels. (cid:63) Speaker, e-mail: [email protected] a r X i v : . [ h e p - l a t ] A p r Operator setup
We consider two types of operators here, namely a tetraquark operator with two quarks and two anti-quarks having the desired quantum numbers and a two meson operator corresponding to the quantumnumbers of that of the tetraquark state. The construction of the tetraquark operator employs a productof a diquark and antidiquark as suggested by Ja ff e [4]. The diquarks and antidiquarks are constructedwith the so called “good diquark" [4] prescription. We would like to construct a tetraquark operatorwith I ( J P ) = + ) in the diquark-antidiquark picture with two antibottom and two light quarks in thefollowing configuration:( uq ) → ( c , , F A ) , (¯ b ¯ b ) → ( c , , F s ) , D ( x ) = u a α ( x ) ( C γ ) αβ q b β ( x ) ¯ b a κ ( x )( C γ i ) κρ ¯ b b ρ ( x ) . (1)where the braces indicate the Color, Spin and Flavor (C,S,F) degrees of freedom. For the case of lightquarks F A indicates antisymmetric flavour combination which in this case will be in f . For the doubleantibottom quarks, the flavour symmetry is manifestly symmetric F S . In the light diquark, the flavour q ∈ ( d , s , c ) allows for studying di ff erent flavours of tetraquark states. The tetraquark operator shownon line two, indicated by D ( x ) (keeping consistent notation with Ref [6]), is constructed by taking adot product of the aforementioned diquark and antidiquark in color space. A two meson operator withthe quantum numbers as I ( J P ) = + ) can be constructed with di ff erent flavours q ∈ ( d , s , c ) as : M d ( x ) = B + ( x ) B ∗ ( x ) − B ( x ) B + ∗ ( x ) → ( q = d ) M s ( x ) = B + ( x ) B ∗ s ( x ) − B s ( x ) B + ∗ ( x ) → ( q = s ) M c ( x ) = B + ( x ) B ∗ c ( x ) − B + c ( x ) B + ∗ ( x ) → ( q = c ) (2)With these operators, correlation functions were computed on the lattices which will be described inthe next section. The calculations presented at the conference were performed on MILC ensembles employing HISQgauge action and N f = + + m s / m l is fixed to 5. The details can be found in Ref [8, 9]. Table 1.
Ensemble parameters used in this work V β a(fm) m π (MeV) m π L N confs ×
64 6.00 0.1207(14) 305 4.54 23648 ×
144 6.72 0.0582(5) 319 4.51 70For the propagator computation, we have used wall sources and the configurations were gaugefixed in Coulomb gauge and the links were then HYP smeared. In the valence sector, we employ anoverlap action where the details can be found in Refs [10–12]. The use of overlap action eliminates O ( a ) lattice artifacts. In addition with the use of a multimass algorithm, a range of input bare masses Propagators were also computed on point sources although the results are not shown here. an be accommodated. The strange quark mass is tuned by equating the fictitious pseudoscalar ¯ ss to685 MeV [12]. The charm quark mass was tuned by setting its spin averaged kinetic mass ( m η c + m J /ψ ) / am c = . , .
290 were used for the24 ×
64 and 48 ×
144 lattices respectively.The bottom sector employs a NRQCD action as shown in Ref [13]. In this set up, all terms up to1 / M and leading term in 1 / M are included in the action where M = am b corresponds to the barebottom quark mass. The interaction part of NRQCD Hamiltonian includes O ( a ) improved derivativesand also six improvement coe ffi cients c .. c . The details of the action can be found in Ref [11]. Weuse the non-perturbative determination of these coe ffi cients as done by the HPQCD collaboration [14]for the coarser lattices. For the finer lattice, they are set to their tree level values. The bottom quarkmass was tuned by setting its lattice spin averaged mass of (1S) bottomonium: M kin (1 S ) = aM kin ( Υ ) + aM kin ( η b ) , M kin = a p − ( a ∆ E ) a ∆ E (3)to its experimental value. The kinetic mass is computed as shown in the right equation above. In obtaining the ground states of the correlation functions, we employ the variational method [15, 16].With the tetraquark operator D ( x ) and the two meson operator M ( x ), we compute a correlator matrix: C i j ( t ) = (cid:88) x (cid:104) |O i ( x , t ) O † j (0 , | (cid:105) O i ( x , t ) ∈ (cid:8) D ( x , t ) , M ( x , t ) (cid:9) (4)The correlator matrix is a 2 × C i j ( t + ∆ t ) v j ( t ) = λ ( t ) C i j ( t ) v j ( t ) m e ff ( t ) = − log λ ( t ) ∆ t (5)where λ ( t ) , v j ( t ) are the eigenvalue and eigenvectors from the solution of the GEVP. It is convenientto construct e ff ective masses as shown in right equation above. ud ¯ b ¯ b The results for the ud ¯ b ¯ b tetraquarks are shown in Fig 1. In the case of the ud ¯ b ¯ b , the threshold statesare those of two free mesons namely B and B ∗ . The plot in the left panel shows the e ff ective massesof the two free meson states obtained from the correlator C = C B C B ∗ (shown in green). The data inblue corresponds to the lowest level of the GEVP solution of the 2 × ff ective mass of the threshold correlator with the binding indicated onthe plot. The results are also seen to be noisy for t >
25. The calculation has been extended to smallerpion masses and the preliminary results are shown in the right panel. At the pion mass close to theSU(3) point, a comparison can be made with the results at the finer lattice spacing (right panel, datain blue) where the binding energies results are seen to be consistent. As the pion mass is lowered,the binding is seen to get deeper, albeit with higher uncertainties. This trend is found to be consistentwith observations made in Ref [6]. t . . . M eff E =-103.56 ± m π = 550 MeV, a = 0.0583 fm BB ∗ ud ¯ b ¯ b
150 300 450 600 m π − − − ∆ E ud ¯ b ¯ b a = 0 . a = 0 . Figure 1.
Left panel: Preliminary results for ud ¯ b ¯ b tetraquark state. See text for the description of e ff ectivemasses. Right panel : Binding energies for a = . a = . t . . . . M eff E =-109.2 ± us ¯ b ¯ b m π = 497 MeV us ¯ b ¯ bB s B ∗
150 300 450 600 m π − − − ∆ E us ¯ b ¯ b a = 0 . Figure 2.
Preliminary results for us ¯ b ¯ b . Left panel : E ff ective energies for the threshold state and the lowest levelof the GEVP solution. Right panel : Summary of binding energies at lower pion masses at a = . us ¯ b ¯ b The results for us ¯ b ¯ b tetraquarks are shown in Fig 2. The threshold here is that of B s meson and B ∗ meson. As before, the left panel indicates e ff ective masses of the product of the correlators of B s and B ∗ (shown in green) and the lowest level of the GEVP solution (data in blue). A clear indication ofa level below the threshold state in seen for m π =
497 MeV with the binding indicated on the plot.The results shown here are computed on 24 ×
64 lattice with a = . ud ¯ b ¯ b whichossibly indicates a shallower binding at the physical point. These results however are preliminaryand will be improved upon in a future publication. sc ¯ b ¯ b t . . . . . M eff E =-19.19 ± sc ¯ b ¯ b sc ¯ b ¯ bB c B ∗ s Figure 3.
Preliminary results for sc ¯ b ¯ b with all flavours at physical quark mass. We have also included the tetraquark state sc ¯ b ¯ b in our calculation and the results shown in theFig 3. As before the data in green indicates the e ff ective mass of the threshold correlator which inthis case is two free mesons namely B c meson and B ∗ s meson. We also note that this calculation wasdone with quark masses for all flavours at their physical value. The main systematic in case will bethe lattice spacing dependence of the binding energy which is currently ongoing. Due to the shallowresult of the binding energy, study of finite volume e ff ects in this case may also be important. ud ¯ c ¯ c The charm analogue of the doubly bottom tetraquark state is the ud ¯ c ¯ c state. The two meson thresholdsin this case are the D and D ∗ mesons. The results of this calculation are shown in Fig 4. As beforethe data in green are the e ff ective masses of the threshold correlator and the data in blue are those ofthe ud ¯ c ¯ c . The results in this case are seen lie below but close to the threshold of DD ∗ . The resultspresented here are at lattice spacing a = . In this work, we have explored heavy light tetraquarks in the bottom and charm sector. The results on a = . ud ¯ b ¯ b sector, we find a very clear indicationof deeply bound levels and the binding energy increases as the pion mass is lowered. The results An error was later found in computation of D ∗ mass. The corrected threshold is shown in Fig. 4. t . . . . . M eff E =-15.06 ± m π = 550 MeV, a = 0.0583 fm DD ∗ udcc
550 600 650 700 m π (MeV) − − − ∆ E ( M e V ) ud ¯ c ¯ c a = 0 . Figure 4.
Preliminary results for ud ¯ c ¯ c . The color notation is the same as in previous plots. shown here are preliminary and with added statistics these may change. The results at lattice spacings a = . a = . us ¯ b ¯ b are also seen to provide a clearindication of levels below the threshold state at various pion masses. The trend in the slope of thebinding energy approaching to the physical point is not as pronounced as that of ud ¯ b ¯ b indicating thatthe binding may be shallower. This has also been noted in Ref [6]. The e ff ective mass of sc ¯ b ¯ b state isseen to be closer to the threshold at the physical values of the quark masses. The results on ud ¯ c ¯ c areseen to lie below but close to the threshold indicating a shallow bound level in this channel. The calculations are performed using computing resources of the Indian Lattice Gauge Theory Ini-tiative and the Department of Theoretical Physics, TIFR. We thank A. Salve, K. Ghadiali and P.Kulkarni for technical supports. P.M.J. and N.M. acknowledge support from the Department of Theo-retical physics, TIFR. M. P. acknowledges support from Deutsche Forschungsgemeinschaft Grant No.SFB / TRR 55 and EU under grant no. MSCA-IF-EF-ST-744659 (XQCDBaryons). We are grateful tothe MILC collaboration and in particular to S. Gottlieb for providing us with the HISQ lattices.
References [1] A. Bondar et al. (Belle), Phys. Rev. Lett. , 122001 (2012), [2] R. Aaij et al. (LHCb), Phys. Rev. Lett. , 222002 (2014), [3] R.L. Ja ff e, Phys. Rev. D15 , 267 (1977)[4] R. Ja ff e, Physics Reports , 1 (2005)[5] P. Bicudo, J. Scheunert, M. Wagner, Phys. Rev. D95 , 034502 (2017), [6] A. Francis, R.J. Hudspith, R. Lewis, K. Maltman, Phys. Rev. Lett. , 142001 (2017),
7] P. Bicudo, M. Cardoso, A. Peters, M. Pflaumer, M. Wagner, Phys. Rev.
D96 , 054510 (2017), [8] A. Bazavov et al. (MILC), Phys. Rev.
D87 , 054505 (2013), [9] S. Basak, S. Datta, A.T. Lytle, M. Padmanath, P. Majumdar, N. Mathur, PoS
LATTICE2013 ,243 (2014), [10] S. Basak, S. Datta, N. Mathur, A.T. Lytle, P. Majumdar, M. Padmanath (ILGTI), PoS
LAT-TICE2014 , 083 (2015), [11] N. Mathur, M. Padmanath, R. Lewis, PoS
LATTICE2016 , 100 (2016), [12] S. Basak, S. Datta, M. Padmanath, P. Majumdar, N. Mathur, PoS
LATTICE2012 , 141 (2012), [13] G.P. Lepage, L. Magnea, C. Nakhleh, U. Magnea, K. Hornbostel, Phys. Rev.
D46 , 4052 (1992), hep-lat/9205007 [14] R.J. Dowdall et al. (HPQCD), Phys. Rev.
D85 , 054509 (2012), [15] M. Luscher, U. Wol ff , Nucl. Phys. B339 , 222 (1990)[16] B. Blossier, M. Della Morte, G. von Hippel, T. Mendes, R. Sommer, JHEP , 094 (2009),, 094 (2009),