Heavy light tetraquarks from Lattice QCD

We present preliminary results from a lattice calculation of tetraquark states in the charm and bottom sector of the type $ud\bar{b}\bar{b}$, $us\bar{b}\bar{b}$, $ud\bar{c}\bar{c}$ and $sc\bar{b}\bar{b}$. These calculations are performed on $N_f = 2 + 1 + 1$ MILC ensembles with lattice spacing of $a = 0.12~\mathrm{fm} $ and $a=0.06~\mathrm{fm} $. A relativistic action with overlap fermions is employed for the light and charm quarks while a non-relativistic action with non-perturbatively improved coefficients is used in the bottom sector. Preliminary results provide a clear indication of presence of energy levels below the relevant thresholds of different tetraquark states. While in double charm sector we find shallow bound levels, our results suggest deeply bound levels with double bottom tetraquarks.


Introduction
The discovery of the resonances Z b (10607) & Z b (10650) by BELLE [1] in 2012 has shown the existence 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 existence of a new bound state of matter have generated a lot of interest in exploring its hadronic structure with the leading candidate being that of a tetraquark state.A tetraquark state, first employed by Jaffe [3] in the context of light scalar mesons and later for exotic spectroscopy [4], is a colour neutral state formed as a bound system of diquarks and antidiquarks.The tetraquark structure has been recently employed to identify favourable flavour, spin channels in the bottom sector.However, the tools employed in such searches are typically sum rule type calculations.A first principles approach of lattice QCD is more desirable for such searches and in the past year, two lattice studies [5][6][7] have identified a promising channel with the flavor content ud bb .The calculation in Ref [5] computed a potential of two heavy static antiquarks in presence of two light quarks using lattice QCD.This was then used to solve a coupled non-relativistic Schrödinger equation to find a binding energy of ∆E = 90 +43 −36 MeV in the ud bb channel with I(J P ) = 0(1 + ).In a later extension to this work in Ref [7], a resonance prediction 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 bb and us bb with the binding of 189 (10) MeV and 98 (7) MeV respectively.In the work presented at the conference, we explore the tetraquarks of type ud bb , as in Ref [6], confirming a presence of a level well below the threshold state and explore other flavour channels.
Speaker, e-mail: parikshit@theory.tifr.res.inarXiv:1712.08400v2[hep-lat] 30 Apr 2018 We consider two types of operators here, namely a tetraquark operator with two quarks and two antiquarks having the desired quantum numbers and a two meson operator corresponding to the quantum numbers of that of the tetraquark state.The construction of the tetraquark operator employs a product of a diquark and antidiquark as suggested by Jaffe [4].The diquarks and antidiquarks are constructed with the so called "good diquark" [4] prescription.We would like to construct a tetraquark operator with I(J P ) = 0(1 + ) in the diquark-antidiquark picture with two antibottom and two light quarks in the following configuration: where the braces indicate the Color, Spin and Flavor (C,S,F) degrees of freedom.For the case of light quarks F A indicates antisymmetric flavour combination which in this case will be in 3 f .For the double antibottom quarks, the flavour symmetry is manifestly symmetric F S .In the light diquark, the flavour q ∈ (d, s, c) allows for studying different flavours of tetraquark states.The tetraquark operator shown on line two, indicated by D(x) (keeping consistent notation with Ref [6]), is constructed by taking a dot product of the aforementioned diquark and antidiquark in color space.A two meson operator with the quantum numbers as I(J P ) = 0(1 + ) can be constructed with different flavours q ∈ (d, s, c) as : With these operators, correlation functions were computed on the lattices which will be described in the next section.

Lattice setup
The calculations presented at the conference were performed on MILC ensembles employing HISQ gauge action and N f = 2 + 1 + 1 flavours.The ensemble parameters are shown in Table 1.For both the ensembles the charm and strange quark masses are tuned to their physical values, while the ratio m s /m l is fixed to 5. The details can be found in Ref [8,9].For the propagator computation, we have used wall sources1 and the configurations were gauge fixed in Coulomb gauge and the links were then HYP smeared.In the valence sector, we employ an overlap action where the details can be found in Refs [10][11][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 can be accommodated.The strange quark mass is tuned by equating the fictitious pseudoscalar ss to 685 MeV [12].The charm quark mass was tuned by setting its spin averaged kinetic mass (m η c + 3m J/ψ )/4 to its physical value [10] and the bare values of the am c = 0.529, 0.290 were used for the 24 3 × 64 and 48 3 × 144 lattices respectively.
The bottom sector employs a NRQCD action as shown in Ref [13].In this set up, all terms up to 1/M 2 0 and leading term in 1/M 3 0 are included in the action where M 0 = am b corresponds to the bare bottom quark mass.The interaction part of NRQCD Hamiltonian includes O(a) improved derivatives and also six improvement coefficients c 1 ..c 6 .The details of the action can be found in Ref [11].We use the non-perturbative determination of these coefficients 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 quark mass was tuned by setting its lattice spin averaged mass of (1S) bottomonium: to its experimental value.The kinetic mass is computed as shown in the right equation above.

Results
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: The correlator matrix is a 2 × 2 matrix and we solve a generalised eigenvalue problem to obtain the principle (ground state) correlation function as: where λ(t), v j (t) are the eigenvalue and eigenvectors from the solution of the GEVP.It is convenient to construct effective masses as shown in right equation above.

Results for ud bb
The results for the ud bb tetraquarks are shown in Fig 1 .In the case of the ud bb , the threshold states are those of two free mesons namely B and B * .The plot in the left panel shows the effective masses of the two free meson states obtained from the correlator C = C B C B * (shown in green).The data in blue corresponds to the lowest level of the GEVP solution of the 2 × 2 correlator matrix constructed from the correlation functions of the operators mentioned in section 2. The solution of the GEVP yields two levels, the excited level is found to be noisy and is not shown here.With the use of wall sources, the ground states are seen to approach a plateau from below.The data in blue provides a clear indication of a level below the effective mass of the threshold correlator with the binding indicated on the plot.The results are also seen to be noisy for t > 25.The calculation has been extended to smaller pion masses and the preliminary results are shown in the right panel.At the pion mass close to the SU(3) point, a comparison can be made with the results at the finer lattice spacing (right panel, data in 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 consistent with observations made in Ref [6].

Results for us bb
The results for us bb tetraquarks are shown in Fig 2 .The threshold here is that of B s meson and B * meson.As before, the left panel indicates effective 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 of a 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 3 × 64 lattice with a = 0.1207 fm.The right panel in fig 2 shows the results of the pion mass dependence of the binding energies where the slope of the binding energies with respect to pion masses is not as pronounced in comparison with the of ud bb which possibly indicates a shallower binding at the physical point.These results however are preliminary and will be improved upon in a future publication.We have also included the tetraquark state sc bb in our calculation and the results shown in the Fig 3 .As before the data in indicates the effective mass of the threshold correlator which in this case is two free mesons namely B c meson and B * s meson.We also note that this calculation was done with quark masses for all flavours at their physical value.The main systematic in case will be the lattice spacing dependence of the binding energy which is currently ongoing.Due to the shallow result of the binding energy, study of finite volume effects in this case may also be important.

Results for ud cc
The charm analogue of the doubly bottom tetraquark state is the ud cc state.The two meson thresholds in this case are the D and D * mesons.The results of this calculation are shown in Fig 4 .As before the data in green are the effective masses of the threshold correlator and the data in blue are those of the ud cc.The results in this case are seen lie below but close to the threshold of DD *2 .The results presented here are at lattice spacing a = 0.0583 fm and at heavier pion masses.The extension to lower pion masses and another lattice spacing in currently underway.

Conclusions and outlook
In this work, we have explored heavy light tetraquarks in the bottom and charm sector.The results on a = 0.1207 fm is the progress since the conference.In the ud bb sector, we find a very clear indication of deeply bound levels and the binding energy increases as the pion mass is lowered.The results  shown here are preliminary and with added statistics these may change.The results at lattice spacings a = 0.0583 fm and a = 0.1207 fm are seen to be consistent indicating no significant lattice spacing dependence.The work is being currently extended to improve statistics and include results at lower pion masses for at least one more lattice spacing.The results on us bb are also seen to provide a clear indication of levels below the threshold state at various pion masses.The trend in the slope of the binding energy approaching to the physical point is not as pronounced as that of ud bb indicating that the binding may be shallower.This has also been noted in Ref [6].The effective mass of sc bb state is seen to be closer to the threshold at the physical values of the quark masses.The results on ud cc are seen to lie below but close to the threshold indicating a shallow bound level in this channel.

Figure 1 .
Figure 1.Left panel: Preliminary results for ud bb tetraquark state.See text for the description of effective masses.Right panel : Binding energies for a = 0.0583 fm (data in blue) and a = 0.1207 fm (data in green)

Figure 2 .
Figure 2. Preliminary results for us bb .Left panel : Effective energies for the threshold state and the lowest level of the GEVP solution.Right panel : Summary of binding energies at lower pion masses at a = 0.1207 fm.

Figure 3 .
Figure 3. Preliminary results for sc bb with all flavours at physical quark mass.

Figure 4 .
Figure 4. Preliminary results for ud cc.The color notation is the same as in previous plots.

Table 1 .
Ensemble parameters used in this work