Numerical solution of the Schr¨ odinger equation

We define U(r) = V(r)|_V0=0,p=2 with V(r) from of Equation (5.5). U(r) with a set of fit parameters α and d from Table 5.3 corresponds to the ground state energy of aqq¯b¯b four-quark system in a specific channel minus the energy of a pair of far separatedB_(s,c)^(∗) mesons.

qq spin α din fm (ud−du)/√

2 scalar 0.35^+0.04_−0.04 0.42^+0.08_−0.08 uu,(ud+du)/√

2,dd vector 0.29^+0.04_−0.06 0.16^+0.02_−0.01 (s⁽¹⁾s⁽²⁾−s⁽²⁾s⁽¹⁾)/√

2 scalar 0.27^+0.08_−0.05 0.20^+0.10_−0.10 ss vector 0.18^+0.09_−0.02 0.18^+0.11_−0.05 (c⁽¹⁾c⁽²⁾−c⁽²⁾c⁽¹⁾)/√

2 scalar 0.19^+0.12_−0.07 0.12^+0.03_−0.02

Table 5.3.: Parametersα and dobtained from χ² minimizing fits of (5.5) to Lattice QCD¯b¯b potential results.

Thus, the corresponding Hamiltonian for the relative coordinate of the¯b¯bquarks is H = p²

2µ+ 2m_H+U(r), (5.7)

whereµ = m_H/2 is the reduced mass. At large separations, each¯b quark carries the mass of aB_(s,c)^(∗) meson because of screening, and thusm_H =m

B_(s,c)^(∗) . At small separations,m_H = mb could be more appropriate. Throughout this section, we always consider two choices, mH =mB_(s,c) andmH =m_b, which yield qualitatively identical results. Any dependence on the heavy¯bspins is neglected, becauseV(r)has been computed in the static limit m_b → ∞.

Since the¯bquarks are quite heavy, we expect the static limit to be a reasonable approximation.

Note, however, that a similar study [60] with infinitely heavy¯bquarks that takes into account the heavy quark spin finds the same bound state as we report on here with a slightly reduced binding energy.

To investigate the existence of a bound state, we solve the Schr¨odinger equation with the Hamiltonian (5.7) numerically. The strongest binding is expected in an s-wave, for which the radial equation is

− 1 2µ

d²

dr² +U(r)

R(r) =

E−2mH

| {z }

=EB

R(r) (5.8)

with the wave functionψ=ψ(r) =R(r)/r. IfE_B =E−2m_H <0,−E_B can be interpreted as the binding energy. We proceed as explained in [61] and solve this equation by imposing Dirichlet boundary conditionsR(r=∞) = 0and using 4th order Runge-Kutta shooting.

For the scalaru/dchannel, the lowest eigenvalueE_B<0, which implies the existence of a bound four-quark state. For all other channels, i.e. the vectoru/dand thesandcchannels, EB>0, i.e. the correspondingqq¯b¯btetraquarks will most likely not exist in these channels¹. The central value and the combined systematic and statistical error for the binding energy E_B of the tetraquark state in the scalaru/dchannel is obtained by the method discussed in

1As mentioned previously in Section 5.4.3, the Lattice QCD results for the vectorcchannel are not sufficient to perform a quantitative analysis. The¯b¯bpotential in this channel is, however, much less attractive than in the other channels, e.g. the scalarcchannel. Therefore, a bound four-quark state in the vectorcchannel can be excluded.

5.4. Investigation of attractive ground state potentials

Figure 5.7.:¯b¯b potentials in the presence of two lighter quarks qq (qq flavor: up/down in green, strange in blue, charm in red; qq spin: jz = 0, i.e. scalar, in the upper line,j= 1, i.e. vector, in the lower line). The plotted curves with the error bands correspond to eq. (5.5) with the parameter sets from Table 5.3. Vertical lines indicate lattice separationsr = 2a,3a, . . .of Lattice QCD potential resultsV^lat(r) used to generate the parameter sets from Table 5.3 viaχ² minimizing fits.

Section 5.4.3 (generating a distribution forE_Bfrom the fits listed in the same section):

E_B=−90⁺⁴⁶₋₄₂MeV (form_H =m_B), (5.9) EB=−93⁺⁴⁷₋₄₃MeV (formH =m_b). (5.10) These binding energies are roughly twice as large as their combined systematic and statisti-cal errors. In other words, the confidence level for this ud¯b¯b tetraquark state is around2σ.

The corresponding histogram formH =mB representing the determination of the errors as explained above is shown in Figure 5.8.

To quantify also the non-existence of bound four-quark states in the remaining channels, we determine numerically by which factors the heavy masses mH in the Schr¨odinger equation (5.8) have to be increased to obtain bound states, i.e. tiny but negative energies E_B (the potentialsU(r)are kept unchanged, i.e. we stick to the medians forαanddfrom Table 5.3).

The resulting factors are collected in Table 5.4. While the scalar schannel is quite close to be able to host a bound state, the scalar c channel and the vector channels are rather far away, since they would require¯bquarks approximately1.6. . .3.3 times as heavy as they are in nature. Note that the factors listed in Table 5.4 could also be relevant for quark models aiming at studying the binding of tetraquarks quantitatively.

In Figure 5.9, we present our results in an alternative graphical way. Binding energy isolines EB(α, d) = constant are plotted in the α-d-plane starting at a tiny energy EB = −0.1 MeV up to rather strong binding, EB = −100 MeV (gray dashed lines have been computed with

Figure 5.8.: Histogram used to estimate the systematic error for the binding energyEBfor the scalaru/dchannel andmH =mB(green, red and blue bars represent systematic, statistical and combined errors, respectively).

qq spin m_H =m_B(s,c) m_H =m_b

(ud−du)/√

2 scalar 0.46 0.49

uu,(ud+du)/√

2,dd vector 1.49 1.57

(s⁽¹⁾s⁽²⁾−s⁽²⁾s⁽¹⁾)/√

2 scalar 1.20 1.29

ss vector 2.01 2.18

(c⁽¹⁾c⁽²⁾−c⁽²⁾c⁽¹⁾)/√

2 scalar 2.57 3.24

Table 5.4.: Factors by which the massm_H has to be multiplied to obtain a tiny but negative energyE_B. The factor1indicates a strongly bound state, while for values1 bound states are essentially excluded.

m_H = m_B(s,c), gray solid lines with m_H = m_b). The three plots correspond to u/d, s and c light quarks qq, respectively. For the detailed discussion about systematic error estimation for α and d, cf. Section 5.4.3. The extensions of the point clouds represent the systematic uncertainties with respect toαandd. If a point cloud is localized above or left of the isoline withE_B =−0.1 MeV (approximately the binding threshold), the corresponding four quarks

¯b¯bqq will not form a bound state. A localization below or right of that isoline is a strong indication for the existence of a tetraquark. In case the point cloud is intersected by that isoline, the estimated systematic error is too large to make a definite statement regarding the existence or non-existence of a bound four-quark state. The big red and green bars in horizontal and vertical direction represent the combined systematic and statistical errors ofα andd, as quoted in Table 5.3. One can observe and conclude the following from Figure 5.9:

• There is clear evidence for a tetraquark state in the scalaru/dchannel.

• The scalar schannel is close to binding/unbinding. For a further investigation of this channel, cf. [62] where a corresponding resonance treatment using techniques from