4. Heavy-light four-quark systems 37
5.6. Summary
In this chapter, we presented an investigation of¯b¯bqq potentials for infinitely heavybquarks and lighter quark flavors q ∈ {u, d, s, c} for different isospin channels I = 0 and I = 1.
For s and c quarks with finite mass or isospin I = 1 we found no evidence for a bound state. We found strong evidence for a¯b¯budbound state in theI = 0channel with quantum numbersI(JP) = 0(1+). An extrapolation to the physical pion mass yielded a binding energy ofEB = −90+43−36 MeV. With this result we predict a new tetraquark state that has not been measured experimentally, yet.
6. ¯ b ¯ bud systems in Nonrelativistic QCD
In this chapter we investigate ¯b¯bud systems with b quarks of finite mass. We treat the b quarks in the framework of Nonrelativistic QCD (NRQCD). In Chapter 5 we studied¯b¯bud four-quark systems with static b quarks. In the I(JP) = 0(1+) channel we found evidence for a bound four-quark state. The binding energy of this state was found to beEbinding = 90+36−43MeV. Furthermore, in another study [64] using bottom quarks of finite mass similar results were obtained. Such a tetraquark candidate can be a very important prediction to give directions for future experimental research. However, it is crucial to investigate the candidate system under conditions that are as realistic as possible. Therefore usingbquarks of finite mass instead of infinitely heavybquarks and taking into account the spin of the heavy quarks in addition to the spin of the dynamical quarks is the next logical step. However, as we state below, both the static-light as well as finite-mass approach have advantages and disadvantages. We investigate the¯b¯budsystem by taking into account various structures of the four-quark state. We present techniques to explore the bound state predicted in the previous section. Furthermore, we discuss not yet resolved discrepancies of results computed with NRQCD from results computed in the static approximation. Results of this chapter have been published in [9].
6.1. ¯ b ¯ bud systems with b quarks of finite mass – Expectations
6.1.1. Properties of the¯b¯budsystem
Using heavy quarks of finite mass instead of static quarks has a particular advantage: In the former case, relativistic effects and the spin of the heavy quarks contributes to the energy of the system (cf. Section 5.1). Therefore,B and B∗ mesons are not any more degenerate in mass. A bound¯b¯budsystem with the quantum numbersI(JP) = 0(1+)can be realized either by a mesonicBB∗ orB∗B∗ molecule or by a diquark-antidiquark structure.
Possible structures
The ¯b¯bud state with four quarks of finite mass in the I(JP) = 0(1+) channel can exist in different structures. In the following, we present a selection of these structures, which we take into account in the further analysis by means of their creation operators. Note that we suppress the space-time argument of the operators.
Mesonic moleculeBB∗: TheBB∗operator reads:
OBB∗= ¯baAΓ1ABuaB¯bbCΓ2CDdbD−¯baAΓ1ABdaB¯bbCΓ2CDubD (6.1)
where the antisymmetric flavor combinationud−duresults in isospinI = 0. To realize quan-tum numbersJP = 1+, we chooseΓ1 =γ5 andΓ2=γi.
Mesonic moleculeB∗B∗: TheB∗B∗ operator reads:
OB∗B∗= ¯baAΓ1ABuaB¯bbCΓ2CDdbD−¯baAΓ1ABdaB¯bbCΓ2CDubD. (6.2) To realize quantum numbersJP = 1+, we chooseεilm Γ1 =γl,Γ2 =γm
, whereεilm is the Levi-Civita tensor.
Diquark-antidiquarkDd:
TheDdoperator reads:
ODd=εabc¯bbAΓ1AB¯bcBab0c0
ubC0Γ2CDdcD0 −dbC0Γ2CDucD0
. (6.3)
To realize quantum numbersJP = 1+, we choose the gamma matrix combinationsΓ1 =Cγi andΓ2 =Cγ5 with the charge conjugation matrixC.
Momentum projections
In addition to different four-quark structures, we take into account different possible momen-tum projections. On the one hand, we consider the system to have total momenmomen-tum zero.
On the other hand we consider the individualB mesons to have zero momentum each. To illustrate how the different projections are realized, we start with the creation operator of a Bmeson with constituent quarks of finite mass on the lattice:
OB(p)(t) = 1
√Vs X
x∈Vs
¯b(x, t)Γu(x, t) e−ix·p (6.4) withp = 2πLn and ni = 0,1, ..., L/a−1 (with L the spatial lattice extent and a the lattice spacing). For simplicity, we suppress color and spin indices. Vs = L3 is the spacial lattice volume. To realize zero momentum, we setp= 0and expression (6.4) reads:
OB(0)(t) = 1
√Vs X
x∈Vs
¯b(x, t)Γu(x, t). (6.5)
Note that in contrast to the case of static lightB mesons (cf. Chapters 5 and 7) the sum over the spatial lattice volume is mandatory because the position of thebquarks is not fixed. Now we consider the creation operator of a mesonic molecule of B mesons where each meson carries a specified arbitrary momentum:
OB(p)B(q)(t) = 1
√Vs X
x∈Vs
¯b(x, t)Γu(x, t) e−ix·p 1
√Vs X
y∈Vs
¯b(y, t)Γu(y, t) e−iy·q. (6.6) To realize a momentum projection where the full system has overall zero momentum, we have to constrain the individual momenta according top +q = 0. Expression (6.6) then reads:
O[BB](0)(t) = 1
√Vs X
x∈Vs
¯b(x, t)Γu(x, t) 1
√Vs X
y∈Vs
¯b(y, t)Γu(y, t) e−i(x−y)·p. (6.7)
6.1. ¯b¯budsystems withbquarks of finite mass – Expectations In order to find a mesonic molecule, bothB mesons have to be located at the same spatial position, i.e. we demandx=y. Thus we find
O[BB](0)(t) = 1
√Vs X
x∈Vs
¯b(x, t)Γu(x, t)¯b(x, t)Γu(x, t). (6.8) In case of individualB mesons projected to zero momentum each, we set p = q = 0 and (6.6) reads:
OB(0)B(0)(t) = 1
√Vs X
x∈Vs
¯b(x, t)Γu(x, t) 1
√Vs X
y∈Vs
¯b(y, t)Γu(y, t). (6.9) Note that the summation over spatial pointsxandyhas to be performed before the individual mesons are multiplied. Therefore we cannot setx=y here. The operator corresponds to a meson scattering state.
All operators considered in this thesis
In case of the BB∗ structure (cf. Equation (6.1)) we take into account the two different momentum projections:
• B(0)B∗(0) – Each meson is separately projected to zero momentum. For reasons de-scribed below we will only use this operator at the sink.
• [BB∗] (0)– The whole four-quark system is projected to zero momentum.
The different momentum projections give two different operators. We call themOB(0)B∗(0) andO[BB∗](0).
As in the previous case, there are two different momentum projections to be taken into account for theB∗B∗structure (cf. (6.2)):
• B∗(0)B∗(0)– Each meson is separately projected to zero momentum. For reasons de-scribed below we will only use this operator at the sink.
• [B∗B∗] (0)– The whole four-quark system is projected to zero momentum.
The different momentum projections give the two different operatorsOB∗(0)B∗(0)andO[B∗B∗](0). For the diquark-antidiquark structure (cf. (6.3)) there is only one reasonable momentum projection. Diquark and antidiquark are jointly projected to zero momentum: [Dd] (0). We refer to the corresponding operator asO[Dd](0).
6.1.2. The correlation matrix for all different structures
We can build a correlation matrix(Cij)using the operators we defined in Section 6.1.1. The matrix elements read:
Cij(tsink−tsource) =hΩ|Oi(tsink)Oj†(tsource)|Ωi (6.10) withOi, Oj ∈ {OB(0)B∗(0), O[BB∗](0), OB∗(0)B∗(0), O[B∗B∗](0), O[Dd](0)}. Due to the fact that we use point sources to compute the light quark propagators it is not possible to project the mesons separately to zero momentum at the source. The reason is that one needs to sum
over all positions of the individual mesons to project each of them to zero momentum, cf.
Equation (6.9). However, the point source is fixed at position xsource. This leads to some restrictions to the correlation matrix as we will point out below. We give an overview of the correlation matrix in Table 6.1. Most of the matrix elements can be exactly computed
sink
source
O†[BB∗](0) O†[B∗B∗](0) O[Dd](0)† OB(0)B† ∗(0) O†B∗(0)B∗(0)
O[BB∗](0) d.c. d.c. d.c. n.d.c. n.d.c.
O[B∗B∗](0) d.c. d.c. d.c. n.d.c. n.d.c.
O[Dd](0) d.c. d.c. d.c. n.d.c. n.d.c.
OB(0)B∗(0) d.c. d.c. d.c. n.c. n.c.
OB∗(0)B∗(0) d.c. d.c. d.c. n.c. n.c.
Table 6.1.: Schematic representation of the correlation matrix. The meaning of the abbre-viations is the following: d.c.: exactly directly computable, n.d.c.: not directly computable but can be obtained indirectly, n.c.: neither exactly computable di-rectly nor indidi-rectly.
directly. These elements are tagged by the label d.c. (directly computable). Elements that take into account separately projected mesons at the source cannot be computed directly. In case they have off-diagonal counterparts that can be obtained by complex conjugation of their off-diagonal counterparts. The label of these elements is n.d.c. (not directly computable). All other elements cannot be computed using point sources directly nor indirectly. We refer to them as n.c.-elements (not computable). From Table 6.1 we can see that there are four elements of this type. Using the GEP, one can only obtain the overlap factors and energies for the2×2and3×3submatrices of the5×5correlation matrix represented by Table 6.1 which do not contain n.c.-elements. In the following section, we present a way to approximately compute the missing overlaps and energies.
Computing the missing overlaps and energies
In the following we consider a matrix built of only two operators for simplicity. Extending the method to a larger matrix is straightforward. Consider the operators
O1 =X
x
B(x)X
y
B(y) (6.11)
and
O2 =X
x
B(x)B(x). (6.12)
withB(x) = ¯b(x)Γu(x). OperatorO1 represents a mesonic molecule where each meson is separately projected to zero momentum. Operator O2 describes a mesonic molecule with both mesons jointly projected to zero momentum. The correlation matrix reads:
Cij(tsink−tsource) =hΩ|Oi(tsink)O†j(tsource)|Ωi. (6.13) In this example we can only exactly compute C12 and C22 directly. We cannot take into accountO†1at the source because we only have light-quark propagators with the fixed starting