$b\bar b u\bar d$ four-quark systems in the Born-Oppenheimer approximation: prospects and challenges

We summarize previous work on $\bar b \bar bud$ four-quark systems in the Born-Oppenheimer approximation and discuss first steps towards an extension to the theoretically more challenging $b\bar b u\bar d$ system. Strategies to identify a possibly existing $b\bar b u\bar d$ bound state are discussed and first numerical results are presented.

Four-quark states bbqq are object of experimental and theoretical research. The theoretical investigation of such systems is, for reasons which will be described below, very challenging. Four-quark statesbbqq with q ∈ {u, d, s, c, } are theoretically more straightforward to investigate. These states have not yet been measured experimentally. However, it is interesting to study them e.g. to be able to make predictions for experiment and to gain conceptual insight of four-quark states. Previous studies ofbbqq and bbqq systems can be found e.g. in [2][3][4][5][6][7][8][9][10].
To investigate binding of four-quarksbbud and bbud, respectively, we work in the so-called Born-Oppenheimer approximation [11]. We first consider the b quarks to be static. Consequently, the heavy quark spin decouples from the system. We consider u/d quarks with a finite mass. The central element of the investigation is the potential of the two static quarks in the presence of the lighter quarks which we compute using Lattice QCD. We extract potentials V(r) from correlation functions Speaker, e-mail: peters@th.physik.uni-frankfurt.de O denotes a four-quark creation operator with defined quantum numbers (cf. [12] for details). It is possible to consider many different flavor/parity/angular momentum channels. For large separations between the static quarks, the potentials correspond to static-light BB potentials and BB potentials, respectively, cf. Figure 1 for the BB case. Once the potentials are computed, we insert them into the Schrödinger equation and check for bound states or resonances.

3bbud systems in the Born-Oppenheimer approximation: recent results
We first consider potentials of two static antiquarksbb in the presence of lighter quarks ud. An example plot is shown in Figure 2. The green line corresponds to a three parameter fit with respect to α, d and p of the function V(r) = − α r exp (− (r/d) p ) to the lattice data, cf. [16]. The investigation predicts two tetraquark states that have not yet been measured experimentally: Abbud bound state in the I(J P ) = 0(1 + ) channel can be identified. The binding energy with respect to the m B +m B * threshold is E B = −90 +43 −36 MeV [13] not considering effects from the heavy quark spin and E B = −59 +30 −38 MeV [14] taking into account heavy quark spin effects. Furthermore, a resonance in the I(J P ) = 0(1 − ) channel is found. The resonance energy with respect to the 2m B threshold is E R = 17 +4 −4 MeV and the resonance width is Γ = 112 +90 −103 MeV [15].

bbud systems in the Born-Oppenheimer approximation
The recently measured states Z b (10610) and Z b (10650) are experimentally interesting examples for a bbqq four-quark candidate. They have the quantum numbers I(J P ) = 1(1 + ). In the following we consider the positively charged state Z + b (cf. e.g. [1,17,18]). However, all results also hold for Z − b . Isospin I = 1 corresponding to the positive electric charge is realized by light quark flavours ud. Parity P = + is consistent with a possible loosely bound B ( * )B * structure, since both B ( * ) andB * have P = − and hence in combination result in P = +. Note, however, that in the static approximation B and B * mesons are degenerate in mass. Therefore we will denote both, B and B * as B in the following. Numerically we find a rather deep and wide potential for light total angular momentum j = 0. In the static approximation the different spin alignments of the static quarks are degenerate, i.e. we cannot distinguish j b = 0 or j b = 1. This means, the total angular momentum can either be J = 0 or J = 1, i.e. all our statements apply to bbud four-quark system not only with I(J P ) = 1(1 + ), but also with I(J P ) = 1(0 + ). Up to now, only the I(J P ) = 1(1 + ) channel has been measured experimentally. To measure the I(J P ) = 1(0 + ) channel a different experimental setup than realized in current experiments would be necessary.

bbud systems: possible structures
bbud states in the I(J P ) = 1(1 + ) channel may have different structures. We distinguish between fourquark structures such as the mesonic molecule BB and two-particle states such as a bottomonium state and a pion, QQ + π. Examples of the different and frequently discussed structures and their descriptions can be found in Table 1. label description sketch BB tetraquark or two-particle state A bound four-quark state made of a loosely bound BB meson pair (a so-called mesonic molecule) or two far separated and essentially non-interacting B mesons BB (QQ) * π tetraquark or two-particle state A bound four-quark state made of an excited bottomonium state and a loosely bound pion π + with zero momentum or two far separated and essentially non-interacting mesons. In the static approximation, a bottomonium state is realized by the static quark Q and the static antiquark Q connected by a gluonic string.

QQ π
QQπ p tetraquark or two-particle state A bound four-quark state made of the bottomonium ground state Υ(1S )/η b (1S ) and a loosely bound pion π + with momentum p (including p = 0) or two far separated and essentially non-interacting mesons.
QQ π p tetraquark state A bound four-quark state made of a diquark (color antitriplett) and an anti-diquark (color triplett).dQ uQ Table 1. Examples for frequently discussed structures of a bbud state in the I(J P ) = 1(1 + ) channel. In the text, the structures are referred to by their labels.
In the following, we list the potentials that belong to the bbud ground state and higher excited states in the I(J P ) = 1(1 + ) channel.
• Ground state (denoted as V 0 (r)): -As numerical results indicate (cf. Section 4.2), one can identify V 0 (r) with the bottomonium ground state Υ(1S )/η b (1S ) and a pion at zero momentum, QQ + π. The bottomonium state is represented by two static quarks connected by a gluonic string.
• First excited state (denoted as V 1 (r)): -For small separations r of b andb one can distinguish different cases: -A two-particle state B +B where two mesons are far separated and basically not interacting or a four-quark state BB where four quarks form a hadron, -a diquark-antidiquark state, -an excited bottomonium state and a pion, realized as a two-particle state (QQ) * + π where the pion is essentially equally distributed over space or a four-quark state (QQ) * π where the pion is close to the QQ pair or -a two-particle state corresponding to a bottomonium state and a pion with nonzero momentum QQ + π p or a four-quark state QQπ p .
Of course in QCD the state can also correspond to a mixture of the above mentioned structures or a combination of four quarks without any manifest structure. -For large separations r of b andb the first excited state and the ground state swap places (cf. Figure  3). The first excited state corresponds to a bottomonium state and a pion at zero momentum. The ground state corresponds to a two-particle state of a B meson and aB meson, B +B. This can be understood as follows: The gluonic string between the two heavy quarks will not persist for large separations, because its energy increases linearly. Therefore all structures different from B +B are excited states.
One possible scenario is sketched in Figure 3. The blue curve is the ground state potential V 0 which is expected to have the same shape as the static quark-antiquark potential shifted by the mass of the pion. The ordering of the next excitations is, however, less clear. In particular in case of non-vanishing pion momentum, it depends on the light quark mass as well as on the spatial lattice extent L: momentum values are quantized on the lattice, p = 2π L n with n i = 0, 1, ..., L/a − 1 (with a the lattice spacing). So for larger lattice extents, the potentials that correspond to non-zero momentum pions will move closer and closer together and more states will be below a potential corresponding to a four quark bound state. This behavior is indicated by the fading green curves in the Figure. The red curve accounts for a possible bbud four-quark potential which one would like to identify in order to investigate a possibly existing tetraquark state. The yellow curve is the potential of the first excitation of a bottomonium state and a pion at rest. It is an open and interesting question whether it is above the four-quark potential or below.

Numerical investigation of bbud ground state and first excited state in the I(J P ) = 1(1 + ) channel
We investigate the ground state and first excited bbud state by considering creation operators of QQ+π (respectively QQπ) and BB (respectively B +B) structures. The aim is to check whether the first excited bb potential in the presence of lighter quarks ud is still attractive enough to host a bound state if contributions from the QQ + π state have been removed. We compute the correlation matrix C jk (t, r) which can be expressed in terms of contributions of potentials and overlaps A n jk : with |x − y| = r. The operators read: Γ appearing in both operators is a combination of Dirac matrices that realizes either j b = 0 or j b = 1. It does not affect the potentials V j (r) since the spin of the heavy quarks is irrelevant as mentioned above. The matrix Γ is a combination of Dirac matrices leading to the same quantum numbers ( j z , P • C, P x ) as the QQ + π operator (4) (for details on ( j z , P, C, P x ), cf. e.g. [13]). Among the possible choices for Γ, the combination Γ = (1 − γ 0 )γ 5 yields the strongest QQ attraction, if one takes into account only the operator O BB . Therefore, we consider this combination as the most promising to search for a stable bbud tetraquark. U ab (x; y) denotes a product of gauge links connecting the two static quarks. We determine the first excited state potential by solving the Generalized Eigenvalue Problem [19]. This potential as well as the ground state potential and the static-light quark-antiquark potential for comparison can be found in Figure 4. Solving the Schrödinger equation yields a binding energy of with respect to the m B + m B * threshold. This value might be a vague indication for a tetraquark state.

Future plans to investigate the first excited bbud state
To investigate the structure of the first excited state |1 we plan to use the overlap of tetraquark and two-particle trial states, respectively, and |1 . The overlaps can be computed using Equation ( and e.g. the operators given in Equations (3) and (4). The volume dependence of the overlap can be estimated by a simple quantum mechanical calculation: The static quarks are located at fixed positions x 0 and y 0 , respectively. The lighter quarks have no fixed location. Their positions are referred to as u and v. The quantum mechanical wavefunctions of the tetraquark and the two-particle trial state each are composed of the wavefunctions of the heavy quarks which are Dirac δ functions as well as of the wavefunction of the light quarks ψ 4q/2p (x). The wavefunctions of the heavy quarks decouple from the system because the heavy quark positions are fixed. In the following, we only consider the wavefunctions of the light quarks ψ 4q and ψ 2p . The tetraquark trial state can be modeled as: and the two-particle trial state can be modeled as: where we use center-of-mass coordinates r = u − v and R = u+v 2 . p is the pion momentum. The normalization 1 √ V s makes the state independent of the spatial volume V s . We introduce the functions f (r) and g(r) that are 0, if |r| > d hadron , else nonzero functions (where the details are irrelevant). d hadron is the typical extent of a hadron, i.e. d hadron 1fm. We consider the overlap of the trial states with the first excited bbud state |1 : Case 1: |1 is a two-particle state: with g (r) = 0 if |r| > d hadron , else a nonzero function.
The conclusion is that the volume dependence of the overlap can provide information whether the first excited bbud state is a two-particle state or a four-quark state (tetraquark).

Summary and outlook
We investigate the static-light bbud four-quark state in the I(J P ) = 1(1 + ) channel. A bbud bound state must have two properties: The light quarks must be close to the heavy quarks and the corresponding potential must be sufficiently attractive to host a bound state. We take into account different possible structures of the bbud state and identify a candidate for an attractive bbud potential. By calculating the corresponding binding energy we find signatures consistent with a bbud tetraquark. The same methods applied in case of the Z b states described here could be applied to less well understood states, e.g. the X(3872) [20], or used to predict new states. However, it is questionable whether it is sensible to treat the c quark within the Born-Oppenheimer approximation. In the future, the first excited bbud state should be studied in more detail, e.g. by investigating the volume dependence of the state with sufficiently large statistics. We present a possible strategy to investigate this volume dependence by means of the overlap of the first excited state with different trial states. This way one can find out whether the first excited state is a tetraquark or a two-particle state. This can be an important step for a solid interpretation of the resulting potential and thus for any statement about the bbud tetraquark.