# Masses and Mixing of Tetraquarks Using Glozman-Riska Hyperfine Interaction

## Abstract

In this paper we perform a detailed study of the masses and mixing of the single charmed scalar tetraquarks: . We also give a systematic analysis of these tetraquark states by weight diagrams, quantum numbers and flavor wave functions. Tetraquark masses are calculated using four different fits. The following SU(3) representations are discussed: , , and . We use the flavor-spin Glozman-Riska interaction Hamiltonian with SU(3) flavor symmetry breaking. There are 27 different tetraquarks composed of a charm quark and of the three light flavors : 11 cryptoexotic (3 D, 4 D, 4 D) and 16 explicit exotic states. We discuss D and its isospin partners in the same multiplet, as well as all the other four-quark states. Some explicit exotic states appear in the spectrum with the same masses as D(2632) in and with the same masses as D(2317) in representation, which confirm the tetraquark nature of these states.

###### pacs:

12.39.-x, 12.39.Pn, 12.40.Yx, 14.65.Bt^{1}

## I Introduction

The meson D(2317), discovered in 2003. year in high energy electron-positron collisions at SLAC (Stanford Linear Accelerator Center) by the BABAR group (1) and confirmed by BELLE experiments (2), possesses a mass of 2317 MeV, some 170 MeV lighter than expected, at least according to prevalent theories of quark interactions. Hence physicists need a new explanation of how a charm quark attached to an antistrange quark should have this particular mass. In general, D and D mesons are a class of particles, each consisting of a charm quark attached to a light antiquark. The BABAR detection group at SLAC (1) responsible for the experimental discovery suggests that the D(2317) might be a novel particle made of four quarks. Meson D(2308) was discovered by BELLE group (3). The mass difference between strange D(2317) and nonstrange D(2308) meson (9 MeV) is at least ten times below the expected value of - mass difference. Also experimentally state D(2632) (discovered in SELEX experiments (4)) does not fit into former theoretical predictions because it is too light to be an (radial) excitation of the D(2317).

Jaffe (5) suggested the possible existence of four-quark states for light flavor dimesons and made predictions for tetraquark spectroscopy. In Ref. (6) it is also provided a framework for a quark-model classification of the many two-quark-two-antiquark states.

In Refs. (7); (8); (9) the
D(2317) meson is explained as a scalar
system: van Beveren *et al.* claim that in their model,
assuming that the meson is indeed a charm-antistrange combination,
the mass comes out in the right range if the strong-nuclear-force
interactions responsible for the creation and annihilation of extra
quark-antiquark pairs are taken into account. According to van
Beveren and Rupp (10) and Barnes *et al.*
(11), the D(2632) resonance, being 0.52
GeV heavier than the D ground state, could turn out to
be the first radial excitation of the D(2112) meson. On
the other hand, Terasaki and Hayashigaki
(13); (14); (12) have assigned the
D(2317) to the T = 0 member of the isotriplet
which belong to the lighter class of four-quark
mesons and have investigated the decay rates of the members of the
same multiplet. Also in Refs. (15); (16); (17) it
is shown that it might be expected that the measured
D(2317) is an exotic state with the structures of a
four-quark. Liu *et al.* (18) argue that the
D(2632) resonance may be a member of a scalar
tetraquark multiplet. The possible tetraquark nature of the three
mentioned mesons is discussed e.g. in Refs.
(19); (20); (21); (22); (23).

The three charmed scalar mesons: D(2317),
D(2308) and D(2632) does not fit well into
predictions of the quark model because of these three reasons:

(i) absolute mass of the D(2317) is 170 MeV below
mass predicted from the quark model for the scalar
meson,

(ii) the small mass gap between D(2317) and
D(2308) is puzzling and leads to a new model for these states.

(iii) the state D(2632) does not fit into former
theoretical predictions because it is too light to be an (radial)
excitation of the D(2317).

These three dissimilarities influenced giving some theoretical proposals about the possible structure of the mesons D(2317), D(2308) and D(2632). According to this, we analyze the possibility that these three states (or some of them) are tetraquarks.

In this work we perform a schematic study of the mass splitting of the single charmed tetraquarks in the SU(3) flavor representations. In Section II we construct the wave functions of mentioned tetraquarks. Then we present the flavor-spin Glozman-Riska interaction Hamiltonian. The formalism of calculating SU(3) flavor symmetry breaking corrections to the flavor-spin interaction energy is presented in Section III. Also it discusses meson and baryon fit and numerical analysis. The light and heavy meson and baryon experimental masses are fitted with aim to calculate the constituent quark masses and then to calculate tetraquark masses from our theoretical model. We discuss masses with Glozman-Riska (GR) (24) hyperfine interaction (HFI). Equations that correspond to our theoretically predicted masses are given for all 27 states, as well as their numerical values. The quark model of confinement cannot reproduce the spin-dependent hyperfine splitting in the hadron spectra without additional contributions from a hyperfine interaction. That is why we take into account GR hyperfine interaction. We include mass mixing effects for particles with the same quantum numbers and show it in mass spectra. The last section is a short summary.

## Ii Analysis and Method

Tetraquarks with charm quantum number C = 1 and with three light flavors are grouped by the same properties, into multiplets with the same baryon number, spin and intrinsic parity. If a particle belongs to a given multiplet, all of its isospin partners (the same isotopic spin magnitude T and different 3-components T) belong to the same multiplet.

The flavor SU(3) decomposition of the 27 possible combinations is given in Figure 1 by the Young diagrams. Under the transformation of SU(3), the charm quark is singlet. The numbers 15, 3, 3 and 6 are dimensions of Young diagrams and they designate the number of particles in the same group i.e. SU(3) flavor multiplet. Particles belonging to the same multiplet have the same baryonic number, spin and intrinsic parity. They also have similar masses.

Weight diagrams which represent the following product: are given in Figures .

In these weight diagrams ordinate shows hypercharge Y:

(1) |

where B is baryonic number (1/3 for quark, -1/3 for antiquark), S is strangeness (-1 for quark, 1 for quark) and C is charm (1 for quark, -1 for quark). So, for tetraquarks with one quark attached to one light quark and two light antiquarks, we have: and C = 1. Also, for electric charge of a particle, we have:

(2) |

We plot the eigenvalues of T and Y that occur for the quarks in a representation as points in the T - Y plane. We first combine two of the antiquarks. The quantum numbers Y and T are additive and thus their values for a state are obtained by simply adding the values for and . The points in the weight diagram for the -representation are thus obtained by taking every point of one antiquark diagram to be the origin of another antiquark diagram. Figure 2 shows that the nine combinations arrange themselves into two SU(3) multiplets, where the 3 is symmetric and the is antisymmetric under interchange of the two antiquarks. Then we add the third quark triplet. The final decomposition is displayed in Figures 3 and 4. The subscripts S and A on the multiplets indicate that the flavor states are symmetric ( and ) or antisymmetric ( and ) under interchange of the last two antiquarks.

Knowing quantum numbers for the set of 27 scalar tetraquarks, they are classified in groups as shown in Figure 5. We denote the states with strangeness S = 2 as , with S = 1 as (T = 1) and D (T = 0), with S = 0 as (T = 3/2) and D (T = 1/2) and with S = -1 as .

From the weight diagrams we read off the quark content of the tetraquarks. The four-quark content, as well as quantum numbers, calculated for all 27 states in the following SU(3) representations: , , and are given in Table 1. There is mixing between states from symmetric multiplets and , and also between antisymmetric multiplets and , while symmetric and antisymmetric multiplets do not mix with each other. Mixing is due to the same quantum numbers: electric charge Q(), third isospin projection T, isospin T and strangeness S. According to Table 1, mixed states are D, D and D from symmetric multiplets and D and D from antisymmetric multiplets. Which mixed state belongs to the and which to the is arbitrary at present and, in fact, the physical particle may be some superposition of the two states. The same applies for mixed states from and . The flavor wave functions, requiring orthogonality between each state, are given in Table 2. It is known that a meson is composed of a quark and an antiquark, but as can be seen from Table 2, experimentally detected states D(2317) and D(2632) in addition to also have , and combinations whose probability is determined by the square of the coefficient in front of each combination. In case of D(2308), besides there are also and combinations. These facts clearly indicate the tetraquark components in wave functions of the three mentioned states.

multiplet | tetraquark | quark | electrical charge | isospin projection | isospin | strangeness |

label | content | Q | T | T | S | |

2 | 1/2 | 1/2 | 2 | |||

1 | -1/2 | 1/2 | 2 | |||

2 | 1 | 1 | 1 | |||

1 | 0 | 1 | 1 | |||

0 | -1 | 1 | 1 | |||

D | 1 | 0 | 0 | 1 | ||

2 | 3/2 | 3/2 | 0 | |||

1 | 1/2 | 3/2 | 0 | |||

0 | -1/2 | 3/2 | 0 | |||

-1 | -3/2 | 3/2 | 0 | |||

D | 1 | 1/2 | 1/2 | 0 | ||

D | 0 | -1/2 | 1/2 | 0 | ||

1 | 1 | 1 | -1 | |||

0 | 0 | 1 | -1 | |||

-1 | -1 | 1 | -1 | |||

D | 1 | 0 | 0 | 1 | ||

D | 1 | 1/2 | 1/2 | 0 | ||

D | 0 | -1/2 | 1/2 | 0 | ||

2 | 1 | 1 | 1 | |||

1 | 0 | 1 | 1 | |||

0 | -1 | 1 | 1 | |||

D | 1 | 1/2 | 1/2 | 0 | ||

D | 0 | -1/2 | 1/2 | 0 | ||

0 | 0 | 0 | -1 | |||

D | 1 | 0 | 0 | 1 | ||

D | 1 | 1/2 | 1/2 | 0 | ||

D | 0 | -1/2 | 1/2 | 0 |

multiplet | tetraquark | flavor wave function |
---|---|---|

and mixed states | D( – ) | ; |

D( – ) | ; | |

D( – ) | ; | |

D | ||

and mixed states | D( – ) | ; |

D( – ) | ; |

The interaction we use is given by the Hamiltonian operator (24):

(3) |

where are Gell-Mann matrices for flavor SU(3), are the Pauli spin matrices and C is a constant. We employ this schematic flavor-spin interaction between quarks and antiquarks which leads to Glozman-Riska HFI contribution to tetraquark masses (24):

(4) |

where m are the constituent quark effective masses: = and - flavor wave functions. With the addition of tetraquark masses without influence of Glozman-Riska HFI, finally for tetraquark masses we have:

(5) |

multiplet | tetraquark | ( = | ( = |
---|---|---|---|

+ 2 + | |||

2+ + | |||

3 + | |||

2 + + | |||

and mixed states | D( – ) | 2 + + ; 3 + | ; |

D( – ) | 3 + ; + 2 + | ; | |

2 + + | |||

2 + + | |||

D | 2 + + | ||

and mixed states | D( – ) | 3 + ; + 2 + | ; |

## Iii Results

Using the obtained flavor wave functions of scalar tetraquarks (see Table 2), the tetraquark masses without influence of GR HFI are determined. GR HFI contributions to the tetraquark masses are calculated according to the relation (4) and the total tetraquark masses by relation (5). The corresponding results are given in Table 3.

The fit of hadron masses is used to determine masses of constituent quarks. We performed mass fit for: light mesons (, , , , , , , ), heavy mesons (D, D, D, D, D, D, , J/), light baryons (N, , , , , , , ) and heavy baryons (, , , , , ). Applying the Hamiltonian (3) to the constituent quarks, we obtained the theoretical meson and baryon masses with the GR contribution included. Consequently, we have the set of equations (6)–(13) for theoretical masses of light pseudoscalar mesons, light vector mesons, charmed mesons, strange charmed mesons, double charmed mesons, light baryons - octet, light baryons - decuplet and heavy baryons, respectively. The corresponding experimental masses, taken from ”Particle Data Group” site: http://pdg.lbl.gov (25), are appended to the right side of each equation:

(6) |

(7) |

(8) |

(9) |

(10) |

(11) |

(12) |

(13) |

Masses , and are the results of the hadron fit. The constant C is set so that the lightest tetraquark from multiplet has equal mass as D(2317). We performed these calculations using theoretical and experimental masses of all particles listed in equations (6)–(13), except for mixed states (6) and (13) because the meson octet and singlet mix and the flavor functions of mixed states are given only in a first approximation (see (5)).

The values for each set of equations for masses are evaluated as:

(14) |

where is the model prediction for the hadron mass, is the experimental hadron mass and is the uncertainty of the mass. After the values of parameters (, , ) were obtained by fitting meson and baryon experimental masses, they were used for calculation of the tetraquark masses.

The hadron mass fits resulted in the parameter values given in the Table 4. From the meson fit, calculated masses for and quarks are smaller ( 311 MeV, 487 MeV) than those from the baryon fit ( 388 MeV, 556 MeV). Due to smaller value of the masses obtained from the meson fit are more reliable. Both fits gave the similar values for constant C ().

With parameters from Table 4 we calculated tetraquark masses. The tetraquark masses calculated from meson fit parameters are given in Table 5. The results from Table 5 show that the isotriplet from has the same mass as D(2632) and that the isotriplet from has the same mass as D(2317).

hadrons | (MeV) | (MeV) | (MeV) | C (10 MeV) | |
---|---|---|---|---|---|

mesons | 311 | 487 | 1592 | 1.04 | 7.30 |

baryons | 388 | 556 | 1267 | 2.29 | 7.60 |

multiplet | tetraquark | (MeV) | (MeV) | (MeV) |

2877 | -146 | 2731 | ||

2701 | -228 | 2473 | ||

2525 | -301 | 2224 | ||

2701 | -228 | 2473 | ||

and mixed states | D( – ) | 2701; 3053 | -228; 615 | 2473; 3668 |

D( – ) | 2525; 2877 | -301; 877 | 2224; 3754 | |

2701 | -384 | 2317 | ||

2701 | -601 | 2100 | ||

D | 2701 | -384 | 2317 | |

and mixed states | D( – ) | 2525; 2877 | -601; -384 | 1924; 2493 |

GR contribution is positive or negative due to signs of the and products. It is negative in , and -plets, and for – mixed states one of the mixed states has negative and the other one has positive GR contribution. The positive GR contribution for two mixed states (see Table 3) comes out because of the mixing of the states: it changes the properties and shifts masses from the theoretical predictions.

For experimentally detected states (D(2317), D(2308) and D(2632)), hadron fits resulted in theoretical masses with relatively significant statistical uncertainties. These uncertainties are mainly due to inaccuracies in constitutive quark masses obtained using hadron fits (see Table 4). For example, experimental masses for D and D mesons are 1968 MeV and 2112 MeV, but they have the same quark content (see eq. (9)) and therefore their constituent quarks have different theoretical masses. Besides, D(2308) and D(2632) are mixed states (see Table 2) and therefore their flavor wave functions are given only in a first approximation. In spite of the uncertainties, none of the found states with strangeness equal to zero have mass around = 2405 MeV, which agrees with the conclusion obtained in Refs. (20) and (26) that the state D(2405) (found by the FOCUS collaboration (27)) is not a tetraquark, but a normal state.

Tetraquark mass spectrum from the meson fit, without and with GR HFI influence and with SU(3) symmetry breaking is presented in Figure 6. The general conclusion is that tetraquarks are arranged in the same way in both spectra: from meson and baryon fits. The spectra obtained from different fits have a similar arrangement of particles and if the values of parameters are changed, the whole spectrum could be shifted towards higher or lower masses and it could be shrunk or broadened. In both spectra it is possible to identify D(2317) as the lowest state in multiplet and D(2632) as a mixed state from mixing of multiplets and . Also, in both spectra (for example see Table 5), GR HFI mostly reduces the obtained masses except for one of D mixed states and one of D mixed states from – mixing.

tetraquark | ( = | ( = | (MeV) | (MeV) | (MeV) |
---|---|---|---|---|---|

1244 | -744 | 500 | |||