Possible f Quark Model of Tetraquarks and Pentaquarks

Selected tetraquark and pentaquark candidates are compared to meson and baryon states populated by ~2.9 GeV/c 2 f quarks postulated by Chapman. The meson and baryon states incorporating an f quark are modeled using a first-order mass formula based on the methodology developed by Zel'dovich and Sakharov. The resulting states have masses below the experimental tetraquark and pentaquark values. First-order mesons and baryons mass formula results containing an f quark improve as the mass of the postulated quark increases.


Model Formulation
Zel'dovich and Sakharov 18,19 proposed a semiempirical mass formula that provides a prediction of mesons and baryons masses in terms of effective quark masses. Within this formulation, quark wave functions are assumed to reside in their lowest S state. The meson (m) mass (M) formula of Refs. 18 and 19 is: where m 1 (m 2 ) are the mass of the first (second) quark comprising the meson, m o is the average mass of a first generation quark 20,21 , and the σ i (i = 1 and 2) are the spin vectors for the quarks incorporated into the meson. The parameters δ m and b m are 40 MeV and 615 MeV, respectively 19 .
The last term in Eq. 1 represents the spin-spin interaction of the quarks and σ 1 ·σ 2 is the scalar product of the quark spin vectors. σ 1 ·σ 2 has the value -3/4 and +1/4 for pseudoscalar and vector mesons, respectively.
In a similar manner, the baryon (b) mass formula 18,19 is: where the m i labels the three baryon quarks (i = 1, 2, and 3) and δ b and b b are 230 MeV and 615 MeV, respectively 19 .
In formulating the meson and baryon mass formulae, effective quark masses provided by Griffiths 20 21 . The three generations are specified by the square brackets and the quark charges are given within parenthesis in terms of the proton charge e.
The first-order meson model only permits a primitive L = 0 coupling structure between the quarks J π meson = J π (quark 1) x L x J π (quark 2) = 1/2 + x 0 x 1/2 + (5) where L is the angular momentum between the clusters. The allowed meson J π values are 0 + for pseudoscalar mesons and 1 + for vector mesons.

Results and Discussion
First-order mass formulas are compared to the mesons and baryons containing the postulated f quark 2 . Calculations are provided for f d-bar and d f-bar mesons (Table 1), f u-bar and u f-bar mesons (Table 2), f s-bar and s f-bar mesons (Table 3), f f-bar mesons (Table 4), and the fuu baryon (Table 5). Both pseudoscalar 0 + and vector meson 1 + states are presented in Tables 1-4. For baryons, Table 5 lists both 1/2 + and 3/2 + states. Given the uncertainties in the models, further matching with f meson and baryon states is not addressed. Moreover, the discussion is limited to possible tetraquark and pentaquark states that match the possible J π values derived from the first-order mass formula.

f d-bar and d f-bar Mesons
Chapman proposed a number of f d-bar and d f-bar mesons 2 . Following the previous discussion, the first-order mass formula results are compared to states with J π = 1 + states (χ c1 (3872), Z c 0 (3900), χ c1 (4274), and χ c1 (4685)) and a J π = 0 + state (χ c0 (4700)) in Table 1. In general, the first-order mass formula results for f d-bar and d f-bar mesons underestimate the experimental multiquark (i.e., tetraquark and pentaquark) results summarized in Table 1. All first-order mass formula results lie below the experimental multiquark values. However, the results improve as the f quark mass is increased.
Qeios, CC-BY 4.0 · Article, June 6, 2023 Qeios ID: 8T3IVE · https://doi.org/10.32388/8T3IVE 3/9    Table 3 provides a summary of first-order mass formula results for f s-bar and s f-bar mesons. The predicted first-order mass formula results are below the values for the χ c1 (4140), X(3960) and χ c0 (4500) states 2 . Consistent with the f d-bar and d f-bar (Table 1) and f u-bar and u f-bar (

f f-bar Mesons
f f-bar meson projections using the first-order mass formula are summarized in Table 4. Chapman 2 provides f f-bar meson results for the X(6900) 0 + state. First-order mass formula results approach the experimental mass value 2 as the assumed f quark mass increases beyond the 2900 MeV/c 2 value utilized by Chapman.

fuu Baryon
In addition to mesons containing f quarks, Ref. 2 also addressed the fuu baryon and compared it to J π = 1/2 + P c + (4380) and J π = 3/2 + (P c + (4457) states. The baryon results are similar to meson calculations. In particular, the first-order mass formula underestimates the experimental values 2 . In addition, the first-order mass formula results approach the experimental mass values 2 as the assumed f quark mass increases beyond the 2900 MeV/c 2 value utilized by Chapman.

Conclusions
The f quark assumption proposed by Chapman 2 offers an alternative explanation for tetraquark and pentaquark structures. Chapman proposes a 2900 MeV/c 2 mass for the f quark.
First-order mass formula calculations incorporating these f quark meson and baryon structures yield values below the experimental results. The results approach the experimental tetraquark and pentaquark masses as the f quark mass increases beyond the 2900 MeV/c 2 value suggested by Chapman 2 .