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REVIEW 2 major objections 125 references

Four Regge series for the quadruply heavy pentaquark bbūcc are fixed by diquark and triquark structure, not by pure fitting.

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2026-07-14 10:20 UTC pith:QDHSFOZS

load-bearing objection Clean extension of the authors’ own Regge program to bbūcc; the four-series formula and mass tables are new and usable, but the “structure fixes the form” claim is only approximate because nested reduced-mass terms are absorbed by later fitting. the 2 major comments →

arxiv 2607.10631 v1 pith:QDHSFOZS submitted 2026-07-12 hep-ph

λ, rho, and σ Regge trajectories for the quadruply heavy pentaquark bbbar{u}cc in the diquark-triquark picture

Xin-Ru Liu , Qi Liu , Jiao-Kai Chen This is my paper
classification hep-ph
keywords Regge trajectoriesquadruply heavy pentaquarkdiquark-triquarkbbūccλ ρ σ modesspin-averaged masses
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper claims that the mass of the quadruply heavy pentaquark bbūcc can be written as a sum of quark masses, a constant, and four independent excitation terms whose powers are fixed by whether each mode is heavy-heavy or heavy-light. Two of those terms scale as quantum number to the two-thirds power and one as its square root; the fourth also scales as two-thirds. Because the pentaquark is assembled from a diquark and a triquark, the functional form of the three internal trajectories is inherited from the known Regge relations of those sub-clusters rather than being free parameters that must be fitted from scratch. The authors compute rough spin-averaged masses for the radial and orbital excitations in both (bb)(ū(cc)) and (cc)(ū(bb)) configurations, and show that the sub-cluster trajectories control the slopes even though they do not map one-to-one onto the pentaquark series. The result supplies a systematic way to organize the excited spectrum of any multi-heavy pentaquark once its internal clustering is fixed.

Core claim

The mass formula M = 2mb + 2mc + mu + 5C/2 + βλ(xλ + c0λ)^{2}/^{3} + βρ1(xρ1 + c0ρ1)^{2}/^{3} + βρ2√(xρ2 + c0ρ2) + βσ(xσ + c0σ)^{2}/^{3} completely organizes the four series of Regge trajectories of bbūcc; the powers and the guidance for the coefficients come directly from the diquark and triquark Regge relations once the internal structure is taken into account.

What carries the argument

The composite Regge relation obtained by nesting the heavy-heavy (M ~ x^{2}/^{3}) and heavy-light (M ~ √x) formulas of the diquark and triquark inside the pentaquark, which forces the λ, ρ1 and σ series to scale as x^{2}/^{3} and the ρ2 series as √x.

Load-bearing premise

Only color-antitriplet diquarks are kept and the mixed (bc)(ū(bc)) configuration is discarded; if either the repulsive color-sextet channel or mode mixing contributes substantially, the four-series decomposition and the quoted mass formula no longer hold.

What would settle it

A lattice or experimental determination of several spin-averaged masses of λ-, ρ1-, ρ2- and σ-excited states of bbūcc that cannot be reproduced by the four-term formula with the predicted powers 2/3 and 1/2.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 0 minor

Summary. The manuscript derives a four-term Regge mass formula for the quadruply heavy pentaquark bbūcc in the diquark-triquark picture, M = 2mb + 2mc + mu + 5C/2 + βxλ(xλ + c0xλ)2/3 + βxρ1(xρ1 + c0xρ1)2/3 + βxρ2√(xρ2 + c0xρ2) + βxσ(xσ + c0xσ)2/3, by nesting the authors’ earlier diquark and triquark Regge relations obtained from the spinless Salpeter equation via Bohr-Sommerfeld quantization. Two configurations, (bb)(ū(cc)) and (cc)(ū(bb)), are retained; the mixed (bc)(ū(bc)) channel is discarded. Four series of trajectories (λ, ρ1, ρ2, σ) are constructed, spin-averaged masses of the corresponding excited states are tabulated (Tables III, V, VI), and simple two-parameter fits of the same functional form are presented (Table IV). The central claim is that the internal diquark/triquark structure is indispensable for fixing the functional forms of the ρ1, ρ2 and σ trajectories, which would otherwise be obtainable only by pure phenomenological fitting.

Significance. If the nested construction is accepted, the work supplies the first systematic four-series Regge analysis for a quadruply heavy pentaquark and yields concrete, falsifiable mass estimates that can be compared with future lattice or experimental results. The algebraic reduction from the Salpeter equation through the Bohr-Sommerfeld condition to the explicit four-term formula (Eqs. 22–25) is fully written out and internally consistent once the input diquark/triquark trajectories and the empirical cf, c0 relations are granted. The demonstration that the complete ρ1 and σ expressions (e.g., Eq. 28) are not identical to the pure diquark/triquark trajectories, yet share the same leading power-law behavior, is a useful clarification for the multiquark Regge literature.

major comments (2)
  1. The claim that internal structure “fixes the functional form” of the ρ1, ρ2 and σ trajectories (Abstract and §III.E) is only partially realized. Because βxλ and c0xλ depend on the reduced mass μλ = Md1 Mt/(Md1 + Mt), and Md1 (Mt) already contains the ρ1 (σ) excitation, the complete ρ1 trajectory (explicitly written for the radial case in Eq. 28) is not a pure (x + c0)2/3 term. The authors replace it by a two-parameter fit of that form (Eq. 29, Table IV). The same uncontrolled fitting step is required for the σ trajectories of the (cc)(ū(bb)) configuration, where an alternative ν = 7/12 or √x form actually fits better (Table VII, Fig. 5). The residual nested dependence should be quantified (e.g., by the size of the difference between the complete expression and the simple fit across the tabulated range) so that the reader can judge how strongly the structure really constrains the form.
  2. Only the color-antitriplet diquark channel is retained and the mixed (bc)(ū(bc)) configuration is discarded (Sec. II.A). Both choices are stated without quantitative estimate of the neglected contributions. If either the repulsive sextet channel or the mixed configuration contributes appreciably, the four-series decomposition and the quoted mass formula cease to apply. A short estimate of the expected mass shift or a reference to a calculation that bounds these effects would strengthen the central claim.

Circularity Check

3 steps flagged

Pentaquark mass formula and trajectory forms obtained by nesting self-cited diquark/triquark Regge relations whose parameters were previously fitted; simple ρ₁/σ forms recovered only after re-fitting the nested expressions.

specific steps
  1. self citation load bearing [Sec. II.C–D, Eqs. (9), (14), (21)–(25); citations [67–69, 86–89, 96]]
    "Using the diquark and triquark Regge trajectory relations, we propose the Regge trajectory relations for the quadruply heavy pentaquark bb¯ucc: M = 2mb + 2mc + mu + 5C/2 + βxλ(xλ + c0xλ)^{2/3} + βxρ_{1}(xρ_{1} + c0xρ_{1})^{2/3} + βxρ_{2}√xρ_{2} + c0xρ_{2} + βxσ(xσ + c0xσ)^{2/3}."

    The base heavy-heavy and heavy-light formulas, the unified ansatz (21), the reduced-mass coefficients, and the nesting procedure itself are taken verbatim from the authors’ own preceding papers. The pentaquark relation is therefore an algebraic rearrangement of those self-cited inputs; once they are granted, Eq. (25) follows by construction with no new dynamical content.

  2. fitted input called prediction [Sec. III.A–C, Tables III–VI, Eqs. (26)–(29), Table IV]
    "Circles represent the predicted data calculated via the complete forms of the Regge trajectory relations in Eqs. (25) and (23). The black lines represent the fitted formulas, which are obtained by fitting the calculated data in Table III … Fitting the calculated results with Eq. (28) yields the simple form M = 13.0941 + 0.3309(0.0099 + nr1)^{2/3}."

    All numerical inputs (quark masses, σ, C, cf x, c0x and the empirical relations (26)–(27)) are imported from earlier fits by the same group. The tabulated masses are therefore evaluations of those fitted formulas; the subsequent two-parameter re-fits are then presented as the λ/ρ/σ trajectories. The “predictions” inherit the prior phenomenology by construction.

  3. other [Abstract; Sec. III.C–E, Eqs. (28)–(29), Table VII, Fig. 5]
    "We demonstrate that accounting for the internal structure and substructure of pentaquarks is indispensable for constructing the ρ_{1}-, ρ_{2}-, and σ-trajectories; without such structural considerations, functional form and trajectory parameters can only be obtained via pure fitting … for the (cc)(¯u(bb)) configuration, M = 13.0979 + 0.25979(0.0519 + l3)^{7/12} is better than M = 13.1498 + 0.213174(0.0009 + l3)^{2/3}."

    The “demonstration” consists of writing the nested formula that already encodes the structure. The complete ρ_{1} expression (Eq. 28) still contains residual λ-mode dependence through μλ(Md1); the authors discard it and recover a pure (x + c0)^{2/3} form only by an extra fit. For one σ series an alternative exponent fits better, showing that the claimed structure-fixed functional form is not uniquely determined and is restored only after the fitting step the structure was supposed to render unnecessary.

full rationale

The central mass formula (Eq. 25) and the four series of trajectories are constructed by algebraic nesting of the heavy-heavy (M ∼ x^{2/3}) and heavy-light (M ∼ √x) Regge relations that the same authors derived and parametrized in prior works. Once those inputs (including the empirical cf, c0 relations and numerical values of m_q, σ, C) are accepted, the pentaquark expressions, the tabulated spin-averaged masses, and the statement that internal structure is indispensable follow by direct substitution; no independent first-principles derivation is supplied. The authors themselves then replace the lengthy nested ρ₁ and σ expressions by simple two-parameter fits of the same functional form (and, for one σ series, note that a different power fits better). The work therefore re-applies and re-fits an existing phenomenological framework to a new system rather than producing independent predictions. The non-one-to-one correspondence between substructure trajectories and the four series is a genuine observation, so the circularity is only partial.

Axiom & Free-Parameter Ledger

3 free parameters · 4 axioms · 0 invented entities

The central mass formula and the trajectory behaviors rest on a chain of phenomenological inputs: the spinless Salpeter equation with Cornell potential, the Bohr-Sommerfeld quantization that produces the 2/3 and 1/2 powers, the restriction to color-antitriplet clusters, and a large set of numerical constants and empirical fitting formulas taken from the authors’ previous papers. No new dynamical principle is introduced; the novelty is the assembly of these ingredients for the mixed heavy pentaquark.

free parameters (3)
  • mu, mb, mc, σ, C = mu=0.33 GeV, mb=4.88 GeV, mc=1.55 GeV, σ=0.18 GeV², C=−0.3 GeV
    Constituent masses and Cornell parameters taken directly from Ref. [95]; they set the overall mass scale and the string tension that enters every β coefficient.
  • cf_nr, cf_l and all c0x for diquarks = cf_nr=1, cf_l=1.17; c0nr(bb)=0.01, c0nl(bb)=0.001, c0nr(cc)=0.205, c0nl(cc)=0.337
    Normalization and intercept parameters for the σ-mode trajectories, taken from the authors’ earlier diquark paper [67]; they control the absolute height of the σ series.
  • empirical formulas for cf_L, c0L, cf_Nr, c0Nr, cf_nr2, c0nr2, cf_l2, c0l2 = coefficients listed in Eqs. (26)–(27)
    Polynomial or power-law expressions (Eqs. (26)–(27)) that map reduced masses onto trajectory coefficients; they were fitted in prior works of the same group and are used without re-fitting.
axioms (4)
  • domain assumption Spinless Salpeter equation with Cornell potential V=−(3/4)[Vc+σr+C](Fi·Fj) describes the bound-state mass of diquarks, triquarks and the pentaquark.
    Invoked at the opening of Sec. II.B; standard in the authors’ series but not derived from QCD.
  • domain assumption Bohr-Sommerfeld quantization applied to the heavy-heavy and heavy-light limits yields the universal powers ν=2/3 and ν=1/2 that appear in every trajectory.
    Used to obtain Eqs. (9) and (14); the quantization condition itself is an approximation whose accuracy for multiquarks is unproven.
  • ad hoc to paper Only color-antitriplet diquarks and color-triplet triquarks are retained; the color-sextet channel is discarded.
    Stated after Eq. (1) following Refs. [72,73]; if the repulsive channel mixes appreciably the four-series decomposition fails.
  • ad hoc to paper The (bc)(ū(bc)) configuration can be neglected because of mode mixing.
    Explicitly declared in Sec. II.A; removes an entire class of states whose trajectories would otherwise interfere.

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Systematic investigations of four series of Regge trajectories for quadruply heavy pentaquarks are still lacking. Using the diquark and triquark Regge trajectory relations, we propose the Regge trajectory relations for the quadruply heavy pentaquark ${bb\bar{u}cc}$: $M=2m_{b}+2m_{c}+m_{u}+5C/2 +\beta_{x_{\lambda}}(x_{\lambda}+c_{0x_{\lambda}})^{2/3}$$+\beta_{x_{\rho_1}}(x_{\rho_1}+c_{0x_{\rho_1}})^{2/3}+\beta_{x_{\rho_2}}\sqrt{x_{\rho_2}+c_{0x_{\rho_2}}} +\beta_{x_{\sigma}}(x_{\sigma}+c_{0x_{\sigma}})^{2/3}$. Four series of Regge trajectories, namely the $\lambda$-, $\rho_1$-, $\rho_2$-, and $\sigma$-trajectories, are investigated. We demonstrate that accounting for the internal structure and substructure of pentaquarks is indispensable for constructing the $\rho_1$-, $\rho_2$-, and $\sigma$-trajectories; without such structural considerations, functional form and trajectory parameters can only be obtained via pure fitting against theoretical or experimental data. We further prove that the Regge trajectories of diquark 1, triquark, and diquark 2 (emdedded within the triquark) do not correspond one-to-one to the $\rho_1$-, $\rho_2$-, and $\sigma$-trajectories. Nevertheless, these trajectories govern the behaviors of the respective $\rho_1$-, $\rho_2$-, and $\sigma$-trajectories. For both configurations $(bb)(\bar{u}(cc))$ and $(cc)(\bar{u}(bb))$, the $\lambda$-, $\rho_1$-, and $\sigma$-trajectories exhibit behavior of $M{\sim}x^{2/3}$ ($x=n_{r_1},n_{r_3},l_1,l_3,N_{r},L$), whereas the $\rho_2$-trajectories exhibit behavior of $M{\sim}\sqrt{x}$ ($x=n_{r_2},\,l_2$). The functional behavior of Regge trajectories for diquarks and triquark offers guidance for fitting the pentaquark Regge trajectories. Additionally, we provide rough estimates for spin-averaged masses of the $\lambda$-, $\rho_1$-, $\rho_2$-, and $\sigma$-excited states.

Figures

Figures reproduced from arXiv: 2607.10631 by Jiao-Kai Chen, Qi Liu, Xin-Ru Liu.

Figure 1
Figure 1. Figure 1: FIG. 1: Schematic diagram of a pentaquark in the diquark [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: The [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Comparison of two approximation results. Cir [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗

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Reference graph

Works this paper leans on

125 extracted references · 19 canonical work pages · 12 internal anchors

  1. [1]

    The circles stand for quarks and antiquarks

    Diquark 2 contains quarks q4 and q5. The circles stand for quarks and antiquarks. σ separate the quarks within diquark 1 and diquark 2, respectively. λ corresponds to the separation between diquark 1 and the triquark, while ρ2 corresponds to the separation between the antiquark ¯q3 and diquark 2. Four excited modes exist: the ρ1-mode describes radial and ...

  2. [2]

    The di- quark (q1q2) is either {q1q2} or [q1q2], where {q1q2} and [q1q2] represent the permutation symmetric and antisym- metric flavor wave functions, respectively

    can also be written as |n2s1+1 1 l1j1,n 2s2+1 2 l2j2,n 2s3+1 3 l3j3,N 2S+1LJ ⟩. The di- quark (q1q2) is either {q1q2} or [q1q2], where {q1q2} and [q1q2] represent the permutation symmetric and antisym- metric flavor wave functions, respectively. N = Nr + 1, whereNr = 0, 1, · · ·. n1, 2, 3 =nr1, 2, 3 + 1, where nr1, 2, 3 = 0, 1, · · ·. Nr, nr1 , nr2 and nr3...

  3. [3]

    Regge trajectory relations for heavy-heavy and heavy-light systems For heavy-heavy systems, since m1,m 2≫| p|, Eq

    (6) 3 C. Regge trajectory relations for heavy-heavy and heavy-light systems For heavy-heavy systems, since m1,m 2≫| p|, Eq. ( 3) reduces to M Ψ d,t,p (r) = [ (m1 +m2) + p2 2µ +Vd,t,p ] Ψ d,t,p (r), (7) where µ = m1m2 m1 +m2 . (8) By employing the Bohr-Sommerfeld quantization ap- proach [ 46] and using Eqs. ( 5) and ( 7), we obtain the parametrized relatio...

  4. [4]

    The second reads M =mR + √ β 2x(x +c0x) +κ xm3/ 2 2 (x +c0x)1/ 4, (17) valid under the condition m2≪ M , with mR =m1 +C′, κ x = 4 3 √ πβ x, (18) where βx is in ( 10)

    with mR in (10), where m2 denotes the mass of the light constituent. The second reads M =mR + √ β 2x(x +c0x) +κ xm3/ 2 2 (x +c0x)1/ 4, (17) valid under the condition m2≪ M , with mR =m1 +C′, κ x = 4 3 √ πβ x, (18) where βx is in ( 10). Equation ( 14) with ( 10) generalizes the relation [ 50] M =m1 +m2 + √ a(nr +αl +b) (19) and the formula [ 86] (M − m1 − ...

  5. [5]

    Although they give different behavior of m2, Eq

    and ( 16). Although they give different behavior of m2, Eq. ( 14) with ( 10) and Eq. (17) with ( 18) produce consistent results for l, nr < 10 and have the same behavior M ∼x1/ 2 [88]. TABLE I: The coefficients for heavy-heavy systems (HHS) and heavy-light systems (HLS). HHS HLS ν 2/ 3 1 / 2 cc ( σ ′2/µ ) 1/ 3 √ σ ′ cl1,l 2,l 3,L 3/ 2 2 cnr1 ,n r2 ,n r3 ,N r...

  6. [6]

    The binding energies of the pen- taquark is ǫ =C +βxλ (xλ +c0xλ )2/ 3

    and ( 23), we have M =Md1 +Mt +C +βxλ (xλ +c0xλ )2/ 3 (xλ =L, Nr) (24) when diquark 1 and the triquark are regarded as struc- tureless constituents. The binding energies of the pen- taquark is ǫ =C +βxλ (xλ +c0xλ )2/ 3. Even though Eq. (

  7. [7]

    shares an identical functional form for configurations (bb)(¯u(cc)) and ( cc)(¯u(bb)), the values of Md1, Mt, βxλ , and c0xλ differ between these two configurations. When the triquark is taken as a bound state composed a di- quark and an antiquark, and diquarks 1 and 2 are each regarded as bound states of two heavy quarks, we have M =2mb + 2mc +mu + 5 2C +βx...

  8. [8]

    ( 23) and ( 25), as the com- plete forms of the Regge trajectories

    and ( 23), or from Eqs. ( 23) and ( 25), as the com- plete forms of the Regge trajectories. The obtained con- stant and the mode under consideration are referred to the main part of the Regge trajectories. For example, when considering the ρ1-trajectories for the configura- tion (bb)(¯u(cc)),βxρ 2 √ xρ 2 +c0xρ 2 andβxσ (xσ +c0xσ )2/ 3 are constants, while ...

  9. [9]

    141µ λ , cf Nr = 1

    540 − 0. 141µ λ , cf Nr = 1 . 008 + 0 . 008µ λ , and c0Nr = 0 . 334 −

  10. [10]

    087µ λ . In Ref. [ 69], we correct this and introduce m0 into Eq. (26). 5 by the following relations [ 69] cf nr2 = ( mB mc +mu ) 0. 3 (

  11. [11]

    2835mB m0 +0

    4115 − 0. 2835mB m0 +0. 03927 ( mB m0 ) 2 − 0. 00192 ( mB m0 ) 3) , c0nr2 =mc +mu mB ( − 3. 2445 + 3. 3231mB m0 − 1. 0442 ( mB m0 ) 2 + 0. 1333 ( mB m0 ) 3 − 0. 005737 ( mB m0 ) 4) , cf l2 = ( mB mc +mu ) 0. 3 (

  12. [12]

    3188mB m0 +0

    4697 − 0. 3188mB m0 +0. 04506 ( mB m0 ) 2 − 0. 002205 ( mB m0 ) 3) , c0l2 =mc +mu mB ( − 5. 1878 + 5. 2976mB m0 − 1. 6633 ( mB m0 ) 2 + 0. 2127 ( mB m0 ) 3 − 0. 009171 ( mB m0 ) 4) . (27) Here, mB = mu +mH . mH represents the mass of di- quark for doubly heavy triquark (¯u(cc)) or (¯u(bb)). TABLE II: Parameter values [ 67, 95]. mu = 0. 33 GeV, mb = 4. 88 ...

  13. [13]

    ( 25) takes the simplest form

    and ( 25), we can see that terms corresponding to ρ1-, ρ2-, and σ -trajectories are con- stant when consideringλ-trajectories; therefore, Eq. ( 25) takes the simplest form. The λ-trajectories are the sim- plest among four series of Regge trajectories for the pen- taquark bb¯ucc. The Regge trajectory behavior can read directly from Eq. ( 25),M ∼x2/ 3 λ , w...

  14. [14]

    The black lines repre- sent the fitted formulas, which are obtained by fitting the calculated data in Table III

    and ( 23). The black lines repre- sent the fitted formulas, which are obtained by fitting the calculated data in Table III. For the λ-mode, the fitted formulas are identical to the complete forms, which are listed in Table IV. C. ρ-trajectories for the pentaquark bb¯ucc The ρ1- and ρ2-trajectory relations are obtained by using Eqs. ( 25), ( 23), ( 26), ( 27)...

  15. [15]

    The calculated results are listed in Table V

    Spin-averaged masses of radially and orbitally ρ1- and ρ2-excited states can be calculated by using the ρ1- and ρ2-trajectory relations. The calculated results are listed in Table V. In Table V, entries labeled with ( × ) represent nonexist states: the 1 3p2 and 1 3f4 diquark states of (bb) and (cc) are forbidden by the Pauli principle [ 67]. From Eqs. ( ...

    2032

  16. [16]

    7637 + 0

    4019 + l2 M = 12. 7637 + 0. 5776 √

  17. [17]

    4699 + l2 σ M = 12. 923 + 0. 4411(0. 2829 + nr3 )2/ 3 M = 13. 1627 + 0. 2612(0. 0009 + nr3 )2/ 3∗ M = 12. 9401 + 0. 3428(0. 3969 + l3)2/ 3 M = 13. 1498 + 0. 2132(0. 0009 + l3)2/ 3∗ 13 13.2 13.4 13.6 13.8 14 14.2 14.4 0 1 2 3 4 (bb)(¯u(cc)) Nr M (GeV) (a) 13 13.2 13.4 13.6 13.8 14 14.2 14.4 0 1 2 3 4 (bb)(¯u(cc)) L M (GeV) (b) 13 13.2 13.4 13.6 13.8 14 14....

  18. [18]

    332799(0

    2779 + 0. 332799(0. 01 +nr1)2/ 3 ) − 1/ 3 × [

  19. [19]

    332799(0

    61 + 0. 332799(0. 01 +nr1)2/ 3)

  20. [20]

    332799(0

    2779 + 0. 332799(0. 01 +nr1)2/ 3 ] 2/ 3 [

  21. [21]

    332799(0

    61 + 0. 332799(0. 01 +nr1 )2/ 3)

  22. [22]

    332799(0

    2779 + 0. 332799(0. 01 +nr1)2/ 3 ] . (28) Fitting the calculated results with Eq. ( 28) yields the simple form M = 13. 0941 + 0. 3309(0. 0099 +nr1)2/ 3. (29) 7 13 13.2 13.4 13.6 13.8 14 14.2 14.4 0 1 2 3 4 (bb)(¯u(cc)) ρ1 nr1 M (GeV) (a) 13 13.2 13.4 13.6 13.8 14 14.2 14.4 0 1 2 3 4 (bb)(¯u(cc)) ρ1 l1 M (GeV) (b) 13 13.2 13.4 13.6 13.8 14 14.2 14.4 0 1 2 ...

  23. [23]

    By fitting the calculated masses, we obtain the fitted formulas M =

    and ( 25) (CF). By fitting the calculated masses, we obtain the fitted formulas M =

  24. [24]

    213174(0

    1498 + 0 . 213174(0. 0009 + l3)2/ 3 (A) and M = 13 . 0979 +

  25. [25]

    0519 + l3)7/ 12 (B)

    25979(0. 0519 + l3)7/ 12 (B). |n2s1+1 1 l1j1 , n 2l2, n 2s3+1 3 l3j3 , N L ⟩ CF A B |13s1, 1s, 13s1, 1S⟩ 13.1408 13.1518 13.1441 |13s1, 1s, 13p2, 1S⟩(× ) 13.3646 13.3631 13.3655 |13s1, 1s, 13d3, 1S⟩ 13.4923 13.4883 13.4930 |13s1, 1s, 13f4, 1S⟩(× ) 13.5962 13.5933 13.5959 |13s1, 1s, 13g5, 1S⟩ 13.6860 13.6871 13.6855 D. σ -trajectories for the pentaquark bb...

  26. [26]

    for function approxima- tion, even though other functional forms can sometimes yield decent or even superior approximation results. For the (bb)(¯u(cc)) configuration, the complete form of radial and orbital σ -trajectories can be well approximated by 8 13 13.2 13.4 13.6 13.8 14 14.2 0 1 2 3 4 (bb)(¯u(cc)) σ nr3 M (GeV) (a) 13 13.2 13.4 13.6 13.8 14 14.2 0...

  27. [27]

    0009+ l3 )2/ 3

    213174(0. 0009+ l3 )2/ 3. The dashed line represents the fitted formula M = 13. 0979 + 0. 25979(0. 0519 + l3)7/ 12. the simple formulas which take the functional form in Eq. ( 25). In contrast, for the ( cc)(¯u(bb)) configuration, M =

  28. [28]

    41051√ 0

    9774 + 0. 41051√ 0. 1979 +nr3 provides a better fit to the complete form than M = 13. 1627 + 0. 26124(0. 0009 + nr3)2/ 3. Likewise, M = 13. 0979+0. 25979(0. 0519+l3)7/ 12 is better than M = 13. 1498 + 0. 213174(0. 0009 +l3)2/ 3. Table VII and Fig. 5 show comparison between the two sets of fitting results. Observing Eq. ( 25), we find that for the ρ- and σ - ...

    1979

  29. [29]

    Takahashi et al

    F. Takahashi et al. (Particle Data Group), Int. J. Mod. Phys. A 41 , 2630011 (2026)

    2026

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