--- title: Tetraquark Regge Trajectory Relations url: https://www.emergentmind.com/topics/tetraquark-regge-trajectory-relations type: topic --- # Tetraquark Regge Trajectory Relations Searching arXiv for recent and foundational papers on tetraquark Regge trajectories, including the specified paper and closely related work. Tetraquark Regge trajectory relations are spectral relations that organize tetraquark masses by orbital angular momentum, radial quantum number, or internal excitation coordinates, usually in the \((J,M^2)\) and \((n,M^2)\) planes and, in substructure-aware treatments, by \(\lambda\)- and \(\rho\)-mode variables. In the current literature, these relations appear in several non-equivalent forms: linear radial trajectories used to extrapolate compact \(cc\bar c\bar c\) spectra, quasi-linear orbital and radial trajectories for diquark–antidiquark systems, modified massive-endpoint string relations, nonlinear heavy-light formulas, and mode-dependent power laws that distinguish inter-cluster from intra-cluster excitations [2006.13028, 1606.02732, 2302.06794, 2407.04222, 2508.18899]. ## 1. Conceptual frameworks and tetraquark degrees of freedom Most tetraquark Regge constructions adopt an effective two-body description, but the meaning of the two bodies depends on the framework. In quasi-linear Regge phenomenology, tetraquarks are treated as compact diquark–antidiquark systems, with the four-quark state approximated by two effective color sources. This is the working assumption in the analyses of \(ss\bar b\bar b\), \(bb\bar b\bar b\), \(cc\bar c\bar c\), \(ss\bar c\bar c\), \(qq\bar q\bar q\), \(qq\bar s\bar s\), and \(ss\bar s\bar s\) spectra. In the hidden-heavy \((Qq)(\bar Q\bar q')\) literature, the same approximation is refined by distinguishing the relative coordinate \(\lambda\) between the diquark and antidiquark from the internal coordinates \(\rho_1\) and \(\rho_2\) inside the diquark and antidiquark, respectively. The resulting tetraquark state carries three excitation channels rather than one [2509.23149, 2511.02279, 2510.04496, 2407.04222, 2508.18899]. A different but related picture appears in HISH, where a tetraquark is modeled as a single open string stretched between a baryonic vertex with an attached diquark and an antibaryonic vertex with an attached antidiquark. Classically, this places tetraquarks in the same rotating-string family as mesons and baryons, but with tetraquark-specific endpoint interpretation, natural baryon–antibaryon decay channels, and possibly different intercept structure. In that framework, the proposal that a genuine stringy exotic hadron should lie on a modified Regge trajectory, whereas a molecule has no clear reason to do so, is presented as a criterion rather than a theorem [1606.02732]. A third line of work uses constituent diquark trajectories as building blocks. Heavy-light diquarks and light diquarks are assigned their own Regge relations, with the expectation that these diquark spectra can then be used to investigate \(\rho\)-mode excitations of tetraquarks. This does not by itself produce a full tetraquark trajectory, but it supplies the constituent masses and excitation patterns needed in diquark–antidiquark models [2308.02289, 2310.05131]. ## 2. Canonical trajectory equations The literature does not use a single universal Regge equation for tetraquarks. Instead, several representative relations coexist. | Relation class | Representative equation | Scope | |---|---|---| | Linear radial ansatz | \(M_n^2=\alpha(n-1)+\alpha_0\) | Radial \(cc\bar c\bar c\) extrapolation | | Quasi-linear orbital/radial ansatz | \(J=\beta(0)+\beta' M^2\), \(n=\alpha(0)+\alpha' M^2\) | Diquark–antidiquark orbital and radial spectra | | Heavy-light nonlinear relations | \(M=m_1+m_2+C'+\beta_x\sqrt{x+c_{0x}}\) | Heavy-light diquarks, mesons, baryons, tetraquarks | | Heavy-light corrected relation | \(M=m_1+C'+\sqrt{\beta_x^2(x+c_{0x})+\frac{4}{3}\sqrt{\pi}\,\beta_x m_2^{3/2}(x+c_{0x})^{1/4}}\) | Heavy-light systems with light-mass correction | | Mode-separated tetraquark relations | \(M=m_{R\lambda}+\beta_{x_\lambda}(x_\lambda+c_{0x_\lambda})^{2/3}\), \(M_{\rho_i}=m_{R\rho_i}+\beta_{x_{\rho_i}}\sqrt{x_{\rho_i}+c_{0x_{\rho_i}}}\) | Hidden-heavy and bottom-charm \(\lambda/\rho\) spectra | In the fully charmed QCD-sum-rule analysis, the radial ansatz \(M_n^2=\alpha(n-1)+\alpha_0\) is used with \(n=1,2,3\) for the ground state, first radial excitation, and second radial excitation. A direct consequence is \(M_3^2=2M_2^2-M_1^2\), which is the actual extrapolation underlying the tabulated \(n=3\) masses. This construction is channel by channel rather than universal: \(\alpha\) and \(\alpha_0\) are fitted separately for \(0^{++}\), \(1^{+-}\), \(2^{++}\), and \(1^{--}\) using the corresponding \(M_1\) and \(M_2\) values [2006.13028]. In quasi-linear tetraquark phenomenology, the orbital and radial slopes are extracted from adjacent levels. The orbital slope is written as \[ \beta'=\frac{1}{M_{J+1}^2-M_J^2}, \] and the radial slope as \[ \alpha'=\frac{1}{M_{2S}^2-M_{1S}^2}. \] These relations are then used to generate higher states through expressions of the form \[ M_{J+k}=\sqrt{M_J^2+\frac{k}{\beta'}}, \] with analogous radial constructions in the \((n,M^2)\) plane. This is the formal basis of the \(ss\bar b\bar b\), \(bb\bar b\bar b\), \(cc\bar c\bar c\), \(ss\bar c\bar c\), \(qq\bar q\bar q\), \(qq\bar s\bar s\), and \(ss\bar s\bar s\) spectra obtained from Regge phenomenology [2509.23149, 2511.02279, 2510.04496]. The heavy-light relations are structurally different. They describe \(M\) itself rather than \(M^2\) as a square-root function of either orbital or radial excitation. In this family of models, tetraquarks do not share a universal numerical slope with mesons and baryons, even when they satisfy the same functional form; instead, tetraquarks have their own fitted \(c_{fx}\) or \(\beta_x\) parameters [2302.06794]. ## 3. Modified string relations and explicit nonlinearity The massive-endpoint string formulation makes the departure from naive linearity explicit. In HISH, for generic endpoint masses \(m_1,m_2\), the hadron energy \(E\), angular momentum \(J\), and endpoint velocities \(\beta_i\) satisfy coupled relations rather than a closed linear law. For the symmetric case \(m_1=m_2=m\), the working formulas become \[ E = 2m\left(\frac{\beta\arcsin(\beta)+\sqrt{1-\beta^2}}{1-\beta^2}\right), \] \[ J+n-a = 2\pi \alpha' m^2\frac{\beta^2}{(1-\beta^2)^2}\left(\arcsin(\beta)+\beta\sqrt{1-\beta^2}\right). \] The idealized linear quantum relation \(J+n=\alpha' E^2+a\) is therefore replaced by a massive-endpoint modification in which the endpoint masses induce nonlinear corrections. In this setting, the tetraquark slope is not claimed to be uniquely tetraquark-specific, but the intercept may be, and a conjectured relation \(\tilde a_t=\tilde a_m+2(\tilde a_b-\tilde a_m)\) is proposed as a qualitative expectation rather than a derivation [1606.02732]. The flux-tube model with finite quark masses also produces nonlinear Regge behavior. There the tetraquark is represented as a rotating string with clustered quark masses at its ends, and the actual trajectory is parametric: \[ M=M(l,f; m_{q_i},K),\qquad J=J(l,f; m_{q_i},K). \] Because the formulas contain \(\sin^{-1}(f)\), \(\sin^{-1}(\eta f)\), \(\sqrt{1-f^2}\), and Lorentz factors, the \(J\)-versus-\(M^2\) relation is not reduced to a single fitted straight line. That analysis states explicitly that tetraquark Regge trajectories are nonlinear and become significantly nonlinear as the rotational speed rises [2307.13284]. Nonlinearity also appears in the heavy-light trajectory program. There the two proposed relations, \[ M=m_1+m_2+C'+\beta_x\sqrt{x+c_{0x}}, \] and \[ M=m_1+C'+\sqrt{\beta_x^2(x+c_{0x})+\frac{4}{3}\sqrt{\pi}\,\beta_x m_2^{3/2}(x+c_{0x})^{1/4}}, \] lead to trajectories that are concave downward in the \((M^2,n_r)\) and \((M^2,l)\) planes. In this formulation, tetraquarks satisfy the same functional family as heavy-light diquarks, mesons, and baryons, but with smaller fitted coefficients than the corresponding meson and baryon cases [2302.06794]. ## 4. \(\lambda\)- and \(\rho\)-trajectory separation The \(\lambda/\rho\) decomposition is the sharpest statement in the modern tetraquark Regge literature. For hidden-bottom and hidden-charm tetraquarks \((Qq)(\bar Q\bar q')\), the \(\lambda\)-mode is the relative motion of the diquark against the antidiquark, while the \(\rho_1\)- and \(\rho_2\)-modes are internal excitations of the diquark and antidiquark. The corresponding trajectory laws are \[ M=m_{R\lambda}+\beta_{x_\lambda}(x_\lambda+c_{0x_\lambda})^{2/3},\qquad x_\lambda=L,N_r, \] \[ M_{\rho_1}=m_{R\rho_1}+\beta_{x_{\rho_1}}\sqrt{x_{\rho_1}+c_{0x_{\rho_1}}},\qquad x_{\rho_1}=l_1,n_{r_1}, \] \[ M_{\rho_2}=m_{R\rho_2}+\beta_{x_{\rho_2}}\sqrt{x_{\rho_2}+c_{0x_{\rho_2}}},\qquad x_{\rho_2}=l_2,n_{r_2}. \] The resulting asymptotic behaviors are \[ M\sim x_\lambda^{2/3},\qquad M\sim x_\rho^{1/2}. \] The stated reason is dynamical: the \(\lambda\)-mode behaves as an effective heavy-heavy problem, whereas each \(\rho\)-mode behaves as a heavy-light subsystem [2407.04222]. For bottom-charm tetraquarks \((bq)(\bar c\bar q')\) and \((cq)(\bar b\bar q')\), the same distinction is retained, but the full \(\rho_1\)- and \(\rho_2\)-trajectory formulas become lengthy because the reduced mass of the \(\lambda\)-mode, \[ \mu_\lambda=\frac{M_{\rho_1}M_{\rho_2}}{M_{\rho_1}+M_{\rho_2}}, \] depends on the excited diquark and antidiquark masses. That dependence is the explicit reason given for why \(\rho_1\)- and \(\rho_2\)-trajectories cannot be obtained by simply imitating meson Regge trajectories. In the substructure-aware form, the tetraquark mass is written as \[ M=m_b+m_q+m_c+m_{q'}+2C+\beta_{x_\lambda}(x_\lambda+c_{0x_\lambda})^{2/3}+\beta_{x_{\rho_1}}\sqrt{x_{\rho_1}+c_{0x_{\rho_1}}}+\beta_{x_{\rho_2}}\sqrt{x_{\rho_2}+c_{0x_{\rho_2}}}, \] although the authors then show that the complete \(\rho\)-trajectories are well approximated by simpler fitted square-root formulas. All three trajectory families are stated to be concave downward in the \((M^2,x)\) plane when the confining potential is linear [2508.18899]. The diquark-only studies provide the constituent counterpart of this mode decomposition. Heavy-light diquark trajectories require light-quark-mass and Cornell-\(C\) corrections, while light diquark trajectories are described by provisional square-root formulas with \(V_d=V_m/2\). Both studies explicitly state that these diquark trajectories can be used to investigate \(\rho\)-mode excitations of tetraquarks, but they stop short of constructing the full tetraquark trajectory [2308.02289, 2310.05131]. ## 5. Additivity rules, inequalities, and flavor dependence The quasi-linear tetraquark program imports two standard relations into the diquark–antidiquark sector: \[ \beta_{i\bar i}(0)+\beta_{j\bar j}(0)=2\beta_{i\bar j}(0), \] \[ \frac{1}{\beta'_{i\bar i}}+\frac{1}{\beta'_{j\bar j}}=\frac{2}{\beta'_{i\bar j}}. \] Eliminating intercepts yields \[ \beta'_{i\bar i}M_{i\bar i}^2+\beta'_{j\bar j}M_{j\bar j}^2=2\beta'_{i\bar j}M_{i\bar j}^2. \] From the requirement that the slope ratios be real and positive, the mixed-flavor state is constrained by the linear and quadratic mass inequalities \[ 2M_{i\bar j}>M_{i\bar i}+M_{j\bar j}, \qquad 2M_{i\bar j}^2\beta'_{ss\bar b\bar b}>\beta'_{bb\bar b\bar b}, \] with analogous relations in the charm sector and in radial slopes. In the light–strange sector, the reported ordering is \[ \alpha'_{qq\bar q\bar q}>\alpha'_{qq\bar s\bar s}>\alpha'_{ss\bar s\bar s}, \] and similarly for the radial \(\beta'\) values. The shared phenomenological lesson is that heavier flavor content reduces both orbital and radial slopes [2509.23149, 2511.02279, 2510.04496]. The heavy-light universal-description program reaches a different but compatible conclusion. It argues that heavy-light tetraquarks satisfy the same two functional formulas as heavy-light diquarks, mesons, and baryons, yet the fitted slopes differ distinctively for mesons, baryons, and tetraquarks. The tetraquark mean values \(\langle c_{fn_r}\rangle=0.553\), \(\langle c_{fl}\rangle=0.579\) for the first relation, and \(\langle c_{fn_r}\rangle=0.647\), \(\langle c_{fl}\rangle=0.676\) for the second, are explicitly stated to be the smallest among the three hadron classes. This suggests flatter tetraquark trajectories in the shifted-mass variables used there, but it does not imply universal tetraquark slopes across all flavor sectors [2302.06794]. ## 6. Spectroscopic applications, proposed assignments, and limitations The most direct application to experiment is the fully charmed \(cc\bar c\bar c\) study of the LHCb di-\(J/\psi\) spectrum. In that analysis, QCD sum rules provide the first radial excitations \[ 0^{++}: 6.48\pm0.08~\mathrm{GeV},\quad 1^{+-}: 6.52\pm0.08~\mathrm{GeV},\quad 2^{++}: 6.56\pm0.08~\mathrm{GeV},\quad 1^{--}: 6.58\pm0.09~\mathrm{GeV}, \] and the Regge extrapolation gives the second radial excitations \[ 0^{++}: 6.94\pm0.08~\mathrm{GeV},\quad 1^{+-}: 6.96\pm0.08~\mathrm{GeV},\quad 2^{++}: 7.00\pm0.08~\mathrm{GeV},\quad 1^{--}: 7.02\pm0.09~\mathrm{GeV}. \] These masses are used to argue that the broad structure from \(6.2\) to \(6.8\) GeV is compatible with the first radial excited states of scalar, axialvector, vector, or tensor \(cc\bar c\bar c\) tetraquarks, while the narrow structure near \(6.9\) GeV is compatible with the second radial excited scalar or axialvector states. The same work notes that the mass gaps \(M_3-M_1=0.91\sim0.95\) GeV are consistent with the charmonium gap \(m_{\psi^{\prime\prime}}-m_{J/\psi}=0.94\) GeV [2006.13028]. In HISH, the \(Y(4630)\) is proposed as the main tetraquark example because it decays predominantly to \(\Lambda_c\bar\Lambda_c\), which is the natural decay channel of the stringy tetraquark configuration. Using \(m_c=1490\) MeV, \(\alpha'_J=0.86~\mathrm{GeV}^{-2}\), and \(\alpha'_n=0.59~\mathrm{GeV}^{-2}\), the fitted intercepts are \(a=-5.8\) for the orbital trajectory and \(a=-4.0\) for the radial trajectory if \(Y(4630)\) is the first state. The authors interpret the largeness of these negative intercepts as a sign that \(Y(4630)\) may itself be an excited radial state. The same framework predicts higher radial and orbital states, as well as analogous \(Y_b\) and \(Y_s\) trajectories [1606.02732]. Other frameworks are more organizational than assignment-driven. The massive flux-tube model reproduces masses for \(X(3872)\), \(Z_c(3900)\), \(X(4140)\), \(X(4430)\), \(X(6900)\), \(Z_b(10610)\), and \(Z_b(10650)\) by selecting state-dependent string lengths and endpoint speeds, but it does not extract a universal fitted \(\alpha\) and \(\alpha_0\) for tetraquarks. The quasi-linear \(ss\bar c\bar c\) and \(cc\bar c\bar c\) study suggests \(\psi(4660)\) as a possible \(ss\bar c\bar c\) \(1^1P_1\) state and \(\chi_{c0}(4700)\) as a possible \(ss\bar c\bar c\) \(2^1S_0\) state. The light-flavor Regge study proposes assignments such as \(X(2075)\) as \(qq\bar s\bar s\) \(1^3S_1\) and \(a_2(2175)\) as \(qq\bar s\bar s\) \(1^5S_2\) [2307.13284, 2511.02279, 2510.04496]. Several limitations recur across the literature. In the fully charmed radial analysis, only two levels per channel are known before extrapolation, the inputs are theoretical rather than experimental, no explicit uncertainty on \(\alpha\) and \(\alpha_0\) is quoted, and no alternative trajectory forms are explored [2006.13028]. In the string and flux-tube approaches, the tetraquark is not treated as a full four-body problem but as an effective two-end or diquark–antidiquark system, often with pointlike diquarks or averaged endpoint configurations [1606.02732, 2307.13284]. In the heavy-light and light-diquark programs, the authors explicitly describe some formulas as provisional and note that experimental information and higher \(\lambda\)-excited theoretical states are scarce [2302.06794, 2308.02289, 2310.05131]. A plausible implication is that “tetraquark Regge trajectory relations” presently denote a family of model-dependent organizing principles rather than a single settled law of exotic-hadron spectroscopy. Source: https://www.emergentmind.com/topics/tetraquark-regge-trajectory-relations