Relativistic Screened Potential Models

• Relativistic screened potential models are effective descriptions that replace strictly long-range interactions with screened forms to simulate phenomena like string breaking, Debye shielding, and Vainshtein screening.
• They employ specific screening functions—such as Cornell-like saturation and Yukawa potentials—to refine spectral predictions and account for threshold effects in quarkonium, plasma, and gravitational contexts.
• Parameter fitting and piecewise approximations in these models enable detailed evaluations of spin splittings, mass shifts, and observable transitions, while highlighting limitations when nonrelativistic dynamics are applied.

A relativistic screened potential model is a class of effective descriptions in which a screened interaction is embedded into a relativistic or relativized dynamical equation. In hadron spectroscopy, screening usually softens the long-distance confining interaction through a saturation term such as $\lambda(1-e^{-\mu r})/\mu$, representing string breaking from virtual light $q\bar q$ creation; in plasma spectroscopy, it replaces Coulomb kernels by Yukawa forms characterized by $\mu=1/\lambda_D$; and in massive-gravity studies of relativistic stars, screening refers to the Vainshtein mechanism rather than to saturation of an interparticle potential (Bokade et al., 6 Jan 2025, Bokade et al., 2024, Das et al., 2014, Yamazaki et al., 2018). The literature also shows that the phrase is not synonymous with relativistic kinematics: several screened-potential constructions used for charmonium or strange baryons remain explicitly nonrelativistic, even when they address observables usually discussed alongside relativistic models (0903.5506, González, 2015, Menapara et al., 2024).

1. Domain-specific meaning and common structure

The cited literature uses the expression for several related but non-identical constructions. What unifies them is the replacement of a strictly long-range interaction by a screened one, with the screening parameter controlling either flattening of confinement, Debye shielding, or recovery of general-relativistic behavior inside a Vainshtein region.

Domain	Dynamical framework	Screening form
Bottomonium and charmonium	Spinless Salpeter or relativized/nonrelativistic Schrödinger equations	$\lambda(1-e^{-\mu r})/\mu$
Plasma-embedded atoms and ions	Dirac–Coulomb Hamiltonian solved with RCC	$e^{-\mu r}/r$
Threshold-aware charmonium GSPM	Piecewise Schrödinger problem between open-charm thresholds	Flat regions outside threshold crossing radii
Relativistic stars in dRGT	Modified Einstein–TOV system with algebraic constraint	Screened and unscreened Vainshtein branches

In the quarkonium setting, the screened form is Cornell-like at short distance and saturates at large $r$. In the plasma setting, the same general idea appears as Debye–Hückel screening of one-body and two-body Coulomb terms. In the dRGT star problem, the relevant distinction is between a branch that connects to Schwarzschild space-time and another that implies significant deviation from asymptotically flat space-time (Bokade et al., 6 Jan 2025, Das et al., 2014, González, 2015, Yamazaki et al., 2018).

2. Relativistic quarkonium formulations

The most explicit relativistic screened potential implementations in the cited material are the bottomonium and charmonium studies based on the spinless Salpeter equation. For equal quark masses, the Hamiltonian is written as

$H=2\sqrt{\mathbf{p}^{\,2}+m_Q^2}+V(r),$

with

$$V(r)=V_V(r)+V_S(r),\qquad V_V(r)=-\frac{4}{3}\frac{\alpha_s(r)}{r},\qquad V_S(r)=\lambda\frac{1-e^{-\mu r}}{\mu}+V_0.$$

The coordinate-space running coupling is modeled as

$$\alpha_s(r)=\sum_{i=1}^3 \alpha_i\,\mathrm{erf}(\gamma_i r),$$

with

$$\alpha_1=0.15,\ \alpha_2=0.15,\ \alpha_3=0.20,\qquad \gamma_1=\frac12,\ \gamma_2=\frac{\sqrt{10}}2,\ \gamma_3=\frac{\sqrt{1000}}2.$$

Spin splittings are added perturbatively through

$q\bar q$0

with a Gaussian-smeared contact term and the usual spin–orbit and tensor structures derived from $q\bar q$1 and $q\bar q$2 (Bokade et al., 6 Jan 2025, Bokade et al., 2024).

The numerical solution strategy is also explicit. The Salpeter equation is reduced to a radial integral equation, the reduced radial wavefunction is expanded in a spherical-Bessel basis on a finite interval $q\bar q$3, and truncation at $q\bar q$4 basis functions yields a matrix eigenvalue problem. The eigenvalues are interpreted as spin-averaged masses and the eigenvectors as normalized radial wavefunctions, after which spin-dependent splittings are evaluated perturbatively (Bokade et al., 6 Jan 2025, Bokade et al., 2024).

Parameterization is system dependent. In the bottomonium analysis the fitted values are

$q\bar q$5

with $q\bar q$6 absorbed in the fit. In the charmonium analysis the corresponding fitted values are

$q\bar q$7

again with $q\bar q$8 included but not separately listed (Bokade et al., 6 Jan 2025, Bokade et al., 2024).

Phenomenologically, screening compresses higher excitations relative to an unscreened Cornell potential. In bottomonium this improves the placement of $q\bar q$9 and $\mu=1/\lambda_D$0 multiplets relative to $\mu=1/\lambda_D$1, but discrepancies remain for $\mu=1/\lambda_D$2–$\mu=1/\lambda_D$3 and the $\mu=1/\lambda_D$4 levels, which the paper attributes to coupled-channel and threshold effects not included explicitly. The same study treats $\mu=1/\lambda_D$5 as a $\mu=1/\lambda_D$6–$\mu=1/\lambda_D$7 mixed state with $\mu=1/\lambda_D$8, $\mu=1/\lambda_D$9 as a $\lambda(1-e^{-\mu r})/\mu$0–$\lambda(1-e^{-\mu r})/\mu$1 mixed state with $\lambda(1-e^{-\mu r})/\mu$2, $\lambda(1-e^{-\mu r})/\mu$3 as a pure $\lambda(1-e^{-\mu r})/\mu$4 state, and $\lambda(1-e^{-\mu r})/\mu$5 and $\lambda(1-e^{-\mu r})/\mu$6 as a $\lambda(1-e^{-\mu r})/\mu$7–$\lambda(1-e^{-\mu r})/\mu$8 mixed pair with $\lambda(1-e^{-\mu r})/\mu$9 (Bokade et al., 6 Jan 2025).

In charmonium the screened Salpeter model reproduces low-lying levels and uses $e^{-\mu r}/r$0–$e^{-\mu r}/r$1 mixing to organize the vector sector above open-charm threshold. The fitted mixing pattern assigns $e^{-\mu r}/r$2 to the $e^{-\mu r}/r$3–$e^{-\mu r}/r$4 system with $e^{-\mu r}/r$5, $e^{-\mu r}/r$6 and $e^{-\mu r}/r$7 to the $e^{-\mu r}/r$8–$e^{-\mu r}/r$9 system with $r$0, $r$1 and $r$2 to the $r$3–$r$4 system with $r$5, and $r$6 and $r$7 to the $r$8–$r$9 system with $H=2\sqrt{\mathbf{p}^{\,2}+m_Q^2}+V(r),$0; $H=2\sqrt{\mathbf{p}^{\,2}+m_Q^2}+V(r),$1 is favored as a predominantly $H=2\sqrt{\mathbf{p}^{\,2}+m_Q^2}+V(r),$2 state with $H=2\sqrt{\mathbf{p}^{\,2}+m_Q^2}+V(r),$3 keV (Bokade et al., 2024).

3. Relativized and threshold-dependent screened charmonium models

Earlier charmonium applications implement screening in forms that are not fully relativistic in the Salpeter sense. The 2009 screened charmonium study uses a nonrelativistic Schrödinger equation with a screened scalar confinement term,

$H=2\sqrt{\mathbf{p}^{\,2}+m_Q^2}+V(r),$4

together with perturbative Breit–Fermi spin-dependent terms. Its fitted parameters are

$H=2\sqrt{\mathbf{p}^{\,2}+m_Q^2}+V(r),$5

$H=2\sqrt{\mathbf{p}^{\,2}+m_Q^2}+V(r),$6

The model is termed “relativized” because first-order relativistic corrections to wave functions are included in E1 transition calculations, not because the bound-state equation itself is Salpeter-like (0903.5506).

That study emphasizes that screening lowers higher excitations by $H=2\sqrt{\mathbf{p}^{\,2}+m_Q^2}+V(r),$7–$H=2\sqrt{\mathbf{p}^{\,2}+m_Q^2}+V(r),$8 relative to unscreened potentials. It assigns $H=2\sqrt{\mathbf{p}^{\,2}+m_Q^2}+V(r),$9 to $$V(r)=V_V(r)+V_S(r),\qquad V_V(r)=-\frac{4}{3}\frac{\alpha_s(r)}{r},\qquad V_S(r)=\lambda\frac{1-e^{-\mu r}}{\mu}+V_0.$$0, favors $$V(r)=V_V(r)+V_S(r),\qquad V_V(r)=-\frac{4}{3}\frac{\alpha_s(r)}{r},\qquad V_S(r)=\lambda\frac{1-e^{-\mu r}}{\mu}+V_0.$$1 rather than $$V(r)=V_V(r)+V_S(r),\qquad V_V(r)=-\frac{4}{3}\frac{\alpha_s(r)}{r},\qquad V_S(r)=\lambda\frac{1-e^{-\mu r}}{\mu}+V_0.$$2, proposes $$V(r)=V_V(r)+V_S(r),\qquad V_V(r)=-\frac{4}{3}\frac{\alpha_s(r)}{r},\qquad V_S(r)=\lambda\frac{1-e^{-\mu r}}{\mu}+V_0.$$3, $$V(r)=V_V(r)+V_S(r),\qquad V_V(r)=-\frac{4}{3}\frac{\alpha_s(r)}{r},\qquad V_S(r)=\lambda\frac{1-e^{-\mu r}}{\mu}+V_0.$$4, $$V(r)=V_V(r)+V_S(r),\qquad V_V(r)=-\frac{4}{3}\frac{\alpha_s(r)}{r},\qquad V_S(r)=\lambda\frac{1-e^{-\mu r}}{\mu}+V_0.$$5, and $$V(r)=V_V(r)+V_S(r),\qquad V_V(r)=-\frac{4}{3}\frac{\alpha_s(r)}{r},\qquad V_S(r)=\lambda\frac{1-e^{-\mu r}}{\mu}+V_0.$$6, and interprets $$V(r)=V_V(r)+V_S(r),\qquad V_V(r)=-\frac{4}{3}\frac{\alpha_s(r)}{r},\qquad V_S(r)=\lambda\frac{1-e^{-\mu r}}{\mu}+V_0.$$7 as a mostly $$V(r)=V_V(r)+V_S(r),\qquad V_V(r)=-\frac{4}{3}\frac{\alpha_s(r)}{r},\qquad V_S(r)=\lambda\frac{1-e^{-\mu r}}{\mu}+V_0.$$8 state with some $$V(r)=V_V(r)+V_S(r),\qquad V_V(r)=-\frac{4}{3}\frac{\alpha_s(r)}{r},\qquad V_S(r)=\lambda\frac{1-e^{-\mu r}}{\mu}+V_0.$$9 admixture. The paper explicitly states that, in the “quark–meson duality” sense, screening effectively mimics the net mass shifts from continuum loops (0903.5506).

A different implementation appears in the generalized screened potential model. The GSPM is nonrelativistic and threshold-aware: for a given energy region $$\alpha_s(r)=\sum_{i=1}^3 \alpha_i\,\mathrm{erf}(\gamma_i r),$$0, the potential is piecewise,

$$\alpha_s(r)=\sum_{i=1}^3 \alpha_i\,\mathrm{erf}(\gamma_i r),$$1

with crossing radii fixed by

$$\alpha_s(r)=\sum_{i=1}^3 \alpha_i\,\mathrm{erf}(\gamma_i r),$$2

The charmonium parameters are

$$\alpha_s(r)=\sum_{i=1}^3 \alpha_i\,\mathrm{erf}(\gamma_i r),$$3

No Breit–Fermi terms are included, so fine structure is not predicted (González, 2015).

The GSPM generates a threshold-induced proliferation of near-threshold $$\alpha_s(r)=\sum_{i=1}^3 \alpha_i\,\mathrm{erf}(\gamma_i r),$$4 states. Its $$\alpha_s(r)=\sum_{i=1}^3 \alpha_i\,\mathrm{erf}(\gamma_i r),$$5 level $$\alpha_s(r)=\sum_{i=1}^3 \alpha_i\,\mathrm{erf}(\gamma_i r),$$6 at $$\alpha_s(r)=\sum_{i=1}^3 \alpha_i\,\mathrm{erf}(\gamma_i r),$$7 MeV is identified with $$\alpha_s(r)=\sum_{i=1}^3 \alpha_i\,\mathrm{erf}(\gamma_i r),$$8 and has $$\alpha_s(r)=\sum_{i=1}^3 \alpha_i\,\mathrm{erf}(\gamma_i r),$$9 fm, compared with $$\alpha_1=0.15,\ \alpha_2=0.15,\ \alpha_3=0.20,\qquad \gamma_1=\frac12,\ \gamma_2=\frac{\sqrt{10}}2,\ \gamma_3=\frac{\sqrt{1000}}2.$$0 fm for the Cornell $$\alpha_1=0.15,\ \alpha_2=0.15,\ \alpha_3=0.20,\qquad \gamma_1=\frac12,\ \gamma_2=\frac{\sqrt{10}}2,\ \gamma_3=\frac{\sqrt{1000}}2.$$1 state. It assigns a $$\alpha_1=0.15,\ \alpha_2=0.15,\ \alpha_3=0.20,\qquad \gamma_1=\frac12,\ \gamma_2=\frac{\sqrt{10}}2,\ \gamma_3=\frac{\sqrt{1000}}2.$$2 state $$\alpha_1=0.15,\ \alpha_2=0.15,\ \alpha_3=0.20,\qquad \gamma_1=\frac12,\ \gamma_2=\frac{\sqrt{10}}2,\ \gamma_3=\frac{\sqrt{1000}}2.$$3 at $$\alpha_1=0.15,\ \alpha_2=0.15,\ \alpha_3=0.20,\qquad \gamma_1=\frac12,\ \gamma_2=\frac{\sqrt{10}}2,\ \gamma_3=\frac{\sqrt{1000}}2.$$4 MeV to $$\alpha_1=0.15,\ \alpha_2=0.15,\ \alpha_3=0.20,\qquad \gamma_1=\frac12,\ \gamma_2=\frac{\sqrt{10}}2,\ \gamma_3=\frac{\sqrt{1000}}2.$$5, predicts $$\alpha_1=0.15,\ \alpha_2=0.15,\ \alpha_3=0.20,\qquad \gamma_1=\frac12,\ \gamma_2=\frac{\sqrt{10}}2,\ \gamma_3=\frac{\sqrt{1000}}2.$$6, places $$\alpha_1=0.15,\ \alpha_2=0.15,\ \alpha_3=0.20,\qquad \gamma_1=\frac12,\ \gamma_2=\frac{\sqrt{10}}2,\ \gamma_3=\frac{\sqrt{1000}}2.$$7 at $$\alpha_1=0.15,\ \alpha_2=0.15,\ \alpha_3=0.20,\qquad \gamma_1=\frac12,\ \gamma_2=\frac{\sqrt{10}}2,\ \gamma_3=\frac{\sqrt{1000}}2.$$8 MeV, and predicts an additional $$\alpha_1=0.15,\ \alpha_2=0.15,\ \alpha_3=0.20,\qquad \gamma_1=\frac12,\ \gamma_2=\frac{\sqrt{10}}2,\ \gamma_3=\frac{\sqrt{1000}}2.$$9 state $q\bar q$00 at $q\bar q$01 MeV. Between $q\bar q$02 and $q\bar q$03 GeV it produces at least four $q\bar q$04 states versus a single Cornell $q\bar q$05 state near $q\bar q$06 MeV (González, 2015). This suggests that “screening” in quarkonium can mean either smooth saturation or threshold-induced piecewise flattening.

4. Relativistic plasma-screened atomic models

In plasma spectroscopy the relativistic screened potential model is built on a Dirac–Coulomb Hamiltonian with Debye-screened interactions. For an $q\bar q$07-electron system,

$q\bar q$08

with the one-body Yukawa potential

$q\bar q$09

and the ideal two-body Debye form

$q\bar q$10

The plasma regime is explicitly weakly coupled, with $q\bar q$11, so the Debye–Hückel description is appropriate. The paper varies the Debye length $q\bar q$12 directly and uses $q\bar q$13 in atomic units (Das et al., 2014).

Two screening prescriptions are contrasted. Model A screens only the electron–nucleus term, leaving the electron–electron interaction unscreened. Model B screens both, using the approximation

$q\bar q$14

The many-body calculation is carried out with one-valence Fock-space RCC,

$q\bar q$15

truncated at CCSD:

$q\bar q$16

Screening modifies the one- and two-electron integrals entering the normal-ordered Hamiltonian and therefore changes both the Dirac–Fock reference and the correlation corrections (Das et al., 2014).

The main spectroscopic consequences are explicit. Ionization potentials decrease monotonically as $q\bar q$17 decreases. Critical Debye lengths for loss of the ground state are, in Model A, Li I $q\bar q$18 a.u., Ca XVIII $q\bar q$19 a.u., and Ti XX $q\bar q$20 a.u.; in Model B they shift to Li I $q\bar q$21 a.u., Ca XVIII $q\bar q$22 a.u., and Ti XX $q\bar q$23 a.u., showing that two-body screening stabilizes the system more than nuclear-only screening (Das et al., 2014).

Excitation energies exhibit a model-dependent line-shift pattern. In Model A, Li I shows red shifts for the studied lines, while in Li-like Ca XVIII and Ti XX the $q\bar q$24 $q\bar q$25 lines tend to blue-shift as $q\bar q$26 decreases and the $q\bar q$27 lines red-shift. In Model B, $q\bar q$28 transitions universally red-shift with decreasing $q\bar q$29, whereas $q\bar q$30 transitions show a hump-like behavior: blue-shift at larger $q\bar q$31, turning to red-shift at smaller $q\bar q$32 (Das et al., 2014).

The same framework predicts level crossings near the continuum, especially for higher-$q\bar q$33 states, and substantial screening dependence of lifetimes. For example, under Model B the Li I lifetime $q\bar q$34 changes from $q\bar q$35 ns at $q\bar q$36 a.u. to $q\bar q$37 ns at $q\bar q$38 a.u., whereas $q\bar q$39 changes from $q\bar q$40 ns to $q\bar q$41 ns over the same range. In Ca XVIII, $q\bar q$42 decreases from $q\bar q$43 ps at $q\bar q$44 a.u. to $q\bar q$45 ps at $q\bar q$46 a.u., while $q\bar q$47 increases from $q\bar q$48 ps to $q\bar q$49 ps (Das et al., 2014).

Radiative diagnostics are built from RCC matrix elements and standard E1/M1/E2 Einstein coefficients, with line intensity ratios evaluated under LTE and optically thin conditions. For the ratio of $q\bar q$50 to $q\bar q$51, the theory gives $q\bar q$52 versus experiment $q\bar q$53 for Ca XVIII at $q\bar q$54 and $q\bar q$55 eV, and $q\bar q$56 versus $q\bar q$57 for Ti XX at $q\bar q$58 and $q\bar q$59 eV (Das et al., 2014).

5. Screened strange-baryon spectroscopy and the nonrelativistic contrast

A useful counterexample is the screened-potential study of $q\bar q$60 and $q\bar q$61 baryons. It employs a hypercentral constituent-quark model for a three-quark system, but the treatment is explicitly nonrelativistic:

$q\bar q$62

with hyperradius

$q\bar q$63

The spin-independent sector is

$q\bar q$64

$q\bar q$65

The constituent masses are $q\bar q$66 GeV and $q\bar q$67 GeV, and the spectra shown use $q\bar q$68 (Menapara et al., 2024).

The effective spin-dependent interaction is written as

$q\bar q$69

but the paper does not provide explicit functional forms or coefficients for $q\bar q$70, $q\bar q$71, or $q\bar q$72. It also does not specify numerical values for the string tension $q\bar q$73 or the coupling $q\bar q$74, and it does not report a parameter fit, $q\bar q$75, or RMS deviation (Menapara et al., 2024).

Even with those omissions, the phenomenology is clear. The low-lying S-wave ground states match exactly: $q\bar q$76 MeV and $q\bar q$77 MeV. First excitations in S and lower P/D states are generally within $q\bar q$78–$q\bar q$79 MeV of PDG values. By contrast, higher-$q\bar q$80 states tend to be underpredicted by more than $q\bar q$81 MeV; for example, the model gives $q\bar q$82 MeV versus PDG $q\bar q$83 and $q\bar q$84 MeV versus PDG $q\bar q$85 (Menapara et al., 2024).

The same work reports that screening compresses the spectrum relative to the authors’ earlier linear confinement, lowers higher excitations, reduces hyperfine splittings, and yields Regge trajectories described by

$q\bar q$86

which are stated to be “linear in nature” (Menapara et al., 2024). The relevance to the broader topic is methodological: it shows that screened-potential phenomenology can be imported into baryon spectroscopy without relativistic kinematics, so the label must always be read together with the dynamical equation actually used.

6. Relativistic screening in dRGT massive gravity

A structurally different use of the terminology appears in the study of static, spherically symmetric relativistic stars in de Rham–Gabadadze–Tolley massive gravity by Yamazaki, Katsuragawa, Odintsov, and Nojiri. Here the starting point is the dRGT action with a flat reference metric and graviton-mass potential terms,

$q\bar q$87

together with the physical metric

$q\bar q$88

and a nontrivial embedding of the flat reference metric through a radial function $q\bar q$89 (Yamazaki et al., 2018).

The modified TOV system is

$q\bar q$90

$q\bar q$91

supplemented by energy-momentum conservation and, crucially, by an algebraic constraint for $q\bar q$92 derived from $q\bar q$93. In the non-minimal model this becomes a quartic equation in $q\bar q$94 (Yamazaki et al., 2018).

The minimal model is defined by

$q\bar q$95

In that case the analysis yields

$q\bar q$96

in the near zone $q\bar q$97, so the Vainshtein mechanism fails and the physical metric cannot approach the Schwarzschild solution outside the star. The paper identifies this as the absence of screening in the minimal model (Yamazaki et al., 2018).

For the non-minimal model, where $q\bar q$98, the quartic admits two branches. One is a screened branch that can be cast as

$q\bar q$99

which connects to Schwarzschild in the Vainshtein region. The other is an unscreened or strongly modified branch,

$\mu=1/\lambda_D$00

which implies large deviations from asymptotic flatness (Yamazaki et al., 2018). In this context, screening therefore means recovery of general-relativistic behavior rather than flattening of a potential.

7. Interpretive issues, limitations, and recurrent misconceptions

A first recurring misconception is to equate screening with relativistic dynamics. The bottomonium and one charmonium model are genuinely Salpeter-based (Bokade et al., 6 Jan 2025, Bokade et al., 2024), but the higher-charmonium screened model solves a nonrelativistic Schrödinger equation with first-order relativistic corrections only in E1 matrix elements (0903.5506), the GSPM is explicitly nonrelativistic and spin-independent (González, 2015), and the $\mu=1/\lambda_D$01 study is explicitly nonrelativistic in a hypercentral three-body formulation (Menapara et al., 2024).

A second issue is the status of screening as an effective proxy. In higher charmonium, screening is presented as modeling string breaking due to light-quark pair creation and, in the “quark–meson duality” sense, as mimicking net mass shifts from continuum loops (0903.5506). The relativistic charmonium and bottomonium Salpeter studies likewise note that explicit open-flavor thresholds, meson loops, and coupled-channel dynamics are not dynamically incorporated, even though these effects are likely important for states such as $\mu=1/\lambda_D$02–$\mu=1/\lambda_D$03, $\mu=1/\lambda_D$04, $\mu=1/\lambda_D$05, and $\mu=1/\lambda_D$06 (Bokade et al., 6 Jan 2025, Bokade et al., 2024).

A third issue is parameter transparency. The Salpeter quarkonium studies determine parameters by $\mu=1/\lambda_D$07 fits, but the additive constant $\mu=1/\lambda_D$08 is not separately tabulated (Bokade et al., 6 Jan 2025, Bokade et al., 2024). The strange-baryon screened model does not give numerical values for $\mu=1/\lambda_D$09 or $\mu=1/\lambda_D$10, does not specify the radial dependence of the spin-dependent pieces, and does not report $\mu=1/\lambda_D$11 or RMS deviations (Menapara et al., 2024). In the plasma application, the approximation $\mu=1/\lambda_D$12 is explicitly adopted to reduce computational cost, so Model B is adequate for trends but not quantitatively exact for all two-body screening effects (Das et al., 2014). In the GSPM, the abrupt saturation at the threshold crossing radii neglects threshold widths, overlaps, and degeneracies (González, 2015).

A fourth issue concerns observables. Screened potentials often improve mass systematics at high excitation, but widths remain more sensitive. In bottomonium the model describes many masses and several E1 transitions well, yet M1 transitions are noted to be highly sensitive to wavefunctions and relativistic corrections (Bokade et al., 6 Jan 2025). In charmonium, the model reproduces much of the low-lying spectrum and a broad E1 pattern, but several decay constants and annihilation widths overshoot experiment, including $\mu=1/\lambda_D$13, $\mu=1/\lambda_D$14, and the $\mu=1/\lambda_D$15 widths of $\mu=1/\lambda_D$16 and $\mu=1/\lambda_D$17 (Bokade et al., 2024). The plasma RCC study reproduces the qualitative behavior of line-intensity ratios but attributes residual discrepancies to Debye-model simplifications and the approximate treatment of two-body screening (Das et al., 2014).

Taken together, these works suggest that the relativistic screened potential model is best understood not as a single formalism but as a family of screened effective descriptions whose meaning depends on the microscopic mechanism assigned to screening. In quarkonium it usually encodes string breaking and spectral compression; in plasma spectroscopy it encodes Debye shielding in a relativistic many-electron Hamiltonian; in dRGT gravity it encodes the existence or failure of a screened Vainshtein branch. The shared mathematical motif is the replacement of an unscreened long-range interaction by a screened one, but the physical content and the degree of relativistic fidelity are model-specific (Bokade et al., 6 Jan 2025, Das et al., 2014, Yamazaki et al., 2018).