---
title: Relativistic Screened Potential Models
url: https://www.emergentmind.com/topics/relativistic-screened-potential-model
type: topic
---

# Relativistic Screened Potential Models

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 [2501.03147], [2408.06759], [1402.3094], [1812.10239]. 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], [1507.02397], [2412.00344].

## 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 [2501.03147], [1402.3094], [1507.02397], [1812.10239].

## 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
$$
V_{SD}(r)=V_{SS}(r)\,\vec S_q\!\cdot\!\vec S_{\bar q}+V_{LS}(r)\,\vec L\!\cdot\!\vec S+V_T(r)\,S_{12},
$$
with a Gaussian-smeared contact term and the usual spin–orbit and tensor structures derived from $V_V$ and $V_S$ [2501.03147], [2408.06759].

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 $r\in[0,L]$, and truncation at $N$ 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 [2501.03147], [2408.06759].

Parameterization is system dependent. In the bottomonium analysis the fitted values are
$$
m_b=4.744~\text{GeV},\quad \sigma=4.967~\text{GeV}^2,\quad
\lambda=0.240~\text{GeV},\quad \mu=0.039~\text{GeV},\quad
\Lambda_{\text{QCD}}=0.17~\text{GeV},
$$
with $V_0$ absorbed in the fit. In the charmonium analysis the corresponding fitted values are
$$
m_q=1.319~\text{GeV},\quad \sigma=1.281~\text{GeV}^2,\quad
\lambda=0.297~\text{GeV},\quad \mu=0.141~\text{GeV},\quad
\Lambda=0.17~\text{GeV},
$$
again with $V_0$ included but not separately listed [2501.03147], [2408.06759].

Phenomenologically, screening compresses higher excitations relative to an unscreened Cornell potential. In bottomonium this improves the placement of $nP$ and $nD$ multiplets relative to $nS$, but discrepancies remain for $\Upsilon(4S)$–$\Upsilon(6S)$ and the $3P$ levels, which the paper attributes to coupled-channel and threshold effects not included explicitly. The same study treats $\Upsilon(10355)$ as a $3S$–$2D$ mixed state with $\theta=19.28^\circ$, $\Upsilon(10580)$ as a $4S$–$3D$ mixed state with $\theta=-30.72^\circ$, $\Upsilon(10753)$ as a pure $\Upsilon_1(3D)$ state, and $\Upsilon(10860)$ and $\Upsilon(11020)$ as a $5S$–$4D$ mixed pair with $\theta=44.55^\circ$ [2501.03147].

In charmonium the screened Salpeter model reproduces low-lying levels and uses $S$–$D$ mixing to organize the vector sector above open-charm threshold. The fitted mixing pattern assigns $\psi(3770)$ to the $2S$–$1D$ system with $\theta=-6.41^\circ$, $\psi(4040)$ and $\psi(4160)$ to the $3S$–$2D$ system with $\theta=45.13^\circ$, $\psi(4230)$ and $\psi(4360)$ to the $4S$–$3D$ system with $\theta=-24.15^\circ$, and $\psi(4415)$ and $Y(4500)$ to the $5S$–$4D$ system with $\theta=-16.11^\circ$; $Y(4660)$ is favored as a predominantly $5D$ state with $\Gamma_{ee}=0.16$ keV [2408.06759].

## 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,
$$
V_{\text{scr}}(r)=V_V(r)+V_S(r),\qquad
V_V(r)=-\frac{4}{3}\frac{\alpha_C}{r},\qquad
V_S(r)=\lambda\frac{1-e^{-\mu r}}{\mu},
$$
together with perturbative Breit–Fermi spin-dependent terms. Its fitted parameters are
$$
\alpha_C=0.5007,\quad \lambda=0.21~\text{GeV}^2,\quad \mu=0.0979~\text{GeV},
$$
$$
m_c=1.4045~\text{GeV},\quad \sigma_{\text{smear}}=1.362~\text{GeV},\quad \alpha_S=0.26.
$$
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 $O(50$–$200~\text{MeV})$ relative to unscreened potentials. It assigns $Z(3930)$ to $\chi_{c2}(2P)$, favors $\psi(4415)\equiv\psi(5S)$ rather than $\psi(4S)$, proposes $Y(4008)\sim\psi(3S)$, $Y(4260)\sim\psi(4S)$, $Y(4320/4360)\sim\psi(3D_1)$, and $Y(4660)\sim\psi(6S)$, and interprets $X(3872)$ as a mostly $\chi_{c1}(2P)$ state with some $D^0\bar D^{*0}$ 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 $[M_{T_{j-1}},M_{T_j}]$, the potential is piecewise,
$$
V_{[M_{T_{j-1}},M_{T_j}]}(r)=
\begin{cases}
M_{T_{j-1}}-2m_c, & r<r_{T_{j-1}},\\[4pt]
\sigma r-\kappa/r, & r_{T_{j-1}}\le r\le r_{T_j},\\[4pt]
M_{T_j}-2m_c, & r\ge r_{T_j},
\end{cases}
$$
with crossing radii fixed by
$$
\sigma r_{T_i}-\kappa/r_{T_i}=M_{T_i}-2m_c.
$$
The charmonium parameters are
$$
\sigma=850~\text{MeV/fm},\quad \kappa=100~\text{MeV}\cdot\text{fm},\quad m_c=1348.6~\text{MeV}.
$$
No Breit–Fermi terms are included, so fine structure is not predicted [1507.02397].

The GSPM generates a threshold-induced proliferation of near-threshold $J^{++}$ states. Its $1^{++}$ level $2p[T_0,T_1]$ at $3871.7$ MeV is identified with $X(3872)$ and has $r_{\rm rms}\approx 3.6$ fm, compared with $1.1$ fm for the Cornell $2P$ state. It assigns a $0^{++}$ state $1p[T_1,T_2]$ at $3897.9$ MeV to $X(3915)$, predicts $\Gamma(X(3915)\to\gamma\gamma)/\Gamma(\chi_{c0}(1P)\to\gamma\gamma)\approx0.02$, places $\chi_{c2}(2P)$ at $3903.0$ MeV, and predicts an additional $1^{++}$ state $C(4017)$ at $4017.3$ MeV. Between $4.0$ and $4.4$ GeV it produces at least four $J^{++}$ states versus a single Cornell $3P$ state near $4295$ MeV [1507.02397]. 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 $N_e$-electron system,
$$
H=\sum_{i=1}^{N_e}\left[c\,\alpha_i\cdot p_i+\beta_i m_e c^2+V_{\text{nuc}}(r_i)\right]+\sum_{i<j}V_{ee}(r_{ij}),
$$
with the one-body Yukawa potential
$$
V_{\text{nuc}}(r)=-\frac{Ze^{-\mu r}}{r},
$$
and the ideal two-body Debye form
$$
V_{ee}(r_{ij})=\frac{e^{-\mu r_{ij}}}{r_{ij}}.
$$
The plasma regime is explicitly weakly coupled, with $\Gamma<1$, so the Debye–Hückel description is appropriate. The paper varies the Debye length $D$ directly and uses $\mu=1/D$ in atomic units [1402.3094].

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
$$
V_{\text{eff}}(r_i)=V_{\text{nuc}}(r_i)+e^{-\mu r_i}\sum_{j\neq i}\frac{1}{r_{ij}}.
$$
The many-body calculation is carried out with one-valence Fock-space RCC,
$$
|\Psi_v\rangle=e^T(1+S_v)|\Phi_v\rangle,
$$
truncated at CCSD:
$$
T=T_1+T_2,\qquad S_v=S_{1v}+S_{2v}.
$$
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 [1402.3094].

The main spectroscopic consequences are explicit. Ionization potentials decrease monotonically as $D$ decreases. Critical Debye lengths for loss of the ground state are, in Model A, Li I $\approx14$ a.u., Ca XVIII $\approx0.30$ a.u., and Ti XX $\approx0.25$ a.u.; in Model B they shift to Li I $\approx0.40$ a.u., Ca XVIII $\approx0.043$ a.u., and Ti XX $\approx0.040$ a.u., showing that two-body screening stabilizes the system more than nuclear-only screening [1402.3094].

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 $\Delta n=0$ $ns\to np$ lines tend to blue-shift as $D$ decreases and the $\Delta n\neq0$ lines red-shift. In Model B, $\Delta n\neq0$ transitions universally red-shift with decreasing $D$, whereas $\Delta n=0$ transitions show a hump-like behavior: blue-shift at larger $D$, turning to red-shift at smaller $D$ [1402.3094].

The same framework predicts level crossings near the continuum, especially for higher-$l$ states, and substantial screening dependence of lifetimes. For example, under Model B the Li I lifetime $\tau(2p_{1/2})$ changes from $26.60$ ns at $D=200$ a.u. to $20.73$ ns at $D=9$ a.u., whereas $\tau(3s_{1/2})$ changes from $30.38$ ns to $147.75$ ns over the same range. In Ca XVIII, $\tau(2p_{1/2})$ decreases from $741$ ps at $D=7.52$ a.u. to $91$ ps at $D=0.50$ a.u., while $\tau(4s_{1/2})$ increases from $1.64$ ps to $556$ ps [1402.3094].

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 $(2p\,{}^2P_{1/2}\to3d\,{}^2D_{3/2})$ to $(2p\,{}^2P_{3/2}\to3s\,{}^2S_{1/2})$, the theory gives $\approx9.42$ versus experiment $\approx7.8$ for Ca XVIII at $n_e=2\times10^{21}\,\text{cm}^{-3}$ and $T_e=600$ eV, and $\approx9.60$ versus $\approx8.4$ for Ti XX at $n_e=1\times10^{21}\,\text{cm}^{-3}$ and $T_e=800$ eV [1402.3094].

## 5. Screened strange-baryon spectroscopy and the nonrelativistic contrast

A useful counterexample is the screened-potential study of $\Lambda$ and $\Sigma$ baryons. It employs a hypercentral constituent-quark model for a three-quark system, but the treatment is explicitly nonrelativistic:
$$
H=\frac{P^2}{2m}+V_{\text{SI}}(x)+V_{\text{SD}}(x),\qquad
m=\frac{2m_\rho m_\lambda}{m_\rho+m_\lambda},
$$
with hyperradius
$$
x=\sqrt{\rho^2+\lambda^2}.
$$
The spin-independent sector is
$$
V_{\text{SI}}(x)=V_{\text{conf}}(x)+V_{\text{Col}}(x),
$$
$$
V_{\text{conf}}(x)=a\left(\frac{1-e^{-\mu x}}{\mu}\right),\qquad
V_{\text{Col}}(x)=-\frac23\frac{\alpha_s}{x}.
$$
The constituent masses are $m_u=m_d=0.290$ GeV and $m_s=0.500$ GeV, and the spectra shown use $\mu=0.3$ [2412.00344].

The effective spin-dependent interaction is written as
$$
V_{\text{SD}}(x)=V_{SS}(x)\,(\vec S_\rho\cdot\vec S_\lambda)+V_{\gamma S}(x)\,(\vec\gamma\cdot\vec S)+V_T(x)\left[S^2-\frac{3(\vec S\cdot\vec x)(\vec S\cdot\vec x)}{x^2}\right],
$$
but the paper does not provide explicit functional forms or coefficients for $V_{SS}(x)$, $V_{\gamma S}(x)$, or $V_T(x)$. It also does not specify numerical values for the string tension $a$ or the coupling $\alpha_s$, and it does not report a parameter fit, $\chi^2$, or RMS deviation [2412.00344].

Even with those omissions, the phenomenology is clear. The low-lying S-wave ground states match exactly: $\Lambda\,1S(1/2^+)=1115$ MeV and $\Sigma\,1S(1/2^+)=1193$ MeV. First excitations in S and lower P/D states are generally within $\sim20$–$50$ MeV of PDG values. By contrast, higher-$L$ states tend to be underpredicted by more than $100$ MeV; for example, the model gives $\Lambda\,1^2F_{7/2}(7/2^-)=1965$ MeV versus PDG $\sim2100$ and $\Lambda\,1^2G_{9/2}(9/2^+)=2179$ MeV versus PDG $\sim2350$ [2412.00344].

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
$$
J=aM^2+a_0,\qquad n=bM^2+b_0,
$$
which are stated to be “linear in nature” [2412.00344]. 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,
$$
G_{\mu\nu}+m_0^2 I_{\mu\nu}=\kappa^2 T_{\mu\nu},
$$
together with the physical metric
$$
ds^2=-e^{2\nu(r)}dt^2+e^{2\lambda(r)}dr^2+r^2d\Omega^2
$$
and a nontrivial embedding of the flat reference metric through a radial function $\chi(r)$ [1812.10239].

The modified TOV system is
$$
GM'(r)=4\pi G\,\rho\,r^2+\frac12 m_0^2 r^2 I^t{}_t,
$$
$$
\nu'=\frac{4\pi G\,p\,r^3+GM-\frac12 m_0^2 r^3 I^r{}_r}{r(r-2GM)},
$$
supplemented by energy-momentum conservation and, crucially, by an algebraic constraint for $\chi(r)$ derived from $\nabla_\mu I^{\mu\nu}=0$. In the non-minimal model this becomes a quartic equation in $\chi(r)$ [1812.10239].

The minimal model is defined by
$$
\beta_0=3,\qquad \beta_1=-1,\qquad \beta_2=\beta_3=0.
$$
In that case the analysis yields
$$
\frac{\chi}{r}\sim \left(\frac{r_V}{r}\right)^3
$$
in the near zone $M_s\ll r\ll r_V$, 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 [1812.10239].

For the non-minimal model, where $\beta_2,\beta_3\neq0$, the quartic admits two branches. One is a screened branch that can be cast as
$$
\frac{\chi}{r}=1+\mathcal{O}\!\left(\frac{M_s}{r}\right),
$$
which connects to Schwarzschild in the Vainshtein region. The other is an unscreened or strongly modified branch,
$$
\frac{\chi}{r}\sim \pm\sqrt{\frac{B_0}{D_0}}\left(\frac{r_V}{r}\right)^{3/2},
$$
which implies large deviations from asymptotic flatness [1812.10239]. 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 [2501.03147], [2408.06759], 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 [1507.02397], and the $\Lambda/\Sigma$ study is explicitly nonrelativistic in a hypercentral three-body formulation [2412.00344].

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 $\Upsilon(4S)$–$\Upsilon(6S)$, $X(3872)$, $\psi(4040)$, and $\psi(4160)$ [2501.03147], [2408.06759].

A third issue is parameter transparency. The Salpeter quarkonium studies determine parameters by $\chi^2$ fits, but the additive constant $V_0$ is not separately tabulated [2501.03147], [2408.06759]. The strange-baryon screened model does not give numerical values for $a$ or $\alpha_s$, does not specify the radial dependence of the spin-dependent pieces, and does not report $\chi^2$ or RMS deviations [2412.00344]. In the plasma application, the approximation $V_{\text{eff}}(r_i)=V_{\text{nuc}}(r_i)+e^{-\mu r_i}\sum_{j\neq i}1/r_{ij}$ is explicitly adopted to reduce computational cost, so Model B is adequate for trends but not quantitatively exact for all two-body screening effects [1402.3094]. In the GSPM, the abrupt saturation at the threshold crossing radii neglects threshold widths, overlaps, and degeneracies [1507.02397].

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 [2501.03147]. 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 $f_{J/\psi}$, $f_{\eta_c}$, and the $ggg$ widths of $J/\psi$ and $\psi(2S)$ [2408.06759]. 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 [1402.3094].

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 [2501.03147], [1402.3094], [1812.10239].

Source: https://www.emergentmind.com/topics/relativistic-screened-potential-model