Fractional Power Scalar Potential Overview

• Fractional power scalar potentials are defined by non-integer exponents (e.g., V(φ)=V₀φᵖ) and model phenomena in cosmology, quantum mechanics, and nonlocal electrodynamics.
• They emerge naturally through dimensional transmutation, field redefinitions, and exact integrability methods, improving analytic control across different frameworks.
• Applications include designing inflationary scenarios, solving singular potentials in the Dirac equation, and deforming the Coulomb law via fractional operators in nonlocal media.

Searching arXiv for the cited papers to ground the article in the current literature. arxiv_search(query="(Socorro et al., 18 Jun 2026) OR (Harigaya et al., 2012) OR (Harigaya et al., 2014) OR (Ashrafzadeh et al., 2023) OR (Zahoor et al., 24 Jul 2025) OR (Agboola et al., 2013) OR (Tarasov, 2013) OR (Malkawi, 2015)", max_results=10) arxiv_search(query="Fractional power-law potential inflation Gauss-Bonnet arXiv", max_results=5) Fractional power scalar potentials are potentials whose dependence on the relevant variable involves a non-integer exponent. In scalar-field cosmology, the standard forms are monomials such as $V(\phi)=V_0\phi^p$ with rational $p$, and inverse powers such as $V(\phi)=V_0\phi^{-\lambda}$; in relativistic quantum mechanics, the same phrase may denote Lorentz-scalar central interactions with singular fractional radial dependence; and in nonlocal electrodynamics it appears through fractional operators that deform the Coulomb potential into a power law. Across these settings, fractional powers are used for distinct purposes: dynamical generation by dimensional transmutation, exact integrability after field redefinition, quasi-exact solvability in the Dirac equation, and effective descriptions of nonlocal screening (Socorro et al., 18 Jun 2026, Harigaya et al., 2012, Agboola et al., 2013, Tarasov, 2013).

1. Functional forms and domain-specific meanings

The phrase “fractional power scalar potential” is not tied to a single formalism. In the cosmological literature represented here, it most often denotes a scalar-field potential of the form

$V(\phi)=V_0\phi^p$

with rational $p$, or an inverse-power form

$V(\phi)=V_0\phi^{-\lambda}.$

In the relativistic Dirac problem, by contrast, “scalar potential” refers to a Lorentz-scalar interaction $S(r)$, and the fractional-power family is written as

$V_s(r)=\sum_{p=1}^{2N-1}\frac{a_p}{r^{p/N}}.$

In media with power-law spatial dispersion, the scalar electrostatic potential $\Phi(r)$ is not chosen directly as a monomial; instead it emerges from a fractional Poisson equation and behaves as $\Phi(r)\propto r^{-(2-\alpha)}$ for $p$0 (Agboola et al., 2013, Tarasov, 2013).

Context	Representative form	Role
Scalar-field cosmology	$p$1	Inflationary dynamics
K-essence-like cosmology	$p$2	Exact classical and quantum solutions
Dirac equation	$p$3	Bound-state spectra
Nonlocal electrostatics	$p$4	Power-law screening

A recurrent misconception is that a fractional exponent must be inserted by hand as a purely phenomenological ansatz. That is not generally correct in the literature considered here. In strongly coupled supersymmetric gauge theories, the exponent can be calculable from color and flavor data, while in the Sáez–Ballester–K-essence construction an inverse power becomes an exponential after a field redefinition (Harigaya et al., 2012, Harigaya et al., 2014, Socorro et al., 18 Jun 2026). Another source of confusion is terminological: the same phrase covers scalar-field potentials in cosmology, Lorentz-scalar couplings in the Dirac equation, and electrostatic scalar potentials in nonlocal media, which are mathematically related only in a broad sense.

2. Negative-power scalar potentials and exact integrability in Sáez–Ballester–K-essence theory

A particularly explicit realization is the K-essence-like cosmological model built from a negative power-law Sáez–Ballester potential,

$p$5

with

$p$6

where $p$7 corresponds to quintessence and $p$8 to phantom behavior. The crucial step is the field redefinition $p$9, under which $V(\phi)=V_0\phi^{-\lambda}$0 and the potential becomes

$V(\phi)=V_0\phi^{-\lambda}$1

In these variables the scalar field has a standard kinetic energy and an exponential potential. The details emphasize that this simple exponential form is what makes the subsequent analysis completely integrable in the flat FLRW ansatz and in minisuperspace quantum cosmology (Socorro et al., 18 Jun 2026).

In the flat FLRW gauge $V(\phi)=V_0\phi^{-\lambda}$2, $V(\phi)=V_0\phi^{-\lambda}$3, the Einstein–Klein-Gordon system is

$V(\phi)=V_0\phi^{-\lambda}$4

$V(\phi)=V_0\phi^{-\lambda}$5

$V(\phi)=V_0\phi^{-\lambda}$6

For the special case $V(\phi)=V_0\phi^{-\lambda}$7 and $V(\phi)=V_0\phi^{-\lambda}$8, the Hamilton–Jacobi or canonical method yields the closed-form solution

$V(\phi)=V_0\phi^{-\lambda}$9

$V(\phi)=V_0\phi^p$0

and therefore

$V(\phi)=V_0\phi^p$1

The deceleration parameter,

$V(\phi)=V_0\phi^p$2

approaches $V(\phi)=V_0\phi^p$3 as $V(\phi)=V_0\phi^p$4, because the exponential term dominates and gives $V(\phi)=V_0\phi^p$5 and $V(\phi)=V_0\phi^p$6. In the paper’s interpretation this is precisely a de Sitter attractor, and the exponential potential obtained after field redefinition is identified as the textbook example of a scaling or inflationary model (Socorro et al., 18 Jun 2026).

The same construction is carried into quantum cosmology. With the lapse choice $V(\phi)=V_0\phi^p$7, the minisuperspace Hamiltonian is

$V(\phi)=V_0\phi^p$8

Upon quantization, one obtains the Wheeler–DeWitt equation with Hartle–Hawking factor ordering parameter $V(\phi)=V_0\phi^p$9,

$p$0

For $p$1 and $p$2, the separated normal-mode solution is

$p$3

while for general $p$4 the wavefunctions are expressed in terms of modified Bessel functions $p$5 when $p$6 or ordinary Bessel functions $p$7 when $p$8. The paper states that, in this formalism, the scalar field remains as a cosmic background where the universe unfolds, a point that is glimpsed from the quantum solution perspective (Socorro et al., 18 Jun 2026).

3. Dynamical origin of fractional exponents in strongly coupled gauge theories

A second major line of research treats the fractional exponent as a dynamical output of strongly coupled supersymmetric gauge theory rather than as an arbitrary model parameter. In the $p$9 construction of chaotic inflation, integrating out the strongly coupled sector in its $V(\phi)=V_0\phi^{-\lambda}.$0-confining regime yields the effective superpotential

$V(\phi)=V_0\phi^{-\lambda}.$1

where $V(\phi)=V_0\phi^{-\lambda}.$2 is the dynamical scale and $V(\phi)=V_0\phi^{-\lambda}.$3 is the inflaton multiplet. Supersymmetry breaking by $V(\phi)=V_0\phi^{-\lambda}.$4 then gives

$V(\phi)=V_0\phi^{-\lambda}.$5

Identifying the inflaton with the canonically normalized imaginary part of $V(\phi)=V_0\phi^{-\lambda}.$6,

$V(\phi)=V_0\phi^{-\lambda}.$7

one obtains the familiar fractional-power form

$V(\phi)=V_0\phi^{-\lambda}.$8

with $V(\phi)=V_0\phi^{-\lambda}.$9 for $S(r)$0 (Harigaya et al., 2012).

In this framework the exponent traces directly back to the solution of the quantum-deformed moduli constraint. The same analysis states that the amplitude of the scalar power spectrum fixes the dynamical scale to a typical value

$S(r)$1

so that the inflationary energy $S(r)$2 lies hierarchically below the Planck scale. For the monomial $S(r)$3, the slow-roll parameters are

$S(r)$4

the e-fold estimate is $S(r)$5, and the standard predictions become

$S(r)$6

For $S(r)$7 and $S(r)$8, the paper quotes $S(r)$9–$V_s(r)=\sum_{p=1}^{2N-1}\frac{a_p}{r^{p/N}}.$0 and $V_s(r)=\sum_{p=1}^{2N-1}\frac{a_p}{r^{p/N}}.$1–$V_s(r)=\sum_{p=1}^{2N-1}\frac{a_p}{r^{p/N}}.$2 (Harigaya et al., 2012).

The later dynamical chaotic inflation program generalizes this mechanism substantially. Starting from an $V_s(r)=\sum_{p=1}^{2N-1}\frac{a_p}{r^{p/N}}.$3-confining supersymmetric gauge theory that is deformed at large inflaton field values into a dynamical supersymmetry-breaking model, one matches the one-loop running above and below the heavy-quark threshold $V_s(r)=\sum_{p=1}^{2N-1}\frac{a_p}{r^{p/N}}.$4. The effective strong scale becomes

$V_s(r)=\sum_{p=1}^{2N-1}\frac{a_p}{r^{p/N}}.$5

and the vacuum energy in the DSB phase is

$V_s(r)=\sum_{p=1}^{2N-1}\frac{a_p}{r^{p/N}}.$6

This makes the exponent calculable from beta-function coefficients. The details list explicit cases: minimal $V_s(r)=\sum_{p=1}^{2N-1}\frac{a_p}{r^{p/N}}.$7 gives $V_s(r)=\sum_{p=1}^{2N-1}\frac{a_p}{r^{p/N}}.$8; $V_s(r)=\sum_{p=1}^{2N-1}\frac{a_p}{r^{p/N}}.$9 gives $\Phi(r)$0 in the minimal model; $\Phi(r)$1 gives $\Phi(r)$2; the $\Phi(r)$3 model gives $\Phi(r)$4; and $\Phi(r)$5 with $\Phi(r)$6 gives $\Phi(r)$7 in the minimal case (Harigaya et al., 2014).

The same program also formulates a set of self-consistency constraints. Among them are the planckian field-value bound $\Phi(r)$8, the vacuum-energy bound $\Phi(r)$9, the perturbative decoupling condition $\Phi(r)\propto r^{-(2-\alpha)}$0, and the approximate shift symmetry $\Phi(r)\propto r^{-(2-\alpha)}$1 used to remove the supergravity $\Phi(r)\propto r^{-(2-\alpha)}$2-problem. A central significance of this literature is therefore methodological: fractional exponents can be UV-motivated, tied to holomorphy and dimensional transmutation, and embedded in explicit gauge sectors rather than postulated as isolated low-energy monomials (Harigaya et al., 2014).

4. Gauss–Bonnet-coupled fractional potentials, CMB observables, and primordial black holes

Fractional-power scalar potentials also play a central role in inflationary models coupled to the Gauss–Bonnet invariant. In one formulation, the action is

$\Phi(r)\propto r^{-(2-\alpha)}$3

with

$\Phi(r)\propto r^{-(2-\alpha)}$4

and $\Phi(r)\propto r^{-(2-\alpha)}$5. The exact FLRW background equations imply that a transient ultra-slow-roll phase appears when the balance condition

$\Phi(r)\propto r^{-(2-\alpha)}$6

is satisfied. Around this point, $\Phi(r)\propto r^{-(2-\alpha)}$7 while $\Phi(r)\propto r^{-(2-\alpha)}$8 becomes large, and the curvature perturbation spectrum can grow by several orders of magnitude, up to $\Phi(r)\propto r^{-(2-\alpha)}$9, which the paper identifies as sufficient to seed primordial black holes (Ashrafzadeh et al., 2023).

The same analysis gives explicit CMB-scale predictions. For the three fractional exponents it reports

$p$00

all of which can lie within the Planck 2018 confidence contours once $p$01 is adjusted. On small scales, the model is used to engineer three separate peaks in $p$02, giving asteroid-mass primordial black holes with $p$03 and $p$04, Earth-mass primordial black holes with $p$05 and $p$06, and stellar-mass primordial black holes with $p$07 and $p$08. The corresponding secondary gravitational waves are described by a present-day spectrum with peaks in the LISA/BBO/DECIGO and PTA/SKA bands, with $p$09–$p$10 and a broken power-law shape (Ashrafzadeh et al., 2023).

A later Einstein–Gauss–Bonnet treatment places the same class of fractional-power potentials against ACT constraints. There the action is

$p$11

with

$p$12

and either

$p$13

In slow roll, the paper introduces

$p$14

together with the standard EGB hierarchies $p$15 and $p$16, and evaluates

$p$17

Using the ACT DR6 constraints

$p$18

the paper reports that for the hyperbolic coupling, $p$19 and $p$20 with $p$21 lie well inside the $p$22 ACT contour for choices such as $p$23, that $p$24 remains within $p$25 but moves toward the boundary, and that $p$26 is excluded at $p$27 (Zahoor et al., 24 Jul 2025).

The same work extends the discussion to reheating. For

$p$28

it gives

$p$29

for $p$30. Since all cases satisfy $p$31, the paper concludes that the model can meet the lower bound from Big Bang nucleosynthesis while remaining compatible with ACT-favored $p$32 values (Zahoor et al., 24 Jul 2025). This suggests that the observational status of fractional monomials is highly model-dependent: in pure Einstein gravity some exponents may be disfavored, whereas EGB couplings can reopen viable parameter space.

5. Fractional-power scalar potentials in the Dirac equation

In the relativistic single-particle problem, the relevant object is a central scalar interaction $p$33 entering the Dirac Hamiltonian

$p$34

and the analysis of Agboola and Zhang specializes to equally mixed interactions,

$p$35

The fractional-power family is

$p$36

with $p$37, $p$38, rational exponents in $p$39, and $p$40. After separation into radial variables, the upper component satisfies a Schrödinger-like equation,

$p$41

where

$p$42

This establishes a direct route from fractional singular powers to exactly solvable bound-state problems (Agboola et al., 2013).

For $p$43, corresponding to the “square-root” family

$p$44

the exact bound-state energy equation is

$p$45

For $p$46, the “third-root” family

$p$47

gives

$p$48

In both cases the energies are determined implicitly in closed form, and for small $p$49 can be inverted to radicals according to the paper’s tables (Agboola et al., 2013).

The radial wave functions are finite polynomials times known exponentials. For $p$50, with $p$51,

$p$52

and for $p$53, with $p$54,

$p$55

The lower component follows from

$p$56

Normalizability requires signs of the $p$57 such that $p$58 as $p$59 and $p$60 decays to zero as $p$61, together with real positive Bethe-ansatz roots and additional algebraic constraints on the highest-order couplings (Agboola et al., 2013).

The nonrelativistic limit produces an ordinary Schrödinger Hamiltonian with the same fractional powers. For $p$62, defining $p$63, the paper gives

$p$64

and then derives $p$65, $p$66, and $p$67 by the Hellmann–Feynman theorem. The significance is that fractional-power scalar potentials are not restricted to inflationary model building; they also furnish analytically tractable singular interactions in relativistic spectral theory (Agboola et al., 2013).

6. Fractional operators, nonlocal media, and deformations of the Coulomb potential

A further extension arises when the fractional structure is moved from the algebraic form of the potential to the operators that generate it. In media with power-law spatial dispersion, the longitudinal permittivity derived from a fractional Liouville equation is

$p$68

with Debye radius

$p$69

Combining this with electrostatic constitutive relations yields the fractional Poisson equation

$p$70

For a point charge, the solution is

$p$71

where

$p$72

As $p$73, one recovers the Coulomb form $p$74; as $p$75, the decay becomes $p$76, which the paper interprets as strong nonlocal screening (Tarasov, 2013).

A related but distinct analysis computes the Riemann–Liouville fractional integral and derivative of the Newtonian or Coulomb potential $p$77 for all real orders $p$78. Along the $p$79 direction, with lower limit at $p$80, both the fractional integral for $p$81 and the fractional derivative for $p$82 take the same unified form by analytic continuation,

$p$83

The paper emphasizes that the identical functional form for $p$84 and $p$85 follows from analytic continuation of Beta and Gamma functions. It interprets the fractional integral as a potential generated by a fractional mass or charge distribution smeared along the $p$86 axis, and the fractional derivative as a nonlocal fractional gradient that modifies the inverse-square law and adds an infinite tower of Legendre-multipole corrections (Malkawi, 2015).

These operator-based constructions clarify an important conceptual point. Fractional-power behavior need not originate from a literal monomial $p$87 or $p$88. It can instead emerge from nonlocal constitutive laws or fractional differentiation acting on a conventional $p$89 kernel. A plausible implication is that “fractional power scalar potential” is best understood as a broader analytic class characterized by non-integer scaling, rather than by a single mechanism or field-theoretic interpretation.

7. Conceptual themes and recurrent points of interpretation

Across the literature surveyed here, three themes recur. First, fractional exponents are often structurally generated rather than arbitrarily imposed. In strongly coupled supersymmetric gauge theories, $p$90 or more general rational values follow from dimensional transmutation and threshold matching; in the Sáez–Ballester–K-essence model, an inverse power in $p$91 is mathematically equivalent to an exponential in $p$92 after $p$93 (Harigaya et al., 2012, Harigaya et al., 2014, Socorro et al., 18 Jun 2026).

Second, fractional powers frequently improve analytic control. The inverse-power Sáez–Ballester potential becomes exactly solvable at both the classical and Wheeler–DeWitt levels because the field redefinition produces an exponential potential; the Dirac equation with singular fractional radial couplings admits closed-form energy equations and explicit radial wave functions; and operator-based nonlocal models produce explicit Green functions and Legendre expansions (Socorro et al., 18 Jun 2026, Agboola et al., 2013, Tarasov, 2013, Malkawi, 2015).

Third, observational and physical interpretation depends strongly on the framework. In pure monomial inflation, the predictions are set directly by $p$94 through $p$95 and $p$96; with Gauss–Bonnet couplings, the same fractional potentials can enter ultra-slow-roll regimes, produce primordial black holes and secondary gravitational waves, or satisfy ACT $p$97 constraints in parameter regions that do not exist without the extra curvature coupling (Harigaya et al., 2012, Ashrafzadeh et al., 2023, Zahoor et al., 24 Jul 2025). The literature therefore does not support a single universal verdict on fractional-power scalar potentials. Instead, it presents a family of mathematically related but physically diverse constructions whose properties are controlled by the surrounding kinetic structure, gauge origin, gravitational sector, or nonlocal operator content.