Vacuum-induced current density from a magnetic flux threading a cosmic dispiration in $(D+1)$-dimensional spacetime

Abstract: We investigate the vacuum-induced current density for a charged scalar field in a $(D+1)$-dimensional cosmic dispiration spacetime threaded by a magnetic flux. This background combines a cosmic string and a screw dislocation, yielding a nontrivial helical geometry. By constructing the normalized mode functions of the Klein--Gordon equation, we evaluate the Wightman function and obtain the vacuum expectation value of the current density. We show that, in addition to the azimuthal component describing a persistent current around the defect, a nonvanishing axial component is induced as a direct consequence of the helical structure of the spacetime. Both components are periodic functions of the magnetic flux, depending only on its fractional part, reflecting the Aharonov--Bohm nature of the effect. Closed expressions are obtained for both massive and massless fields in arbitrary dimensions. We demonstrate that the screw dislocation parameter plays a crucial role in the behavior of the induced currents, leading to the regularization of the axial component at the origin and controlling its magnitude. The asymptotic behavior of both components is analyzed in detail. Our results reduce to known expressions in the absence of the screw dislocation, providing a consistency check. In particular, we examine the physically relevant $(3+1)$-dimensional case, where numerical analysis reveals nontrivial features arising from the interplay between topology and gauge effects.

• The paper derives arbitrary-dimensional expressions for azimuthal and axial vacuum currents of a charged scalar around a flux-threaded cosmic dispiration, using mode sums, Wightman functions, and Poisson resummation.
• The screw-dislocation parameter induces a persistent axial current absent for cosmic strings, while also regularizing the azimuthal current at the defect core and controlling its power-law or exponential scaling.
• Both currents are periodic in the fractional magnetic flux, vanish at integer and half-integer flux values, decrease exponentially for large mass or separation, and recover cosmic-string results when the screw parameter vanishes.

Overview and physical setting

This paper computes the vacuum expectation value (VEV) of the current density for a charged scalar field propagating in a $(D+1)$-dimensional cosmic dispiration spacetime threaded by a magnetic flux along the defect core (2604.07648). The background combines two topological features: a conical deficit encoded by the parameter $q$ (with $q^{-1} = 1 - 4G\mu_0$ for a string of linear mass density $\mu_0$), and a screw dislocation encoded by $\kappa$, which couples the angular coordinate to the longitudinal one through the line element $$ds^2 = dt^2 - dr^2 - r^2 d\phi^2 - (dz + \kappa\, d\phi)^2 - \sum_i (dx^i)^2$$. The helical pitch is $p = 2\pi\kappa/q$, and the geometry can equivalently be described as a locally flat spacetime with the identification $(r,\phi,Z) \sim (r, \phi + 2\pi/q, Z + p)$.

The motivation is twofold. First, VEVs of the four-current act as source terms in Maxwell's equations in semiclassical gravity/electrodynamics, so they quantify how vacuum polarization responds to topology and gauge potentials. Second, induced currents are sensitive probes of Aharonov–Bohm-type physics: here the Bessel order of the radial modes is shifted as $n \to q(n - h + \alpha)$, where $h = \kappa\nu/q$ and $q$0, meaning that both the geometric parameters and the gauge flux modify the effective angular momentum quantum number even where the field strength vanishes.

Mode functions and Wightman function

The Klein–Gordon equation is solved by separation of variables, exploiting the conserved quantities associated with $q$1, $q$2, $q$3, and the extra dimensions. Regularity at the origin discards the Neumann contribution (whose order depends on the continuous quantum number $q$4 and is non-square-integrable), leaving normalized modes proportional to $q$5 with $q$6 and normalization $q$7. The analysis is restricted to minimal curvature coupling ($q$8); this is a deliberate simplification, since the idealized zero-core defect yields a delta-function Ricci scalar at $q$9 that would otherwise introduce localized contact terms.

The positive-frequency Wightman function is obtained via Wick rotation and a Schwinger-type integral representation, reducing the mode sum to a single integral over $q^{-1} = 1 - 4G\mu_0$0 involving modified Bessel functions $q^{-1} = 1 - 4G\mu_0$1 and an auxiliary function $q^{-1} = 1 - 4G\mu_0$2 containing the sum over $q^{-1} = 1 - 4G\mu_0$3 and an integral over $q^{-1} = 1 - 4G\mu_0$4. The key technical step, carried out in appendices, is the evaluation of these structures using the Poisson summation formula together with the integral representation of $q^{-1} = 1 - 4G\mu_0$5, which converts the mode sums into rapidly convergent expressions built from Macdonald functions $q^{-1} = 1 - 4G\mu_0$6.

Azimuthal current

The azimuthal component $q^{-1} = 1 - 4G\mu_0$7 describes a persistent vacuum current circulating around the defect. For a massive scalar it takes the closed form

$q^{-1} = 1 - 4G\mu_0$8

where $q^{-1} = 1 - 4G\mu_0$9, $\mu_0$0, $\mu_0$1, and $\mu_0$2. Several structural properties follow directly:

• Flux periodicity: the result depends only on the fractional part $\mu_0$3 of $\mu_0$4, confirming the Aharonov–Bohm character; the currents vanish identically at $\mu_0$5 and $\mu_0$6.
• Role of $\mu_0$7: for $\mu_0$8 the discrete sum is absent; at $\mu_0$9 only the screw-dislocation contribution survives.
• Regularization at the core: unlike the pure cosmic-string case, for $\kappa$0 the azimuthal current remains finite at $\kappa$1, with massless behavior $\kappa$2. The paper notes explicitly that a divergence does arise when both $\kappa$3 and $\kappa$4 vanish simultaneously.
• Asymptotics: for large $\kappa$5 or $\kappa$6 the current decays exponentially via $\kappa$7; the massless limit exhibits long-range power-law falloff.

Setting $\kappa$8 recovers the known cosmic-string results of Bragança, Mota, and Bezerra de Mello, providing a consistency check; the appendix demonstrates that the series $\kappa$9 admits a closed hyperbolic form precisely in this limit.

For the physically relevant case $$ds^2 = dt^2 - dr^2 - r^2 d\phi^2 - (dz + \kappa\, d\phi)^2 - \sum_i (dx^i)^2$$0, numerical plots show the sinusoidal dependence on $$ds^2 = dt^2 - dr^2 - r^2 d\phi^2 - (dz + \kappa\, d\phi)^2 - \sum_i (dx^i)^2$$1 and a systematic suppression of the current magnitude as $$ds^2 = dt^2 - dr^2 - r^2 d\phi^2 - (dz + \kappa\, d\phi)^2 - \sum_i (dx^i)^2$$2 increases — the screw dislocation reduces the amplitude relative to the pure string configuration.

Axial current

The central new result is a nonvanishing axial component $$ds^2 = dt^2 - dr^2 - r^2 d\phi^2 - (dz + \kappa\, d\phi)^2 - \sum_i (dx^i)^2$$3, which has no analogue in the pure cosmic-string problem and arises entirely from the helical mixing of $$ds^2 = dt^2 - dr^2 - r^2 d\phi^2 - (dz + \kappa\, d\phi)^2 - \sum_i (dx^i)^2$$4 and $$ds^2 = dt^2 - dr^2 - r^2 d\phi^2 - (dz + \kappa\, d\phi)^2 - \sum_i (dx^i)^2$$5. Its structure parallels the azimuthal expression but with weight factors $$ds^2 = dt^2 - dr^2 - r^2 d\phi^2 - (dz + \kappa\, d\phi)^2 - \sum_i (dx^i)^2$$6 instead of $$ds^2 = dt^2 - dr^2 - r^2 d\phi^2 - (dz + \kappa\, d\phi)^2 - \sum_i (dx^i)^2$$7 and with the coefficient function $$ds^2 = dt^2 - dr^2 - r^2 d\phi^2 - (dz + \kappa\, d\phi)^2 - \sum_i (dx^i)^2$$8:

$$ds^2 = dt^2 - dr^2 - r^2 d\phi^2 - (dz + \kappa\, d\phi)^2 - \sum_i (dx^i)^2$$9

Key properties: the axial current vanishes identically at $p = 2\pi\kappa/q$0 (the coordinates decouple), vanishes at $p = 2\pi\kappa/q$1 and $p = 2\pi\kappa/q$2, and — notably — remains finite at the origin with no divergence even when both $p = 2\pi\kappa/q$3 and $p = 2\pi\kappa/q$4 vanish, in contrast to the azimuthal component. In the massless case at $p = 2\pi\kappa/q$5 it scales as $p = 2\pi\kappa/q$6, i.e., more slowly than the azimuthal component's $p = 2\pi\kappa/q$7. For small $p = 2\pi\kappa/q$8 the current grows linearly in $p = 2\pi\kappa/q$9, implying an intermediate regime where the axial current is enhanced as $(r,\phi,Z) \sim (r, \phi + 2\pi/q, Z + p)$0 departs from zero before being exponentially suppressed at large $(r,\phi,Z) \sim (r, \phi + 2\pi/q, Z + p)$1; this non-monotonic behavior is visible in the $(r,\phi,Z) \sim (r, \phi + 2\pi/q, Z + p)$2 numerics.

Limitations and open questions

The paper concedes several restrictions on its scope. The defect core is treated as idealized (zero thickness), so curvature is concentrated in a delta distribution at $(r,\phi,Z) \sim (r, \phi + 2\pi/q, Z + p)$3; only minimal coupling is considered, leaving the effect of a non-minimal $(r,\phi,Z) \sim (r, \phi + 2\pi/q, Z + p)$4 term unexamined. The sums over $(r,\phi,Z) \sim (r, \phi + 2\pi/q, Z + p)$5 in the general ($(r,\phi,Z) \sim (r, \phi + 2\pi/q, Z + p)$6) expressions do not admit closed analytic forms and must be evaluated numerically. Backreaction of the induced currents on the gauge field is not treated, although the author identifies it, along with fermionic fields, finite-temperature effects, and non-minimal coupling, as concrete open directions. Whether the regularization of the core currents by $(r,\phi,Z) \sim (r, \phi + 2\pi/q, Z + p)$7 survives a finite-core model remains unaddressed.

Conclusion

The paper derives closed-form expressions for the vacuum-induced current density of a charged scalar field in a $(r,\phi,Z) \sim (r, \phi + 2\pi/q, Z + p)$8-dimensional cosmic dispiration threaded by magnetic flux, valid for arbitrary dimension and for both massive and massless fields. Both components are periodic in the fractional flux $(r,\phi,Z) \sim (r, \phi + 2\pi/q, Z + p)$9, exhibit exponential suppression at large $n \to q(n - h + \alpha)$0 or $n \to q(n - h + \alpha)$1, and reduce correctly to known cosmic-string results when $n \to q(n - h + \alpha)$2. The qualitative advance is the demonstration that helical torsion induces a persistent axial vacuum current, regularizes the current density at the defect core, and controls its magnitude — establishing the screw-dislocation scale $n \to q(n - h + \alpha)$3 as the governing length scale of vacuum fluctuations near the defect.