Elastic scattering of electron by a Yukawa potential in non-commutative spacetime

Technical Summary: Elastic Scattering of Electrons by a Yukawa Potential in Non-Commutative Spacetime

Problem Statement
This study investigates the elastic scattering of electrons by a Yukawa potential within the framework of non-commutative (NC) geometry. While standard quantum mechanics treats spacetime coordinates as commuting variables, NC geometry posits a fundamental modification where coordinates satisfy a non-commutative algebra, $[x^\mu, x^\nu] = i\Theta^{\mu\nu}$. This approach serves as a promising avenue for exploring quantum gravity effects and the quantization of spacetime at small scales. The paper addresses the gap in understanding how such non-commutative corrections modify standard scattering potentials and cross-sections, specifically for the Yukawa potential, which models screened Coulomb interactions in atomic and nuclear physics.

Methodology
The authors employ a perturbative approach based on the first-order Born approximation. The core methodology involves the following steps:

1. Coordinate Transformation: The study utilizes Bopp's shift transformation to map the non-commutative coordinates $\hat{x}^\mu$ to commutative coordinates $x^\mu$. The shift is defined as $\hat{x}_\mu = x_\mu - \frac{1}{2}\Theta_{\mu\nu}P^\nu$, where $P^\nu$ is the momentum operator.
2. Potential Derivation: By applying this shift to the standard Yukawa potential $V(r) = V_0 \frac{e^{-\alpha r}}{r}$ and expanding to the leading order in the NC parameter $\Theta$, the authors derive a modified NC potential.
3. Scattering Amplitude Calculation: Using the derived NC potential, the scattering amplitude $\hat{f}(\theta)$ is calculated via the Born approximation integral.
4. Cross-Section Analysis: The differential ($d\hat{\sigma}/d\Omega$) and total ($\hat{\sigma}$) cross-sections are computed. The study analyzes these quantities across various regimes, including different values of the NC parameter $\Theta$, screening parameter $\alpha$, atomic number $Z$, and incident electron energy $E$.
5. Bound Estimation: The paper applies the theoretical results to electron scattering by specific molecules (Hydrogen $H_2$ and Scandium Hydride $ScH$) to estimate lower bounds on the NC parameter $\Theta$ across a wide range of incident energies (from eV to GeV).

Key Contributions and Results

• Modified Potential Structure: The primary theoretical finding is that the NC correction to the Yukawa potential, to leading order, results in a potential structurally identical to a screened Kratzer potential:
  $$\hat{V}(r) = \left( \frac{V_1}{r} + \frac{V_2}{r^2} \right) e^{-\alpha r}$$
  where $V_1$ and $V_2$ depend on the NC parameter $\Theta$ and the particle energy $E$. In the limit where the screening parameter $\alpha \to 0$, this reduces to the standard Kratzer potential, and further to the NC Coulomb potential. This suggests that Kratzer-type potentials may be geometric consequences of spacetime non-commutativity.
• Scattering Amplitude and Cross-Sections: The authors derived explicit expressions for the NC scattering amplitude and cross-sections. The results indicate that non-commutativity enhances the differential cross-section at small scattering angles.
• Energy Dependence and Bounds on $\Theta$: A significant result is the established relationship between the system's energy level and the bounds on the NC parameter.
  • For non-relativistic electrons scattering off an $H_2$ molecule, the lower bound on $\sqrt{\Theta}$ is estimated at the order of $10^{-11}$ m.
  • For ultra-relativistic electrons (up to 100 GeV) scattering off $H_2$, the bound tightens to $\sim 10^{-26}$ m.
  • For ultra-relativistic electrons scattering off a heavier molecule ($ScH$, $Z=22$), the study derives a new lower bound of $\sim 10^{-28}$ m.
• Comparison with Existing Bounds: The paper notes that these bounds, particularly the $10^{-28}$ m limit derived from heavy molecule scattering, represent a significant improvement over previous bounds derived from NC field theory ($10^{-17}$ to $10^{-20}$ m) and hydrogen atom spectroscopy ($10^{-15}$ to $10^{-22}$ m).

Significance and Claims
The paper claims that its work provides a direct theoretical link between the quantum nature of spacetime and observable scattering phenomena. By demonstrating that the NC correction naturally yields the screened Kratzer and Kratzer potentials, the authors argue that these well-known potentials in molecular physics may have a geometric origin rooted in non-commutative spacetime.

The authors interpret the non-commutativity of spacetime as a form of "tension" that responds to the presence of energy at the quantum level, analogous to how spacetime curvature responds to mass in general relativity. The study concludes that higher energy levels in scattering experiments allow the NC parameter to approach the Planck scale more closely, thereby offering a viable pathway for probing quantum gravity effects through subatomic scattering experiments. The derived bounds, particularly those involving heavy molecules at ultra-relativistic energies, are presented as robust constraints on the NC parameter within quantum systems.