Coulomb Potential in Podolsky-Carroll-Field-Jackiw Electrodynamics

This paper investigates the Coulomb potential within a combined Podolsky and Carroll-Field-Jackiw electrodynamics framework, demonstrating that while the Podolsky term regularizes short-distance divergences, the Lorentz-violating CFJ background can reintroduce these divergences and significantly alter the interaction potential through both spatial anisotropy and timelike dispersion effects.

The Gist

Imagine the universe as a giant, invisible web of forces. For over a century, physicists have used a set of rules called Maxwell's Electromagnetism to describe how electric charges (like electrons) talk to each other. It's a brilliant theory, but it has a tiny, annoying glitch: if you try to calculate the force between two electrons when they are exactly on top of each other, the math explodes. The force becomes infinite. It's like trying to divide by zero.

This paper is about two different "patches" scientists have tried to fix this glitch, and what happens when they try to wear both patches at the same time.

The Two Patches

1. The Podolsky Patch (The "Softener")
Think of an electron as a point of light. In the old theory, this point is infinitely small and infinitely sharp. Podolsky Electrodynamics suggests that maybe, just maybe, the electron isn't a sharp point, but a tiny, fuzzy cloud.

• The Analogy: Imagine trying to poke someone with a needle (the old theory). It hurts a lot if you get too close. Podolsky says, "Let's replace the needle with a soft, fuzzy ball."
• The Result: When two fuzzy balls get close, they don't hit a sharp point; they squish against each other. The force stops growing infinitely and stays finite. It "regularizes" the math, making the universe safe at very small distances.

2. The CFJ Patch (The "Compass")
Now, imagine the universe isn't just a blank canvas; imagine it has a hidden wind blowing in a specific direction. This is the Carroll-Field-Jackiw (CFJ) model. It suggests that the laws of physics might look slightly different depending on which way you are facing. This breaks a fundamental rule called "Lorentz symmetry" (the idea that physics is the same everywhere and in every direction).

• The Analogy: Imagine you are walking in a forest. In a normal forest, the trees look the same no matter which way you turn. In a CFJ forest, there is a giant, invisible arrow pointing North. If you walk North, the trees feel different than if you walk East. The universe has a "preferred direction."

The Big Experiment: Mixing the Patches

The authors of this paper asked a fascinating question: What happens if we combine the "Fuzzy Ball" (Podolsky) with the "Invisible Wind" (CFJ)?

They did the math to see how two electrons would interact in this new, weird universe. Here is what they found, translated into everyday terms:

1. The "Fuzzy" Fix Gets Ruined
The Podolsky patch was great at stopping the infinite explosion at the center. It made the force finite. But, the CFJ "wind" was too strong.

• The Result: When the "wind" blows, it pushes the fuzzy balls back together so hard that the "softening" effect breaks. The infinite explosion returns! The math goes back to blowing up at the center, but this time, it's because the universe has a preferred direction. It's like having a soft pillow, but someone is standing on it with a heavy boot; the pillow can't do its job.

2. The "Compass" Changes the Shape of the Force
The interaction between the electrons isn't just a simple push or pull anymore. It depends on how the electrons are oriented relative to that invisible "wind."

• The Analogy: Imagine two magnets. Usually, they attract or repel based on how close they are. But in this new theory, if you rotate the magnets relative to the "wind," the strength of the pull changes. The force becomes "lopsided."

3. The Long-Distance Surprise
Usually, electric forces get weaker the farther apart you are (like a flashlight beam spreading out).

• The Result: In this combined theory, at very, very large distances, the force doesn't just fade away. It starts to grow linearly.
• The Analogy: Imagine a rubber band. Usually, if you let go, it snaps back. But here, the further you pull the electrons apart, the harder they pull back, eventually becoming a force that grows forever. It's as if the universe is trying to glue everything together over vast distances, but only if you are aligned with the "wind."

Why Does This Matter?

You might ask, "We haven't seen this 'wind' or 'fuzziness' yet, so why study it?"

1. Testing the Limits: Physics is built on the idea that the universe is symmetrical (looks the same everywhere). If we find even a tiny crack in that symmetry (the "wind"), it could be the key to unlocking New Physics—perhaps explaining gravity or the Big Bang.
2. The "What If" Game: By calculating exactly what would happen if these theories were true, scientists can design better experiments. If we build a detector and don't see this weird "linear growth" or "lopsided force," we can rule out these theories and narrow down what the universe actually is.

The Bottom Line

This paper is a theoretical "what-if" story. It says:

• Podolsky's theory tries to fix the math by making particles fuzzy.
• CFJ's theory tries to fix the universe by giving it a direction.
• Together, they create a chaotic mix where the "fuzziness" fails to hide the infinite explosion, and the force between particles starts growing wildly over long distances, depending on which way you face.

It's a reminder that while our current laws of physics work beautifully, the universe might have hidden layers of complexity that we are just beginning to imagine.

Technical Summary

1. Problem Statement

The paper addresses the intersection of two theoretical extensions to standard Maxwell electrodynamics:

• Podolsky Electrodynamics: A higher-derivative generalization of Maxwell's theory designed to regularize short-distance (ultraviolet) divergences. It introduces a parameter $\lambda = 1/m$ (where $m$ is an effective photon mass) that prevents the electric field of a point charge from diverging at the origin ($r \to 0$), yielding a finite potential.
• Carroll-Field-Jackiw (CFJ) Model: A Lorentz-violating extension of Maxwell's theory that introduces a background four-vector field ($v_\mu$). This term breaks Lorentz symmetry and parity while preserving gauge invariance, often used to model effects predicted by quantum gravity theories (e.g., String Theory) near the Planck scale.

The Core Question: How does the introduction of a Lorentz-violating background (CFJ term) affect the regularization of the Coulomb potential provided by the Podolsky term? Specifically, does the Lorentz violation reintroduce the short-distance singularities that Podolsky's theory was intended to remove?

2. Methodology

The authors employ a quantum field theoretical approach within the framework of the Standard Model Extension (SME).

• Lagrangian Formulation: They construct a total Lagrangian density combining the Podolsky higher-derivative term and the CFJ Lorentz-violating term:
  $$\mathcal{L} = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu} + \frac{\lambda^2}{2}\partial_\mu F^{\mu\nu}\partial^\alpha F_{\alpha\nu} + \frac{1}{4}\varepsilon_{\mu\nu\alpha\rho}v^\mu A^\nu F^{\alpha\rho} - \frac{1}{2\xi}(\partial_\mu A^\mu)^2 + \mathcal{L}_{\text{matter}}$$
• Photon Propagator Derivation:
  • The authors derive the equations of motion in momentum space.
  • They construct the inverse of the kinetic operator to obtain the generalized photon propagator $D_{\mu\nu}(k)$.
  • This involves solving a tensor equation using projectors ($\theta_{\mu\nu}$, $\omega_{\mu\nu}$) and the Lorentz-violating tensor $S_{\mu\nu}$.
  • The resulting propagator contains a complex denominator representing a modified dispersion relation involving both $\lambda$ and $v_\mu$.
• Møller Scattering Analysis:
  • The derived propagator is applied to calculate the scattering amplitude for electron-electron (Møller) scattering ($e^- e^- \to e^- e^-$).
  • The calculation focuses on the t-channel diagram, as the u-channel contribution is purely relativistic and does not contribute to the non-relativistic potential.
• Non-Relativistic Limit & Potential Calculation:
  • The transition amplitude is expanded in the non-relativistic limit ($p \ll m$).
  • The interaction potential $V(r)$ is obtained via the Fourier transform of the scattering amplitude.
  • The integration is performed using regularization techniques (splitting the integral into regions and using residue calculus) to handle apparent divergences arising from the Lorentz-violating terms.

3. Key Contributions

• Generalized Propagator: The paper derives, for the first time (according to the authors), the explicit form of the photon propagator in a theory combining Podolsky electrodynamics and the CFJ Lorentz-violating term. This propagator exhibits a non-trivial pole structure distinct from pure Podolsky or pure CFJ theories.
• Dispersion Relation: They establish the modified dispersion relation:
  $$k^4(1 - \lambda^2 k^2)^2 + k^2 v^2 - (k \cdot v)^2 = 0$$
  This relation shows how the energy spectrum is influenced by both the higher-derivative mass scale and the Lorentz-violating background.
• Reintroduction of Divergence: A critical theoretical finding is that while Podolsky electrodynamics alone regularizes the Coulomb singularity at $r=0$, the coupling with the CFJ term reintroduces a short-distance divergence if the background vector has a non-vanishing time-like component ($v_0 \neq 0$).

4. Results

The calculated Coulomb potential $V(r)$ in the non-relativistic limit reveals distinct behaviors in different regimes:

• Short-Distance Behavior ($r \to 0$):

  • Standard QED: $V(r) \sim 1/r$ (Divergent).
  • Pure Podolsky: $V(r) \sim \text{constant}$ (Finite/Regularized).
  • Podolsky-CFJ: The potential behaves as $V(r) \approx \frac{3e^2 |\vec{v}|^2 \vartheta^2 \lambda^2}{4\pi r} + \text{const}$.
  • Conclusion: The Lorentz-violating contribution introduces a new pole of the form $\sim 1/r$, effectively destroying the regularization provided by the Podolsky term if the time-like component of the background vector is non-zero.

• Long-Distance Behavior ($r \to \infty$):

  • The potential initially decays similarly to the Podolsky case but eventually transitions to a linearly growing potential:
    $$V(r) \approx \frac{e^2 |\vec{v}|^2 [5 - \cos(2\alpha)]}{64\pi} r$$
  • This linear growth implies a confining-like behavior at extremely large scales, dominated by the CFJ contribution.
  • The authors estimate that for a typical Lorentz-violating bound ($|\vec{v}| \sim 10^{-41}$ GeV), this linear growth becomes significant only at astronomical distances ($\sim 10^{62}$ light-years).

• Anisotropy: The potential depends on the angle $\alpha$ between the background vector $\vec{v}$ and the position vector $\vec{r}$, explicitly breaking spatial isotropy.

5. Significance

• Theoretical Consistency: The work highlights a tension between regularization mechanisms and Lorentz violation. It suggests that while higher-derivative terms can cure UV divergences, they may be fragile against Lorentz-violating backgrounds, which can restore singularities.
• Phenomenology: The results provide specific constraints and predictions for experimental searches. The reintroduction of the $1/r$ singularity implies that precision tests of the Coulomb law at very short distances could potentially constrain the time-like component of the CFJ vector.
• Cosmological Implications: The linear growth of the potential at large scales offers a theoretical mechanism for long-range forces that could be relevant in cosmological contexts, although the scale is currently far beyond observational reach.
• Methodological Advancement: The derivation of the combined propagator provides a necessary tool for future calculations in generalized electrodynamics involving both higher-derivative and Lorentz-violating terms (e.g., in Casimir effect calculations or thermal field theory).

In summary, the paper demonstrates that the "fix" for the Coulomb singularity provided by Podolsky electrodynamics is not robust; the presence of a Lorentz-violating background (specifically its time-like component) can undo this regularization, restoring the divergence at the origin while introducing novel long-range linear interactions.