On the ultraviolet behavior of the invariant charge in quantum electrodynamics

This paper investigates the ultraviolet behavior of the invariant charge in quantum electrodynamics, demonstrating that a modified real-part definition avoids the Landau pole singularity and utilizing $1/N$ perturbation theory in an imaginary-charge model to derive specific asymptotic corrections that likely extend to standard and other non-asymptotically free QED models.

The Gist

The Big Problem: The "Speed Limit" That Doesn't Exist

Imagine you are driving a car (representing an electron) on a highway (representing the universe). In the world of Quantum Electrodynamics (QED)—the physics of light and electricity—there is a rule about how fast you can interact with other cars. This rule is called the invariant charge (or effective coupling).

In our current understanding of this theory, as you drive faster and faster toward the "speed of light" (or in physics terms, as you look at smaller and smaller distances), the interaction between cars gets stronger and stronger.

Here is the scary part: According to the old math, if you go fast enough, the interaction strength hits a "wall" called the Landau Pole.

• The Analogy: Imagine your speedometer doesn't just go to 200 mph; it keeps going up until it hits a number so high that the needle snaps off, the engine explodes, and the car ceases to exist.
• The Physics: This "explosion" is a mathematical singularity. It suggests that our theory of electricity breaks down completely at very high energies, meaning the theory might not be a true description of reality.

The Paper's Solution: Taking a Detour Through "Imaginary" Space

The author, N.V. Krasnikov, proposes a clever way to fix this broken speedometer without throwing away the whole car. He suggests we stop looking at the road straight ahead and instead look at it from a weird angle.

1. The "Complex" Detour

In math, numbers can be "real" (like 1, 2, 3) or "complex" (involving imaginary numbers, like $i$).

• The Analogy: Imagine the road is a flat map. If you drive straight, you hit a cliff (the Landau Pole). But if you take a detour into a parallel dimension (complex momenta), the cliff disappears. The road just curves smoothly.
• The Result: The author shows that if you calculate the interaction strength using these "complex" numbers, the singularity vanishes. The "speedometer" never breaks; it just behaves differently.

2. The "Real Part" Compromise

Since we live in a "real" world, we can't drive in imaginary dimensions. So, the author suggests a new rule: Take the "Real Part" of the complex result.

• The Analogy: Imagine you are looking at a reflection in a funhouse mirror. The reflection is distorted (complex), but if you take the average of the reflection and the real object, you get a new, stable image.
• The Result: This "New Invariant Charge" is capped. It has a maximum limit. It never explodes. It behaves nicely, even at the highest speeds.

The "Imaginary Charge" Experiment

To prove this works, the author uses a thought experiment involving a "fake" version of the universe.

• The Analogy: Imagine a video game where gravity works backwards. It's not a real world, and you can't live there, but it's a great place to test physics engines.
• The Physics: The author studies a version of QED where the electric charge is "imaginary" (mathematically negative). In this fake world, the theory is "Asymptotically Free." This means that at high speeds, the particles actually stop interacting so much that they become free and easy to calculate.
• The Discovery: The author calculates exactly how the interaction behaves in this "fake" world. Then, he shows that the math for this fake world is almost identical to our real world, just with a different label.
• The Conclusion: Because the "fake" world behaves perfectly at high speeds, our real world likely behaves the same way. The "explosion" (Landau Pole) is just an artifact of using the wrong math tools, not a real physical problem.

The "Large Crowd" Trick ($1/N$ Expansion)

The paper also uses a method called the $1/N$ expansion.

• The Analogy: Imagine trying to predict the behavior of a single person in a crowd. It's chaotic and hard. But if you have a stadium full of 1,000,000 people, you can predict the crowd's behavior by looking at the average. The "noise" of individuals cancels out.
• The Physics: The author imagines a universe with a huge number ($N$) of different types of electrons. By doing the math for this huge crowd, the messy, infinite problems disappear. The results show that the interaction strength gets smaller and smaller at high energies, confirming that the theory is stable.

The Bottom Line

The paper argues that the famous "Landau Pole" (the idea that QED breaks down at high energies) is likely a mathematical illusion caused by looking at the problem the wrong way.

1. Complex Momenta: If you look at the math from a "complex" angle, the singularity disappears.
2. New Definition: By defining the charge as the "real part" of this complex result, we get a theory that is safe, stable, and has a maximum limit.
3. Imaginary Models: By testing a "fake" version of the theory where things are easier to calculate, we see that the real theory probably behaves just as nicely.

In short: The universe doesn't explode at high speeds. We just need to use a slightly different map to see that the road goes on forever without hitting a wall.

Technical Summary

1. Problem Statement

The paper addresses the long-standing issue of the Landau pole singularity in Quantum Electrodynamics (QED).

• The Context: In standard QED, the effective coupling constant (invariant charge, $\bar{\alpha}$) increases with energy (momentum $p^2$). Perturbation theory predicts that at a specific finite energy scale ($p^2 = \Lambda^2$), the coupling diverges to infinity. This "Landau pole" suggests that QED may not be a self-consistent local quantum field theory at short distances (ultraviolet regime).
• The Goal: The author investigates the ultraviolet (UV) behavior of the invariant charge to determine if the Landau pole is an artifact of real-momentum perturbation theory or a fundamental feature, and to explore alternative definitions of the charge that remain well-behaved in the UV.

2. Methodology

The author employs three primary theoretical approaches:

1. Analytic Continuation to Complex Momenta: Instead of restricting analysis to real, positive $p^2$, the invariant charge is evaluated at complex momenta ($p^2 = e^{i\phi}|p^2|$). This allows the investigation of the singularity structure in the complex plane.
2. $1/N$ Expansion: The paper utilizes the large-$N$ expansion (where $N$ is the number of identical massless fermion flavors). This technique sums specific classes of Feynman diagrams (bubble chains) to all orders, providing a non-perturbative handle on the UV behavior.
3. Imaginary Charge Model (Exotic QED): The author analyzes a non-physical model where the electric charge is imaginary ($e' = ie$). In this model, the coupling constant $\alpha' < 0$, rendering the theory asymptotically free. The results from this asymptotically free model are then mapped back to standard QED, based on the observation that the $1/N$ expansion structure is identical for both, differing only by the scale parameter $\Lambda$.

3. Key Contributions and Results

A. Absence of Landau Pole at Complex Momenta

• One-Loop Analysis: The standard one-loop solution for the invariant charge is $\bar{\alpha} \sim [1 - \alpha\beta_2 \ln(p^2/\mu^2)]^{-1}$. For real $p^2$, this diverges at $p^2 = \Lambda^2$.
• Complex Extension: When $p^2$ is complex ($p^2 = e^{i\phi}|p^2|$), the denominator becomes complex. The author demonstrates that the singularity is avoided for any $\phi \neq 0$.
• Real Part Definition: The author proposes defining a new invariant charge as the real part of the standard charge evaluated at complex momenta (specifically the time-like region where $\phi = \pi$, corresponding to $p^2 < 0$).
  • This new charge, $\bar{\alpha}_{Re}$, is bounded from above.
  • It does not possess a Landau pole singularity.
  • In the time-like region, it exhibits a maximum/minimum rather than a divergence.

B. $1/N$ Expansion and UV Asymptotics

The paper derives the precise UV asymptotic behavior of the $(1/N)^k$ corrections to the invariant charge $\bar{\alpha}_k(p^2, \alpha)$:

• For $k > 1$: The corrections decay as $\bar{\alpha}_k \sim (\ln(p^2/\mu^2))^{-k-1}$.
• For $k = 1$: The first correction behaves as $\bar{\alpha}_1 \sim \frac{\ln(\ln(p^2/\mu^2))}{\ln^2(p^2/\mu^2)}$.
• Leading Log Approximation: The total invariant charge in the UV limit is dominated by the leading log term: $\bar{\alpha} \sim (\ln(p^2/\mu^2))^{-1}$. This implies that the charge vanishes as $p^2 \to \infty$, suggesting the theory becomes weakly coupled in the deep UV if the sign of the charge flips (a necessary condition to avoid the contradiction of a positive, increasing charge).

C. Modified $1/N$ Perturbation Theory

• The author identifies that standard $1/N$ corrections involving fermion bubbles inside loops can be UV divergent.
• A modified effective propagator is proposed:
  $$D_{eff}^{imp}(p^2, \Lambda^2) = \frac{1}{p^2(-\beta_2 \ln(p^2/\Lambda^2) + \frac{1}{N}\Pi_2(p^2/\Lambda^2))}$$
• This modification incorporates the divergent $1/N$ subdiagrams into the propagator definition, rendering the resulting perturbation theory ultraviolet finite.

D. The Imaginary Charge Analogy

• By studying QED with imaginary charge (which is asymptotically free), the author calculates reliable UV asymptotics.
• Key Insight: The $1/N$ expansion structure is identical for standard QED (real charge) and Exotic QED (imaginary charge), differing only by the scale parameter ($\Lambda$ vs. $\Lambda_f$).
• Implication: The UV behavior derived for the asymptotically free imaginary model applies to standard QED. This supports the hypothesis that in other non-asymptotically free models (e.g., Scalar QED, Supersymmetric QED, Wess-Zumino model), the invariant charge's UV asymptotics coincides with the leading log approximation.

4. Significance and Conclusion

• Resolution of the Singularity: The paper suggests that the Landau pole is an artifact of restricting the analysis to real momenta. By considering complex momenta or redefining the charge as the real part of the complex-extended charge, the singularity disappears, and the charge remains finite and bounded.
• Self-Consistency: The results imply that QED might be self-consistent in the UV if the invariant charge changes sign at high energies (becoming negative), allowing it to approach zero asymptotically.
• Generalizability: The methodology provides a framework for analyzing the UV behavior of other non-asymptotically free theories, suggesting that their high-energy behavior is governed by leading log approximations similar to the modified QED analysis.
• Technical Advancement: The proposal of a modified, UV-finite $1/N$ perturbation theory offers a robust calculational tool for studying the photon propagator and invariant charge beyond standard perturbation theory.

In summary, Krasnikov argues that the "triviality" or inconsistency of QED at short distances is not a proven fact but a consequence of specific approximations. By utilizing complex momenta and $1/N$ expansions, he demonstrates a pathway where the invariant charge remains well-behaved and finite in the ultraviolet limit.