Neutral and charged pion Form Factors in the intermediate-energy region from double-dilaton HQCD model

Using a double-dilaton Holographic QCD model with a non-perturbative running strong coupling, this study computes neutral and charged pion form factors in the intermediate-energy region to explain experimental deviations from asymptotic behavior and demonstrate that non-perturbative effects persist at higher energy scales than previously thought.

The Gist

Imagine the universe is built out of tiny, invisible Lego bricks called quarks. These bricks snap together to form larger structures called hadrons, the most famous of which are protons and neutrons. But there's a lighter, more fragile Lego structure called the pion.

Scientists have been trying to understand exactly how these pions hold together and how they react when hit by high-energy particles. To do this, they measure something called a "Form Factor." Think of a Form Factor as a fingerprint or a shadow that tells you the shape and internal structure of the pion.

For a long time, physicists believed they understood the rules of the game. They had two main rulebooks:

1. The Low-Energy Rulebook: For slow, gentle interactions, the rules are messy and complex (Non-perturbative).
2. The High-Energy Rulebook: For fast, violent collisions, the rules become simple and predictable (Perturbative QCD).

The scientists in this paper were looking at the "Intermediate Zone"—the middle ground between slow and fast. They expected the "High-Energy Rulebook" to take over here. But the experimental data (the actual fingerprints from the lab) didn't match the predictions. The shadows looked weird.

The New Tool: The "Double-Dilaton" Model

To fix this mismatch, the authors (Héctor and Pere) used a new, fancy tool called a Double-Dilaton Holographic QCD model.

Here is a simple analogy for what they did:
Imagine you are trying to predict how a rubber band stretches.

• Old Theory: You assume the rubber band is perfectly elastic and follows a straight line forever.
• The Problem: When you stretch it a little bit (the intermediate zone), the real rubber band behaves strangely. It doesn't follow the straight line yet, but it's not totally chaotic either.
• The New Model: The authors realized that the "glue" holding the quarks together (the strong force) doesn't just switch from "messy" to "simple" at a specific line. Instead, the strength of the glue changes smoothly, like a dimmer switch rather than a light switch.

They used a mathematical formula (the "running coupling constant") that describes this dimmer switch. It says: "Even at energies we thought were 'simple,' the glue is still acting a bit like the messy, complex glue from the low-energy world."

The Results: What They Found

1. The Neutral Pion (The "Ghost" Pion)
The neutral pion is like a ghost; it's hard to catch and study.

• The Puzzle: Experiments showed its shadow was getting bigger than the "High-Energy Rulebook" predicted.
• The Fix: The authors used their new "dimmer switch" glue. They found that the shadow does eventually settle down to the predicted size, but it takes much longer (higher energy) than we thought. The "messy" glue keeps influencing the pion longer than expected.
• The Analogy: It's like a car slowing down. We thought it would hit the speed limit at 60 mph. But this new model says, "No, it's still coasting at 65 mph because the engine is still revving a bit."

2. The Charged Pion (The "Heavy" Pion)
The charged pion is easier to study.

• The Puzzle: Its shadow also didn't match the simple rules.
• The Fix: Using the same "dimmer switch" glue, they created a hybrid model. They used a mathematical trick called a Padé Approximant (think of this as a bridge).
  • On one side of the bridge (low energy), they used the messy, real-world data.
  • On the other side (high energy), they used the simple theoretical rules.
  • The bridge connects them smoothly.
• The Result: The model showed that the charged pion's shadow actually has a "hump" (a maximum point) before it settles down. This explains why the data looked weird before. It wasn't wrong; we just didn't have the right bridge to connect the two sides.

3. The "Isospin" Difference (The Twin Test)
Finally, they looked at the difference between the neutral and charged pions. Think of them as twins who look almost identical but have a tiny difference in weight.

• By comparing the "shadows" (Form Factors) of these twins, they calculated the tiny difference in their mass.
• The Result: Their calculation matched the real-world measurements almost perfectly. This proved their "dimmer switch" glue model is accurate.

The Big Takeaway

The main message of this paper is that nature is more patient than we thought.

We used to think that once particles got fast enough, the complex, messy rules of the strong force would disappear, replaced by simple math. This paper suggests that the "messy" rules stick around much longer, influencing particles even at energies we thought were "safe" and simple.

In everyday terms:
Imagine you are walking from a muddy field (low energy) onto a paved highway (high energy).

• Old View: You step onto the pavement, and your shoes instantly become clean.
• New View (This Paper): You step onto the pavement, but your shoes stay muddy for a few more steps. The "mud" (non-perturbative physics) lingers longer than we expected, affecting how you walk even on the clean road.

This discovery helps scientists build better maps of the subatomic world, showing us that the transition from chaos to order is a smooth, gradual journey, not a sudden jump.

Technical Summary

1. Problem Statement

The paper addresses the discrepancy between experimental data and theoretical predictions for the neutral ($\pi^0$) and charged ($\pi^\pm$) pion form factors in the intermediate-energy region (transition between Infrared/IR and Ultraviolet/UV physics).

• Neutral Pion ($F_0$): Experimental data from BABAR and BELLE collaborations show a deviation from the perturbative QCD (pQCD) asymptotic limit, suggesting the data lies above the predicted curve. It is unclear if this implies pQCD is only valid at higher energies than previously thought or if experimental uncertainties are underestimated.
• Charged Pion ($F_\pi$): Experimental data also deviates from the expected perturbative asymptotic decay, and this sector remains less explored compared to the neutral pion.
• Core Question: Can non-perturbative physics, specifically a running strong coupling constant ($\alpha_s$) defined across all energy scales, explain these deviations in the intermediate region where pQCD is traditionally assumed to dominate?

2. Methodology

The authors employ a non-perturbative running strong coupling constant ($\hat{\alpha}_s$) derived from a double-dilaton Holographic QCD (HQCD) model (based on their previous work [30]).

• Strong Coupling Model ($\hat{\alpha}_s$):

  • Unlike standard pQCD, this model features an infrared fixed point and a smooth matching to pQCD at $Q_0 = 3.79$ GeV.
  • The functional form is:
    $$\hat{\alpha}_s(Q^2) = \frac{c}{1 - a \, \text{sech}^{4/5}(b(Q^2 + 1))}$$
  • This allows the coupling to remain non-perturbative even at energy scales traditionally considered perturbative ($> \Lambda_{QCD}$).

• Pion Distribution Amplitudes (DAs):

  • The form factors are calculated using the Pion Distribution Amplitude (DA) formalism, expanded in terms of Gegenbauer polynomials $C^{3/2}_{2n}$.
  • The DAs ($\phi_{\pi^0}$ and $\phi_{\pi^\pm}$) are expanded using the non-perturbative $\hat{\alpha}_s(Q^2)$:
    $$\phi(x, Q^2) = \phi_{\text{asym}}(x) \sum_{n=0}^{\infty} b_n \hat{\alpha}_s(Q^2)^{\gamma_n} C^{3/2}_{2n}(2x-1)$$
  • This leads to parametrizations for the form factors $Q^2 F(Q^2)$ as series expansions in $\hat{\alpha}_s$.

• Hybrid Modeling & Padé Approximants:

  • To bridge the gap between low-energy data and high-energy asymptotic limits, the authors use a hybrid approach:
    1. Low Energy: Data is fitted using Padé Approximants (PA) ($P^N_M$).
    2. High Energy: Theoretical expansion using Eq. (9) and (10) with $\hat{\alpha}_s$.
    3. Matching: A matching point $Q_0^2$ is determined where the PA and the high-energy expansion meet, enforcing continuity and differentiability.
  • Model selection is guided by the Akaike Information Criterion (AIC) to avoid overfitting while minimizing $\chi^2$.

3. Key Contributions

1. Application of All-Energy $\hat{\alpha}_s$: The paper demonstrates that a strong coupling constant with an IR fixed point, derived from holographic QCD, can successfully describe pion form factors across the entire $Q^2$ spectrum, including the intermediate region.
2. Resolution of Intermediate-Energy Deviations: The study suggests that the observed deviations from pQCD asymptotic limits are not necessarily experimental errors but rather signatures of non-perturbative effects persisting into the intermediate-energy regime.
3. Isospin-Breaking Analysis: The authors introduce a novel method to quantify isospin-breaking effects by analyzing the difference between the charged and neutral pion DAs ($\phi_{\pi^\pm} - \phi_{\pi^0}$). This allows for the calculation of the quadratic pion mass difference ($\Delta m_\pi^2$) directly from form factor data.

4. Results

A. Neutral Pion Form Factor ($F_0$)

• Naive Prediction: A purely theoretical prediction using the asymptotic limit and $\hat{\alpha}_s$ (without fitting low-energy data) fails to reproduce the experimental data, though it covers the full $Q^2$ range.
• Hybrid Fit: Using a $P^2_1$ Padé approximant for low energies matched to the high-energy expansion:
  • The matching point was found at $Q_0^2 = 4.35 \text{ GeV}^2$.
  • The fit yields coefficients $c_0 \approx 3.62$ and $c_1 \approx -3.2$.
  • Behavior: The model predicts the form factor approaches the asymptotic limit from below, consistent with the data trend, rather than exceeding it as some pQCD interpretations might suggest.
  • Statistical Fit: The reduced $\chi^2$ is 2.7, driven largely by the BABAR collaboration data ($\tilde{\chi}^2_{BABAR} = 5.14$), indicating tension between different experimental datasets.

B. Charged Pion Form Factor ($F_\pi$)

• Hybrid Fit: Using a $P^4_1$ Padé approximant for low energies matched to the high-energy expansion:
  • The matching point was found at $Q_0^2 = 2.21 \text{ GeV}^2$.
  • The fit yields coefficients $c_0 \approx 3.08$ and $c_1 \approx -2.66$.
  • Behavior: The model successfully reproduces the data with a reduced $\chi^2 \approx 1.00$.
  • Key Feature: The curve exhibits a maximum at low energies before decreasing toward the asymptotic limit. Crucially, the curve remains above the standard pQCD prediction at high energies, suggesting missing non-perturbative resummation effects in standard pQCD.

C. Isospin-Breaking Effects

• The difference between the charged and neutral DAs ($\phi_{\pi^\pm} - \phi_{\pi^0}$) was integrated to define an isospin-breaking form factor $F_{Iso}$.
• Using a dispersive integral and assuming negligible isospin breaking in the continuum, the authors derived a relation for the pion mass difference:
  $$\Delta m_\pi^2 = F_{Iso}(0) \frac{m_{\pi^0}^2 m_{\pi^\pm}^2}{f_\pi^2}$$
• Result: The calculated value is $\Delta m_\pi^2 = (1.1 \pm 0.1) \times 10^{-3} \text{ GeV}^2$, which compares reasonably well with the experimental value of $1.3 \times 10^{-3} \text{ GeV}^2$.

5. Significance and Conclusion

• Redefining the Perturbative Regime: The results imply that the transition to the perturbative regime occurs at higher energy scales than the traditional $\Lambda_{QCD} \sim 1$ GeV. Non-perturbative physics remains relevant well into the intermediate-energy region (up to $\sim 4$ GeV$^2$).
• Validation of HQCD Models: The double-dilaton HQCD model provides a consistent framework for the strong coupling that bridges IR and UV physics, successfully describing form factor data where standard pQCD fails.
• Phenomenological Impact: The study opens a new avenue for studying isospin-breaking effects and hadron structure using distribution amplitudes derived from holographic models, suggesting that "high-energy" data may still require non-perturbative corrections.

In summary, the paper argues that the "puzzling" deviations in pion form factors are not anomalies but evidence of non-perturbative strong coupling effects persisting at intermediate scales, successfully modeled by a double-dilaton HQCD approach.