Multiple dispersive bounds. II) Sub-threshold branch-cuts

This paper extends a multiple dispersive bounds strategy to sub-threshold branch-cuts by modifying the standard $z$-expansion to simultaneously incorporate pair-production and sub-threshold constraints, demonstrating through numerical analysis of the charged kaon form factor that this double-bound approach yields more precise extrapolations and greater stability against outer function choices than existing single-bound methodologies.

The Gist

Imagine you are trying to draw a map of a mysterious island. You have some satellite photos of the coastline (experimental data), but the photos are blurry, and you only have a few of them. You want to predict what the island looks like far out at sea, where you have no photos at all.

In the world of particle physics, this "island" is a Hadronic Form Factor. It's a mathematical map that tells us how particles like protons or kaons (which are made of quarks) behave when they are hit by other particles. Physicists need this map to understand the fundamental forces of nature.

However, drawing this map is tricky because the "ocean" has hidden rules called Unitarity and Analyticity. These rules say that the map cannot just be any shape; it must follow strict laws of physics, like conservation of energy.

This paper, written by Silvano Simula and Ludovico Vittorio, introduces a new, smarter way to draw this map, especially when there are "underwater reefs" (sub-threshold branch-cuts) that make the terrain complicated.

Here is the breakdown of their strategy using everyday analogies:

1. The Problem: The "Hidden Reef"

Usually, physicists use a standard method (called the BGL z-expansion) to draw the map. Think of this method as using a flexible ruler to connect the dots you have.

However, in the case of particles like the Kaon, there is a "hidden reef" below the surface.

• The Pair-Production Threshold: Imagine a cliff edge. Above this edge, you can easily see what's happening (particles can be created and observed).
• The Sub-Threshold Cut: Below this cliff, there is a hidden underwater reef. You can't see it directly with your eyes (experiments can't reach there), but you know it's there because it affects the waves above it.

Older methods tried to draw the map by only looking at the "cliff edge" and ignoring the hidden reef, or by lumping everything together into one big, vague rule. This often led to maps that were either too wide (uncertain) or that broke the laws of physics when you tried to predict the shape far out at sea.

2. The Solution: The "Double Fence" Strategy

The authors propose a Double Dispersive Bound.

Imagine you are building a fence to keep a garden safe.

• The Old Way: You build one giant fence around the whole property. It's loose and wobbly. If you try to guess where the flowers are at the back of the garden, your guess could be all over the place.
• The New Way (Double Bound): You build two separate fences.
  1. Fence A: A tight, precise fence around the part of the garden you can see (the pair-production region).
  2. Fence B: A second fence around the hidden underwater reef (the sub-threshold region).

By applying both fences at the same time, you force the map to stay within a much tighter, more accurate corridor. Even though you can't see the reef directly, Fence B uses a clever "resonance model" (a mathematical guess based on known physics, like how a bell rings) to estimate the reef's shape.

3. The "Outer Function" Ambiguity

There is one tricky part. When building Fence B (the one for the hidden reef), you have to choose a "template" or a "shape" for the fence. The authors realized that there isn't just one correct template; you could choose different shapes, and they would all technically fit the rules.

• The Analogy: Imagine you are building a bridge over a river. You know the bridge must touch the banks, but the middle could be a flat plank, a curved arch, or a zigzag.
• The Discovery: The authors tested different bridge shapes (called "outer functions"). They found that if you use the Double Fence strategy, the final map looks almost the same no matter which bridge shape you picked. It's stable.
• The Contrast: If you used the old "Single Fence" method, the shape of your bridge would change the final map wildly. One bridge shape might say the island is flat; another might say it's a mountain. That's bad for science!

4. The Result: A Sharper Map

The authors tested this new method on the Charged Kaon (a specific type of particle). They used real data from experiments and supercomputer simulations (Lattice QCD).

• Precision: The new method produced a much sharper, more precise map, especially for areas far away from the data points (high momentum transfer).
• Stability: The results didn't wiggle around when they changed the "bridge shape" (the outer function).
• The Kaon Radius: They calculated the "radius" of the Kaon (how big it is). Their new, model-independent calculation gave a slightly different size and a more honest error bar than previous estimates. They showed that previous estimates were too confident because they relied on too many assumptions.

Summary

Think of this paper as upgrading from a crayon sketch to a laser-guided blueprint.

• Old Method: "Let's draw a line through these dots and hope it doesn't break physics." (Result: Wobbly, uncertain).
• New Method: "Let's build two strict fences—one for what we see and one for what we know is there but can't see. This forces the line to stay exactly where physics says it must be." (Result: Precise, stable, and trustworthy).

This new approach allows physicists to make much better predictions about how particles behave in high-energy collisions, which is crucial for understanding the fundamental building blocks of our universe.

Technical Summary

1. Problem Statement

The study of hadronic form factors (e.g., electromagnetic form factors of kaons or pions) relies heavily on the principles of unitarity and analyticity. The standard approach, known as the Boyd-Grinstein-Lebed (BGL) $z$-expansion, maps the complex momentum transfer plane ($t$) onto the unit disk using a conformal variable $z$.

However, a significant challenge arises when sub-threshold branch cuts exist. These occur when intermediate on-shell states ($H'_1 H'_2$) can be produced and scatter into the physical channel ($H_1 H_2$) at an invariant mass threshold ($t_{th}$) below the physical pair-production threshold ($t_+$).

• The Issue: In the standard BGL approach, the unitarity bound is often derived solely from the pair-production region ($t \ge t_+$). When a sub-threshold cut exists ($t_{th} < t < t_+$), the form factor acquires an imaginary part in this unphysical region.
• The Consequence: Ignoring this sub-threshold contribution leads to an incomplete application of unitarity constraints. Existing methods that attempt to handle this often rely on single, global dispersive bounds or specific expansions (like Szegő polynomials) that may not strictly enforce unitarity over the entire unit circle, potentially leading to unstable extrapolations at high momentum transfer.

2. Methodology

The authors propose a rigorous framework to handle sub-threshold cuts by implementing double dispersive bounds.

A. Conformal Mapping and Analytic Structure

• They introduce two conformal variables:
  • $z_+$: Mapped to the pair-production threshold $t_+$.
  • $z$: Mapped to the lowest threshold $t_{th}$ (sub-threshold).
• The form factor $f(z)$ is analytic inside the unit disk $|z| < 1$, but the presence of the sub-threshold cut means the imaginary part is non-zero for $t_{th} \le t < t_+$.
• The global unitarity bound $\chi[f]$ is defined as an integral over the full unit circle $|z|=1$. This integral is decomposed into two semi-inclusive parts:
  1. Pair-production arc ($\chi_{pair}$): Corresponds to $t \ge t_+$.
  2. Extra region ($\chi_{extra}$): Corresponds to the sub-threshold region $t_{th} \le t < t_+$.

B. The Double Dispersive Bound

Instead of imposing a single global constraint $\chi[f] \le \chi^U$, the authors impose two simultaneous constraints:

1. $\chi_{pair}[f] \le \chi^U_+$ (Derived from vacuum polarization susceptibilities).
2. $\chi_{extra}[f] \le \chi^U_{extra}$ (Derived from a model of above-threshold resonances).

This "double filter" approach ensures that the coefficients of the $z$-expansion satisfy unitarity constraints for both the physical and sub-threshold regions independently, which is mathematically more constraining than a single global bound.

C. Modeling the Sub-threshold Contribution

Since $\chi^U_{extra}$ cannot be calculated directly from first principles (like lattice QCD) without knowledge of the form factor in the unphysical region, the authors develop a resonance model:

• They model the effects of poles (resonances) located in the second Riemann sheet (above $t_+$) on the form factor in the sub-threshold region.
• The model uses a specific ansatz (Eq. 43) based on the location of conjugate poles $z_R$ derived from resonance masses ($M_R$) and widths ($\Gamma_R$).
• This model is validated against the $\rho(770)$ resonance in the pion form factor and applied to the $\rho(770)$, $\omega(782)$, and $\phi(1020)$ resonances in the kaon form factor.

D. Outer Function Ambiguity

The authors highlight that the choice of the outer function $\phi(z)$ (which weights the unitarity integral) outside the pair-production arc is not unique. They analyze different choices (parameterized by powers $q_1, q_2$) and demonstrate how this choice significantly impacts the value of $\chi^U_{extra}$ and the resulting extrapolation stability.

3. Key Contributions

1. Formalism of Double Bounds: They rigorously derive how to apply multiple dispersive bounds to the BGL expansion, splitting the unitarity constraint into pair-production and sub-threshold components.
2. Resonance Model for $\chi^U_{extra}$: They provide a practical method to estimate the sub-threshold dispersive bound using a resonance model, validated against the pion form factor.
3. Critique of Existing Expansions: They demonstrate that expansions based on Szegő polynomials (which only constrain the pair-production arc) fail to guarantee unitarity on the full circle, potentially leading to unphysical results.
4. Unitarity Filtering: They apply a "unitarity filter" to input data (experimental and Lattice QCD) to remove non-unitary events before fitting, ensuring the input dataset is consistent with the theoretical bounds.

4. Results

The methodology was applied to the electromagnetic form factor of the charged kaon ($F^{(em)}_{K^\pm}$) using:

• Experimental data from FNAL and CERN (spacelike region, low $Q^2$).
• Lattice QCD data from the HotQCD Collaboration (spacelike region, up to $Q^2 \approx 28$ GeV$^2$).

Key Findings:

• Precision at High Momentum Transfer: The $z$-expansion using the double dispersive bound provides significantly more precise extrapolations at large momentum transfer ($Q^2$) compared to the standard single global bound. The uncertainty bands are roughly 2 times narrower.
• Stability: The results obtained with the double bound are highly stable with respect to the choice of the outer function $\phi(z)$ (i.e., the parameter $q$). In contrast, the single bound method shows large variations depending on the choice of $\phi(z)$.
• Unitarity Violation in Other Methods: The expansion using Szegő polynomials (single arc constraint) was found to violate the total unitarity bound ($\sum b_k^2 \gg 1$), confirming the necessity of the double bound approach.
• Kaon Radius:
  • Using experimental data: $r_{K^\pm} = 0.538 \pm 0.066 \pm 0.004$ fm. This suggests the PDG average underestimates the uncertainty by a factor of $\sim 2$ due to model-dependent assumptions.
  • Using Lattice QCD data: $r_{K^\pm} = 0.641 \pm 0.022$ fm. This is consistent with recent Lattice results but with a more robust, model-independent error estimation.

5. Significance

• Theoretical Rigor: This work resolves a long-standing issue in hadronic physics regarding how to properly incorporate sub-threshold branch cuts into model-independent analyses. It establishes that a single global bound is insufficient when sub-threshold cuts are present.
• Phenomenological Impact: The improved precision and stability of the double-bound method are crucial for extracting fundamental parameters (like CKM matrix elements or hadron radii) from experimental and lattice data, particularly when extrapolating to regions where data is scarce or non-existent.
• Future Applications: The framework is general and can be applied to other processes involving sub-threshold cuts, such as weak semileptonic decays of heavy hadrons ($B \to D^{(*)}$, etc.), where similar analytic structures exist.

In summary, Simula and Vittorio demonstrate that a double dispersive bound is the necessary and proper tool for analyzing hadronic form factors in the presence of sub-threshold branch cuts, offering superior precision and stability over existing single-bound or incomplete unitarity methods.