Direct determination of the structure functions $F_L$, $F_S$ and $G$ from $F_2$ and $dF_2/dQ^2$ to $O(\alpha_s^2)$

This paper extends previous work by deriving consistent $O(\alpha_s^2)$ expressions for the longitudinal, singlet, and gluon structure functions ($F_L$, $F_S$, and $G$) directly from the measured $F_2$ and its logarithmic derivative with respect to $Q^2$, while treating small non-singlet corrections at low $x$.

The Gist

Imagine you are trying to figure out the ingredients of a secret soup, but you can only taste the final broth. In the world of particle physics, that "soup" is a proton, and the "broth" is a measurement called $F_2$. Scientists have been trying to reverse-engineer the recipe to find out how much "glue" (gluons) and how much "flavor" (quarks) is actually in the proton, but it's been like trying to guess the recipe of a cake just by tasting the frosting.

This paper presents a new, more precise way to solve that puzzle. Here is the breakdown of what the authors did, using simple analogies:

1. The Problem: A Messy Recipe

In the past, a team led by Lappi tried to figure out the recipe by looking at the broth ($F_2$) and a second measurement called $F_L$ (which tells us how the soup behaves when stirred in a specific way). They found a way to guess the ingredients, but they had to make a big simplification: they ignored the "spices" (complex quantum effects) and only looked at the main ingredients. It was like trying to bake a cake using a recipe that only listed "flour" and "sugar," ignoring the eggs and butter.

2. The Solution: A Mathematical "Magic Lens"

The authors of this paper, Boroun, Durand, and Ha, decided to upgrade that method. They used a mathematical tool called a Laplace Transform.

Think of the relationship between the ingredients (gluons and quarks) and the measurements ($F_2$ and $F_L$) as a complicated knot of tangled strings. In the old method, trying to untangle the knot was messy and required cutting corners (ignoring important physics).

The authors used their "magic lens" (the Laplace Transform) to look at the knot from a different angle. Suddenly, the tangled strings untangled themselves. The complex math that usually requires messy "convolutions" (a type of mathematical mixing) turned into simple multiplication. This allowed them to solve for the ingredients directly without having to guess or ignore the "spices."

3. The Result: A Complete Recipe Book

By using this new lens, they derived a set of formulas that can calculate the Gluon (the glue holding the proton together) and the Singlet (the total flavor) directly from the measured data.

• What they fixed: They corrected the previous work by including the "spices" (higher-order quantum corrections) up to a very high level of precision ($O(\alpha_s^2)$).
• The Catch: To get the full picture, you need to know the "spices" (the non-singlet corrections). However, the authors note that at very small scales (very tiny particles), these spices are so faint they can be ignored or estimated easily.

4. The Test: Predicting the Flavor

To prove their method works, they applied it to real data from the HERA particle accelerator. They took the known measurements of the broth ($F_2$) and its rate of change, and used their new formulas to predict what the "stirring behavior" ($F_L$) should look like.

• The Outcome: Their predictions (the solid lines in their graph) matched the actual experimental data very well.
• The Comparison: When they compared their result to the older, simplified method (the dashed lines), they found that while the old method was "okay," the new method was significantly more accurate. It was the difference between a rough sketch and a high-definition photograph.

Summary

In short, this paper says: "We found a better way to look at the data from particle collisions. By using a specific mathematical trick, we can now calculate the hidden parts of the proton (gluons and quarks) directly from what we measure, with much higher precision than before. We tested it, and it works."

They didn't invent a new particle or change the laws of physics; they just built a better calculator to read the existing data more accurately.

Technical Summary

Technical Summary: Direct Determination of Structure Functions $F_L$, $F_S$, and $G$ from $F_2$ and $dF_2/dQ^2$ to $O(\alpha_s^2)$

Problem Statement
Recent work by Lappi et al. [1] demonstrated that the gluon ($G = xg$) and singlet ($F_S = x\Sigma$) structure functions in deep inelastic scattering (DIS) could be determined directly from the physical distributions $F_2$ and $F_L$. However, their derivation relied on significant approximations: neglecting non-singlet contributions and retaining only the first non-zero terms in the QCD coefficient functions. Consequently, their results involved multiple convolutions that did not generalize simply to higher orders in the strong coupling constant $\alpha_s$. Furthermore, while Kaptari et al. [4, 5] derived $F_L$ to $O(\alpha_s^2)$ using a different method, their approach involved approximations to the $F_2$ parametrization that were difficult to assess systematically. The challenge remains to develop a systematic expansion in powers of $\alpha_s$ that includes complete QCD coefficient functions and handles non-singlet effects, allowing for the direct extraction of $F_L$, $F_S$, and $G$ from measured data without uncontrolled approximations.

Methodology
The authors extend the framework of Lappi et al. by employing Laplace transforms to factorize the Mellin convolutions inherent in the QCD evolution equations. The methodology proceeds as follows:

1. Transformation to Laplace Space: The structure function equations relating $F_2$ and $F_L$ to the singlet, gluon, and non-singlet distributions are converted from the momentum fraction variable $x$ to the Laplace variable $v = \ln(1/x)$. This transformation converts the convolution integrals into simple products in Laplace space.
2. Matrix Inversion: The coupled equations for $F_2$ and $F_L$ are expressed in a compact matrix form involving the Laplace transforms of the coefficient functions. By inverting this matrix, the authors derive explicit expressions for the Laplace transforms of the singlet ($\tilde{F}_S$) and gluon ($\tilde{G}$) distributions directly in terms of the transforms of the measurable quantities ($\tilde{F}_2$, $\tilde{F}_L$) and the non-singlet correction ($\tilde{\Delta}$).
3. Systematic Expansion to $O(\alpha_s^2)$: Unlike previous works that truncated coefficient functions at leading order, this analysis expands the coefficient functions and their inverses systematically to $O(\alpha_s^2)$. This involves retaining products and ratios of lower-order terms to ensure the final results reproduce the original evolution equations to the specified order.
4. Derivation of $F_L$ from $F_2$: Utilizing the evolution equation for $F_2$, the authors express the gluon distribution $G$ in terms of $F_2$ and its logarithmic derivative $dF_2/d\ln Q^2$. Substituting this back into the inverted matrix equations yields an expression for $F_L$ that depends solely on $F_2$, $dF_2/d\ln Q^2$, and known QCD coefficients, effectively decoupling $F_L$ from the need for independent gluon fits.
5. Inverse Transformation: The final results in Laplace space are converted back to $x$-space via inverse Laplace transforms. The authors calculate these transforms analytically, identifying poles and residues to derive explicit convolution kernels ($J_1$ and $J_2$) that relate $F_L$ to $F_2$ and its derivative.

Key Contributions

• Systematic $O(\alpha_s^2)$ Formalism: The paper provides the first systematic derivation of $F_L$, $F_S$, and $G$ in terms of measured $F_2$ and $F_L$ (or $dF_2/dQ^2$) that includes complete QCD coefficient functions up to next-to-leading order ($O(\alpha_s^2)$).
• Resolution of Convolution Complexity: By using Laplace transforms, the authors eliminate the complex multi-fold convolutions present in the original Lappi et al. formulation, providing a cleaner algebraic structure that generalizes to higher orders.
• Correction of Mixed-Order Results: The analysis corrects the mixed-order results of Lappi et al., specifically addressing the missing $O(\alpha_s)$ contributions in the coefficient of $F_L$ that arose from their truncation of coefficient functions.
• Analytic Kernels: The authors derive explicit analytic forms for the inverse Laplace transforms of the coefficient functions, resulting in convolution kernels involving exponential, trigonometric, and polylogarithmic functions (specifically dilogarithms).

Results

• General Expressions: Equations (43) and (46) present the general expansions for the Laplace transforms of $G$ and $F_S$ in terms of $\tilde{F}_2$, $\tilde{F}_L$, and the QCD coefficient functions to $O(\alpha_s^2)$.
• $F_L$ Determination: Equation (72) provides the explicit expression for $F_L(x, Q^2)$ as a convolution of $F_2$ and $dF_2/d\ln Q^2$ with the derived kernels $J_1$ and $J_2$.
• Numerical Application: The authors applied the $O(\alpha_s)$ version of their formalism to the Block-Durand-Ha (BDH) parametrization of HERA $F_2$ data. As shown in Figure 1, the results for $F_L$ (solid curves) agree well with existing H1 data and show small but significant deviations from the approximate leading-order results of Boroun and Ha [2] (dashed curves), which relied on the truncated Lappi et al. method.
• Non-Singlet Handling: The formalism explicitly includes non-singlet terms. The authors note that at very small $x$, these terms are negligible, but the framework allows for their inclusion using existing quark distributions for comparisons at moderate $x$.

Significance and Claims
The paper claims to provide a rigorous, systematic tool for determining the longitudinal and singlet/gluon structure functions directly from physical observables without relying on iterative fits that assume specific parton distribution shapes. The authors emphasize that their method:

• Avoids the approximations regarding coefficient functions used in previous leading-order analyses.
• Corrects the structural inconsistencies in the mixed-order results of Lappi et al.
• Offers a pathway to predict $F_L$ in kinematic regions (very small $x$, large $Q^2$) where direct measurement is difficult, provided a precise parametrization of $F_2$ exists.

However, the authors remain modest regarding immediate practical application. They note that current experimental measurements of $F_L$ are not yet extensive or accurate enough to fully validate the $O(\alpha_s^2)$ approach. They suggest that while their method avoids the approximations forced upon Kaptari et al., extending the current numerical implementation to $O(\alpha_s^2)$ would be a valuable future step to improve agreement with data and fully exploit the systematic expansion.