Transverse momentum dependent gluon density in a proton at low $x$ in the Laplace transform method

This paper employs the Laplace transform method to derive compact analytical expressions for both integrated and transverse momentum dependent gluon densities in a proton at very low $x$, demonstrating that these simplified formulas accurately capture the essential features of more complex calculations while closely matching results from other analytical and numerical approaches.

The Gist

Imagine the proton not as a solid marble, but as a bustling, chaotic city inside a tiny sphere. In this city, the most important residents are gluons—the particles that act like the glue holding everything together.

Physicists usually try to map this city by looking at how much "momentum" (speed and direction) these gluons have moving forward, like cars driving down a straight highway. This is called the "integrated" view. But in the high-speed collisions happening at modern particle accelerators, the gluons also wiggle side-to-side. To understand the full picture, scientists need a map that shows both the forward speed and the side-to-side wiggles. This is called the Transverse Momentum Dependent (TMD) gluon density.

The problem is that calculating this side-to-side movement, especially when the gluons are moving very slowly relative to the proton's total energy (a state physicists call "low x"), is incredibly difficult. It's like trying to predict the exact path of a leaf swirling in a hurricane using complex, messy math that requires supercomputers.

The Paper's Solution: The "Laplace Transform" Shortcut

The authors of this paper, a team from Iran, the US, Russia, and the UK, propose a clever shortcut. Instead of wrestling with the messy, complex equations directly, they use a mathematical tool called the Laplace transform.

Think of the Laplace transform as a special pair of glasses or a translator.

• Without the glasses: The math looks like a tangled knot of spaghetti. It's hard to see the pattern.
• With the glasses: The knot untangles. The complex equations turn into simple, neat lines that are easy to read and solve.

By putting their equations through this "translator," the team was able to derive simple, compact formulas that describe how these gluons behave. They didn't just look at the simplest version; they included the "next-to-leading" corrections, which are like adding the fine details to a sketch to make it look like a realistic painting.

What They Found

1. Accuracy with Simplicity: When they tested their simple formulas against the results from massive supercomputer simulations and other complex methods used by major physics groups (like CTEQ and NNPDF), their results matched very closely.
   • Analogy: It's like they built a simple, hand-drawn map of the city that turned out to be just as accurate as a GPS system that took a supercomputer hours to generate.
2. The "Soft" and "Hard" Zones: They found that at very low side-to-side speeds (the "soft" zone), the gluons behave in a way that needs to be guessed or modeled (like a foggy area on a map). But once the speed picks up (the "hard" zone), their simple formulas work perfectly.
3. The "Sudakov" Effect: They also looked at a factor called the "Sudakov form factor." You can think of this as a safety net or a braking system. It accounts for the fact that gluons don't just fly off randomly; they have a tendency to avoid radiating energy in certain ways. The authors showed that adding this "braking system" to their simple formulas changes the results only slightly, mostly in the low-speed zone.

Why This Matters

The main achievement of this paper isn't discovering a new particle or a new law of physics. Instead, it's about efficiency and clarity.

In the world of high-energy physics, researchers often have to run incredibly complex, time-consuming computer simulations to get a prediction for an experiment. This paper says, "You don't always need the supercomputer." You can use these new, simple analytical formulas. They capture the essential features of the complex calculations but are much easier to use and understand.

In Summary

The authors took a very complicated problem—mapping the side-to-side movement of gluons inside a proton at low energies—used a mathematical "translator" (the Laplace transform) to simplify the equations, and produced a set of easy-to-use formulas. These formulas work just as well as the heavy-duty computer simulations, making it easier for physicists to interpret data from particle colliders like the LHC without getting lost in the mathematical weeds.

Technical Summary

Technical Summary: Transverse Momentum Dependent Gluon Density in a Proton at Low x in the Laplace Transform Method

Problem Statement
Modern and future high-energy colliders (such as the LHC, LHeC, EIC, EicC, NICA, and CEPC) rely heavily on Quantum Chromodynamics (QCD) for predicting hard processes in $ep$, $eA$, $pp$, and $pA$ collisions. While the collinear factorization scheme, governed by DGLAP evolution equations, successfully describes processes with a single hard scale by neglecting parton transverse momenta ($k_T$), it becomes unreliable for processes involving multiple hard scales or small longitudinal momenta. In these regimes, Transverse Momentum Dependent (TMD) parton distributions (or unintegrated parton densities, uPDFs) are required.

Despite their importance, proton TMDs remain less well-known than their collinear counterparts. Various approaches exist, including BFKL, CCFM, BK, and JIMWLK evolution equations, as well as prescriptions like Kimber-Martin-Ryskin (KMR) and Watt-Martin-Ryskin (WMR) that derive TMDs from collinear PDFs. However, a need persists for simple, analytical expressions that capture the essential features of these complex calculations, particularly in the asymptotic limit $x \to 0$ where gluon contributions dominate.

Methodology
The authors investigate the gluon distribution in a proton at very low $x$, deriving both integrated and transverse momentum dependent densities using the Laplace transform technique. The approach proceeds as follows:

1. Framework: The study focuses on the asymptotic limit $x \to 0$, where quark contributions are neglected, and the gluon density obeys the DGLAP evolution equation. The TMD gluon distribution, $f_g(x, k_T^2)$, is approximated via the relation $f_g(x, k_T^2) \simeq \frac{\partial}{\partial \mu^2} G(x, \mu^2) \big|_{\mu^2=k_T^2}$, where $G(x, \mu^2) = xg(x, \mu^2)$.
2. Variable Transformation: The DGLAP equation is rewritten using variables $\upsilon = \ln(1/x)$ and $t = \ln \mu^2$. The convolution integral in the evolution equation is transformed into a product in Laplace space.
3. Laplace Transform Application: The evolution equation is solved in the Laplace domain ($s$-space). The solution involves the Laplace transforms of the splitting functions, $h_{gg}^{(n)}(s)$, and the initial gluon density. The authors include Leading Order (LO) and main Next-to-Leading Order (NLO) contributions.
4. Perturbative Expansion: The solution is expanded in powers of the strong coupling constant $\alpha_s$. The authors retain terms up to $O(\alpha_s^2)$, neglecting higher-order terms.
5. Inverse Transform: Analytical inverse Laplace transforms are performed to return to $x$-space. This involves handling terms like $F(s)/s$ and $F(s)/s^2$, which correspond to integrals of the initial distribution. The initial gluon distribution is parametrized in the soft region ($k_T^2 < k_0^2$) using a general form involving free parameters $N, a, b$, and $k_0$.
6. Sudakov Form Factor: The authors also consider the inclusion of the Sudakov form factor, $T_g(k_T^2, \mu^2)$, which resums virtual contributions. Analytic expressions for the Sudakov factor under both angular and strong ordering conditions are derived in the Appendix.

Key Contributions and Results

• Analytical Expressions: The primary contribution is the derivation of compact, analytical expressions for the integrated gluon density $xg(x, \mu^2)$ and the TMD gluon density $f_g(x, k_T^2)$ valid in the asymptotic $x \to 0$ limit. These expressions include LO and NLO corrections.
• Validation: The derived analytical results for the collinear gluon density are compared with numerical solutions of the DGLAP equations from major global analysis groups (CTEQ-TEA, NNPDF4.0, MSHT'2020, and IMP). The authors report "reasonable well agreement" for moderate and large $\mu^2$ values, particularly with CTEQ-TEA fixed-flavor-number-scheme (FFNS) predictions.
• TMD Comparison: The calculated TMD gluon densities are compared with predictions from the KMR/WMR approach (KL'2025) and a CCFM-based scenario (LLM'2024). The authors find "notable agreement" between their LO results and the LLM'2024 predictions for moderate gluon transverse momenta ($k_T^2 \leq 100 \text{ GeV}^2$) across a wide $x$ range.
• Sudakov Effects: The inclusion of the Sudakov form factor is shown to have a small effect, visible primarily at small transverse momenta, while the factor largely disappears in the large-$k_T^2$ region.

Significance and Claims
The paper claims that the Laplace transform technique offers a distinct advantage through the simplicity of the derived analytical expressions. These formulas are capable of reproducing the main features of more complicated numerical calculations and other analytical approaches.

The authors assert that their method provides "correct behaviors" for extracted gluon densities and demonstrates comparability with established parametrization groups. The work is presented as a further investigation into gluon dynamics, providing a tool that could impact present LHC measurements and the preparation for future facilities. The authors remain modest regarding the scope, noting that the restriction of the perturbative expansion limits the range of $t$ values to be close to the starting scale $t_0$, though they argue this restriction is not very strong for hard scales. They also acknowledge that discrepancies at very large transverse momenta or in the soft region arise from differences in QCD evolution assumptions and soft-region modeling, respectively.

The study concludes that the developed Laplace transform technique is applicable to small-$x$ QCD processes and that the agreement between their simple formulas and more complex considerations is "very promising" for future, more detailed investigations.