New Solutions of RG Equations for \alpha_s and y_{top}

Technical Summary: New Solutions of RG Equations for $\alpha_s$ and $y_{top}$

Problem Statement
The paper addresses the challenge of obtaining explicit analytical solutions for the Renormalization Group (RG) equations governing the running of the strong coupling constant ($\alpha_s$) and the top quark Yukawa coupling ($y_{top}$) within the Standard Model. While previous work [1] established a strategy for summing leading, next-to-leading (NL), and higher-order logarithms for a single coupling (e.g., in QCD), extending this approach to a system with two interacting couplings presents significant mathematical complications. Specifically, the non-linear nature of the RG equations in higher orders of Perturbation Theory (PT) typically precludes analytical solutions, often resulting in expressions involving special functions or mixing leading and subleading logarithmic terms in a way that obscures the specific order of approximation.

Methodology
The authors propose a strategy that avoids solving the full non-linear RG equations exactly at a given order. Instead, they employ a linearization technique followed by a loop expansion. The methodology proceeds as follows:

1. Loop Expansion: The running couplings $\alpha$ and $y$ are expanded as a series $\sum \alpha_\ell$ and $\sum y_\ell$, where $\ell$ denotes the loop order. The boundary conditions are set such that $\alpha_1(0)=\alpha_0$, $y_1(0)=y_0$, and higher-order terms vanish at the initial scale.
2. Linearization: The RG equations are linearized order-by-order. The first-order terms ($\alpha_1, y_1$) satisfy the standard one-loop equations. Higher-order terms ($\alpha_\ell, y_\ell$ for $\ell > 1$) satisfy linear differential equations where the source terms depend only on lower-order solutions ($\ell' < \ell$).
3. Analytical Integration: Because the equations for $\ell > 1$ are linear, they admit explicit integral solutions. The authors derive these solutions, noting that while the $\alpha$ sector remains purely logarithmic, the $y$ sector introduces a "tricky" solution even at one-loop, leading to higher-order terms that contain a small component described by a hypergeometric function (denoted $y^H_\ell$).
4. Asymptotic Approximation: In the asymptotic regime, the authors argue that the hypergeometric components are numerically suppressed and can be neglected, allowing the solutions to be expressed primarily through elementary logarithms.
5. RG Invariant Parametrization: To improve stability and clarity, the solutions are rewritten in terms of RG invariants: $\hat{L}$ (related to the scale $\Lambda$) and $K$ (a parameter characterizing the trajectory in the coupling phase space). This re-parametrization absorbs initial conditions into the definitions of the invariants.
6. Vertical Summation: A key innovation is the "vertical summation" of infinite series of logarithms. By identifying a recursive structure where higher-order corrections repeat the functional form of lower-order ones but with shifted arguments, the authors construct an iterative procedure. This allows for the resummation of leading logarithms to all orders, effectively improving the approximation by iterating the solution structure.

Key Contributions and Results

• Explicit Analytical Forms: The paper provides explicit analytical formulas for the running of $\alpha_s$ and $y_{top}$ up to three-loop order. Unlike traditional methods that yield complex special functions, these solutions contain only logarithms (and simple powers of couplings), facilitating easier analysis and summation.
• Handling Two Couplings: The authors successfully extend the single-coupling resummation strategy to the two-coupling case of the Standard Model. They demonstrate that while the coupling $y_{top}$ affects the running of $\alpha_s$ negligibly, the effect of $\alpha_s$ on $y_{top}$ is essential and requires higher-order corrections for accuracy.
• Negligible Special Functions: The analysis reveals that the non-elementary (hypergeometric) parts of the solution, which arise from the complexity of the two-coupling system, are numerically insignificant in the asymptotic regime. This justifies their omission for practical analytical purposes.
• RG Invariant Formulation: The solutions are recast in terms of RG invariants $\hat{L}$ and $K$. The authors show that as the loop order increases, the values of these invariants converge to fixed values, indicating the stability of the approach.
• Vertical Resummation Effects: The application of vertical summation significantly improves the behavior of the solutions near the infrared (IR) pole. While fixed-order solutions (without resummation) exhibit irregularities or non-monotonic behavior near the pole, the vertically resummed solutions become smoother and extend further into the region near the pole, suggesting a tendency toward convergence.

Significance and Claims
The authors claim that their strategy offers a "relatively simple scheme" for obtaining explicit analytical forms of running couplings that sum up infinite series of leading, next-to-leading, and higher-order logarithms. The primary significance lies in the ability to perform these summations analytically without resorting to numerical integration or special functions, which is a departure from standard approaches to non-linear RG equations.

The paper modestly notes that the case of two couplings is more complicated than the single-coupling case, requiring the neglect of certain non-elementary terms. However, within the asymptotic regime considered, these terms are deemed inessential. The proposed method allows for further iterative improvement of the approximation by summing "vertical" series, a process that can be continued as long as new terms in the beta-function are available. The results suggest that this approach yields more stable and regular asymptotic solutions compared to standard perturbative expansions, particularly in the vicinity of the Landau pole where perturbation theory typically fails.