TaF-Dataset: Total Asymptotic Freedom in QFT

• TaF-Dataset is a structured classification of 4D renormalizable quantum field theories that remain perturbative with all gauge, Yukawa, and scalar quartic couplings vanishing in the UV.
• It employs fixed-flow renormalization-group methods and coupling rescaling to isolate universal 1/t falloff, ensuring a stable Gaussian fixed point in the UV regime.
• The dataset underpins models like SU(5)-TAF, Pati–Salam-TAF, and trinification-TAF, providing concrete IR predictions and strict constraints for UV-complete model building.

The TaF-Dataset, as defined in the context of high-energy theoretical physics, is a structured classification of four-dimensional, renormalisable quantum field theories (QFTs) whose renormalisation-group (RG) evolution remains perturbative at all energy scales, such that every gauge, Yukawa, and scalar quartic coupling flows to zero in the ultraviolet (UV) limit. This property, termed Total Asymptotic Freedom (TAF), extends the notion of asymptotic freedom beyond simple gauge theories and is of central interest for models that could potentially be ultraviolet (UV) complete without introducing new strong-coupling phenomena or Landau poles. The principal reference and canonical construction of the TaF-Dataset is presented in (Giudice et al., 2014).

1. Definition and General Structure

A theory belongs to the TaF-Dataset if all dimensionless couplings $g_i$ (gauge), $y_a$ (Yukawa), and $\lambda_m$ (scalar quartic) approach zero as the renormalization scale $\mu \to \infty$, while remaining within the perturbative regime (i.e., $|g_i|, |y_a|, |\lambda_m| \lesssim O(1)$ throughout).

At one-loop accuracy, which suffices because only the Gaussian fixed point exists as a UV attractor for these systems, the following differential conditions must be satisfied:

• For each gauge coupling $g_i$: $dg_i^2/dt = -b_i g_i^4$, with $b_i > 0$ for asymptotic freedom. ($t \equiv (4\pi)^{-2} \ln(\mu^2/\mu_0^2)$)
• For each Yukawa coupling $y_a$: $d y_a^2/dt = y_a^2(f_y y_a^2 - f_g g^2)$, with $f_g$ and $f_y$ determined by matter content.
• For each quartic scalar coupling $\lambda_m$: $$d \lambda_m/dt = s_\lambda \lambda_m^2 + s_{\lambda y} \lambda_m y^2 - s_{\lambda g} \lambda_m g^2 - s_y y^4 + s_g g^4$$.

The TAF requirement implies not only $b_i > 0$ (non-Abelian gauge factors) but also stricter algebraic conditions relating gauge, Yukawa, and quartic couplings—typically necessitating specific "fixed-flow" solutions.

2. Classification Algorithm

The systematic construction of the TaF-Dataset proceeds through several steps:

• Rescaling: All couplings are rescaled to isolate the universal $1/t$ asymptotic falloff, defining $\tilde g_i^2(t)$, $\tilde y_a^2(t)$, and $\tilde \lambda_m(t)$ so that, e.g., $g_i^2(t) = \tilde g_i^2(t)/t$ as $t \to \infty$.
• Autonomous RGEs: Express the one-loop RGEs as autonomous polynomial ordinary differential equations (ODEs) in the rescaled variables, $d x_I/d\ln t = V_I(x)$.
• Fixed-Flows: Identify special RG trajectories (fixed-flows) characterized by $V_I(x_\infty) = 0$, where all rescaled couplings approach constant, non-zero values as $t \to \infty$.
• Stability Analysis: Compute the stability (Jacobian) matrix $M_{IJ} = \partial V_I / \partial x_J|_{x=x_\infty}$. Each positive eigenvalue yields an infrared (IR) prediction; negative eigenvalues correspond to free, UV-attractive directions.
• Attraction Basin: A theory is included in the TaF-Dataset if all couplings flow into a single fixed-flow without encountering a Landau pole, and the IR boundary conditions that successfully flow to the fixed-flow constitute its attraction basin.

3. Main TAF Theories and Model Content

The TaF-Dataset consists of a finite set of explicit models, each characterized by its gauge group, matter content, interaction structure, and IR predictions:

Model	Gauge Group	Key Features
SM (TAF limit)	$SU(3)_c \times SU(2)_L$ ($g_1 \to 0$)	Unphysical boundary (e.g., $g_1=0$, $M_t=186$ GeV, $M_h=163$ GeV), serves as instructive limiting case; 5 IR predictions.
SU(5)-TAF	$SU(5)$	Three generations plus vector-like ($5+\bar5$, $24$), one stable quartic solution, two Yukawa and three quartic IR predictions.
Pati–Salam-TAF	$SU(2)_L \times SU(2)_R \times SU(4)_{PS}$	Additional lepton and vectorlike quark content, 15 possible quartic TAF solutions (all metastable), four Yukawa and 6–10 quartic predictions.
Trinification-TAF	$SU(3)_L \times SU(3)_R \times SU(3)_c$	Minimal (one scalar representation), single quartic TAF solution (unstable potential), one quartic prediction.

In all models, existence of TAF relies on sufficient non-Abelian structure and carefully tuned matter content and couplings. Yukawa and quartic sectors must mutually support the decay of all couplings to zero in the UV.

4. TAF Conditions and IR Predictions

The defining property of TAF models is that algebraic relations at the fixed-flow lead to UV-irrelevant (IR-predictive) constraints among low-energy physical parameters. For example:

• In SU(5)-TAF, fixed-flow conditions set $y_t/g_5$ and multiple quartic coupling ratios to definite values, leading to explicit mass and coupling predictions at low scale, modulo threshold corrections from the high-scale spectrum.
• In Pati–Salam-TAF, the process yields gauge unification-like relations for $g_L, g_R, g_4$ at TeV scale, along with complex intra-family mass relations depending on IR-attractive directions in the Yukawa and quartic sectors.

For each TAF model, the number of positive eigenvalues of $M_{IJ}$ indicates the count of IR predictions. For the SM-TAF limit, five predictions are obtained (including $g_1=0$, $M_t=186$ GeV, $M_h=163$ GeV); SU(5) and Pati–Salam-TAF variants each provide several IR-predictive mass or coupling ratios; the trinification model has a single quartic prediction.

5. Explicit Dataset Entries

Each TAF-Dataset entry is characterized explicitly by:

• Gauge group and $\beta$-function coefficients ($b_i$ for each factor).
• Matter content: Chiral fermion and scalar fields with transformation properties.
• Nonzero interactions: Yukawa terms, quartic terms, and their fixed-flow values.
• Fixed-flow algebraic values: e.g., $\tilde y_{t\infty}^2$, $\tilde \lambda_{\infty}$, giving ratios at the IR.
• Stability spectrum: Number and identity of IR-predictive directions.
• Vacuum stability: Some solutions (e.g., SM-TAF, trinification-TAF) yield unstable potentials or require metastable vacuum structure.

A detailed list of the principal models, their couplings, and corresponding IR predictions can be found in (Giudice et al., 2014), including all fixed-flow equations, stability matrices, and group-theoretic normalization.

6. Reproducibility and Computational Framework

All matrix equations and analytic TAF criteria (e.g., for $b_i, f_y, s_i$ in terms of Casimir invariants and Dynkin indices) are given in closed form, so the TaF-Dataset is reproducible using algebraic computing environments by solving $V_I(x_\infty)=0$ with positive eigenvalues. The dataset does not constitute a data file but rather an enumerated set of models and parameters, derived through analytic and algebraic procedures that exhaust the space of four-dimensional renormalizable field theories with total asymptotic freedom.

7. Significance and Limitations

The TaF-Dataset provides an exhaustive basis for exploring fully asymptotically free extensions of the Standard Model and serves as a tool both for UV-complete model building and for elucidating which low-energy phenomena might arise as IR predictions of asymptotically trivial (Gaussian) UV dynamics. The SM itself does not admit TAF with physical values, but SU(5), Pati–Salam, and trinification extensions offer partial realizations. Nevertheless, some entries are phenomenologically excluded or face vacuum stability/metastability constraints. This dataset has critical implications for the program of constructing natural, UV-complete quantum field theories (Giudice et al., 2014).