Functional truncations for the solution of the nonperturbative RG equations
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Bibliographic record
Abstract
Abstract We consider the Wetterich exact renormalization group (RG) equation. Approximate closed equations are obtained from it, applying certain truncation schemes for the effective average action. These equations are solved either purely numerically or by certain extra truncations for the potential and related quantities, called the functional truncations. Traditionally, the functional truncations consist of truncated expansions in powers of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mover> <mml:mi>ρ</mml:mi> <mml:mo stretchy="false">˜</mml:mo> </mml:mover> <mml:mo>−</mml:mo> <mml:msub> <mml:mover> <mml:mi>ρ</mml:mi> <mml:mo stretchy="false">˜</mml:mo> </mml:mover> <mml:mn>0</mml:mn> </mml:msub> </mml:math> , where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mover> <mml:mi>ρ</mml:mi> <mml:mo stretchy="false">˜</mml:mo> </mml:mover> <mml:mo>∝</mml:mo> <mml:msup> <mml:mi>ϕ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> , φ is the averaged order parameter, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mover> <mml:mi>ρ</mml:mi> <mml:mo stretchy="false">˜</mml:mo> </mml:mover> <mml:mn>0</mml:mn> </mml:msub> </mml:math> corresponds to the minimum of the dimensionless potential <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mover> <mml:mi>ρ</mml:mi> <mml:mo stretchy="false">˜</mml:mo> </mml:mover> <mml:mo stretchy="false">)</mml:mo> </mml:math> , depending on the infrared cut-off scale k . We propose a new approach of functional truncations, using the expansion <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mover> <mml:mi>ρ</mml:mi> <mml:mo stretchy="false">˜</mml:mo> </mml:mover> <mml:mo stretchy="false">)</mml:mo> <mml:mo>−</mml:mo> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>−</mml:mo> <mml:mi>s</mml:mi> <mml:mrow> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mrow> <mml:mo>−</mml:mo> <mml:mi>μ</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:mfenced close=")" open="("> <mml:mrow> <mml:msub> <mml:mi>u</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>k</mml:mi> </mml:mrow> </mml:msub> <mml:mi>s</mml:mi> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>u</mml:mi> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mi>k</mml:mi> </mml:mrow> </mml:msub> <mml:msup> <mml:mi>s</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>+</mml:mo> <mml:mo>⋯</mml:mo> </mml:mrow> </mml:mfenced> </mml:math> , where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>s</mml:mi> <mml:mo>=</mml:mo> <mml:mover> <mml:mi>ρ</mml:mi> <mml:mo stretchy="false">˜</mml:mo> </mml:mover> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mover> <mml:mi>ρ</mml:mi> <mml:mo stretchy="false">˜</mml:mo> </mml:mover> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>+</mml:mo> <mml:mover> <mml:mi>ρ</mml:mi> <mml:mo stretchy="false">˜</mml:mo> </mml:mover> <mml:mo stretchy="false">)</mml:mo> </mml:math> , <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mover> <mml:mi>ρ</mml:mi> <mml:mo stretchy="false">˜</mml:mo> </mml:mover> <mml:mn>0</mml:mn> </mml:msub> </mml:math> is an optimization parameter and µ is the exponent, describing the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mover> <mml:mi>ρ</mml:mi> <mml:mo stretchy="false">˜</mml:mo> </mml:mover> <mml:mo stretchy="false">→</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> </mml:math> asymptotic. The newly developed method provides accurate estimates of the critical exponents η , ν and als
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it