Achieving Effective Renormalization Scale and Scheme Independence via the Principle of Observable Effective Matching
Bibliographic record
Abstract
In this work, we explicate a new approach for eliminating renormalization scale and scheme (RSS) dependence in observables. We develop this approach by matching RSS-dependent observables (such as cross-sections and decay rates) to a theory which is independent of both these forms of dependencies. We term the fundamental basis behind this approach as the principle of observable effective matching (POEM), which entails matching of a scale- and scheme-dependent observable with the fully physical scale (PS) and dynamical scale-dependent theory at loop orders at which RSS independence is guaranteed. This is aimed toward achieving so-called “effective” RSS-independent expressions as the resulting dynamical dependence is derived from a particular order in RSS-dependent perturbation theory. With this matching at a PS at which the coupling (and masses) is experimentally determined at this scale, we obtain an “effective theoretical observable (ETO)”, a finite-order RSS-independent version of the RSS-dependent observable. We illustrate our approach with a study of the cross-section ratio <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m1"><mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>e</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>e</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:mrow></mml:msub></mml:math> for e + e − → hadrons, which is demonstrated to achieve scale and scheme independence utilizing the three- and four-loop order <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m2"><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>M</mml:mi><mml:mi>S</mml:mi></mml:mrow><mml:mo>̄</mml:mo></mml:mover></mml:mrow></mml:math> scheme expression in QCD perturbation theory via matching at both one-loop and two-loop orders for obtaining the ETO. With two-loop matching, we obtain an ETO prediction of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m3"><mml:mfrac><mml:mrow><mml:mn>3</mml:mn></mml:mrow><mml:mrow><mml:mn>11</mml:mn></mml:mrow></mml:mfrac><mml:msubsup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>e</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>e</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mi>e</mml:mi><mml:mi>f</mml:mi><mml:mi>f</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mn>1.05243</mml:mn><mml:msubsup><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>0.0006</mml:mn></mml:mrow><mml:mrow><mml:mo>+</mml:mo><mml:mn>0.0006</mml:mn></mml:mrow></mml:msubsup></mml:math> at Q = 31.6 GeV , which is in excellent agreement with the experimental value of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m4"><mml:mfrac><mml:mrow><mml:mn>3</mml:mn></mml:mrow><mml:mrow><mml:mn>11</mml:mn></mml:mrow></mml:mfrac><mml:msubsup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>e</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>e</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mi>e</mml:mi><mml:mi>x</mml:mi><mml:mi>p</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mn>1.052</mml:mn><mml:msubsup><mml:mrow><mml:mn>7</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>0.005</mml:mn></mml:mrow><mml:mrow><mml:mo>+</mml:mo><mml:mn>0.005</mml:mn></mml:mrow></mml:msubsup></mml:math> . Given its new conceptual basis, ease of use, and performance, we contend that POEM be explored in its application for obtaining ETOs for predicting RSS-independent observables across domains of high-energy theory and phenomenology as well as other areas of fundamental and applied physics, such as cosmology and statistical and condensed matter physics.
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How this classification was reachedexpand
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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".