Precise perturbative predictions from fixed-order calculations

Abstract

Abstract The intrinsic conformality (iCF) is a general property of the renormalizable gauge theory, which ensures the scale invariance of a fixed-order series at each perturbative order. Following the idea of iCF, we suggest a novel single-scale setting approach under the principle of maximum conformality (PMC) with the purpose of removing the conventional renormalization scheme and scale ambiguities. We call this newly suggested single-scale procedure the PMC ∞ -s approach, in which an overall effective α s , and hence an overall effective scale is achieved by identifying the { β 0 } terms at each order. Its resultant conformal series is scale-invariant and satisfies all renormalization group requirements. The PMC ∞ -s approach is applicable to any perturbatively calculable observable, and its resultant perturbative series provides an accurate basis for estimating the contribution from the unknown higher-order (UHO) terms. Using the Higgs decay into two gluons up to five-loop quantum chromondynamics (QCD) corrections as an example, we show how the PMC ∞ -s works, and we obtain <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msubsup> <mml:mrow> <mml:mfenced close="∣" open=""> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi mathvariant="normal">Γ</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">H</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:mfenced> </mml:mrow> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>PMC</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>∞</mml:mo> </mml:mrow> </mml:msub> <mml:mo>−</mml:mo> <mml:mi mathvariant="normal">s</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>PAA</mml:mi> </mml:mrow> </mml:msubsup> <mml:mo>=</mml:mo> <mml:msubsup> <mml:mrow> <mml:mn>334.45</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>7.03</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>+</mml:mo> <mml:mn>7.07</mml:mn> </mml:mrow> </mml:msubsup> <mml:mspace width="0.25em"/> <mml:mi>KeV</mml:mi> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msubsup> <mml:mrow> <mml:mfenced close="∣" open=""> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi mathvariant="normal">Γ</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">H</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:mfenced> </mml:mrow> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>PMC</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>∞</mml:mo> </mml:mrow> </mml:msub> <mml:mo>−</mml:mo> <mml:mi mathvariant="normal">s</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">B</mml:mi> <mml:mi>.</mml:mi> <mml:mi mathvariant="normal">A</mml:mi> <mml:mi>.</mml:mi> </mml:mrow> </mml:msubsup> <mml:mo>=</mml:mo> <mml:msubsup> <mml:mrow> <mml:mn>334.45</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>6.29</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>+</mml:mo> <mml:mn>6.34</mml:mn> </mml:mrow> </mml:msubsup> <mml:mspace width="0.25em"/> <mml:mi>KeV</mml:mi> </mml:math> . Here the errors are squared averages of those mentioned in the body of the text. The <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>Pad</mml:mi> <mml:mover accent="true"> <mml:mrow> <mml:mi>e</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>́</mml:mo> </mml:mrow> </mml:mover> </mml:math> approximation approach (PAA) and the Bayesian approach (BA) have been adopted to estimate the contributions from the UHO terms. We also demonstrate that the PMC ∞ -s approach is equivalent to our previously suggested single-scale setting approach (PMCs), which also follows from the PMC but treats the { β i } terms from a different point of view. Thus, proper use of the renormalization group equation can provide a solid way to solve the scale-setting problem.

Topics & Concepts