Nonperturbative effects have a small but nontrivial impact on the determination of the strong coupling from hadronic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>τ</mi></math> decay data. Several approaches have been proposed to take these into account, the two most important of which are the “truncated Operator Product Expansion (OPE)” approach and “Duality Violation (DV)-model” approach. Recently, Pich and Rodríguez-Sánchez have raised a number of criticisms of the latter approach, including, most notably, claims of the existence of (i) a supposed instability with respect to variations of the model for incorporating quark-hadron duality violations, and (ii) an alleged redundancy in the fitting strategy employed in the DV-model approach. In this paper, we investigate these criticisms and show they fail to survive more detailed scrutiny of the mathematical or numerical arguments that underpin them. We also show that, while the redundancy claim does not apply to the DV-model approach, it does, in fact, apply to the truncated OPE approach. In particular, the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>α</mi><mi>s</mi></msub></math> value determined in the latter turns out to derive purely from perturbation theory, with no role played by the nonperturbative condensates determined in the rest of the analysis. This leads to the conclusion that a revision of the conventional understanding of what is learned from truncated OPE analyses is necessary and that only very limited self-consistency checks are possible within this framework. These observations raise new, nontrivial issues for the truncated OPE approach.

Brazil

Diogo Boito

United States

Maarten Golterman

Australia

Canada

Kim Maltman

Spain

Santiago Peris

Quark-hadron duality and the determination of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>α</mi><mi>s</mi></msub></math> from hadronic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>τ</mi></math> decay: Facts vs myths

In previous work we have combined the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>π</mi><mo>−</mo></msup><msup><mi>π</mi><mn>0</mn></msup></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>2</mn><msup><mi>π</mi><mo>−</mo></msup><msup><mi>π</mi><mo>+</mo></msup><msup><mi>π</mi><mn>0</mn></msup></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>π</mi><mo>−</mo></msup><mn>3</mn><msup><mi>π</mi><mn>0</mn></msup></math> spectral data obtained from hadronic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>τ</mi></math> decays measured by the ALEPH and OPAL experiments, together with electroproduction data for several of the subleading hadronic modes and BABAR data for the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>K</mi><mover accent="true"><mi>K</mi><mo stretchy="false">¯</mo></mover></math> mode to construct an inclusive nonstrange vector spectral function entirely based on experimental data, with no Monte-Carlo generated input. In this paper, we include, for the first time, the Belle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>τ</mi><mo stretchy="false">→</mo><msup><mi>π</mi><mo>−</mo></msup><msup><mi>π</mi><mn>0</mn></msup><msub><mi>ν</mi><mi>τ</mi></msub></math> high-statistics decay data to construct a new inclusive nonstrange vector spectral function that combines more of the world’s available data. As no Belle data are at present available for the two <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>4</mn><mi>π</mi></math> modes, this requires a revised data analysis in comparison with our previous work. From the resulting new spectral function, we obtain a new determination of the strong coupling, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>α</mi><mi>s</mi></msub></math>, using our previously developed strategy based on finite-energy sum rules. We find, at the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>Z</mi></math> mass scale, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>α</mi><mi>s</mi></msub><mo stretchy="false">(</mo><msubsup><mi>m</mi><mi>Z</mi><mn>2</mn></msubsup><mo stretchy="false">)</mo><mo>=</mo><mn>0.1159</mn><mo stretchy="false">(</mo><mn>14</mn><mo stretchy="false">)</mo></math>. We discuss the smaller central value and larger error of our new result compared to our previous result, showing the shifts to be due mainly to significant changes in updated HFLAV results for the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>π</mi><mo>−</mo></msup><mn>3</mn><msup><mi>π</mi><mn>0</mn></msup></math> decay mode.

Aaron Eiben

Lucas M. Mansur

Strong coupling from hadronic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>τ</mi></math>-decay data including <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>τ</mi><mo stretchy="false">→</mo><msup><mi>π</mi><mo>−</mo></msup><msup><mi>π</mi><mn>0</mn></msup><msub><mi>ν</mi><mi>τ</mi></msub></math> from Belle

A key issue affecting the attempt to reduce the uncertainty on the Standard Model prediction for the muon anomalous magnetic moment is the current discrepancy between lattice-QCD and data-driven results for the hadronic vacuum polarization. Progress on this issue benefits from precise data-driven determinations of the isospin-limit light-quark-connected (lqc) and strange-plus-light-quark-disconnected (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi mathvariant="normal">s</mi><mo>+</mo><mi>lqd</mi></mrow></math>) components of the related RBC/UKQCD windows. In this paper, using a strategy employed previously for the intermediate window, we provide data-driven results for the lqc and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi mathvariant="normal">s</mi><mo>+</mo><mi>lqd</mi></mrow></math> components of the short- and long-distance RBC/UKQCD windows. Comparing these results with those from the lattice, we find significant discrepancies in the lqc parts but good agreement for the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi mathvariant="normal">s</mi><mo>+</mo><mi>lqd</mi></mrow></math> components. We also explore the impact of recent CMD-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>3</mn><mtext> </mtext><mtext> </mtext><msup><mi>e</mi><mo>+</mo></msup><msup><mi>e</mi><mo>−</mo></msup><mo stretchy="false">→</mo><msup><mi>π</mi><mo>+</mo></msup><msup><mi>π</mi><mo>−</mo></msup></math> cross section results, demonstrating that an upward shift in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ρ</mi></math>-peak region of the type seen in the CMD-3 data serves to eliminate the discrepancies for the lqc components without compromising the good agreement between lattice and data-driven <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi mathvariant="normal">s</mi><mo>+</mo><mi>lqd</mi></mrow></math> results.

Genessa Benton

United Kingdom

Alexander Keshavarzi

Data-driven results for light-quark connected and strange-plus-disconnected hadronic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>g</mi><mo>−</mo><mn>2</mn></mrow></math> short- and long-distance windows

The MUonE experiment is designed to extract the hadronic contribution to the electromagnetic coupling in the spacelike region <math><mi>Δ</mi><msub><mi>α</mi><mrow><mi>had</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo></math> from elastic <math><mi>e</mi><mi>μ</mi></math> scattering. The leading-order hadronic vacuum polarization contribution to the muon <math><mi>g</mi><mo>−</mo><mn>2</mn></math>, <math><msubsup><mi>a</mi><mi>μ</mi><mrow><mi>HVP</mi><mo>,</mo><mi>LO</mi></mrow></msubsup></math>, can then be obtained from a weighted integral over <math><mi>Δ</mi><msub><mi>α</mi><mrow><mi>had</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo></math>. This, however, requires knowledge of <math><mi>Δ</mi><msub><mi>α</mi><mrow><mi>had</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo></math> in the whole domain of integration, which cannot be achieved by experiment. In this work, we propose to use Padé and D-Log Padé approximants as a systematic and model-independent method to fit and reliably extrapolate the future MUonE experimental data, extracting <math><msubsup><mi>a</mi><mi>μ</mi><mrow><mi>HVP</mi><mo>,</mo><mi>LO</mi></mrow></msubsup></math> with a conservative but competitive uncertainty, using no or very limited external information. The method relies on fundamental analytic properties of the two-point correlator underlying <math><msubsup><mi>a</mi><mi>μ</mi><mrow><mi>HVP</mi><mo>,</mo><mi>LO</mi></mrow></msubsup></math> and provides lower and upper bounds for the result for <math><msubsup><mi>a</mi><mi>μ</mi><mrow><mi>HVP</mi><mo>,</mo><mi>LO</mi></mrow></msubsup></math>. We demonstrate the reliability of the method using toy datasets generated from a model for <math><mi>Δ</mi><msub><mi>α</mi><mrow><mi>had</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo></math> reflecting the expected statistics of the MUonE experiment.

Cristiane Y. London

Pere Masjuan

Camilo Rojas

Model-independent extrapolation of MUonE data with Padé and D-Log approximants

A number of discrepancies have emerged between lattice computations and data-driven dispersive evaluations of the RBC/UKQCD intermediate-window-hadronic contribution to the muon anomalous magnetic moment. It is therefore interesting to obtain data-driven estimates for the light-quark-connected and strange-plus-disconnected components of this window quantity, allowing for a more detailed comparison between the lattice and data-driven approaches. The aim of this paper is to provide these estimates, extending the analysis to several other window quantities, including two windows designed to focus on the region in which the two-pion contribution is dominant. Clear discrepancies are observed for all light-quark-connected contributions considered, while good agreement with lattice results is found for strange-plus-disconnected contributions to the quantities for which corresponding lattice results exist. The largest of these discrepancies is that for the RBC/UKQCD intermediate window, where, as previously reported, our data-driven result, <math><msubsup><mi>a</mi><mi>μ</mi><mrow><mi>W</mi><mn>1</mn><mo>,</mo><mrow><mi>lqc</mi></mrow></mrow></msubsup><mo>=</mo><mn>198.9</mn><mo>(</mo><mn>1.1</mn><mo>)</mo><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>10</mn></mrow></msup></math>, is in significant tension with the results of 8 different recent lattice determinations. Our strategy is the same as recently employed in obtaining data-driven estimates for the light-quark-connected and strange-plus-disconnected components of the full leading-order hadronic vacuum polarization contribution to the muon anomalous magnetic moment. Updated versions of those earlier results are also presented, for completeness.

Data-driven estimates for light-quark-connected and strange-plus-disconnected hadronic <math><mi>g</mi><mo>−</mo><mn>2</mn></math> window quantities

We present the first data-driven result for <math><mrow><msubsup><mrow><mi>a</mi></mrow><mrow><mi>μ</mi></mrow><mrow><mi>win</mi><mo>,</mo><mi>lqc</mi></mrow></msubsup></mrow></math>, the isospin-limit light-quark connected component of the intermediate-window Hadronic-vacuum-polarization contribution to the muon anomalous magnetic moment. Our result, <math><mrow><mo>(</mo><mn>198.8</mn><mo>±</mo><mn>1.1</mn><mo>)</mo><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>10</mn></mrow></msup></mrow></math>, is in significant tension with eight recent mutually compatible high-precision lattice-QCD determinations, and provides enhanced evidence for a puzzling discrepancy between lattice and data-driven determinations of the intermediate-window quantity, one driven largely by a difference in the light-quark connected component.

Data-Driven Determination of the Light-Quark Connected Component of the Intermediate-Window Contribution to the Muon <math><mi>g</mi><mo>−</mo><mn>2</mn></math>

We describe how recent determinations of exclusive-mode contributions to <math><msubsup><mi>a</mi><mi>μ</mi><mrow><mi>LO</mi><mo>,</mo><mi>HVP</mi></mrow></msubsup></math>, the leading-order hadronic vacuum polarization contribution to the anomalous magnetic moment of the muon, can be used to provide, up to small electromagnetic (EM) corrections accessible from the lattice, a data-based dispersive determination of <math><mrow><msubsup><mrow><mi>a</mi></mrow><mrow><mi>μ</mi></mrow><mrow><mtext>lqc</mtext><mo>;</mo><mi>IL</mi></mrow></msubsup></mrow></math>, the isospin-limit, light-quark-connected contribution to <math><msubsup><mi>a</mi><mi>μ</mi><mrow><mi>LO</mi><mo>,</mo><mi>HVP</mi></mrow></msubsup></math>. Such a determination is of interest in view of the existence of a number of lattice results for this quantity, emerging evidence for a tension between lattice and dispersive determinations of <math><msubsup><mi>a</mi><mi>μ</mi><mrow><mi>LO</mi><mo>,</mo><mi>HVP</mi></mrow></msubsup></math>, and the desire to clarify the source of this tension. Taking as input for the small EM correction that must be applied to the purely data-driven dispersive determination the result <math><mo>−</mo><mn>0.93</mn><mo>(</mo><mn>58</mn><mo>)</mo><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>10</mn></mrow></msup></math> obtained in a recent BMW lattice study, we find <math><msubsup><mi>a</mi><mi>μ</mi><mrow><mtext>lqc</mtext><mo>;</mo><mi>IL</mi></mrow></msubsup></math> to be <math><mn>635.0</mn><mo>(</mo><mn>2.7</mn><mo>)</mo><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>10</mn></mrow></msup></math> if the results of Keshavarzi, Nomura, and Teubner are used for the exclusive-mode contributions and <math><mn>638.4</mn><mo>(</mo><mn>4.1</mn><mo>)</mo><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>10</mn></mrow></msup></math> if instead those of Davier, Höcker, Malaescu, and Zhang are used.

Data-based determination of the isospin-limit light-quark-connected contribution to the anomalous magnetic moment of the muon

We develop a number of sum rules comparing spectral integrals involving judiciously chosen weights to integrals over the corresponding Euclidean two-point function. The applications we have in mind are to the hadronic vacuum polarization that determines the most important hadronic correction <math><msubsup><mi>a</mi><mi>μ</mi><mrow><mi>HVP</mi></mrow></msubsup></math> to the muon anomalous magnetic moment. First, we point out how spectral weights may be chosen that emphasize narrow regions in <math><msqrt><mi>s</mi></msqrt></math>, providing a tool to investigate emerging discrepancies between data-driven and lattice determinations of <math><msubsup><mi>a</mi><mi>μ</mi><mrow><mi>HVP</mi></mrow></msubsup></math>. Alternatively, for a narrow region around the <math><mi>ρ</mi></math> mass, they may allow for a comparison of the dispersive determination of <math><msubsup><mi>a</mi><mi>μ</mi><mrow><mi>HVP</mi></mrow></msubsup></math> with lattice determinations zooming in on the region of the well-known BABAR-KLOE discrepancy. Second, we show how such sum rules can in principle be used for carrying out precision comparisons of hadronic-<math><mi>τ</mi></math>-decay-based data and <math><msup><mi>e</mi><mo>+</mo></msup><msup><mi>e</mi><mo>−</mo></msup><mo>→</mo><mtext>hadrons</mtext><mo>(</mo><mi>γ</mi><mo>)</mo></math>-based data, where lattice computations can provide the necessary isospin-breaking corrections.

Spectral-weight sum rules for the hadronic vacuum polarization

In a previous article, we have shown that the discrepancy between the fixed-order (FOPT) and contour-improved (CIPT) perturbative expansions for τ hadronic spectral function moments, which had affected the precision of α s determinations for many years, may be reconciled by employing a renormalon-free (RF) scheme for the gluon condensate (GC) matrix element. In addition, the perturbative convergence of spectral function moments with a sizeable GC correction can be improved. The RF GC scheme depends on an IR factorization scale R and the normalization N g of the GC renormalon. In the present work, we use three different methods to determine N g , yielding a result with an uncertainty of 40%. Following two recent state-of-the-art strong coupling determination analyses at <math> <mi>O</mi> </math> $$ \mathcal{O} $$ ( <math> <msubsup> <mi>α</mi> <mi>s</mi> <mn>5</mn> </msubsup> </math> $$ {\alpha}_s^5 $$ ), we show that using the renormalon-free GC scheme successfully reconciles the results for α s ( <math> <msubsup> <mi>m</mi> <mi>τ</mi> <mn>2</mn> </msubsup> </math> $$ {m}_{\tau}^2 $$ ) based on CIPT and FOPT. The uncertainties due to variations of R and the uncertainty of N g only lead to a small or moderate increase of the final uncertainty of α s ( <math> <msubsup> <mi>m</mi> <mi>τ</mi> <mn>2</mn> </msubsup> </math> $$ {m}_{\tau}^2 $$ ), and affect mainly the CIPT expansion method. The FOPT and CIPT results obtained in the RF GC scheme may be consistently averaged. The RF GC scheme thus constitutes a powerful new ingredient for future analyses of τ hadronic spectral function moments.

Austria

Miguel Benitez-Rathgeb

André Hoang

Germany

Matthias Jamin

Reconciling the contour-improved and fixed-order approaches for τ hadronic spectral moments. Part II. Renormalon norm and application in α  s  determinations

We propose a simple and easy-to-implement scheme for a renormalon-free gluon condensate (GC) matrix element, which is analogous to implementations of short-distance heavy-quark mass renormalization schemes existing in the literature already for a long time. Because the scheme is based on a perturbative subtraction at the level of the matrix element, with a freely adaptable infrared factorization scale, it can be implemented with little effort for any observable where the GC is relevant. The scheme depends on the renormalon norm of the GC which has to be supplemented independently. We apply the scheme to the fixed-order (FOPT) and contour-improved (CIPT) perturbative expansions of τ hadronic spectral function moments. These expansions exhibit a long-standing discrepancy for moments used in high-precision determinations of the strong coupling in the commonly used GC scheme that is not renormalon-free. We show that the scheme is capable of resolving the FOPT-CIPT discrepancy problem. At the same time, the perturbative behaviour of the moments that previously showed bad convergence properties and for which the non-perturbative corrections from the GC are sizeable, is substantially improved. The new GC scheme may provide a powerful theoretical tool for future phenomenological applications.