# Evidence against naive truncations of the OPE from ${e}^{+}{e}^{-}\to \text{hadrons}$ below charm

## Abstract

The operator product expansion (OPE), truncated in dimension, is employed in many contexts. An example is the extraction of the strong coupling, $$\alpha_s$$, from hadronic $$\tau$$-decay data, using a variety of analysis methods based on finite-energy sum rules. Here, we reconsider a long-used method, which parametrizes non-perturbative contributions to the $I=1$ vector and axial vacuum polarizations with the OPE, setting several higher-dimension coefficients to zero in order to implement the method in practice. The assumption that doing this has a negligible effect on the value of $$\alpha_s$$ is tantamount to the assumption that the low-dimension part of the OPE converges rapidly with increasing dimension near the $$\tau$$ mass. Were this assumption valid, it would certainly have to be valid at energies above the $$\tau$$ mass as well. It follows that the method can be tested using data obtained from $$e^+e^-\to\mbox{hadrons}$$, as they are not limited by the kinematic constraints of $$\tau$$ decays. We carry out such an investigation using a recent high-precision compilation for the $$R$$ ratio, arguing that it provides insights into the validity of the strategy, even if it probes a different, though related channel. We find that $e^+e^-$-based tests call into question the implied assumption of rapid convergence of the low-dimension part of the OPE around the $$\tau$$ mass, and thus underscore the need to restrict finite-energy sum-rule analyses to observables which receive only contributions from lower-order terms in the OPE.

- Authors:

- Publication Date:

- Research Org.:
- San Francisco State Univ., San Francisco, CA (United States); San Francisco State Univ., CA (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), High Energy Physics (HEP); São Paulo Research Foundation (FAPESP)

- OSTI Identifier:
- 1570035

- Alternate Identifier(s):
- OSTI ID: 1569841; OSTI ID: 1594604; OSTI ID: 1901683

- Grant/Contract Number:
- FG03-92ER40711; SC0013682; 2015/20689-9; 309847/2018-4; CICYTFEDER-FPA2017-86989-P

- Resource Type:
- Published Article

- Journal Name:
- Physical Review D

- Additional Journal Information:
- Journal Name: Physical Review D Journal Volume: 100 Journal Issue: 7; Journal ID: ISSN 2470-0010

- Publisher:
- American Physical Society (APS)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; Quantum chromodynamics; Strong interaction

## Citation Formats

```
Boito, Diogo, Golterman, Maarten, Maltman, Kim, and Peris, Santiago. Evidence against naive truncations of the OPE from e + e − → hadrons below charm. United States: N. p., 2019.
Web. doi:10.1103/PhysRevD.100.074009.
```

```
Boito, Diogo, Golterman, Maarten, Maltman, Kim, & Peris, Santiago. Evidence against naive truncations of the OPE from e + e − → hadrons below charm. United States. https://doi.org/10.1103/PhysRevD.100.074009
```

```
Boito, Diogo, Golterman, Maarten, Maltman, Kim, and Peris, Santiago. Thu .
"Evidence against naive truncations of the OPE from e + e − → hadrons below charm". United States. https://doi.org/10.1103/PhysRevD.100.074009.
```

```
@article{osti_1570035,
```

title = {Evidence against naive truncations of the OPE from e + e − → hadrons below charm},

author = {Boito, Diogo and Golterman, Maarten and Maltman, Kim and Peris, Santiago},

abstractNote = {The operator product expansion (OPE), truncated in dimension, is employed in many contexts. An example is the extraction of the strong coupling, $\alpha_s$, from hadronic $\tau$-decay data, using a variety of analysis methods based on finite-energy sum rules. Here, we reconsider a long-used method, which parametrizes non-perturbative contributions to the $I=1$ vector and axial vacuum polarizations with the OPE, setting several higher-dimension coefficients to zero in order to implement the method in practice. The assumption that doing this has a negligible effect on the value of $\alpha_s$ is tantamount to the assumption that the low-dimension part of the OPE converges rapidly with increasing dimension near the $\tau$ mass. Were this assumption valid, it would certainly have to be valid at energies above the $\tau$ mass as well. It follows that the method can be tested using data obtained from $e^+e^-\to\mbox{hadrons}$, as they are not limited by the kinematic constraints of $\tau$ decays. We carry out such an investigation using a recent high-precision compilation for the $R$ ratio, arguing that it provides insights into the validity of the strategy, even if it probes a different, though related channel. We find that $e^+e^-$-based tests call into question the implied assumption of rapid convergence of the low-dimension part of the OPE around the $\tau$ mass, and thus underscore the need to restrict finite-energy sum-rule analyses to observables which receive only contributions from lower-order terms in the OPE.},

doi = {10.1103/PhysRevD.100.074009},

journal = {Physical Review D},

number = 7,

volume = 100,

place = {United States},

year = {2019},

month = {10}

}

https://doi.org/10.1103/PhysRevD.100.074009

#### Figures / Tables:

*R*ratio in the region 2 ≤

*s*≤ 6 GeV

^{2}. The red solid and red dashed lines show the results obtained from perturbation theory with

*α*

_{s}(more »

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*Figures/Tables have been extracted from DOE-funded journal article accepted manuscripts.*