# The generalized scheme-independent Crewther relation in QCD

## Abstract

The Principle of Maximal Conformality (PMC) provides a systematic way to set the renormalization scales order-by-order for any perturbative QCD calculable processes. The resulting predictions are independent of the choice of renormalization scheme, a requirement of renormalization group invariance. The Crewther relation, which was originally derived as a consequence of conformally invariant field theory, provides a remarkable connection between two observables when the β function vanishes: one can show that the product of the Bjorken sum rule for spin-dependent deep inelastic lepton–nucleon scattering times the Adler function, defined from the cross section for electron–positron annihilation into hadrons, has no pQCD radiative corrections. The “Generalized Crewther Relation” relates these two observables for physical QCD with nonzero β function; specifically, it connects the non-singlet Adler function (D ^{ns}) to the Bjorken sum rule coefficient for polarized deep-inelastic electron scattering (C _{Bjp}) at leading twist. A scheme-dependent Δ _{CSB}-term appears in the analysis in order to compensate for the conformal symmetry breaking (CSB) terms from perturbative QCD. In conventional analyses, this normally leads to unphysical dependence in both the choice of the renormalization scheme and the choice of the initial scale at any finite order. However, by applying PMC scale-setting, we can fix the scales of the QCD coupling unambiguously at every order of pQCD. The result is that both D ^{ns} and the inverse coefficient C$$-1\atop{Bjp}$$ have identical pQCD coefficients, which also exactly match the coefficients of the corresponding conformal theory. Thus one obtains a new generalized Crewther relation for QCD which connects two effective charges, $$\hat{α}$$ _{d}(Q)=Σ _{i≥1}$$\hat{α}^i\atop{g1}$$(Qi), at their respective physical scales. This identity is independent of the choice of the renormalization scheme at any finite order, and the dependence on the choice of the initial scale is negligible. Lastly, similar scale-fixed commensurate scale relations also connect other physical observables at their physical momentum scales, thus providing convention-independent, fundamental precision tests of QCD.

- Authors:

- Chongqing Univ. (China). Dept. of Physics
- Univ. of Pittsburgh, PA (United States). Dept. of Physics and Astronomy
- SLAC National Accelerator Lab., Menlo Park, CA (United States)

- Publication Date:

- Research Org.:
- SLAC National Accelerator Lab., Menlo Park, CA (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), High Energy Physics (HEP) (SC-25); National Natural Science Foundation of China (NNSFC)

- OSTI Identifier:
- 1373120

- Report Number(s):
- SLAC-PUB-16845

Journal ID: ISSN 0370-2693; PII: S0370269317303830

- Grant/Contract Number:
- 11625520; 11275280; AC02-76SF00515

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Physics Letters. Section B

- Additional Journal Information:
- Journal Volume: 770; Journal Issue: C; Journal ID: ISSN 0370-2693

- Publisher:
- Elsevier

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS

### Citation Formats

```
Shen, Jian-Ming, Wu, Xing-Gang, Ma, Yang, and Brodsky, Stanley J.
```*The generalized scheme-independent Crewther relation in QCD*. United States: N. p., 2017.
Web. doi:10.1016/j.physletb.2017.05.022.

```
Shen, Jian-Ming, Wu, Xing-Gang, Ma, Yang, & Brodsky, Stanley J.
```*The generalized scheme-independent Crewther relation in QCD*. United States. doi:10.1016/j.physletb.2017.05.022.

```
Shen, Jian-Ming, Wu, Xing-Gang, Ma, Yang, and Brodsky, Stanley J. Wed .
"The generalized scheme-independent Crewther relation in QCD". United States.
doi:10.1016/j.physletb.2017.05.022. https://www.osti.gov/servlets/purl/1373120.
```

```
@article{osti_1373120,
```

title = {The generalized scheme-independent Crewther relation in QCD},

author = {Shen, Jian-Ming and Wu, Xing-Gang and Ma, Yang and Brodsky, Stanley J.},

abstractNote = {The Principle of Maximal Conformality (PMC) provides a systematic way to set the renormalization scales order-by-order for any perturbative QCD calculable processes. The resulting predictions are independent of the choice of renormalization scheme, a requirement of renormalization group invariance. The Crewther relation, which was originally derived as a consequence of conformally invariant field theory, provides a remarkable connection between two observables when the β function vanishes: one can show that the product of the Bjorken sum rule for spin-dependent deep inelastic lepton–nucleon scattering times the Adler function, defined from the cross section for electron–positron annihilation into hadrons, has no pQCD radiative corrections. The “Generalized Crewther Relation” relates these two observables for physical QCD with nonzero β function; specifically, it connects the non-singlet Adler function (Dns) to the Bjorken sum rule coefficient for polarized deep-inelastic electron scattering (CBjp) at leading twist. A scheme-dependent ΔCSB-term appears in the analysis in order to compensate for the conformal symmetry breaking (CSB) terms from perturbative QCD. In conventional analyses, this normally leads to unphysical dependence in both the choice of the renormalization scheme and the choice of the initial scale at any finite order. However, by applying PMC scale-setting, we can fix the scales of the QCD coupling unambiguously at every order of pQCD. The result is that both Dns and the inverse coefficient C$-1\atop{Bjp}$ have identical pQCD coefficients, which also exactly match the coefficients of the corresponding conformal theory. Thus one obtains a new generalized Crewther relation for QCD which connects two effective charges, $\hat{α}$d(Q)=Σi≥1$\hat{α}^i\atop{g1}$(Qi), at their respective physical scales. This identity is independent of the choice of the renormalization scheme at any finite order, and the dependence on the choice of the initial scale is negligible. Lastly, similar scale-fixed commensurate scale relations also connect other physical observables at their physical momentum scales, thus providing convention-independent, fundamental precision tests of QCD.},

doi = {10.1016/j.physletb.2017.05.022},

journal = {Physics Letters. Section B},

number = C,

volume = 770,

place = {United States},

year = {Wed May 10 00:00:00 EDT 2017},

month = {Wed May 10 00:00:00 EDT 2017}

}