Abstract
Estimates of higher-order contributions for perturbative series in QCD, in view of their asymptotic nature, are delicate, though indispensable for a reliable error assessment in phenomenological applications. In this work, the Adler function and the scalar correlator are investigated, and models for Borel transforms of their perturbative series are constructed, which respect general constraints from the operator product expansion and the renormalisation group. As a novel ingredient, the QCD coupling is employed in the so-called C-scheme, which has certain advantages. For the Adler function, previous results obtained directly in the \(\overline{\mathrm{MS}}\) scheme are supported. Corresponding results for the scalar correlation function are new. It turns out that the substantially larger perturbative corrections for the scalar correlator in \(\overline{\mathrm{MS}}\) are dominantly due to this scheme choice, and can be largely reduced through more appropriate renormalisation schemes, which are easy to realise in the C-scheme.
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Notes
For historical reasons, we shall speak about the “large-\(\beta _0\)” approximation, although in the notation employed in this work, the leading coefficient of the \(\beta \)-function is termed \(\beta _1\).
It only depends on the scheme-invariant \(\beta \)-function coefficients \(\beta _1\) and \(\beta _2\).
For notational simplicity, in the ensuing discussion we have dropped the superscript C in the C-scheme coupling \({\hat{a}}_Q^C\).
This can also be inferred from the general form of an invariant, polynomial contribution in the C-scheme, which is presented in appendix B.
References
J.C. Le-Guillou, J. Zinn-Justin, Large Order Behavior of Perturbation Theory (North-Holland, Amsterdam, 1990)
M. Beneke, Renormalons. Phys. Rept. 317, 1–142 (1999). arXiv:hep-ph/9807443
S.L. Adler, Some simple vacuum polarization phenomenology: \(e^+e^-\rightarrow \,\) Hadrons. Phys. Rev. D 10, 3714 (1974)
M. Beneke, M. Jamin, \(\alpha _s\) and the \(\tau \) hadronic width: fixed-order, contour-improved and higher-order perturbation theory. JHEP 0809, 044 (2008). arXiv: 0806.3156 [hep-ph]
M. Beneke, D. Boito, M. Jamin, Perturbative expansion of \(\tau \) hadronic spectral function moments and \(\alpha _s\) extractions. JHEP 01, 125 (2013). arXiv:1210.8038 [hep-ph]
D. Boito, M. Jamin, R. Miravitllas, Scheme variations of the QCD coupling and hadronic \(\tau \) decays. Phys. Rev. Lett. 117, 152001[arXiv: 1606.06175 [hep-ph]] (2016). arXiv:1606.06175 [hep-ph]
M. Jamin, R. Miravitllas, Scalar correlator. Higgs decay into quarks, and scheme variations of the QCD coupling. JHEP 1610, 059 (2016). arXiv:1606.06166 [hep-ph]
W.A. Bardeen, A.J. Buras, D.W. Duke, T. Muta, Deep inelastic scattering beyond the leading order in asymptotically free gauge theories. Phys. Rev. D 18, 3998 (1978)
X.G. Wu, J.M. Shen, B.L. Du, S.J. Brodsky, Novel demonstration of the renormalization group invariance of the fixed-order predictions using the principle of maximum conformality and the \(C\)- scheme coupling. Phys. Rev. D 97, 094030 (2018). arXiv:1802.09154 [hep-ph]
S.G. Gorishnii, A.L. Kataev, S.A. Larin, The \({{\cal{O}}}(\alpha _s^3)\) corrections to \(\sigma _{{\rm tot}}(e^+ e^- \rightarrow {\rm hadrons})\) and \({\Gamma }(\tau ^- \rightarrow \nu _\tau + {\rm hadrons})\) in QCD. Phys. Lett. B 259, 144 (1991)
L.R. Surguladze, M.A. Samuel, Total hadronic cross-section in \(e^+ e^-\) annihilation at the four-loop level of perturbative QCD. Phys. Rev. Lett. 66, 560 (1991)
P.A. Baikov, K.G. Chetyrkin, J.H. Kühn, Hadronic \(Z\)- and \(\tau \)-Decays in Order \(\alpha _s^4\). Phys. Rev. Lett. 101, 012002 (2008). arXiv:0801.1821 [hep-ph]
D. Boito, M. Jamin, R. Miravitllas, Scheme variations of the QCD coupling. EPJ Web Conf. 137, 05007 (2017). arXiv:1612.01792 [hep-ph
S.G. Gorishnii, A.L. Kataev, S.A. Larin, L.R. Surguladze, Corrected three loop QCD correction to the correlator of the quark scalar currents and \(\Gamma _{\rm tot}(H_0\rightarrow {\rm hadrons})\). Mod. Phys. Lett. A 5, 2703 (1990)
K.G. Chetyrkin, Correlator of the quark scalar currents and \(\Gamma _{\rm tot}(H_0\rightarrow {\rm hadrons})\) at \({{\cal{O}}}(\alpha _s^3)\) in pQCD. Phys. Lett. B 390, 309 (1997). arXiv:hep-ph/9608318
P.A. Baikov, K.G. Chetyrkin, J.H. Kühn, Scalar correlator at \({{\cal{O}}}(\alpha _s^4)\), Higgs decay into \(b\)- quarks and bounds on the light quark masses. Phys. Rev. Lett. 96, 012003 (2006). arXiv:hep-ph/0511063
D.J. Broadhurst, Chiral symmetry breaking and perturbative QCD. Phys. Lett. 101B, 423 (1981)
P.A. Baikov, K.G. Chetyrkin, J.H. Kühn, Five-loop fermion anomalous dimension for a general gauge group from four-loop massless propagators. JHEP 1704, 119 (2017). arXiv:1702.01458 [hep-ph]
M. Jamin, R. Miravitllas, Absence of even-integer \(\zeta \)- function values in Euclidean physical quantities in QCD. Phys. Lett. B 779, 452 (2018). arXiv:1711.00787 [hep-ph]
J. Davies, A. Vogt, Absence of \(\pi ^2\) terms in physical anomalous dimensions in DIS: verification and resulting predictions. Phys. Lett. B 776, 189 (2018). arXiv:1711.05267 [hep-ph]
P.A. Baikov, K.G. Chetyrkin, The structure of generic anomalous dimensions and no-\(\pi \)theorem for massless propagators (2019). arXiv:1804.10088 [hep-ph]
D. Boito, D. Hornung, M. Jamin, Anomalous dimensions of four-quark operators and renormalon structure of mesonic two-point correlators. JHEP 1512, 090 (2015). arXiv:1510.03812 [hep-ph]
M. Jamin, The scalar gluonium correlator: large-\(\beta _0\) and beyond. JHEP 1204, 099 (2012). arXiv:1202.1169 [hep-ph]
L.S. Brown, L.G. Yaffe, C.X. Zhai, Large-order perturbation theory for the electromagnetic current current correlation function. Phys. Rev. D 46, 4712 (1992). arXiv: hep-ph/9205213
G. Grunberg, The renormalization scheme invariant Borel transform and the QED renormalons. Phys. Lett. B 304, 183 (1993)
D. Boito, F. Oliani, Renormalons in integrated spectral function moments and \(\alpha _s\) extractions. Phys. Rev. D 101, 074003 (2020). arXiv:2002.12419 [hep-ph]
M. Beneke, Large-order perturbation theory for a physical quantity. Nucl. Phys. B 405, 424 (1993)
D.J. Broadhurst, Large-\(N\) expansion of QED: Asymptotic photon propagator and contributions to the muon anomaly, for any number of loops. Z. Phys. C 58, 339 (1993)
D.J. Broadhurst, A.L. Kataev, C.J. Maxwell, Renormalons and multiloop estimates in scalar correlators, Higgs decay and quark-mass sum rule. Nucl. Phys. B 592, 247 (2001). arXiv:hep-ph/0007152
V.P. Spiridonov, K.G. Chetyrkin, Nonleading mass corrections and renormalization of the operators \(m{\bar{\psi }}\psi \) and \(G_{\mu \nu }^2\). Sov. J. Nucl. Phys. 47, 522 (1988) [Yad. Fiz. 47 (1988) 818]
L.R. Surguladze, F.V. Tkachov, Two-loop effects in QCD sum rules for light mesons. Nucl. Phys. B 331, 35 (1990)
D. Boito, P. Masjuan, F. Oliani, Higher-order QCD corrections to hadronic \(\tau \) decays from Padé approximants. JHEP 08, 075 (2018). arXiv:1807.01567 [hep-ph]
I. Caprini, Higher-order perturbative coefficients in QCD from series acceleration by conformal mappings. Phys. Rev. D 100, 056019 (2019). arXiv:1908.06632 [hep-ph]
M. Jamin, R. Miravitllas, (2021) (work in progress)
O.V. Tarasov, A.A. Vladimirov, Y.A. Zharkov, The Gell–Mann-Low function of QCD in the three-loop approximation. Phys. Lett. B 93, 429 (1980)
T. van Ritbergen, J.A.M. Vermaseren, S.A. Larin, The four-loop \(\beta \)-function in quantum chromodynamics. Phys. Lett. B 400, 379 (1997). arXiv:hep-ph/9701390
M. Czakon, The four-loop QCD \(\beta \)-function and anomalous dimensions. Nucl. Phys. B 710, 485 (2005). arXiv:hep-ph/0411261
P.A. Baikov, K.G. Chetyrkin, J.H. Kühn, Five-loop running of the QCD coupling constant (2019). arXiv:1606.08659 [hep-ph]
J.A.M. Vermaseren, S.A. Larin, T. van Ritbergen, The four-loop quark mass anomalous dimension and the invariant quark mass. Phys. Lett. B 405, 327 (1997). arXiv:hep-ph/9703284
P.A. Baikov, K.G. Chetyrkin, J.H. Kühn, Quark mass and field anomalous dimensions to \({{\cal{O}}}(\alpha _s^5)\). JHEP 1410, 76 (2014). arXiv:1402.6611 [hep-ph]
Acknowledgements
Partial collaboration in this work with Ramon Miravitllas, and interesting discussions with Andre Hoang, are gratefully acknowledged. The author would also like to thank the FWF Austrian Science Fund under the Project No. P28535-N27 for partial support, and the particle physics group at the University of Vienna, where part of this work was completed.
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Appendices
Appendix A: Renormalisation group functions and dependent coefficients
In our notation, the QCD \(\beta \)-function and mass anomalous dimension are defined as:
It is assumed that we work in a mass-independent renormalisation scheme and in this study throughout the modified minimal subtraction scheme \({\overline{\mathrm{MS}}}\) is used. To make the presentation self-contained, below the known coefficients of the \(\beta \)-function and mass anomalous dimension in the given conventions shall be provided. Numerically, for \(N_c=3\) and \(N_f=3\), the first five coefficients of the \(\beta \)-function are given by [35,36,37,38]
and the first five for \(\gamma _m(a)\) are found to be [39, 40]
The dependent perturbative coefficients \(d_{n,k}\) with \(k>1\) can be expressed in terms of the independent coefficients \(d_{n,1}\), and coefficients of the QCD \(\beta \)-function and mass anomalous dimension. In particular, the coefficients \(d_{n,2}\), which are required in Eq. (33), take the form
Appendix B: General scheme-invariant structure
In this appendix, the general scheme-invariant structure of a two-point correlation function in the C-scheme will be provided, which for example has to be obeyed by the polynomial contribution in Eq. (69). Denoting the structure by \(P({\hat{a}}_Q)\), it takes the general form
where \(\widehat{C}\equiv \beta _1/2\,C\). Relations between the coefficients \(y_{n,k}\) can be obtained from the RG equation (6). Up to order \({\hat{a}}_Q^4\), and setting \(\lambda = \beta _2/\beta _1\), those relations read:
If still higher orders are required, it is an easy matter to compute them from the RG equation. Like for the two-point correlators, the coefficients \(y_{n,0}\) cannot be determined from the renormalisation group, and can be considered independent.
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Jamin, M. Higher-order behaviour of two-point current correlators. Eur. Phys. J. Spec. Top. 230, 2609–2624 (2021). https://doi.org/10.1140/epjs/s11734-021-00266-y
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DOI: https://doi.org/10.1140/epjs/s11734-021-00266-y