Abstract
We give a brief review of the current understanding of renormalons of the static QCD potential in coordinate and momentum spaces. We also reconsider estimate of the normalization constant of the \(u=3/2\) renormalon and propose a new way to improve the estimate.
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Notes
If we denote the UV contribution to \(\delta E_\mathrm{US}\) which cancels the IR renormalon by \(\delta E_\mathrm{US}|_{\text {UV contr.}}=V_A^2(r;\mu ) r^2 \mathcal {O}(\mu )\), it should be \(\mu \) independent. To obtain Eq. (5), we use the fact that \(\mathrm{Im} \, V_S(r)_{\pm }|_{u_*=3/2} \propto V_A^2(r;\mu ) r^2 \mathcal {O}(\mu ) =V_A^2(r;\mu _0) r^2 \mathcal {O}(\mu _0)\) and then use Eq. (6). Note that \(\exp [-2 \int _{\alpha _s(\mu _0)}^0 \mathrm{d}x \, \gamma (x)/\beta (x)] \mathcal {O}(\mu _0)\) is \(\mu _0\) independent.
If the minimal sensitivity scale is not found in the range \(1/2< \mu r <5\), we treat \(\mu r=1\) as the minimal sensitivity scale.
The final error is estimated in this way in Ref. [15] and we follow it.
Rigorously speaking, \({d_n^f}^{u_* \mathrm{(asym)}} \propto (\mu ^2 r^2)^{u_*}\) does not exactly hold in general cases because \(c_{k, u_*}\) is a polynomial of \(\log (\mu ^2 r^2)\). When a renormalon uncertainty is exactly proportional to \(\varLambda _{\overline{\mathrm{MS}}}^{2 u_*}\), \(c_{k, u_*}\) does not have \(\log (\mu ^2 r^2)\) dependence and \(d_n^{u_* \mathrm{(asym)}} \propto (\mu ^2 r^2)^{u_*}\) is exact.
In Scheme A, we assume dimensional regularization in calculating the three-loop coefficient. Then we drop the divergent term \(1/\epsilon \) (associated with the IR divergence) and set the renormalization scale to 1/r. (Both of the soft and ultra-soft renormalization scales are set to 1/r.)
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Acknowledgements
The author is grateful to Yukinari Sumino as this work is largely based on Ref. [14], which is done in collaboration with him. This work was supported by JSPS Grant-in-Aid for Scientific Research Grant Number JP19K14711.
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Appendix A: Notation and basic relations
Appendix A: Notation and basic relations
In this appendix we summarize basic knowledge on renormalon and clarify the notation used in this paper. The beta function is given by
The QCD dynamical scale in the \(\overline{\mathrm{MS}}\) scheme is defined by
We denote the dimensionless static QCD potential by v(r),
and the dimensionless QCD force by f(r),
where \(L=\log (\mu ^2 r^2)\). We define the Borel transform of such a perturbative series by
where X is v(r) or f(r) (or momentum-space potential \(\alpha _V(q)\)). Around the singularity at \(t=u_*/b_0>0\), it behaves as
where \(N_{u_*}\), \(\nu _{u_*}\), and \(c_{k, u_*}\) are parameters, and \(\cdots \) denotes a regular function at \(t=u_*/b_0\). The asymptotic behavior of the perturbative coefficient due to the first IR renormalon \(t=u_*/b_0\) follows from the above singular Borel transform as
The renormalon uncertainty of X is defined by the imaginary part of a regularized Borel integral:
This is renormalization scale independent. Writing the renormalon uncertainty as
with \(s_0=1\), we have the following relations,
and
As discussed in Sect. 2, since the renormalon uncertainties in coordinate space are given by
(where \(\mathcal {O}(\alpha _s^2)\) can be zero) one can see that \(\nu _{u_*}\) in Eq. (32) is given by
for \(u_*=1/2\) or 3/2 (where Eq. (25) is used). One can also calculate \(s_{k, u_*}\) and thus \(c_{k, u_*}\) by expanding Eq. (2) or (5) in \(\alpha _s\).
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Takaura, H. Renormalons in static QCD potential: review and some updates. Eur. Phys. J. Spec. Top. 230, 2593–2600 (2021). https://doi.org/10.1140/epjs/s11734-021-00253-3
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DOI: https://doi.org/10.1140/epjs/s11734-021-00253-3