UV contributions to energy of a static quark-antiquark pair in large $\beta_0$ approximation

TL;DR

This paper calculates the ultraviolet contributions to the energy of a static quark-antiquark pair within the large-beta_0 approximation, clarifying the UV part's role in the operator product expansion and renormalon cancellation.

Contribution

It extends a recent method to compute the UV part of the total energy in terms of the MS-bar mass, including the r-independent component, aiding understanding of renormalon effects.

Findings

Computed the genuine UV part of the energy in the large-beta_0 approximation.

Extended a method consistent with the operator product expansion.

Determined the r-independent part of the energy.

Abstract

The total energy of a static quark-antiquark pair $E_{\rm tot}(r)=2m_{\rm pole}+V_{\rm QCD}(r)$ is known to be predictable up to ${\cal O}(\Lambda_{\rm QCD}^3 r^2)$ and ${\cal O}(\frac{\Lambda_{\rm QCD}^3}{\overline{m}^2})$ renormalon uncertainties in the large-$\beta_0$ approximation, after canceling ${\cal O}(\Lambda_{\rm QCD})$ renormalons. We compute the predictable part (genuine UV part) in terms of the $\overline{\rm MS}$ mass $\overline{m}$, extending a recently-proposed method which conforms with OPE. In particular the $r$-independent part is determined. The result would help understanding the nature of $E_{\rm tot}(r)$ in the context of OPE with renormalon subtraction.

Equations

$$V_{\mathrm{QCD}}(r)=V_C(r)+C_0^V+C_1^{V}r+\mathcal{O}({\Lambda }_{\mathrm{QCD}}^3{r}^2)\text{mbox}forr\ll {\Lambda }_{\mathrm{QCD}}^{-1},$$

$$m_{\mathrm{pole}}=\overline{m}+\Delta M_0(\overline{m})+C_0^m+\frac{C_1^m}{\overline{m}}+\mathcal{O}({\Lambda }_{\mathrm{QCD}}^3/{\overline{m}}^2)\text{mbox}for\overline{m}\gg {\Lambda }_{\mathrm{QCD}},$$

$$\delta m(\mu )$$

$$=-4\pi i{C}_F\overline{m}(\mu )\int \frac{d^{D}q}{{\left(2\pi \right)}^D}\frac{2+\left(D-2\right)\frac{p\cdot q}{\overline{m}(\mu {)}^2}}{\left(q^2+2p\cdot q+i0\right)\left(q^2+i0\right)}\frac{{\alpha }_s(\mu ){\overline{\mu }}^{2ϵ}}{1-\Pi (q^2,ϵ)}+(c.t.).$$

$$\frac{1}{q_E^2}={\int }_0^{\infty }\frac{d\tau }{\tau }\delta (\tau -q_E^2)=\mathrm{Im}{\int }_0^{\infty }\frac{d\tau }{\pi \tau }\frac{1}{q_E^2-\tau -i0}.$$

$$\delta \equiv \frac{\delta m}{\overline{m}}=\mathrm{Im}{\int }_0^{\infty }\frac{d\tau }{\pi \tau }{W}_m\left(\frac{\tau }{{\overline{m}}^2},\frac{\overline{\mu }}{\overline{m}};ϵ\right)\frac{{\alpha }_s(\mu )}{1-\Pi (-\tau ,ϵ)}+(c.t.),$$

$$W_m=\frac{C_F}{4\pi }\Gamma (ϵ){\int }_0^{1}dx(2+2x-2ϵx){\left(\frac{4\pi {\bar{\mu }}^2}{{\overline{m}}^2{x}^2+\tau x-\tau -i0}\right)}^{ϵ}.$$

$$\frac{{\alpha }_s(\mu )}{1-\Pi (-\tau ,ϵ)}\to {\alpha }_{{\beta }_0}(\tau )=\frac{4\pi /{\beta }_0}{\log \left(\tau /{\Lambda }^{\prime 2}\right)};{\Lambda }^{\prime }=e^{5/6}{\Lambda }_{\mathrm{QCD}}$$

$$\delta$$

$$G(t;\mu )=\sum _{n=0}^{\infty }{G}_n(\mu )t^n={\left(\frac{e^{5/3}{\mu }^2}{\overline{m}(\mu {)}^2}\right)}^{t}3(1-t)\frac{\Gamma (1+t)\Gamma (1-2t)}{\Gamma (3-t)},$$

$$g(t)=\sum _{n=0}^{\infty }{g}_n{t}^n=\frac{3-2t}{6}\frac{\Gamma (4-2t)}{\Gamma (1+t)\Gamma (2-t{)}^2\Gamma (3-t)}.$$

$${\gamma }_n={\left(\frac{-{\beta }_0}{4}\right)}^n{C}_F{g}_n,i.e.,{\gamma }_m^{({\beta }_0)}({\alpha }_s)=C_{F}g\left(-\frac{{\beta }_0{\alpha }_s}{4\pi }\right),$$

$$d_n=\frac{C_F}{2}{\left(\frac{{\beta }_0}{4}\right)}^n\left[G_{n+1}(\overline{m})n!+\frac{(-1{)}^n}{n+1}{g}_{n+1}\right]\text{mbox}with\delta =\sum _{n=0}^{\infty }{d}_n{\left(\frac{{\alpha }_s(\overline{m})}{\pi }\right)}^{n+1},$$

$$\mu \frac{d}{d\mu }\left(\frac{{\alpha }_s(\mu )}{\pi }\right)=-\frac{{\beta }_0}{2}{\left(\frac{{\alpha }_s(\mu )}{\pi }\right)}^2,$$

$$\mu \frac{d}{d\mu }\overline{m}(\mu )=-{\gamma }_m^{({\beta }_0)}({\alpha }_s(\mu ))\overline{m}(\mu ),{\gamma }_m^{({\beta }_0)}({\alpha }_s(\mu ))=\sum _{n=0}^{\infty }{\gamma }_n{\left(\frac{{\alpha }_s(\mu )}{\pi }\right)}^{n+1}.$$

$$\delta ={\delta }_G+{\delta }_g,$$

$${\delta }_G(\mu =\overline{m})=\frac{C_F{\alpha }_s(\overline{m})}{2\pi }\sum _{n=0}^{\infty }{\left(\frac{{\beta }_0{\alpha }_s(\overline{m})}{4\pi }\right)}^n{G}_{n+1}(\overline{m})\cdot n!,$$

$${\delta }_g(\mu =\overline{m})=\frac{C_F{\alpha }_s(\overline{m})}{2\pi }\sum _{n=0}^{\infty }{\left(\frac{{\beta }_0{\alpha }_s(\overline{m})}{4\pi }\right)}^n\frac{(-1{)}^n}{n+1}{g}_{n+1}.$$

$${\delta }_g=-\frac{2C_F}{{\beta }_0}{\int }_0^{-a}dx\frac{g(x)-g_0}{x};a=\frac{{\beta }_0{\alpha }_s(\overline{m})}{4\pi }.$$

$${\delta }_g(\overline{m})\to \frac{C_F{\alpha }_s(\overline{m})}{2\pi }{g}_1=-\frac{5}{2}\frac{C_F}{{\beta }_0}\frac{1}{\log ({\overline{m}}^2/{\Lambda }_{\mathrm{QCD}}^2)}\text{mbox}for\overline{m}\gg {\Lambda }_{\mathrm{QCD}}.$$

$${\delta }_G(\overline{m})$$

$$={\int }_0^{\infty }\frac{d\tau }{\pi \tau }\left[\mathrm{Im}{\overline{W}}_m\left(\frac{\tau }{{\overline{m}}^2}\right)-\frac{3C_F}{4\pi }\theta (\tau -e^{5/3}{\overline{m}}^2)\right]{\alpha }_{{\beta }_0}(\tau ),$$

$${\overline{W}}_m(s)=-\frac{C_F}{4\pi }{\int }_0^{1}dx\left[(2+2x)\log \left(x^2+sx-s-i0\right)+2x\right],$$

$$f_{\mathrm{UV}}=\frac{3C_F}{{\beta }_0}\log \left[\frac{\log (M^2/{\Lambda }^{\prime 2})}{\log ({\overline{m}}^2/{\Lambda }_{\mathrm{QCD}}^2)}\right].$$

$${\overline{W}}_m=\frac{C_F}{4\pi }\left[4+2\pi i{\left(\frac{\tau }{{\overline{m}}^2}\right)}^{1/2}-3\left(\frac{\tau }{{\overline{m}}^2}\right)-\frac{3\pi i}{4}{\left(\frac{\tau }{{\overline{m}}^2}\right)}^{3/2}+\cdots \right],$$

$${\delta }_{G,UV}$$

$$={\delta }_{G,0}(\overline{m})+\frac{C_0^m({\mu }_f)}{\overline{m}}-\frac{3C_F}{{\beta }_0}\frac{{\Lambda }^{\prime 2}}{{\overline{m}}^2}+\mathcal{O}({\Lambda }_{\mathrm{QCD}}^3/{\overline{m}}^3).$$

$${\delta }_{G,0}(\overline{m})$$

$$=\frac{C_F}{\pi {\beta }_0}{\int }_0^{\infty }dsw(s)A\left(s{\overline{m}}^2/{\Lambda }^{\prime 2}\right)+\frac{3C_F}{{\beta }_0}\log \left[\frac{\log \left({\overline{m}}^2/{\Lambda }_{\mathrm{QCD}}^2\right)}{\pi /e}\right],$$

$$A(x)=\frac{\pi }{2}-\arctan \left[\frac{\log (x)}{\pi }\right],$$

$$w(s)=-2+s\log s-2\left(s^2-2s-2\right)T(s)-\frac{3}{s}\theta \left(s-\frac{{\Lambda }^{\prime 2}}{{\overline{m}}^2}\right),$$

$$T(s)=\theta (4-s)\frac{\arctan \left(\sqrt{\frac{4-s}{s}}\right)}{\sqrt{s(4-s)}}+\theta (s-4)\frac{\mathrm{arctanh}\left(\sqrt{\frac{s-4}{s}}\right)}{\sqrt{s(s-4)}}.$$

$${\delta }_{G,0}(\overline{m})$$

$$C_0^m({\mu }_f)=-\frac{C_F}{2\pi }\mathrm{Re}{\int }_{C_b}\frac{d\tau }{\sqrt{\tau }}{\alpha }_{{\beta }_0}(\tau ),$$

$$\Delta M_0(\overline{m})=({\delta }_g+{\delta }_{G,0})\cdot \overline{m}.$$

Full text

TU–1085

UV contributions to energy of a static quark-antiquark pair

in large-$\beta_{0}$ approximation

Yuuki Hayashi and Yukinari Sumino

*Department of Physics, Tohoku University

Sendai, 980-8578 Japan

The total energy of a static quark-antiquark pair $E_{\rm tot}(r)=2m_{\rm pole}+V_{\rm QCD}(r)$ is known to include ${\cal O}(\Lambda_{\rm QCD}^{3}r^{2})$ and ${\cal O}(\Lambda_{\rm QCD}^{3}/\overline{m}^{2})$ renormalon uncertainties in the large-$\beta_{0}$ approximation, after canceling ${\cal O}(\Lambda_{\rm QCD})$ renormalons. We compute the (renormalon-free) genuine UV part in terms of the $\overline{\rm MS}$ mass $\overline{m}$, extending a recently-proposed method which conforms with OPE. In particular the $r$-independent part is determined. The result would help understanding the nature of $E_{\rm tot}(r)$ in the context of OPE with renormalon subtraction.

Being much smaller than ordinary hadrons, a hadron composed of a heavy quark and its anti-particle (heavy quarkonium) is an ideal system that can be systematically analyzed using solid analysis tools of the strong interaction, such as perturbative QCD and operator product expansion (OPE). In particular various analyses of the leading-order total energy of this system, defined by $E_{\rm tot}(r)\equiv 2m_{\rm pole}+V_{\rm QCD}(r)$, have provided a lot of insight into the theoretical structure of perturbative QCD and OPE. The analyses were brought to a new phase by the discovery of the cancellation of ${\cal O}(\Lambda_{\rm QCD})$ renormalons in $E_{\rm tot}(r)$ [1, 2, 3], which led to a dramatic improvement in convergence of perturbative series of $E_{\rm tot}(r)$ and to much more accurate computation.

Series of studies [4, 5, 6, 7, 8] on the static QCD potential $V_{\rm QCD}(r)$ in perturbative QCD have shown that the potential can be expressed in an expansion in $r$ in the form

[TABLE]

by resumming logarithms by renormalization group (RG) or in the large-$\beta_{0}$ approximation. Here, the $r$-independent constant $C_{0}^{V}$ includes ${\cal O}(\Lambda_{\rm QCD})$ renormalon; $V_{C}(r)$ and $C_{1}^{V}r$ correspond to genuinely ultraviolet (UV) part and can be computed without renormalon uncertainties. $V_{C}(r)$ has a Coulomb-like form with logarithmic corrections at short-distances.111 The logarithmic corrections render the short-distance behavior of $V_{\rm QCD}(r)$ to be consistent with the RG equation. At large $r$, $V_{C}(r)$ approaches a pure Coulomb potential.

The above expansion in $r$ is consistent with OPE performed in an effective field theory (EFT) “potential non-relativistic QCD” (pNRQCD) [9]. In fact, $V_{C}(r)+C_{1}^{V}r$ can be identified with the leading Wilson coefficient of OPE of $V_{\rm QCD}(r)$ after subtraction of renormalons. This OPE of $V_{\rm QCD}(r)$ is formulated beyond the large-$\beta_{0}$ approximation (i.e., including sub-leading logarithms), and recently it has been applied to a precise determination of $\alpha_{s}(M_{Z})$ by comparing the above prediction with lattice computation in an OPE framework, in which the ${\cal O}(\Lambda_{\rm QCD})$ and ${\cal O}(\Lambda_{\rm QCD}^{3}r^{2})$ renormalons are subtracted [10, 11].

Refs. [7, 8] have developed a prescription [referred to as “Contour Deformation (CD) prescription” hereafter], which performs an expansion of a general observable $X(Q)$ in the inverse of the hard scale $Q$, after resummation to all orders in $\alpha_{s}$ within the large-$\beta_{0}$ approximation. ($1/Q=r$ in the case of the QCD potential.) This prescription achieves separation of UV and infrared (IR) contributions in a natural way. Furthermore, a detailed connection of this expansion to OPE is given through the expansion-by-region technique.

Similarly to the QCD potential, we expect that, when expressed in terms of a short-distance mass222 “Short-distance mass” stands for a class of quark mass definitions which contain only contributions from UV degrees of freedom to the quark self-energy in the renormalization. See e.g. [12, 13] for various definitions and their comparisons.

$\overline{m}$, the pole mass of a heavy quark can be expressed in the form

[TABLE]

where the $\overline{m}$-independent constant $C_{0}^{m}$ includes ${\cal O}(\Lambda_{\rm QCD})$ renormalon333 It is suggested that the ${C_{1}^{m}}/{\overline{m}}$ term also includes a renormalon beyond the large-$\beta_{0}$ approximation. We discuss this issue at the end of the paper. ; $\Delta M_{0}(\overline{m})$ denotes the part proportional to $\overline{m}$ with logarithmic corrections at large $\overline{m}$. Nevertheless, even in the case restricting to the large-$\beta_{0}$ approximation, there is a difficulty to apply the CD prescription to carry out this expansion. The difficulty comes from (UV) renormalization of the pole mass, and we need to devise an extension of the prescription to deal with it.

As already mentioned, in the combination $2C_{0}^{m}+C_{0}^{V}$, the ${\cal O}(\Lambda_{\rm QCD})$ renormalons cancel. As a result $E_{\rm tot}(r)$ can be computed up to ${\cal O}(\Lambda_{\rm QCD}^{3}r^{2})$ and ${\cal O}(\Lambda_{\rm QCD}^{3}/\overline{m}^{2})$ renormalon uncertainties. The explicit expression of this computable part of $E_{\rm tot}(r)$ should depend on the definition of the short-distance mass $\overline{m}$ to be used. Analytic or semi-analytic analyses of this computable part have been missing. The purpose of this paper is to give an explicit expression for it and to provide a semi-analytic analysis, in the case we choose the $\overline{\rm MS}$ mass for $\overline{m}$ and within the large-$\beta_{0}$ approximation. An interesting question may be as follows: Is there a constant term proportional to $\Lambda_{\rm QCD}$ (independent of $r$ and $\overline{m}$) included in this computable part? [Noting that $C_{1}^{V}r=(2\pi e^{5/3}C_{F}/\beta_{0})\Lambda_{\rm QCD}^{2}r$ in the large-$\beta_{0}$ approximation, this may not be a completely absurd question. For instance, one may suspect a possibility that different short-distance masses are mutually related by an ${\cal O}(\Lambda_{\rm QCD})$ difference.]

Subtraction of the ${\cal O}(\Lambda_{\rm QCD})$ renormalons from $E_{\rm tot}(r)$ and $m_{\rm pole}$ has also been studied in [14, 15] in connection with pNRQCD EFT. The analyses concern how to subtract the renormalons from finite-order perturbative series (not restricting to the large-$\beta_{0}$ approximation). A major difference of our method is that (in the large-$\beta_{0}$ approximation) we resum the series to all orders and extract a renormalon-free part in the form which conforms with OPE, i.e., expansion in $r$ or $1/\overline{m}$. In this way we can obtain an insight into the analytic structure of $E_{\rm tot}(r)$ which would be useful in the framework of OPE.

In the large-$\beta_{0}$ approximation, the difference of the pole mass and the $\overline{\rm MS}$ mass can be computed as follows.

[TABLE]

Here, (c.t.) denotes contributions of the diagrams including counter terms. $\bar{\mu}=\frac{\mu}{\sqrt{4\pi}}\,e^{\gamma_{E}/2}$, and $\mu$ represents the renormalization scale in the $\overline{\rm MS}$ scheme. The external momentum is fixed as $p^{2}=\overline{m}(\mu)^{2}$. We employ dimensional regularization with $D=4-2\epsilon$. $\Pi(q^{2},\epsilon)$ denotes the one-loop vacuum polarization of gluon in the large-$\beta_{0}$ approximation (in dimensional regularization). It is understood that in the end the limit $\epsilon\to 0$ is taken at each order of the expansion in $\alpha_{s}(\mu)$.

We can perform Wick rotation and the integral can be brought to a one-parameter integral over the modulus-squared of the Euclidean gluon momentum $\tau=q_{E}^{2}=-q^{2}$ [16, 17] by replacing the gluon propagator as

[TABLE]

We obtain

[TABLE]

where

[TABLE]

The difficulty in applying the CD prescription to this integral representation is as follows. The prescription, as it is formulated, requires that we can set

[TABLE]

inside the $\tau$ integral. In eq. (5) this is not possible, however, since we cannot take the limit $\epsilon\to 0$ before $\tau$ integration due to the UV divergent nature of the integral. We will circumvent the difficulty by separating $\delta$ into two parts and introducing a UV cut-off.

We recall the all-order formula of $\delta$ in $\alpha_{s}$ expansion [17]:

[TABLE]

The coefficients $G_{n}$ and $g_{n}$ are given by

[TABLE]

They are related via RG equations to the anomalous dimension of the running mass and the RG-invariant constant terms of ${\delta}$ [18]:

[TABLE]

where, within the large-$\beta_{0}$ approximation, the RG equations read

[TABLE]

Thus, the above series is naturally separated into two parts:

[TABLE]

Here and hereafter, we set $\mu=\overline{m}$. [We are interested in the difference of the RG invariant masses $\delta m(\overline{m})=m_{\rm pole}-\overline{m}(\overline{m})$.] Eq. (11) shows that $g_{n}$’s are determined completely by the mass anomalous dimension within the large-$\beta_{0}$ approximation. Namely, $g_{n}$’s are determined by the UV divergences relevant to the heavy quark $\overline{\rm MS}$ mass renormalization. Hence, it is natural to consider $\delta_{g}$ as a genuinely UV quantity. This series expansion has a non-zero (finite) radius of convergence about $\alpha_{s}(\overline{m})=0$ and can be expressed by $\gamma_{m}^{(\beta_{0})}$ through eq. (11) as

[TABLE]

In Fig. 1 we plot $\delta_{g}$ as a function of $\overline{m}$. The leading behavior of $\delta_{g}$ for large $\overline{m}$ is given by

[TABLE]

$\delta_{g}$ becomes highly oscillatory at $\overline{m}/\Lambda^{\prime}\lesssim 1$ reflecting the oscillatory behavior of $g(t)$ at $t\lesssim-1$.

On the other hand, $\delta_{G}$ includes IR renormalons, hence the series is asymptotic (convergence radius is zero). We can separate a genuine UV part and IR sensitive part of $\delta_{G}$ by the CD prescription. By appropriately subtracting the UV divergence, we can write

[TABLE]

where444 Roughly speaking, to obtain only the $G$ part, we can take the $\epsilon^{0}$-part before $\tau$-integration.

[TABLE]

and

[TABLE]

In the equality of eq. (21) we have expressed $f_{\rm UV}$ in an integral form and taken the limit $M\to\infty$; $\theta(x)$ denotes the unit step function. $\overline{W}_{m}(s)$ can be expressed in terms of elementary functions. By expanding eq. (20) or (21) in $\alpha_{s}(\overline{m})$ one can check that eq. (16) is reproduced.

Now we can apply the CD prescription to the first term of eq. (20). We introduce a factorization scale $\mu_{f}(\gg\Lambda^{\prime})$ and restrict the integral region as $(0,M^{2})\to(\mu_{f}^{2},M^{2})$. Then the integral becomes well defined avoiding the singularity of $\alpha_{\beta_{0}}(\tau)$. We can separate the integral to a genuine UV part (independent of $\mu_{f}$) and IR sensitive part (dependent on $\mu_{f}$). Using the expansion of $\overline{W}_{m}$ for $|\tau/\overline{m}^{2}|\ll 1$,

[TABLE]

we obtain

[TABLE]

The first and third terms of the expansion are independent of $\mu_{f}$. The first term is given by

[TABLE]

where

[TABLE]

and

[TABLE]

The integral contour $C_{a}$ is shown in Fig. 2(a). In eq. (27), we rotate the contour to the negative real $\tau$-axis ($\tau=e^{i\pi}\,s\,\overline{m}^{2}/\Lambda^{\prime 2}$) in order to remove the $M^{2}$ dependence between the second and third terms of eq. (26). The dependence of $\delta_{G,0}(\overline{m})$ on $\overline{m}/\Lambda^{\prime}$ is shown in Fig. 3. Its asymptotic form is given by

[TABLE]

The second term of eq. (25) depends on $\mu_{f}$. If we multiply the term by $\overline{m}$, it is independent of $\overline{m}$ and is given by

[TABLE]

where the integral contour $C_{b}$ is shown in Fig.  2(b). The corresponding term in the QCD potential has exactly the form such that $2C_{0}^{m}(\mu_{f})+C_{0}^{V}(\mu_{f})=0$ in the CD prescription [5], showing the cancellation of renormalons in the $r$ and $\overline{m}$-independent part of $E_{\rm tot}(r)$; c.f., eqs. (1) and (2).

Thus, $\Delta M_{0}(\overline{m})$ in eq. (2) is given by

[TABLE]

The asymptotic form is determined by the sum of eqs. (19) and (31) and reads

[TABLE]

It agrees with the requirement by RG for the leading asymptotic behavior of the mass difference $m_{\rm pole}-\overline{m}(\overline{m})$.

As can be seen from Figs. 1 and 3 (see also Fig. 4 below), the behavior of $\Delta M_{0}(\overline{m})$ at $\overline{m}\gtrsim\Lambda^{\prime}$ is consistent with the expectation that it is proportional to $\overline{m}$ with logarithmic corrections at large $\overline{m}$. At small $\overline{m}(\lesssim\Lambda^{\prime}$), however, $\Delta M_{0}(\overline{m})$ has an oscillatory behavior. This feature is absent in the corresponding expansions of the static potential $V_{\rm QCD}(r)$ and the Adler function [6, 7], whose radiative corrections are dominated by those in Euclidean regions. The oscillatory behavior may reflect the fact that in the pole mass the self-energy corrections close to the on-shell quark configuration involve time-like kinematics. In any case, since this behavior can be concealed by ${\cal O}(\Lambda_{\rm QCD}^{3}/\overline{m}^{2})$ renormalon uncertainties, we cannot make any definite statement about it.

The final result for the genuinely UV (renormalon-free) part of the total energy is given by

[TABLE]

with

[TABLE]

The $\overline{m}$-dependent part, $\Delta M_{0}(\overline{m})+{C_{1}^{m}}/{\overline{m}}$, is shown in Fig. 4 as a function of $\overline{m}/\Lambda^{\prime}$. We see that the contribution of the $C_{1}^{m}/\overline{m}$ term quickly diminishes at $\overline{m}/\Lambda^{\prime}\gtrsim 2$. For completeness we also list the known results for $V_{C}(r)$ and $C_{1}^{V}$ in the large-$\beta_{0}$ approximation:

[TABLE]

Let us present some speculation. $\Delta M_{0}(\overline{m})$ consists of $\overline{m}\cdot\delta_{g}$, the part which can be expanded in the Taylor series in $\alpha_{s}(\overline{m})$, and $\overline{m}\cdot\delta_{G,0}$, the part which cannot be expanded in the Taylor series in $\alpha_{s}(\overline{m})$ [expansion in $1/\log(\overline{m}/\Lambda_{\rm QCD})$ is asymptotic]. The former is tightly connected with the renormalization of the $\overline{\rm MS}$ mass and is intrinsic to this mass scheme. The latter originates from the UV part of the asymptotic series $\sim G_{n+1}\cdot n!$, and hence is likely to be tied to the pole mass, irrespective of the definition of the short-distance mass.

There is no $r$-independent or $\overline{m}$-independent term proportional to $\Lambda_{\rm QCD}$ in the latter part of $E_{\rm tot}(r)$ in expansions in $r$ and $1/\overline{m}$. This would be natural with regard to the fact that the size of a heavy quarkonium bound state is small compared to ordinary hadrons. As a whole we consider that the expression of $E_{\rm tot}(r)$ in eq. (35) has a natural form: It includes only UV contributions; It is composed of the part which conforms with power expansions in $1/\overline{m}$ and $r$ ($\delta_{G,0}$, $C_{1}^{m}$, $V_{C}$ and $C_{1}^{V}$) and the part which can be expanded in the Taylor series in powers of $\alpha_{s}(\overline{m})\sim 1/\log(\overline{m}/\Lambda_{\rm QCD})$ whose expansion coefficients originate from the UV divergences of the short-distance mass ($\delta_{g}$).555 Noting that $\overline{m}$ is much larger than $\Lambda_{\rm QCD}$, in numerical analyses it is not trivial to identify an order $\Lambda_{\rm QCD}$ constant in $\Delta M_{0}(\overline{m})$ which is of order $\overline{m}/\log\overline{m}$. Thus, we see an advantage of performing a semi-analytic analysis.

In turn, this can be taken as an evidence that we have chosen a sensible scheme for separating the UV and IR contributions and performing expansions in $r$ and $1/\overline{m}$.666 The CD prescription is known to have a scheme dependence. The standard scheme choice (“massive gluon scheme”) is favorable from the viewpoint of analyticity [8], and we have chosen this scheme in the above computation.

Finally we comment on possible existence of ${\cal O}(\Lambda_{\rm QCD}^{2}/m)$ renormalon contained in the pole mass, whose properties are as yet not well known. Known properties are as follows [19]. (a) It is induced by the non-relativistic kinetic energy operator $\vec{D}^{2}/(2m)$; (b) It is not forbidden by any symmetry, and corresponds to the singularity at $u=1$ in the Borel plane; (c) It does not appear in the large-$\beta_{0}$ approximation. Thus, it could affect the computable part of $E_{\rm tot}(r)$ beyond the large-$\beta_{0}$ approximation. We leave this issue to future study.777 This renormalon may eventually cancel out due to off-shell effects [20].

Acknowledgements

The authors are grateful to fruitful discussion with H. Takaura. The work of Y.S. was supported in part by Grant-in-Aid for scientific research (No.  17K05404) from MEXT, Japan.