Renormalons in static QCD potential: review and some updates

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.

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

$$\begin{aligned} \mu ^2 \frac{\mathrm{d} \alpha _s(\mu )}{\mathrm{d} \mu ^2}=\beta (\alpha _s) =-b_0 \alpha _s^2-b_1 \alpha _s^3-\cdots . \end{aligned}$$

(24)

The QCD dynamical scale in the $\overline{\mathrm{MS}}$ scheme is defined by

$$\begin{aligned} \varLambda _{\overline{\mathrm{MS}}}^2/\mu ^2= & {} \exp \left[ -\left( \frac{1}{b_0 \alpha _s^2(\mu )}+\frac{b_1}{b_0^2} \log (b_0 \alpha _s(\mu ))\right. \right. \nonumber \\&\left. \left. +\int _0^{\alpha _s(\mu )} \mathrm{d}x \, \left( \frac{1}{\beta (x)}+\frac{1}{b_0 x^2}-\frac{b_1}{b_0^2 x} \right) \right) \right] .\nonumber \\ \end{aligned}$$

(25)

We denote the dimensionless static QCD potential by v(r),

$$\begin{aligned} v(r)=r V_{S}(r)=\sum _{n=0}^{\infty } d^v_n (\mu r) \alpha _s^{n+1}(\mu ) , \end{aligned}$$

(26)

and the dimensionless QCD force by f(r),

$$\begin{aligned} f(r)=r^2 \frac{\mathrm{d} V_{S}}{\mathrm{d} r}=2 \frac{\mathrm{d} v}{\mathrm{d} L}-v=\sum _{n=0}^{\infty } d_n^f (\mu r) \alpha _s^{n+1}(\mu ) , \end{aligned}$$

(27)

where $L=\log (\mu ^2 r^2)$. We define the Borel transform of such a perturbative series by

$$\begin{aligned} B_X(t) :=\sum _{n=0}^{\infty } \frac{d^X_n(\mu r)}{n!} t^n , \end{aligned}$$

(28)

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

$$\begin{aligned} B_X(t)= & {} (\mu ^2 r^2)^{u_*} \frac{N_{u_*}}{(1-b_0 t /u_*)^{1+\nu _{u_*}}} \nonumber \\&\times \sum _{k=0}^{\infty } c_{k, u_*}(\mu r) \left( 1-\frac{b_0 t}{u_*} \right) ^k +\cdots , \quad {} (c_0=1),\nonumber \\ \end{aligned}$$

(29)

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

$$\begin{aligned}&d_n^{u_* (\mathrm{asym})}=N_{u_*} (\mu ^2 r^2)^{u_*} \frac{\varGamma (n+1+\nu _{u_*})}{\varGamma (1+\nu _{u_*})} \left( \frac{b_0}{u_*} \right) ^n \nonumber \\&\quad \times \sum _{k=0}^{\infty } c_{k, u_*}(\mu r) \frac{\nu _{u_*} (\nu _{u_*}-1) \cdots (\nu _{u_*}-k+1)}{(n+\nu _{u_*})(n+\nu _{u_*}-1)\cdots (n+\nu _{u_*}-k+1)}. \end{aligned}$$

(30)

The renormalon uncertainty of X is defined by the imaginary part of a regularized Borel integral:

$$\begin{aligned} \mathrm{Im} X_{\pm }&=\mathrm{Im} \int _{0\pm i0}^{\infty \pm i0} \mathrm{d}t \, B_X(t) e^{-t/\alpha _s(\mu )} \nonumber \\&=\pm \frac{\pi }{b_0} \frac{(\mu ^2 r^2)^{u_*} N_{u_*}}{\varGamma (1+\nu _{u_*})} u_*^{1+\nu _{u_*}} e^{-\frac{u_*}{b_0 \alpha _s(\mu )}} (b_0 \alpha _s(\mu ))^{-\nu _{u_*}}\nonumber \\&\quad \times \sum _k \nu _{u_*} (\nu _{u_*}-1) \cdot \cdots \cdot (\nu _{u_*}-k+1) \nonumber \\&\quad \times \left( b_0/u_* \right) ^{k} c_{k, u_*}(\mu r) \alpha _s^k(\mu ) . \end{aligned}$$

(31)

This is renormalization scale independent. Writing the renormalon uncertainty as

$$\begin{aligned} \mathrm{Im} \, X_{\pm }&=\pm K_{u_*} e^{-u_*/(b_0 \alpha _s(r^{-1}))} (b_0 \alpha _s(r^{-1}))^{-\nu _{u_*}} \nonumber \\&\quad \sum _{k=0}^{\infty } s_{k, u_*} \alpha _s^k(r^{-1}) \end{aligned}$$

(32)

with $s_0=1$, we have the following relations,

$$\begin{aligned} K_{u_*}=\frac{\pi }{b_0} \frac{N_{u_*} }{\varGamma (1+\nu )} u_*^{1+\nu _{u_*}}, \end{aligned}$$

(33)

and

$$\begin{aligned} s_{k, u_*}= & {} \nu _{u_*} (\nu _{u_*}-1) \cdot \cdots \cdot (\nu _{u_*}-k+1)\nonumber \\&\times (b_0/u_*)^k c_k(\mu r=1) \quad \mathrm{for}\quad {} k\ge 1 . \end{aligned}$$

(34)

As discussed in Sect. 2, since the renormalon uncertainties in coordinate space are given by

$$\begin{aligned} K_{u_*} (\varLambda _{\overline{\mathrm{MS}}}^2 r^2)^{u_*} [1+\mathcal {O}(\alpha _s^2(r^{-1}))] , \end{aligned}$$

(35)

(where $\mathcal {O}(\alpha _s^2)$ can be zero) one can see that $\nu _{u_*}$ in Eq. (32) is given by

$$\begin{aligned} \nu _{u_*}=u_* b_1/b_0^2 \end{aligned}$$

(36)

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 $\alpha _s$.