Higher-order behaviour of two-point current correlators

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.

Appendices

Appendix A: Renormalisation group functions and dependent coefficients

In our notation, the QCD $\beta$-function and mass anomalous dimension are defined as:

$$\begin{aligned}&-\,\mu \,\frac{\mathrm{d}a}{\mathrm{d}\mu } \equiv \beta (a) \,=\, \beta _1\,a^2 + \beta _2\,a^3 + \beta _3\,a^4 + \beta _4\,a^5 + \cdots \,, \end{aligned}$$

(93)

$$\begin{aligned}&{\quad }-\,\frac{\mu }{m}\,\frac{\mathrm{d}m}{\mathrm{d}\mu } \equiv \gamma _m(a) \,=\, \gamma _m^{(1)}\,a + \gamma _m^{(2)}\,a^2 \nonumber \\&\qquad \qquad + \gamma _m^{(3)}\,a^3 + \gamma _m^{(4)}\,a^4 + \cdots \,. \end{aligned}$$

(94)

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]

$$\begin{aligned} \beta _1= & {} \frac{9}{2} \,, \quad \beta _2 \,=\, 8 \,, \quad \beta _3 \,=\, \frac{3863}{192} \,, \nonumber \\&\beta _4 \,=\, \frac{140599}{2304} + \frac{445}{16}\,\zeta _3 \,, \nonumber \\ {\quad }\beta _5= & {} \frac{139857733}{663552} + \frac{11059213}{27648}\,\zeta _3\nonumber \\&- \frac{36045}{512}\,\zeta _4 - \frac{534385}{1536}\,\zeta _5 \,, \end{aligned}$$

(95)

and the first five for $\gamma _m(a)$ are found to be [39, 40]

$$\begin{aligned} \gamma _m^{(1)}= & {} 2 \,, \quad \gamma _m^{(2)} \,=\, \frac{91}{12} \,, \quad \gamma _m^{(3)} \,=\, \frac{8885}{288} - 5\,\zeta _3 \,, \nonumber \\ \gamma _m^{(4)}= & {} \frac{2977517}{20736} - \frac{9295}{216}\,\zeta _3 + \frac{135}{8}\,\zeta _4 - \frac{125}{6}\,\zeta _5 \,, \nonumber \\ \gamma _m^{(5)}= & {} \frac{156509815}{248832} - \frac{23663747}{62208}\,\zeta _3 + 170\,\zeta _3^2 + \frac{23765}{128}\,\zeta _4 \nonumber \\&- \frac{22625465}{31104}\,\zeta _5 + \frac{1875}{16}\,\zeta _6 + \frac{118405}{288}\,\zeta _7 \,. \end{aligned}$$

(96)

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

$$\begin{aligned} d_{n,2} \,=\, -\,\frac{1}{2}\,\gamma _m^{(n)} d_{0,1} - \frac{1}{4} \sum \limits _{k=1}^{n-1} \big ( 2\gamma _m^{(n-k)} + k\,\beta _{n-k} \big ) d_{k,1} .\nonumber \\ \end{aligned}$$

(97)

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

$$\begin{aligned} P({\hat{a}}_Q) \,=\, [{\hat{a}}_Q]^\delta \,\biggl \{\, 1 + \sum \limits _{n=1}^\infty ({\hat{a}}_Q)^n \sum \limits _{k=0}^n y_{n,k}\, {\widehat{C}}^{\,k} \,\biggr \} \,, \end{aligned}$$

(98)

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:

$$\begin{aligned} y_{1,1}= & {} \delta \,, \quad y_{2,2} \,=\, \frac{\delta }{2} \,(\delta +1) \,, \nonumber \\ y_{2,1}= & {} (\delta +1)\,y_{1,0} + \lambda \,\delta \,, \nonumber \\ {\qquad }y_{3,3}= & {} \frac{\delta }{6}\,(\delta +1)(\delta +2) \,, \quad y_{3,2} \,=\, \frac{1}{2}\nonumber \\&\times \big [ (\delta +1)(\delta +2)\,y_{1,0} + \lambda \,\delta \,(2\delta +3) \big ] \,, \nonumber \\ {\qquad }y_{3,1}= & {} (\delta +2)\,y_{2,0} + \lambda \,(\delta +1) \,y_{1,0} + \lambda ^2 \delta \,, \nonumber \\ y_{4,4}= & {} \frac{\delta }{24}\,(\delta +1)(\delta +2)(\delta +3) \,, \nonumber \\ {\qquad }y_{4,3}= & {} \frac{1}{6}\,\big [ (\delta +1)(\delta +2)(\delta +3)\,y_{1,0}\nonumber \\&+ \lambda \,\delta \,(3\delta ^2+12\delta +11) \big ] \,, \nonumber \\ {\qquad }y_{4,2}= & {} \frac{1}{6}\,\big [ (\delta +2)(\delta +3)\,y_{2,0} + \lambda \, (\delta +1)(2\delta +5)\nonumber \\&\times y_{1,0} + 3\lambda ^2\delta \,(\delta +2) \big ] \,, \nonumber \\ {\qquad }y_{4,1}= & {} (\delta +3)\,y_{3,0} + \lambda \,(\delta +2) \,y_{2,0} + \lambda ^2 (\delta +1)\,y_{1,0} + \lambda ^3\delta .\nonumber \\ \end{aligned}$$

(99)

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.