Renormalization Group Summation of Laplace QCD Sum Rules for Scalar Gluon Currents

We employ renormalization group (RG) summation techniques to obtain portions of Laplace QCD sum rules for scalar gluon currents beyond the order to which they have been explicitly calculated. The first two of these sum rules are considered in some detail, and it is shown that they have significantly less dependence on the renormalization scale parameter $\mu^2$ once the RG summation is used to extend the perturbative results. Using the sum rules, we then compute the bound on the scalar glueball mass and demonstrate that the 3 and 4-Loop perturbative results form lower and upper bounds to their RG summed counterparts. We further demonstrate improved convergence of the RG summed expressions with respect to perturbative results.


Introduction
When computing the radiative corrections to physical processes, it is necessary to introduce a scale parameter µ 2 in order to remove divergences through the renormalization procedure. Exploiting the fact that any explicit dependence on µ 2 must be cancelled by implicit dependence of physical parameters on µ 2 , one obtains the renormalization group (RG) equation. This equation has been used to fix portions of radiative effects beyond those determined by direct perturbative calculation. (For example, see refs. [1,2,3,4,5,6].) In this letter we apply this approach to Laplace QCD sum rules for scalar gluonic currents. We find that within RG summation, the dependence on the scale parameter µ 2 is significantly diminished; this is expected as any exact solution of the RG equation would necessarily have no dependence on µ 2 . To illlustrate the physical significance of our approach, we do a full QCD sum rule calculation and apply it to determine the mass bound for the scalar glueball. The RG summation approach provided better results with improved convergence and lesser scheme dependence than those obtained via using purely perturbative inputs. This indicates that the RG summation approach can potentially be beneficial to other QCD sum rule applications as well.

The Perturbative Laplace QCD Sum Rules
The scalar gluonic correlation function is expressed as where j G (y) = β(x) αsβ 0 G a µν (y)G a,µν (y), x = α s /π and β(x) is the QCD β-function defined for the evolution of the QCD strong coupling constant, α s .
The perturbative Laplace sum rule L pert k [7,8,9,10] is given by where τ is the inverse square of the Borel mass. The imaginary part of the perturbative scalar gluonic correlator at centre of mass energy s, can be extracted from Im < (G 2 ) 2 >, which has been computed to O(α 4 s ) in [11] and O(α 5 s ) in the QCD coupling α s [12]. One can extract Im Π pert G (s) in the following way using the expression [13] Im Π pert G (s) = T n,m x n L m (s) (3) where L(s) = log(s/µ 2 ). In ref. [13] the results to order O(α 4 s ) appear; here we make use of the following results to order O(α 5 s ) with 3 active quark flavours.

The RG Summed Laplace QCD Sum Rules
We now define by eq. (2). So also by eq.
and so by eqs. (13)(14)(15)) Regrouping terms in the sum in eq. (13), we can write where is the leading-log (LL) contribution to L pert k , S (k) 1 the next-toleading-log (NLL) contribution . . . S (k) p the N p LL contribution. Since the explicit and implicit dependence of L pert k on the unphysical parameter must cancel, we have the RG equation which by eq. (14) becomes where we have the QCD β-function where β 0 = 9/4, β 1 = 4, β 2 = 10.06 and β 3 = 47.23 for 3 active flavours. (Note: The anomalous dimension γ = 0 for scalar gluonic currents.) Order-by-order in powers of x, eqs. (17,20) lead to The boundary conditions for these nested equations are Solving eqs. (22a-c) in turn, we obtain (For S k 3 , see the appendix.) To obtain S An explicit four loop calculation with three quark flavours and taking Λ QCD = 300MeV leads to [14] α s (µ 2 ) = 1 . We plot the purely perturbative L pert 0 and L pert 1 of eqs. (8,11) with the RG improved expressions following from eqs. (14,17,26,27) in figs. 1, 2, 3 and 4. For parametrizing µ dependence, we define µ = ξ √ τ , and plot perturbative and RG-summed expressions for phenomenologically relevant values of ξ = 0.8, 1 and 1.2 respectively. In both sum rules, we note that the RG summed values are remarkably less renormalization scale dependent than the fixed order perturbative results.
To demonstrate the usefulness of our approach, we compute the mass of the scalar glueball using both purely perturbative and RG summation results. Utilizing a standard QCD sum rule approach as in ref. [10], we incorprate non-perturbative parts which include condensate and instanton contributions to the Laplace sum rules. These pieces are combined as follows, where k = 0, 1 and the second and third term are condensate and instanton contributions respectively. We use the provided expressions for L cond k and L cond k in [10] and use the same set of QCD input parameters. The sum-rules provide a robust upper bound on the scalar glueball mass m In Figure 5, we plot the mass bound computed from both perturbative and RG summed Laplace sum rules. We not only find reduced scale dependence for the RG summed expressions, but also note that the purely 3-Loop and 4-Loop estimates are upper bounds to the RG summed mass estimates. This amply demonstrates (using a full QCD sum rule calculation) the benefit of using RG-summed expressions, as compared to using the purely perturbative results.
Towards demonstrating the convergence properties, we plot the 3-loop and 4-loop mass estimates separately, both for perturbative and RG-summed results. Figures 6 and 7 indicate better convergence properties of the RG summed results.
Finally, we also propose an alternate rearrangement of the sum in eq. (13), so that in place of eq. (17) we have where Substitution of eq. (30) into eq. (20) shows that the RG equation is satisfied at each order in L provided If now and Together, eqs. (30-35) show that (Changes in the boundary condition of eq. (34) can be compensated by changes in µ 2 in L.) Eq.
(36) is not unexpected; it shows how all log-dependent contributions to S (k) are fixed by the RG equation to be given in terms of the log-independent contribution to S (k) (i.e., a 0 ).

Discussion
Using the four loop β-function in QCD as well as the four loop contribution to the scalar gluonic correlation function, we have explicitly summed the LL . . . N 3 LL contribution to the corresponding Laplace QCD sum rules. By having incorporated these contributions, the sum rules L pert 0 and L pert 1 have a considerably reduced dependence on the non-physical renormalization scale µ 2 . It is also possible to use the RG equation to show how all log-dependent contributions to the Laplace sum rules are fixed by the log-independent contributions.