A novel demonstration of the renormalization group invariance of the fixed-order predictions using the principle of maximum conformality and the $C$-scheme coupling

As a basic requirement of the renormalization group invariance, any physical observable must be independent of the choice of both the renormalization scheme and the initial renormalization scale. In this paper, we show that by using the newly suggested $C$-scheme coupling, one can obtain a demonstration that the {\it Principle of Maximum Conformality} prediction is scheme-independent to all-orders for any renormalization schemes, thus satisfying all of the conditions of the renormalization group invariance. We illustrate these features for the non-singlet Adler function and for $\tau $ decay to $\nu +$ hadrons at the four-loop level.


I. INTRODUCTION
It is well-known that the fixed-order perturbative predictions for observables in Quantum Chromodynamics (QCD) using conventional methods suffer from an uncertainty in fixing the renormalization scale. It is assumed that at sufficiently high order, one will eventually achieve reliable predictions and minimal dependence on the guessed renormalization scale for global quantities such as a total cross-section. A small scale-dependence for the global quantity is generally caused by accidental cancelations among different orders; the scale uncertainty for contributions at each order could still be very large. It is known that the pQCD series will suffer from divergent renormalon contributions [1,2]. Thus even if a perturbative QCD prediction based on a guessed scale agrees with measurements, one cannot be certain that it is a reliable, accurate representation of the theory.
A valid prediction for any physical observable must be independent of the choice of both the initial renormalization scale and the renormalization scheme; this is the central property of renormalization group invariance (RGI) [3][4][5][6][7]. Thus a primary goal for testing pQCD reliably is how to set the renormalization scale such that one obtains accurate fixed-order predictions with maximum precision while satisfying the principle of RGI [8].
The Principle of Maximum Conformality (PMC) [9][10][11][12][13][14] determines the value of the renormalization scale of the QCD running coupling α s based on the properties of renormalization group equation (RGE). When one applies the PMC, all nonconformal terms are systematically eliminated at each finite order. The PMC satisfies the self-consistency conditions of the renormalization group, * wuxg@cqu.edu.cn † cqusjm@cqu.edu.cn ‡ dblcqu@cqu.edu.cn § sjbth@slac.stanford.edu such as reflectivity, symmetry and transitivity [15]. The PMC scales are achieved by applying the RGE recursively. Specific values for the PMC scales are computed as a perturbative expansion, so they have small uncertainties which vary order-by-order. Since the running coupling sums all of the {β i }-terms, the divergent renormalon terms disappear in the PMC prediction, leading to a more convergent pQCD series. The PMC provides the underlying principle for the Brodsky-Lepage-Mackenzie (BLM) method [16], and it reduces to the standard Gell-Mann and Low (GM-L) scale-setting procedure [17] in the N c → 0 QED Abelian limit [18].
To make the PMC scale-setting procedures simpler and more easy to be automatized, a single-scale approach (PMC-s) has been suggested in Ref. [19]. This method replaces the individual PMC scales at each order by a single scale in the sense of a mean value theorem; moreover, its predictions are explicitly independent of the choice of initial renormalization scale. Since the coefficients obtained using the PMC-s are identical to those of a conformal theory, one can derive all-orders "commensurate scale relations" among physical observables [20,21].
A novel C-scheme coupling has been suggested in the literature, whose scheme-and-scale running behaviors are both governed by a single scheme-independent RGE [22]. In the paper, we shall give the analytic solution for its RGE. Furthermore, by using the C-scheme coupling, we shall show that a strict demonstration on the schemeindependence of PMC prediction to all-orders for any renormalization schemes can be achieved. Thus, by combining the C-scheme coupling with the PMC-s approach, the resulting predictions become completely independent of the choice of the renormalization scheme and the initial renormalization scale, satisfying all of the conditions of RGI.
The remaining parts of the paper are organized as follows. In Sec.II, we define the C-scheme coupling, deduce its RGE and give its solution. We then give the PMC prediction for the C-scheme coupling and demon-strate how scheme-and-scale independent predictions can be achieved. In Sec.III, we present numerical results for two examples, the non-singlet Adler function and τ decays to ν+ hadrons, up to four-loop level. Sec.IV is reserved for a summary.
If one integrates the RGE (1), one obtains where µ 0 is a reference scale and theβ-function is defined It is useful to define an asymptotic scale Λ by collecting all µ 0 -dependent terms on the left-hand-side of Eq.(2) into its definition, leading to the evolution of strong coupling a µ without reference to a specific choice of µ 0 , i.e. ln The asymptotic scale Λ is, by definition, scheme dependent. Eq.(4) implies that the conventional a µ coupling satisfies the following scheme-dependent scale-running behavior 1 As suggested by Ref. [22], one can define a new couplinĝ a µ =α s (µ)/π in the following way: where the phenomenological parameter C is introduced, C = − aµ 0 da/β(a), which incorporates the effects of all scheme-dependent {β i≥2 }-terms. By choosing a specific value for C, the running coupling of the C-scheme will be equivalent to the coupling of any conventional renormalization scheme.
The solution of Eq.(6) is much simpler than the one for the conventional RGE that can only be solved via a perturbative way. The solution can be written in terms of the Lambert W -function, where The function W (z) is a multi-valued function with an infinite number of branches denoted by W n (z) [32]. The correct physical branch can be determined by the requirement thatâ µ must be real and positive for a real positive scale µ. Since in practice n f ≤ 6, we have z < 0, and the physical branch is W −1 (z). One also finds that W −1 (z) monotonically decreases within the region of z ∈ (−1/e, 0), with W −1 (z) ∈ (−∞, −1). The ultraviolet limit corresponds to z → 0 − , W −1 (z) → −∞, leading tô a µ → 0 + , as required by asymptotic freedom.
Using Eq.(6), we can obtain a new RGE for the Cscheme couplingâ µ which has a much simpler form than the standard RGE (1): (8) At the same time, from Eq.(6), one may also observe that Those two equations indicate that • Even though the C-scheme couplingâ µ itself is implicitly scheme-dependent, its scale-running behavior can be scheme-independent.
• Integrating RGE (8) yields a relation ofâ µ for any two scales µ 1 and µ 2 , i.e., Thus ifâ at a scale µ 1 is fixed by a measurement, we can determine its value at any other scale.
• Given the proper choice of C, any coupling constant a µ which is defined in any conventional renormalization scheme can be uniquely expressed by a corresponding C-scheme couplingâ µ .
Transforming Eq.(6) to the following form, we obtain a relation betweenâ µ and the conventional coupling a µ , solving it recursively, we obtain This shows that the conventional coupling a µ at any scale µ can be expanded in terms of the C-scheme couplingâ µ at the same scale; and vice versa. By choosing a suitable C, the new couplingâ µ becomes equivalent to the coupling a µ defined for any corresponding conventional scheme; i.e. a µ =â µ | C . At a different scale µ, a different C needs to be introduced in order to ensure the equivalence of the couplings at the same scale.

B. Scheme-and-scale independent pQCD predictions using PMC scale-setting
The pQCD approximant of an observable up to n thorder level can be generally expressed as where µ is the renormalization scale and Q is the kinematic scale of the process at which it is measured. Without losing generality, we can set the power of the coupling associated with the tree-level term as 1, or equivalently p = 0. The parameters r i andĉ i are the perturbative coefficients for the conventional coupling a µ and the corresponding C-scheme couplingâ µ . Their relations can be obtained by using the relation (11) between the C-scheme couplingâ µ and the conventional coupling a µ , The perturbative coefficients r i for the conventional coupling a µ can be further expanded as {β i }-terms, which satisfy the degeneracy relations among different orders [13,14,33], i.e., The non-conformal coefficients r i,j(≥1) are general functions of µ and Q, which are usually in form of ln µ/Q. For convenience, we identify the coefficients The conformal coefficients are free from µ-dependence, e.g., The authors of Refs. [22,34] have investigated the possibility of obtaining an "optimized" prediction for the truncated pQCD series using the C-scheme coupling by exploiting its scheme dependence. In their treatment, by fixing µ ≡ Q and varying C within a possible domain, an optimal C-value, and thus an optimal scheme, is determined by requiring the absolute value of the last known termĉ n (Q/Q)â n Q to be at its minimum. However, we have noted that the idea of requiring the magnitude of the last known term of the pQCD series to be at its minimum is similar to the postulate of the Principle of Minimum Sensitivity (PMS) [35][36][37], in which the optimal scheme is determined by directly requiring all unknown higher-order terms to vanish. Thus this application of optimization to the C-scheme coupling approach meets the same problems of PMS, such as it does not satisfy the self-consistency conditions of the renormalization group and etc [38,39]. Moreover, the determined optimal value of C shall be different for a different fixed-order prediction, thus it needs to be redetermined when new perturbative terms are known. Although this approach of using the C-scheme coupling could be considered as a practical way to improve pQCD precision, similar to the PMS approach, it cannot be considered as the solution to the conventional scheme-and-scale ambiguities.
In contrast to the PMS, the PMC identifies all the RGinvolved scheme-dependent {β i }-terms in the perturbative series and eliminate them by shifting the scales of the running coupling. After applying the PMC, the coefficients of ρ n match the corresponding conformal series, and thus the prediction is scheme independent in general. We have demonstrated that the PMC leads to schemeindependent pQCD predictions for any dimensional-like renormalization scheme [13,14].
In the following, we shall demonstrate that one can eliminate all scheme-dependent C-terms in a pQCD approximant by applying the PMC-s approach [19]. Since the parameter C identifies any choice of the renormalization scheme, we will achieve a general demonstration of the scheme-independence of the PMC pQCD predictions for any renormalization scheme.
Our demonstration starts from Eq.(13), and we shall consider at least the next-to-leading order pQCD correction to the pQCD prediction; i.e. n ≥ 2. Up to four-loop level, the coefficientsĉ i can be related to the coefficients r i,j for conventional running coupling from Eqs. (14,15,16,17,18,19,20,21). To avoid complexity for determining the coefficients of each {β i }-terms, we adopt the PMC-s approach to eliminate all the RG-involved {β i }terms.
With the help of Eq.(9, 13), the C-dependence of the pQCD approximateρ n can be expressed as which shows that when the non-conformal terms associated with theβ(â µ )-function have been removed, one can achieve a scheme-independent prediction at any fixed order, i.e.β(â µ ) → 0 indicates ∂ρ n /∂C → 0.
Following the standard PMC-s scale-setting procedures, an effective PMC scale Q ⋆ can be obtained by eliminating all nonconformal terms, which can be expanded as a power series inâ Q⋆ via the following way, whose first three coefficients arê It is interesting to find that all high-order coefficientsŜ i (i ≥ 1) are free of the scheme parameter C. By using the definition of C-scheme coupling (6), we obtain The second equation shows that, even though the effective scale Q ⋆ depends on the choice of C, the couplinĝ α Q⋆ is independent to C at any fixed order.
Thus, after fixing the scale Q ⋆ , we achieve a C-scheme independent pQCD serieŝ The pQCD series depends on the choice of scheme via the coefficients r i,j and the {β i≥2 }-functions. Thus, Eq. (25) indicates the scheme-independence of the C-scheme predictions is equivalent to the scheme-independence of the initial choice of scheme, and vice versa. The demonstration of C-scheme independence, as shown by Eq. (22), shows the pQCD predictions are scheme independent for any choice of the initial scheme. By using the above formulas, we obtain pQCD predictions independent of any choice of scheme (represented by any choice of C). This demonstrates to any orders the scheme-independent of the PMC predictions -Given one measurement which sets the value of the coupling at a scale, the resulting PMC predictions are independent of the choice of renormalization scheme.

III. PHENOMENOLOGICAL EXAMPLES
In doing the numerical calculations below, we adopt the world average α The non-singlet Adler function [41] reads where µ is the renormalization scale, a µ = α s (µ)/π, and L = ln µ 2 /Q 2 . γ ns (a µ ) = i≥0 γ ns i a i µ /16π 2 is the nonsinglet part of the photon field anomalous dimension and Π ns (L, a µ ) = i≥0 Π ns i a i µ /16π 2 is the non-singlet part of the polarization function for a flavor-singlet vector current. The pQCD series ofD ns (Q 2 , µ) up to n th -loop level can be written as The perturbative coefficients γ ns i and Π ns i within the MSscheme have been given up to four-loop level [42], and the coefficients r i for µ = Q within the MS-scheme up to four-loop level can be read from Refs. [43,44]. The coefficients at any other choices of the renormalization scale (µ = Q) can be obtained via RGE.

Predictions using conventional scale-setting
Taking Q = M τ , we obtain a four-loop MS-scheme prediction onD ns using conventional scale-setting (Conv.), D ns 4 (M 2 τ , µ = M τ )| Conv. = 0.1286±0.0053±0.0094, (28) where the first error is for ∆α MS s (M Z ) = ±0.0011 and the second error is an estimate of the "unknown" high-order contribution, which is conservatively taken as the maximum value of the last known term of the perturbative series within the possible choices of initial scale [39]. As for the four-loop prediction, we take the maximum value of |r 4 (µ/M τ )a 4 µ | with µ ∈ [M τ , 4M τ ] as the estimated "unknown" high-order contribution.
The unknown fifth-order coefficient has been estimated by several groups, e.g. r 5 ≃ 283 [45] or r 5 ≃ 275 [46]. If using r 5 ≃ 283, Eq.(28) changes tō 1315±0.0057±0.0065. (29) In addition to the scale dependence, the predictions using conventional scale setting is also scheme dependent at any fixed order. We adopt the C-scheme coupling to illustrate this dependence, and we can rewriteD ns n (Q 2 , µ) in terms of the C-scheme couplingâ µ as where the coefficientsĉ i (µ/Q) can be derived by using Eqs. (14,15,16,17). A graphical representation ofD ns (M 2 τ , C) as a function of parameters C and µ is given in Fig. 1, in which we have chosen C ∈ [−2, +2] 1 and µ ∈ [M τ , 4M τ ]. The shaded band shows the four-loop prediction onD ns (M 2 τ , C), which has a large scheme-and-scale dependence. Using a proper C, the prediction using the C-scheme couplinĝ a µ is equivalent to predictions using some of the familiar schemes; e.g. the solid line in Fig.1 is for the MS-scheme and the dashed line is for the MS-scheme. To ensure the equivalence, the value of C should be changed for different scales. By taking C = −0.188 one obtains the conventional MS prediction for µ = M τ ; alternatively it can taken as C = −0.004 for µ = 4M τ .
Requiring the estimated "unknown" high-order contribution, |ĉ n (µ/M τ )â n µ | MAX , to be at its minimum, we can obtain an optimal C-scheme forD ns n (Q 2 , C). At the four-loop level with n = 4, the optimal C-value is −0.972, leading tō D ns 4 (M 2 τ , C = −0.972)| Conv. = 0.1365 ± 0.0069 ± 0.0083, where the central value is for µ = M τ , the first error is for ∆α MS s (M Z ) = ±0.0011 and the second error is an estimate of the "unknown" high-order contribution. As for a five-loop prediction, if using r 5 ≃ 283, the optimal C-value changes to −1.129, and we obtain D ns 5 (M 2 τ , C = −1.129)| Conv. = 0.1338 ± 0.0062 ± 0.0054.

Predictions using PMC scale-setting
Following the standard PMC-s procedures, by resumming all the RG-involved non-conformal {β i }-terms into the running coupling, we obtain ⋆ . The optimal scale Q ⋆ is of perturbative nature: when more loop terms are included, it becomes more accurate. Eq. (24) shows that the C-scheme coupling is independent to the choice of C. By taking Q = M τ , we obtainâ Q⋆ ≡ 0.1056 (41)  where the first error is for ∆α MS s (M Z ) = ±0.0011 and the second error is an estimate of the "unknown" high-order contribution.
In order to compare the scheme dependence before and after applying the PMC, we present the predictions forD ns (M 2 τ , C) for µ = M τ in Fig. 3. At fourloop level, Fig. 3 shows that the scheme-dependence of D ns 4 (M 2 τ , C)| Conv. is rather large, which decreases with increasing values for C; for larger C values, the error band becomes slightly larger. When C = −0.972, the error bar is the minimum, corresponding to the optimal scheme. By using the approximate five-loop term r 5 ≃ 283, we also give the results for the approximate five-loop prediction. Fig. 3 shows a smaller error bar is achieved with a five-loop term,D ns 5 (M 2 τ , C)| Conv. first increases and then decreases with the increment of C, and the optimal scheme is slightly shifted to C = −1.129. It is interesting to find that the predictedD ns (M 2 τ , C)| Conv. with the optimal choice of C are consistent with the PMC predictions within errors. This indicates that the differences caused by various scale-setting approaches could be smeared by including more and more loop terms and be consistent with the scheme-independent predictions. The flat dash-dot line in Fig. 3 shows that the conventional scheme dependence can be eliminated by applying the PMC. Due to the much faster pQCD convergence after applying the PMC, and due to the elimination of the scale dependence, the PMC suggests that the unknown high-order contribution could be quite small in comparison to the present four-loop prediction.
We present the value of each loop-term, LO, NLO, N 2 LO, or N 3 LO, for the four-loop predictionD ns 4 using conventional and PMC scale-settings in Table I pQCD convergence is achieved due to the elimination of divergent renormalon-like terms.

B. τ decays to ν+ hadrons
The ratio of the τ total hadronic branching fraction to its lepton branching fraction can be parameterized as, where S EW is an electroweak correction, |V ud | as well as |V us | are CKM matrix elements. The pQCD correction is encoded in δ (0) and the ellipsis indicate further small subleading corrections. The pQCD correction up to n th - where the coefficients c i for the MS-scheme up to fourloop level can be found in Refs. [45,46].

Predictions using conventional scale-setting
Taking µ = M τ , we obtain a four-loop MS-scheme prediction on δ (0) using conventional scale-setting, where the first error is for ∆α MS s (M Z ) = ±0.0011 and the second error is an estimate of the "unknown" highorder contribution. If we use the predicted five-loop term r 5 ≃ 283 [45] , we obtain The predictions using conventional scale setting are scheme dependent. By using the relation (11), we rewrite δ (0) n (M 2 τ , µ) in terms of the C-scheme couplingâ µ as A graphical representation of δ (0) (M 2 τ , C) as a function of parameters C and µ is given in Fig. 4, in which we have chosen C ∈ [−2, +2] and µ ∈ [M τ , 4M τ ]. The shaded band shows the four-loop prediction on δ (0) (M 2 τ , C), the solid line is MS-scheme prediction and the dashed line is the MS-scheme one.
Requiring the approximate "unknown" high-order contribution, |ĉ n (µ/M τ )â n µ | MAX , minimal, we obtain an optimal C-scheme for δ (0) (M 2 τ , C) at the n th -order level. Using the four-loop prediction with n = 4, the optimal C-value is −1.638, which leads to where the first error is for ∆α MS s (M Z ) = ±0.0011, and the second error is for ±|ĉ 4 (µ/M τ )â 4 µ | MAX . If using the five-loop term r 5 ≃ 283, the optimal C-value changes to −1.813, and we obtain

Predictions for τ decay using PMC scale-setting
Up to four-loop level, the known conformal and nonconformal coefficients for conventional coupling can be read from Ref. [14]. By resumming all the RG-involved non-conformal {β i }-terms into the running coupling, we obtain the PMC prediction for δ  where the first error is for ∆α MS s (M Z ) = ±0.0011 and the second error is an estimate of the "unknown" high-order contribution.
To compare the scheme dependence before and after applying the PMC, we present the predictions on δ (0) (M 2 τ , C) for µ = M τ in Fig. 6. At four-loop level, Fig. 6 shows that the error band shall first increases and then decreases with increasing C; the optimal scheme is obtained for C = −1.638. By using the approximate fiveloop term r 5 ≃ 283, we give the results for the approximate five-loop prediction. Fig. 6 shows a smaller error bar is achieved with a five-loop term,D ns 5 (M 2 τ , C)| Conv. oscillates with the increment of C, and the optimal scheme is slightly shifted to C = −1.813. Similar to the case ofD ns (M 2 τ , C)| Conv. , it is observed that the predicted δ (0) (M 2 τ , C)| Conv. with the optimal choice of C are also consistent with the PMC predictions within errors. The flat dash-dot line in Fig. 6 shows the conventional scheme dependence can be eliminated by applying the PMC. The unknown high-order contribution could be quite small for the present four-loop PMC prediction.
We present the value of each loop-term, LO, NLO, N 2 LO, or N 3 LO, for the four-loop prediction δ (0) 4 using conventional and PMC scale-settings in Table II. The pQCD convergence for the conventional MS-scheme is moderate. The pQCD convergence for the optimal Cscheme (C = −1.638) does not suffer from the usual α s -suppression, |δ |. On the other hand, by applying the PMC, a better pQCD convergence is achieved due to the elimination of divergent renormalon-like terms.

IV. SUMMARY
We have shown that the scheme-and-scale ambiguities introduced by conventional scale-setting are unnecessary by combining the single-scale PMC procedure with the newly suggested scheme-independent C-scheme coupling. We have demonstrated that using the C-scheme, together with the PMC-s scale-setting, leads to perturbative QCD predictions which are independent of the initial renormalization scale and the choice of the renormalization scheme at all orders.
The pQCD predictions based on the PMC satisfy the standard RGI and all the self-consistency conditions of the renormalization group. Because the divergent renormalon terms are eliminated, the convergence of the PMC series depends on conformal coefficients, which are generally more convergent than conventional pQCD series.  The resulting PMC-s approach not only makes the implementation and automation of the PMC simpler and more transparent, but it also achieves precise scheme-and-scale independent predictions simultaneously. This method for eliminating the scale and scheme ambiguities relies heavily on how well we know the precise value and analytic properties of the strong coupling. In conventional RGE, the scale running is entangled with the scheme parameters, which can only be solved perturbatively or numerically. In contrast, the scheme-and-scale running behavior of a C-scheme couplingα s can be exactly separated; it satisfies a RGE free of scheme-dependent {β i }-terms. The resulting schemeindependent RGE for the C-scheme couplingα s provides scheme-independent predictions.
As we have shown, one can utilize a novel C-scheme coupling whose scheme-and-scale running behaviors are governed by a single scheme-independent RGE. The value of the parameter C can be chosen to match any conventional renormalization scheme. By using the C-scheme coupling instead of conventional coupling, we have demonstrated that the C-dependence of the PMC predictions can be eliminated up to any fixed order; since the value of C is arbitrary, it means the PMC prediction is independent of any renormalization scheme. Two four-loop PMC examples confirm those observations. Thus combining the C-scheme coupling with the PMC-s approach, the resulting predictions become completely independent of the choice of the renormalization scheme and the initial renormalization scale, satisfying all of the conditions of RGI. The PMC procedure thus systematically eliminates the scheme-and-scale ambiguities of pQCD predictions, greatly improving the precision of tests of Standard Model and the sensitivity of collider experiments to new physics.