High precision tests of QCD without scale or scheme ambiguities
The 40^th anniversary of the Brodsky–Lepage–Mackenzie method

The renormalization scale and scheme ambiguities are an important source of errors in many processes in perturbative QCD preventing precise theoretical predictions for both standard model (SM) and beyond standard model (BSM) physics. In principle, an infinite perturbative series is devoid of this issue, given the scheme and scale invariance of entire physical quantities [1, 2, 3, 4, 5]; in practice, perturbative corrections are known up to a certain order of accuracy and scale invariance is only approximated in truncated series, leading to scheme and scale ambiguities [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18].

Although perturbative calculations for theoretical predictions are also affected by other sources of errors, e.g. the top- and Higgs-mass uncertainties and the strong-coupling $\alpha_{s}(M_{Z})$ uncertainty, the renormalization scale and scheme ambiguities remain among of the main sources of errors. These scale and scheme ambiguities play an important role for predictions in many fundamental processes in perturbative QCD, also with respect to the other sources of uncertainties. Processes such as gluon fusion in Higgs production [19] or bottom-quark production [20], which are essential for the physics investigated at the Large Hadron Collider (LHC) and at future colliders, are all affected by these ambiguities. In the present era, high-precision predictions are crucial for both SM and BSM physics, in order to test the theory in all sectors and to enhance sensitivity to possible new physics (NP) at colliders.

At the moment two basic strategies exist to deal with this problem. On one hand, according to conventional practice or conventional scale setting (CSS), this problem cannot be avoided and is responsible for part of the theoretical errors. The simple CSS procedure starts from the scale and scheme invariance of a given observable, which translates into complete freedom for the choice of renormalization scale. According to common practice, a first evaluation of the physical observable is obtained by calculating perturbative corrections in a given scheme (commonly used are $\rm MS$ or $\overline{\rm MS}$) and at an initial renormalization scale $\mu_{r}=\mu_{r}^{\rm init}$. Thus, the renormalization scale $\mu_{r}$ is set to one of the typical scales of a process, $Q$, and errors are estimated by varying the scale over a range $[Q/2,2Q]$.

This method is claimed to evaluate uncalculated contributions from higher-order terms and that, due to the perturbative nature of the expansion, the introduction of higher-order corrections would reduce the scheme and scale ambiguities order by order. There is no doubt that the higher the order of the loop corrections calculated, the greater is the precision of the theoretical estimates in comparison with the experimental data, but we cannot determine a priori the level of accuracy necessary for the CSS to achieve the desired precision. And in the majority of cases at present only the NNLO corrections are available. Moreover, the divergent nature of the asymptotic perturbative series and the presence of factorially growing terms (i.e. renormalons [21, 22, 23] severely compromise the theoretical predictions.

However, even though this procedure may give an indication as to the level of conformality and convergence reached by the truncated expansion, it leads to a numerical evaluation of theoretical errors that is quite unsatisfactory and strongly dependent on the value of the chosen scale. Moreover, this method gives predictions with large theoretical uncertainties that are comparable with the calculated order correction; different choices of the renormalization scale may lead to very different results when higher-order corrections are included. For example, the NLO correction to $W+3\,$jets with the BlackHat code [24] can range from negligible to extremely severe, depending on the choice of the particular renormalization scale. One may argue that the proper renormalization scale for a fixed-order prediction may be judged by comparing theoretical results with experimental data, but this method would be strongly process dependent and would compromise the predictivity of the pQCD approach.

Besides the complexity of the higher-order calculations and the slow convergence of the perturbative series, there are many critical points in the CSS method:

• 1.

  In general, the proper renormalization scale value, $Q$, is not known, nor the correct range over which the scale and scheme parameters should be varied in order to obtain the correct error estimate. In fact, in some processes there can be more than one typical momentum scale that may be taken as the renormalization scale according to the CSS procedure; for example, in processes involving heavy quarks typical scales are either the center-of-mass energy $\sqrt{s}$ or also the heavy-quark mass. Moreover, the idea of the typical momentum transfer as the renormalization scale only sets the order of magnitude of the scale, but does not indicate the optimal scale.

• 2.

  No distinction is made among different sources of errors and their relative contributions; e.g. in addition to the errors due to scale-scheme uncertainties there are also errors from missing uncalculated higher-order terms. In such an approach theoretical uncertainties can become quite arbitrary and unreliable.

• 3.

  The convergence of the perturbative series in QCD is affected by uncancelled large logarithms as well as by “renormalon” terms that diverge as $\left(n!\beta_{i}^{n}\alpha_{s}^{n+1}\right)$ at higher orders [21, 22]; this is known as the renormalon problem [23]. Such renormalon terms can give sizable contributions to the theoretical estimates, as shown in $e^{+}e^{-}$ annihilation, $\tau$ decay, deeply inelastic scattering and hard processes involving heavy quarks. These terms are responsible for important corrections at higher orders also in the perturbative region, leading to different predictions according to the different choices of scale (as shown in Ref. [24]). Large logarithms on the other hand can be resummed using the resummation technique [25, 26, 27, 28, 29, 30, 31] and results are IR renormalon free. This does not help for the renormalization scale and scheme ambiguities, which still affect theoretical predictions with or without resummed large logarithms. In fact, as recently shown in Ref. [20] for the $b\bar{b}$-production cross-section at NNLO-order accuracy at hadron colliders, the CSS scale setting leads to theoretical uncertainties that are of the same order as the NNLO corrections $\sim 20-30\%$, taking as the typical momentum scale the $b$-quark mass $m_{b}\sim 4.92$ GeV.

• 4.

  In the Abelian limit $N_{c}\rightarrow 0$ at fixed $\alpha_{em}=C_{F}\alpha_{s}$ with $C_{F}=\left(N_{c}^{2}-1\right)/2N_{c}$, a QCD case effectively approaches the analogous QED case [32, 33]. Thus, to be self-consistent, any QCD scale-setting method should also be extendable to QED and results should be in agreement with the Gell-Mann–Low (GM–L) scheme. This is an important requirement also from the perspective of a grand unified theory (GUT), where only a single global method for setting the renormalization scale may be applied and then it can be considered as a good criterion for verifying if a scale setting is correct or not. CSS leads to incorrect results when applied to QED processes. In the GM–L scheme, the renormalization scale is set with no ambiguity to the virtuality of the exchanged photon/photons, which naturally sums an infinite set of vacuum polarization contributions into the running coupling. Thus, the CSS approach of varying the scale by a factor 2 is not applicable to QED, since the scale is already optimized.

• 5.

  The large amount of forthcoming high-precision experimental data, produced especially by the running at high collision energy and high luminosity of the Large Hadronic Collider (LHC), will require more accurate and refined theoretical estimates. The CSS appears to be more a “guess”; its results are afflicted by large errors and the perturbative series converges poorly with or without large-logarithm resummation or renormalon contributions. Moreover, within such a framework, it is almost impossible to distinguish between SM and BSM signals and in many cases, improved higher-order calculations are not expected to be available in the short term.

To sum up, the conventional scale-setting method assigns an arbitrary range and an arbitrary systematic error to fixed-order perturbative calculations that greatly affects the predictions of pQCD. On the other hand, various strategies for optimization of the truncated expansion have been proposed. We point out that since its very beginning, several attempts have been performed to improve the renormalization procedure and different schemes have been introduced mainly to improve the convergence of the perturbative series. An example is the introduction of the $\overline{\rm MS}$, as suggested in Refs.[34, 35] and later the introduction of the momentum subtraction scheme (MOM) as discussed by Celmaster and Gonsalves in Refs. [36, 6]. The MS scheme, introduced in Ref. [37], is in fact quite arbitrary and related to the particular regularization scheme used, leading to a rather higher second-order contribution in the DIS process, by reabsorbing the scheme term $\ln(4\pi-\gamma_{E})$ into a redefinition of the QCD scale parameter $\Lambda$, Bardeen et al. noticed a significant improvement of the convergence of the perturbative series. Later, the introduction of the MOM scheme, though being gauge dependent, has been shown to also include other scheme terms into $\Lambda$, leading to a scheme which is less dependent on the regularization procedure and tending to a more "physical" scale and nearly to an "optimization procedure" [6]. A recent discussion on the gauge dependence of the MOM scheme has been given in Ref. [38]. We discuss these schemes in more detail in Sec. 2.3.

In general, an optimization or scale-setting procedure is considered reliable if it preserves important self-consistency requirements. All Renormalization Group properties, such as uniqueness, reflexivity, symmetry and transitivity should also be preserved by the scale-setting procedure in order to be generally applied [39]. A fundamental requirement is the scheme independence; other requirements can be suggested by known tested theories (such as QED), by the convergence behavior of the series in particular kinematic regions or by important phenomenological results.

The Principle of Minimal Sensitivity (PMS) is an optimization procedure proposed by Stevenson [11, 10, 9, 8] based on the assumption that, since observables should be independent of the particular RS and scale, their optimal perturbative approximations should be stable under small RS variations. The RS-scheme parameters $\beta_{2},\beta_{3},\dots$ and the scale parameter $\Lambda$ (or the subtraction point $\mu_{r}$) are considered “unphysical” and independent variables; their values are thus set in order to minimize the sensitivity of the estimate to their small variations. This is essentially the core of Optimized Perturbation Theory (OPT) [9], based on the PMS procedure. The convergence of the perturbative expansion is improved by requiring its independence from the choice of RS and $\mu$. Optimization is implemented by identifying the RS-dependent parameters in the truncated series ($\beta_{i}$ for $2\leq i\leq n$ and $\Lambda$), with the request that the partial derivative of the perturbative expansion of the observable with respect to the RS-dependent and scale parameters vanishes. In practice, the PMS scale setting is designed to eliminate the remaining renormalization and scheme dependence in the truncated expansions of the perturbative series.

We would argue that this approach is based more on convergence rather than physical criteria. In particular, the PMS is a procedure that can be extended to higher order and it can be generally applied to calculations obtained in arbitrary initial renormalization schemes. Although this procedure leads to results that are suggested to be unique and scheme independent, it unfortunately violates important properties of the renormalization group, as shown in Ref. [40], such as reflexivity, symmetry, transitivity and also the existence and uniqueness of the optimal PMS renormalization scheme are not guaranteed since they are strictly related to the presence of maxima and minima.

Another optimization procedure, namely the Fastest Apparent Convergence (FAC) criterion was introduced by Grunberg and is based on the idea of effective charges. As pointed out by Grunberg [12, 13, 14], any perturbatively calculable physical quantity can be used to define an effective coupling, or “effective charge”, by entirely incorporating the radiative corrections into its definition. Effective charges can be defined from an observable starting from the assumption that the infinite series of a given quantity is scheme and scale invariant. The effective charge satisfies the same renormalization group equations as the usual coupling. Thus, the running behavior for both the effective coupling and the usual coupling are the same if their RG equations are calculated in the same renormalization scheme. This idea has been discussed in more detail in Refs. [41, 42].

An important suggestion is that all effective couplings defined in the same scheme satisfy the same RG equations. While different schemes or effective couplings would lead to different renormalization group equations. Hence, any effective coupling can be used as a reference for the particular renormalization procedure. In general, this method can be applied to any observable calculated in any RS, also in processes with large higher-order corrections. The FAC scale setting, as has been shown in Ref. [40], preserves the RG self-consistency requirements, although the FAC method can be considered more an optimization approach rather than a proper scale-setting procedure to extend order by order. FAC results depend sensitively on the quantity to which the method is applied. In general, when the NLO correction is large, the FAC proves to be a resummation of the most important higher-order corrections and thus a RG-improved perturbation theory is achieved.

PMS and FAC are procedures commonly in use for scale setting in perturbative QCD together with CSS and an introduction to these methods can be found in Refs. [40, 43]. However, as shown in Refs. [44], these optimization methods not only have the same difficulties of CSS, but they also lead to incorrect and unphysical results in particular kinematic regions.

A solution to the renormalization scale setting problem is offered by the Principle of Maximum Conformality (PMC) [45, 46, 47, 48]. This method is the generalization and extension of the original Brodsky-Lepage-Mackenzie (BLM) method [15] to all theories, to all orders and to all observables and it satisfies all the theoretical requirements of a reliable scale-setting procedure at once, leading to accurate and consistent results. The primary purpose of the PMC method is to solve the scale-setting ambiguity; it has been extended to all orders [49, 50] and it determines the correct running coupling and the correct momentum flow according to RGE invariance [51, 52]. This leads to results that are invariant with respect to the initial renormalization scale and in agreement with the requirement of scale invariance of observables in pQCD [40]. The approach provides a systematic method to eliminate renormalization scheme and scale ambiguities from first principles by absorbing the $\beta$ terms, which govern the behavior of the running coupling via the renormalization group equation. Thus, the divergent renormalon terms cancel, improving convergence of the perturbative QCD series. Furthermore, the resulting PMC predictions do not depend on the particular scheme used, thereby preserving the principles of renormalization group invariance [39, 51]. The PMC procedure is also consistent with the standard Gell-Mann–Low method in the Abelian limit, $N_{c}\rightarrow 0$ [32]. Moreover, in a theory unifying all forces (electromagnetic, weak and strong interactions), such as Grand Unified Theories, one cannot trivially apply a different scale-setting or analytic procedure to different sectors of the theory. The PMC offers the possibility to apply the same method to all sectors of a theory, starting from first principles, eliminating renormalon growth, scheme dependence, scale ambiguity and satisfying the QED Gell-Mann–Low scheme in the zero-color limit $N_{c}\to 0$.

We remark that the fundamental task of the PMC is to solve the renormalization scale and scheme ambiguities and in order to achieve this, it makes use of the RG-equations to reabsorb the $n_{f}$-terms that are related to the UV-divergent diagrams. The procedure of the PMC works with any initial definition of the scheme or scale for the running coupling, MS or $\overline{\rm MS}$ or the ’t Hooft scheme are all equivalent, since the entire scheme dependence of the observable is reabsorbed into the running coupling and into the PMC scale in the final series. In particular, the lower order terms, $\beta_{0},\beta_{1}$, of the $\beta$-function are scheme independent; thus a scheme transformation cannot lead to a prescription for determining the scale. In fact, also in the case of the ’t Hooft scheme (as shown in Refs.[53, 54]) one can eliminate all scheme dependent $\beta_{i}$ coefficients, but there is no prescription for the renormalization scale, which can be considered as the first scheme parameter. Moreover, what seems a good prescription for the running coupling is not necessarily a good prescription for the entire fixed-order calculated quantity. In fact, still considering the ’t Hooft case, the entire series of $\beta_{i},i\geq 2$ is removed from the definition of the coupling, but in a fixed-order calculation these $\beta_{i}$ coefficients would have an impact on the coefficients of the series at each order of calculation and on the convergence of the series. On the other hand, the PMC gives a prescription for fixing the scale and reabsorbing all scheme-dependent terms of a cross-section into the running coupling and into the PMC scale, exposing the perturbative series to a minimization/cancellation of the effects of the scale and scheme uncertainties. An important consequence of the PMC procedure is the RS-invariance of the resulting series. We refer to RS-scheme invariance as the invariance under the extended-renormalization group and its equations (xRGE)[9, 55, 56, 17]. It can be shown that the PMC procedure may be performed either way using the same xRGE¹¹1We will discuss this procedure in detail in a future work soon.. The xRGE show that different scheme definitions can be related at the lowest order by a scale transformation for the case of the minimal subtraction schemes (MS and $\overline{\rm MS}$) and the momentum space subtraction (MOM) scheme, but using the PMC, results would be scheme independent. A recent argument on the scheme dependence of the first conformal coefficient by Stevenson [57] is incorrect. The reason is that he assumes the $v_{1}$ coefficient of the scheme to be totally free with respect to the structure of the color factors and that it can be reabsorbed into the $\Lambda$ parameter. There are several reasons why this certainly leads to wrong results. First, two different couplings in two different schemes at NLO can only be related by a scale transformation according to the RG group, i.e. by a shift of the type: $\beta_{0}\log(\frac{\mu_{1}^{2}}{\mu_{2}^{2}})$ as shown also in Ref. [17]. In fact, given the scheme invariance of the $\beta_{0},\beta_{1}$, the only parameter that can be varied at NLO is the renormalization scale $\mu_{r}$, which can have any real value or at most any value in the complex plane. Thus, the only scheme terms that can be reabsorbed into the scale (either the $\Lambda$ or the renormalization scale $\mu_{r}$) are those related to a shift of the scale using the standard RG group equations. This corresponds to the freedom of subtracting out any finite value together with the pole, in the renormalization procedure (e.g. MS and $\overline{\rm MS}$). If one takes the freedom to vary the scheme according to any other relation that does not correspond to a "proper" RG equation at NLO, this consequently modifies the structure of the color factors for the coupling and thus would obtain a wrong result. If one follows this misleading assumption, one can obtain any result outside of a given initial theory and this is certainly not the aim of the xRGE, which should preserve the invariance of the final result. Moreover, if one changes the color structure for the coupling, one should be aware that, to be consistent, also the structure of the entire fixed-order calculation should be varied accordingly; though we do not agree with this procedure since a change of the color structure correspond to a change of the initial SU(N) theory (e.g. from QCD to QED Ref.[32]). Thus, any relation among couplings in different schemes, which can have perturbative validity in QCD but which cannot be considered as "proper" xRG transformations, should be considered as matching relations among quantities defined in different approximations or obtained using different approaches. Also in this case, results can be improved using the PMC and the residual dependence on the particular implicit definition of the "scheme" can be suppressed perturbatively by adding higher-order calculations. It can be shown that also in this case the results obtained are scheme-independent (e.g. see Ref. [58]). One may point out that, though the LO and the NLO conformal coefficient are scheme invariant, the higher-order conformal coefficients can be scheme dependent, we can answer that in the worst case this scheme dependence for any of the PMCs procedures is highly suppressed. Another argument in Ref. [57] by Stevenson, which is based on the principle of minimum sensitivity (PMS), we hold to be incorrect. Since the PMS is based on the assumption that all the unknown higher-order terms give zero contribution to the pQCD series [9], its prediction directly breaks the standard renormalization group invariance [39, 51], its pQCD series does not have normal perturbative features [59] and can be treated as an effective prediction only when we know the series up to high enough orders and the conventional series has already shown good perturbative features [60]. On the other hand, the PMC respects all features of the renormalization group, and its prediction satisfies all the requirements of standard renormalization group invariance [39, 51, 58, 52]. Moreover, given that the PMC preserves the RG invariance, it is possible to define CSR - Commensurate Scale Relations[50] among the effective charges relating observables in different "schemes" preserving all the group properties. Applying the PMC and the CSRs, one can relate effective couplings, as also conformal coefficients, leading to scheme-independent results for the observables. We remark that in order to apply the PMC correctly, one should be able to distinguish among the nature of the different $n_{f}$-terms: whether they are related to the running of the coupling, to the running of masses or to UV-finite diagrams and in a deeper analysis also to the particular UV-divergent diagram (as discussed in Refs. [61, 48]). Once all $n_{f}$-terms have been associated with the correct diagram or parameter, conformal coefficients are RG invariant and match the coefficients of a conformal theory. Applications of the PMC to different quantities (see Ref. [62, 63]) have recently shown a direct improvement of theoretical predictions. Moreover, a deeper insight into the QCD coupling $\alpha_{s}(Q)$ at all scales, including $Q^{2}=0$, has been recently achieved (e.g. see Refs. [64, 65]), showing results consistent with the PMC.

So far, the PMC approach has been successfully applied to many high-energy processes, including Higgs boson production at the LHC [66], Higgs boson decays to $\gamma\gamma$ [67, 68], $gg$ and $b\bar{b}$ [69, 70, 71, 72, 73, 63] processes, top-quark pair production at the LHC and Tevatron [74, 47, 75, 76, 77, 78, 79], decay process [80, 81], semihard processes based on the BFKL approach [82, 83, 84], electron–positron annihilation to hadrons [49, 50, 51], hadronic $Z^{0}$ boson decays [85, 86], the event shapes in electron–positron annihilation [87, 88, 89, 90, 91], the electroweak parameter $\rho$ [92, 93], $\Upsilon(1S)$ leptonic decay [94, 95], and charmonium production [96, 97, 98] and decay [99, 100, 101, 102]. As shown in Ref. [103], by using the PMC, one can obtain a smooth transition for the running behavior between Bjorken sum rule effective coupling $\alpha_{s}^{g_{1}}(Q)$ in the V-scheme and the Light-Front Holographic QCD (LFHQCD) coupling, going from the perturbative to the non nonperturbative domain respectively. In addition, the PMC provides a possible solution to the $B\to\pi\pi$ puzzle [104] and to the $\gamma\gamma^{*}\rightarrow\eta_{c}$ puzzle [105].

In particular, with regard to the PMC application to top-quark pair hadroproduction, the resulting production cross-sections agree with precise experimental data, and the large discrepancies of the top-quark forward-backward asymmetries between SM estimations and experimental measurements are greatly reduced [47, 74, 75, 76, 77]. Recently, an improved QCD prediction for the top-quark decay process obtained by using the PMC has also been presented in Ref.[80].

A precise top-quark mass can be extracted by the comparison of precise PMC predictions of production cross-sections with experimental data measured at LHC [78, 79]. The determined top-quark pole mass $m_{t}=172.5\pm 1.4$ GeV, from the LHC measurement at $\sqrt{s}=13$ TeV [79] agrees with the world average cited by the Particle Data Group (PDG) [106]. More explicitly, we present a summary of the top-quark pole masses in Fig.(1.1), where our PMC result and previous determinations from collider measurements at different energies and different techniques are presented. Owing to unknown higher-order contributions, this leads to two kinds of residual scale dependence for PMC predictions. However, these two residual scale dependencies are distinct from the conventional scale ambiguities, they are given by the unknown higher-order uncalculated contributions, while the scale dependence of the CSS is related more to the calculated orders. Thus, the residual scale dependence of the PMC results from the unknown higher-order terms and is an unavoidable intrinsic feature of the perturbative approach. It is to be noted that two of the three PMC ambiguities identified in Ref.[107] correspond exactly to two kinds of residual scale dependence. These two residual scale dependencies are highly suppressed; however, if the pQCD convergence of the perturbative series of either the PMC scale or the pQCD approximant is weak, such residual scale dependence could be significant. A recent discussion of the residual scale dependence of PMC predictions can also be found in the review [52].

In this review we display recent developments in solving the renormalization-scale and -scheme ambiguities based on the PMC and present some fundamental applications and results.

With respect to the previous reviews on the PMC (Refs. [40, 52]), in this review we focus on the recently developed method, namely the Infinite-Order Scale-Setting using the Principle of Maximum Conformality (PMC_∞), showing first some of its applications and new features, and then a comparison of this method with the CSS and the other PMC approaches, such as the multi-scale (PMCm), and the single-scale PMC (PMCs), on the Event Shape Variables and on some crucial fully integrated quantities: $R_{e^{+}e^{-}}$, $R_{\tau}$ and $\Gamma(H\to b\bar{b})$.

More in detail: in Section I we give an introduction to the renormalization scale setting problem in QCD; in Section II we recall fundamental equations of the renormalization group and their extended version, starting from the renormalization procedure of the strong coupling $\alpha_{s}(\mu)$ and its renormalization scale dependence; in Section III we summarize formulas and basic concepts of the PMC approach with its features and methods (PMCm, PMCs); in Section IV we introduce the newly developed method PMC_∞ and its new features; in Section V we show results for the application of the PMC_∞ to the event-shape variables: thrust and $C$-parameter, we show particularly new features of the PMC performing interesting limits for thrust (IR conformal and QED limit) and comparing the results under the CSS and the PMC_∞ methods; in this section we also show a novel method to determine the strong coupling and its behavior $\alpha_{s}(Q)$ over a wide range of scales, from a single experiment at a single scale, using the event-shape variable results; in Section VI we present a detailed comparison of the CSS, PMCs, PMCm and PMC_∞ methods tested on fundamental fully integrated quantities at 4 loops: $R_{e^{+}e^{-}}$, $R_{\tau}$ and $\Gamma(H\to b\bar{b})$; finally, in Section VII we summarize and discuss the results.