Duality between QCD Perturbative Series and Power Corrections

We elaborate on the relation between perturbative and power-like corrections to short-distance sensitive QCD observables. We confront theoretical expectations with explicit perturbative calculations existing in literature. As is expected, the quadratic correction is dual to a long perturbative series and one should use one of them but not both. However, this might be true only for very long perturbative series, with number of terms needed in most cases exceeding the number of terms available. What has not been foreseen, the quartic corrections might also be dual to the perturbative series. If confirmed, this would imply a crucial modification of the dogma. We confront this quadratic correction against existing phenomenology (QCD (spectral) sum rules scales, determinations of light quark masses and of \alpha_s from \tau-decay). We find no contradiction and (to some extent) better agreement with the data and with recent lattice calculations.


Introduction
Because of the asymptotic freedom, predictions for short-distance processes are very simple in QCD and essentially reduce to parton model, or to lowest order perturbation theory. This is true, however, only in the leading order approximation. As far as corrections are concerned, there is a double sum which includes expansion in α s (Q 2 ) where Q 2 is a generic large mass parameter and powers of (Λ QCD /Q) k . Consider for example the best studied case of current correlators which determine QCD sum rules [1] (for a review, see e.g. [2]). Then, one usually assumes the following form of the correlator in the x-space: 0|J(x), J(0)|0 ≈ C I (α s (x))I (1) where J(x) is the hadronic current, I is the unit operator and G 2 is the dimension four operator. The coefficient functions C I,G 2 are calculable perturbatively as infinite sums in the running coupling. To appreciate in full the complexity of the general expression in Eq.
These quadratic corrections and their phenomenological significance will be in fact focus of our attention. Let us remind the reader what is understood by these corrections.
Start with the heavy quark potential at short distances. The Cornell version of this potential (which describes the lattice data very well) is very simple: where r is the distance, α s ≡ g 2 /(4π) is the QCD coupling, σ ≈ 0.2 GeV 2 is the string tension. The fit in Eq. (2) works well at all distances. The question is whether such a form of the potential at short distances -let it be only approximate -is acceptable theoretically. There are papers which ascertain a positive answer to this question (see, in particular, [3,4]). The observable (heavy-quark potential in our case) is viewed as represented by a short perturbative series (a single const/r term in our case) plus a leading power correction (quadratic correction, in our case, σ · r). The version used in some other papers (see, in particular, [13]) looks as: lim r→0 V QQ (r) ≈ 1 r n=4 n=1 a n α n s (r) + (const) +σ n · r , (3) where n = 4 is a realistic number of perturbative terms which can be calculated nowadays explicitly and const stands for an infrared renormalon contribution (this could be added to the version in Eq. (2) as well). The last term, proportional toσ n imitates the power correction.
It is quite common [13] to identify the parameters σ from Eq (2) andσ from Eq (3) and compare their numerical values. Our central point is that such an identification is not justified.
There are two dual descriptions: either one uses a short perturbative series and adds the leading quadratic correction by hand, or one uses long perturbative series and then there is no reason to add the quadratic correction [14].
It is worth emphasizing that numerically both approaches in Eqs. (2) and (3) work well (in case of the fit in Eq. (3), the distance r should not be too large). Chronologically, the papers in the series [3,4] appeared first. At that time, the common belief was that the Voloshin-Leutwyler potential is valid nonperturbatively. This would correspond to a cubic correction in Eq. (2) (or (3)). The papers [3,4] established validity of the unconventional (at that time) quadratic correction. The emphasis in later papers [13] was in fact to give another interpretation to the already known quadratic power correction.
The problem of mixing between power-like corrections and perturbative series is not new at all. The standard view is that power corrections are related to divergences in perturbative series due to the factorial growth of the expansion coefficients (for review see, e.g., [15]). This viewpoint formulated long time ago still dominates theoretical thinking. In practice, however, no factorial growth of the expansion coefficients has been observed so far. The reason could be that the ability to calculate the expansion coefficients is limited and the series known explicitly are not long enough.
Here we come actually to a key point. Because in phenomenological applications, one usually assumes, explicitly or tacitly, that large-order asymptote sets in immediately after the terms known explicitly ( for further references and detailed numerical estimates of this type, see [16,17]).
There exists, however, an example of a long perturbative series which allows to check the current ideas on the expansion coefficients. We have in mind the perturbative calculations of the gluon condensate [18,19,20]. This example indicates that, to apply theorems on the asymptotic behaviour of the expansion coefficients, one should have in fact much longer series than are available in reality. Thus, we argue that, in realistic phenomenological fits, one should keep, in particular, the quadratic corrections which are absent from the symbolic expansion in Eq. (1): Thus, our main point is that the properties of the relatively short perturbative series are different from properties of long perturbative series.
Another new point is the impact of the dual models. We will argue, basing on the results of [21], that there exists another source of the quartic corrections, which are usually identified with the infrared-sensitive part of the gluon condensate G 2 . Namely, the same short-distance contributions which control the quadratic correction taken to second order, produce a calculable quartic correction. We confront this insight brought by the dual models with explicit perturbative calculations of papers in Ref. [18,19,20].
In Sect. 2, we discuss an argumentation in favor of the duality between the quadratic correction and long perturbative series. In Sect. 3, we emphasize lessons for the generic structure of perturbative series brought by the explicit calculations of the gluon condensate. In Sect. 4, we propose a simplified generic version of perturbative series. In Sect. 5, we summarize lessons brought by the holographic models. In Sect. 6, we discuss the unexpected duality between the perturbative and quartic power corrections. Sect. 7 is devoted to phenomenology of particular processes. In Sect. 8, we present our conclusions.

Duality expected (quadratic correction)
• Duality between sand t-channels Because of the existing confusion in the literature concerning the duality between long perturbative series and quadratic correction, let us start with the notion of the duality itself.
Consider a hadronic reaction a+b → c+d at relatively low energies. Then the following representation of the amplitude can be reasonable: Such a phenomenology was popular a few decades ago and turned successful. Now, imagine that one starts improving Eq. (4) by summing up the s-channel exchanges: where the sum over the s-channel resonances is taken. Then, if N is large enough one would notice that there is no more space for the t-channel exchanges. The conclusion could be that there are no t-channel particles or that they are decoupled from our hadrons a, b, c, d.
As everybody knows, beginning with the celebrated Veneziano's paper [22], such a conclusion would be wrong. Namely, if one uses sums over the resonances, then it is either s-channel or t-channel exchanges that are allowed but not both.
As was argued in 2003 [14] (if not earlier), similar things happen in the case of the quadratic corrections to the parton model (of which a linear potential is an example). One uses either a short perturbative series and adds a linear term by hand. This is an analogy to the nearest-singularity amplitude in Eq. (4) and corresponds to the form in Eq. (2). Or one uses a long perturbative series and then does not add by hand the linear term. Since it is already included into the perturbative series, by virtue of the general theorems inherent to the Yang-Mills theories. This is then the version in Eq. (3).
Thus, claiming that the parameterσ ≈ 0 in Eq (3) contradicts σ = 0 in Eq (2) is like claiming that summing up the s-channel exchanges proves that there are no t-channel particles in nature.
• Quadratic correction and OPE The proof in [23] that there are no genuine nonperturbative quadratic corrections is simple. Indeed, originally, the quadratic correction was associated with the so called ultraviolet renormalon which corresponds to the following asymptotic series: where Q 2 is a generic large mass parameter inherent to the problem and b 0 ≡ 1 4π (11 − (2/3)n f is the first coefficient in the β-function for n f flavours.
If one treats the expansion in Eq. (6) as an asymptotic series, then its uncertainty is a quadratic correction, Λ 2 QCD /Q 2 . If, on the other hand, one sums up the seriesà la Borel: then there is no uncertainty at all. The crucial observation is that the factorial growth of the expansion coefficients in Eq. (6) are associated with an integration over very large momenta, p 2 ≫ Q 2 . However, because of the asymptotic freedom, this region of integration should not be a source of uncertainty in QCD. Indeed, by introducing a cut off a and using the coupling α s (a) normalized in the UV, one eliminates the integration over momenta p 2 > a −2 and, therefore, there is no ambiguity [23].
From the point of view of the operator product expansion (OPE) in Eq. (1), the quadratic correction we are discussing is hidden in the coefficient function in front of the unit operator, C I (α s (x)) and, in no way, violates the OPE.
However, the QCD (spectral) sum rules were originally based [1] (for a review, see e.g. [2]) on a simplified assumption that the coefficient functions can be approximated by the first terms, while the effect of the confinement is encoded in the power corrections. It is only within this terminology that one might say that the form in Eq. (2) violates the OPE. In a more correct but longer language, what is violated is the assumption that the coefficient functions are approximated by their first terms. More advanced applications of the sum rules are keeping longer and longer perturbative series. Then, the terminology with 'violations of the OPE' due to the quadratic correction becomes obsolete.
Another source of confusion is the observation that, in the Euclidean space, one can ascribe a gauge-invariant meaning to the vacuum matrix element of dimensiontwo operator (A a µ ) 2 [24]. The quantity 0|(A a µ ) 2 |0 min turns to be of significant interest in many applications. This does not change of course the fact that 0|(A a µ ) 2 |0 min does not appear in the operator product expansion (1). None of the papers in Ref [3] claims either violation of the OPE with 'long' perturbative expansions or appearance of 0|(A a µ ) 2 |0 min in the OPE equations. Nevertheless, sometimes one fights just with these would-be-made claims [13]. From the perspective of the perturbative expansion, the most difficult question is: why the quadratic correction could be at all important ? Therefore, it is worth emphasizing that the quadratic correction is required by phenomenology, not yet by the theory. For example, on the lattice, one can give a definition of the 'nonperturbative' heavy quark potential (for review see e.g. [40]). Then, this potential is pure linear starting from the smallest distances available: Moreover, this non-perturbative contribution encodes confinement as well. Thus, it is strongly suggested by the phenomenology that the effects of the confinement are encoded in the quadratic correction which is not explicit in the general OPE in Eq. (1) 1 . As we argue in Section 5, a natural framework for the quadratic correction is provided by the stringy, or holographic formulation of QCD (for a review see, e.g., [25]) .

Lessons from PT calculation of α s GG
The best check of this logic is provided by the beautiful results for perturbative calculation of the gluon condensate on the lattice (the most advanced calculations are due to Rakow et al. [18,19,20]): More precisely, the results refer to perturbative evaluations of the quantity: where a is the lattice spacing and α s (a) is the running coupling normalized at the ultraviolet cut off, p n are the expansion coefficients which are calculated explicitly up to n = N . Finally the difference ∆ N is known numerically since the total value of the l.h.s. of Eq. (9) coïncides with the plaquette action and is known to a very good precision. Phenomenologically, the difference is a fit to power-like corrections: where the coefficients b 2,4 are fitting parameters which depend on the number of perturbative terms calculated explicitly because of the fitting procedure. Explicit results [18,19,20] demonstrate that, indeed, the power corrections in Eq (9) depend strongly on the number N of perturbative terms taken into account explicitly. Namely, up to N ≈ 10 the power corrections are dominated by a quadratic term: 1 A phenomenologically successful fit to the power corrections is provided by the 'short-distance' gluon mass (see [4,5,6,7,8,9,10,11,12]). However, the very notion of the short-distance gluon mass can be introduced only in the Born approximation and only for a certain class of processes [3,4].
That is, the coefficient b N 4 is consistent with zero for such N . However, the numerical value of the coefficient b N 2 in front of the power correction diminishes with increasing N . Thus, perturbative corrections 'eat up' the power correction. In more refined terminology, the perturbative terms are dual to the leading power correction.
At N > 10, a quartic correction emerges as a result of subtracting the perturbative contributions from the total matrix element α s G 2 : And, finally, at about N ≈ 16 one restores the value of the quartic correction which is [18,19]: with a large error, but the result is comparable in magnitude with the standard gluon condensate entering the QCD (spectral) sum rules which we shall discuss in the next section. Another remarkable finding [18,19] is that perturbative coefficients p n entering (9) are well approximated by a simple geometric series: where the fitting parameters u = 0.961 (9), q = 0.99 (7), s = 0.44 (10). The perturbative series with such coefficients is convergent for This simple geometrical series fits explicit calculations of the PT coefficients at least for the first 16 perturbative terms. Extending n → ∞(n ≥ 50), the geometric series reproduces the full answer to an accuracy better than 10 −3 , which is a remarkable result.

Geometric growth of the PT coefficients ?
Physicswise, one can say that the series found in [18,19,20] is determined by the singularity due to the crossover from strong to weak coupling. This is true in pure gluonic channel. This could also be true with account of quarks. Then, we would have, in different channels, geometric series, with approximately the same range of convergence. To see whether a such hypothesis can be ruled out, we compile below the calculated expressions of the Adler-like function in the euclidian region 2 for different channels.
In the vector channel with massless quarks, it reads [30,31,28]:  [26,27,28] indicates a geometric growth, but the effects due to the analytic continuation and to the the β-fuctions induced by the renormalization group equation obscure the exact behaviour of the coefficients. A more appropriate interpretation of a such behaviour requires a more involved analysis [29].
The perturbative corrections to this expression due to the strange quark mass for the neutral vector current, read [26,32]: while it is [26,33]: for the charged current controlling the ∆S = −1 τdecay process. The difference from α 2 s is due to the light by light scattering diagram contributing in the neutral vector two-point correlator.
For the pseudoscalar channel, the QCD expression of the Adler-like function reads for n f = 3 [34,35,36]: Similarly, one can also present the PT expressions of moments in deep-inelastic scatterings. The ones of the well-known Bjorken sum rule for polarized electroproduction or of the Gross-Llewellyn Smith sum rule for neutrino-nucleon scattering, read, for n f = 3 [37,38,39]: One can notice that, in all the cases, the series found, do not show a factorial growth nor an alternate sign but, are consistent with geometric series, with sizable corrections at small n similar to the case of the gluon condensate 3 . Thus, there is an exciting perspective that all the perturbative series are in fact quite simple in large orders.

Insight from dual models
• Holographic quadratic correction In the holographic language, one evaluates the same observables, as within the field theoretic formulation of QCD, but in terms of strings living in extra dimensions. There is no direct derivation of the metrics of the extra dimensions in the QCD case. One rather uses phenomenologically motivated assumptions (see, e.g., [21]). The crucial element is the metrics z in the fifth dimension. Following [21], let us choose the following model: where R 2 is a constant whose explicit definition is not important for us here and the function h(z 2 ) is specified 3 A geometric growth of the PT coefficient has been assumed in [27] for predicting the α 4 s term of the PT series of the D-function in the V+A channel, where the result has been (approximately) confirmed later on by the analytic calculation of [28].
below. Note that, at z → 0, one needs h(z 2 ) → const in order to reproduce an approximate conformal symmetry of the Yang-Mills theories (due to the asymptotic freedom).
We would like to define the function h(z 2 ) in such a way as to ensure confinement at large distances and to reproduce the (leading) quadratic power correction at short distances. The following choice: satisfies these conditions. Note that, while the condition to reproduce confinement, or the area law for the Wilson line is common to all the holographic models, the condition to reproduce the quadratic correction at short distances assumes that it is this correction which encodes the confinement at short distances. One can demonstrate that, assuming Eq. (20), is equivalent to assuming the Cornell potential for the heavy quarks interaction. The numerical value of the constant c can be fixed in terms of the string tension, c 2 = (0.9 GeV) 2 . The simple model in Eqs (19) and (20) turns to be successful phenomenologically (see, in particular, [41] and references therein).
The advantage of using the hologhraphic language, which is quite crucial to our mind, is that, it allows for a perfectly gauge invariant way to introduce and parametrize the quadratic correction. Also, the simple expression (20) looks much more 'natural' than the assumption on approximate equality of the long perturbative series in Eq. (3) and quadratic correction plus short series as can be seen in Eq. (2). What is lacking at this time, is further applications of the same metrics in Eq. (20) to evaluate quadratic corrections to the parton model in other cases, such as the current correlators.
• Holographic quartic correction Presence of the quadratic correction in the string-based approach is an assumption which allows to model the metric in the fifth dimension. However, once the metric is fixed, one can calculate the full answer for the gluon condensate [42].
The model does not account for the running of the coupling but allows to evaluate power corrections. In particular, it produces the value of the 'physical gluon condensate' of the magnitude: which is not unreasonable phenomenologically [1]. What appears even more important is the fact that the dual-model approach produces a new qualitative picture for the power corrections. Namely, in the holographic language α s G 2 ∼ Λ 4 QCD appears as a secondorder effect in the coefficient c introduced in Eq. (20): Since the coefficient c (or the quadratic correction in the holographic language) is associated with short distance, the same is true for the gluon-condensate contribution in Eq. (22). In short, the stringy calculation does not have a counterpart to the infrared-renormalon contribution which is taken for granted in field theoretic approach. This point is worthy to be elaborated.
In both cases of field theory and of stringy calculation, one deals with a propagator, of a particle or a string respectively. In both cases, the leading contribution comes from short distances. If the typical size is of order a, then, in both cases, α s G 2 ∼ a −4 . However, the probability for a (virtual) particle to propagate to the distance of order Λ −1 QCD is power-like suppressed: as revealed by the infrared renormalon (see, e.g., [15]).
In the case of strings, the suppression of the infrared region turns to be exponential [42]: where γ is positive. Intuitively, this much stronger suppression than in Eq. (23) corresponds to the fact that string corresponds to a collection of particles.

Duality unexpected quartic correction
The picture described above was mostly expected on theoretical grounds. In particular, it has been understood since long (see, e.g., [23]) that the quadratic correction, coming from short distances, is calculable, as a matter of principle, through long perturbative series. However, it was expected that the quartic correction in Eq. (11) emerges simultaneously with factorial divergence in the value of the perturbative series expansion coefficients a n (see Eq. (9)): p n+1 p n IR renormalon ∼ n for n ≫ 1 .
This divergence is due to the infrared renormalon (for a review, see [15]). So far [18,19,20], one does not run into the problem of the divergence in Eq. (25): It is even more amusing that, with presently available perturbative terms in Eq (9), one can extract [18,19,20] the 'genuine' gluon condensate in Eq. (11): so that the quartic correction gets disentangled from the infrared renormalon. This observation, if confirmed, is a radical change of dogma.
It is not ruled out that the infrared renormalon still shows up in higher orders of perturbation theory, say, at n ∼ 25, as discussed in [18,19]. However, its contribution will be in any case smaller than the condensate in Eq. (27) determined from perturbative series which looks like a geometric series and exhibits no factorial growth of the coefficients.
It is amusing that the dual models independently provide a mechanism of generating the quartic correction from short distances [see discussion in Eq. (9)]. The condensate in Eq. (9) is not related to any divergence of the perturbative theory either. Thus, two independent approaches result in similar pictures.

Phenomenology of 1/Q 2 corrections
• Tachyonic gluon mass squared λ 2 From the phenomenological point of view, it would be important to relate the quadratic corrections in various channels. However, there is no model independent way to derive such relations. A model which turns successful is the introduction of a tachyonic gluon mass λ 2 at short distances [3,4]. From the calculational point of view one changes the gluon propagator: and checks that the quadratic correction is associated with large momenta k 2 ∼ Q 2 . To the lowest order the analysis is gauge invariant. The model in Eq. (28) is purely heuristic in nature.
• Estimate of α s G 2 and of α s λ 2 One can extract these two power corrections by using ratio of exponential QCD (spectral) sum rules in e + e − → hadrons data [5,6], which is not sensitive to the leading α s corrections. It is worth mentioning that FESR may not be appropriate for extracting such small quantities, as it requires a cancellation of two large numbers which depend on the high-energy parametrization of the spectral function. This feature is signaled by the large range spanned by the determinations of power corrections using FESR [43] and the discrepancies of the estimated quadratic corrections in [44] and [45], which both also differ from the one using the ratio of exponential Borel/Laplace (LSR) used in [5,6]. In addition to previous channels, the gluon condensate can be also obtained using a ratio of LSR for the J/ψ − η c and Υ − η b mass-splittings, which has a minimum sensitivity on the heavy quark mass effects and on the α s corrections [46] The resulting values of the parameters are [5,6,7,46,2]: where a s ≡ α s /π and where the value of the gluon condensate is about 2 times the original SVZ value as expected from Bell-Bertlmann analysis [47] 5 .
One can also use the pseudoscalar LSR for extracting a s λ 2 [4]. Studying the stability of (m u + m d ) with respect to the change of λ 2 , on obtains, at the stability region in λ, a reduction of the value of light quark mass of about 5% and the corresponding λ-value: consistent with previous estimate from e + e → hadrons data though less accurate. Taking into account these uncertainties, we shall consider the conservative value: These results indicate that these power corrections are small though crucial for understanding the nonperturbative properties of QCD. One can also notice that the new quadratic correction can only slightly change the existing QSSR phenomenology because of its smallness.
• QCD (spectral) sum rule scales In other channels the effect of the quadratic correction can be much larger. In particular, such a correction can change drastically the estimate of the scale of violation of asymptotic freedom in the gluonium channel [4]. In the case of the vector quark channel, its contribution to the exponential sum rule is where a s ≡ α s /π. Defining the scale as the mass where the correction to the asymptotic freedom is about 10%, one has: where the quadratic term is a small correction. Using the positivity of the pion contribution in the pseudoscalar channel, one can deduce the constraint: for current values of the light quark running masses [9], where, again, the quadratic term remains a small correction, and, then, does not affect much the predictions of the sum rules without a such term.
In the case of the scalar gluonium channel, however, the unsubtracted sum rule reads: Thus, the tachyonic gluon mass provides a natural explanation of the large scale revealed by the low-energy theorem estimate of the subtracted two-point correlator: which was puzzling to explain without a such term [48].
• Light quark masses Effects of quadratic corrections in the determinations of the light quark masses have been studied in [4,2,8,9]. As anticipated previously, the tachyonic gluon mass tends to lower by 5-6% the values of (m u + m d ) and of m s running masses obtained from (pseudo)scalar sum rules. At the value of λ 2 given in Eq. 31, where a stability of the prediction versus λ 2 is reached, one obtains from pseudoscalar sum rules to order α 3 s [2] the results given in Table 1. The uses of the sum rules in e + e − and in τdecay data indicate that the value of m s tend to increase when |λ 2 | increases. We give in Table 1 the prediction corresponding to the value of λ 2 given in Eq.(31). • α s from τ -decay One of the sensible place where the effect of the quadratic term can be important is the precise extraction of α s from τ -decays [26,27]. The τ -hadronic width can be usually expressed as: where |V ud | = 0.97418 ± 0.00027 [49] is the CKM mixing angle; S EW = 1.0198 ± 0.0006 [50] and δ ′ EW = 0.001 [51]. are known electroweak corrections; δ (0) is the perturbative correction; δ (2) m is the light quark mass corrections; is the sum of the non-perturbative (NP) contributions of dimension D within the SVZ expansion [1]; δ π,a0 is the contribution of the π and a 0 (980) mesons into the longitudinal part of the spectral function, while δ nst are some eventual NP effects not included into δ svz . The sum of the 'standard' corrections from strong interactions are under good control [7]: The 'non-standard' corrections can originate from instantons, duality violations [52] and the quadratic corrections [4]: with: tach | pheno ≃ (46 ± 20)10 −3 [5,6], and where one can notice a partial compensation between δ dv and δ (2) tach . Alternatively, one can use our previous argument on the duality between the long PT series and the short PT series ⊕ Power corrections for determining the quadratic terms. In so doing, one can consider the 1/Q 2 -term as the difference between the prediction of the large β 1approximation (assuming that it approaches the exact result) and the sum of the four calculated α s -corrections. Using δ (0) β = 0.2371 (see,e.g. [17]) corresponding to a typical value of α s (M τ ) = 0.34, one obtains [7]: where we have taken the average of the result obtained using a Fixed Order (FO) [26] and Contour Improved (CI) [27] perturbative series. These estimates agree nicely with the phenomenological fit in Eq. (41). We have estimated the errors using the deviation of the large β 1 -approximation from the sum of the calculated terms of the series truncated at n = 4. Using this semitheoretical result into the expression of the τ -decay rate and the experimental value [53,54]: where the errors are due respectively to the data, to the standard and non-standard corrections and to the evolution from M τ to M Z . This value of α s is in agreement with existing estimates [16,17,28,54,55] obtained using different appreciations of the non-perturbative contributions and of the large order perturbative series. This result agrees with the ones from the Z-width [28] and from a global fit of electroweak data at O(α 4 s ) [54]: and with the most recent world average [56]: α s (M Z )| world = 0.1189 (10) .
One can notice that the 1/Q 2 contribution tends to decrease the value of α s obtained without this term and improves the agreement with the world average.

Conclusions
In conclusion, our main point here is that large-order perturbative and non-perturbative contributions mix up as a matter of principle. The duality between these corrections is expected theoretically and has been predicted prior to the observation of this phenomenon through explicit calculations. The duality, however, was thought to be confined to the quadratic corrections. The most recent and intriguing development is that this perturbative -nonperturbative duality might extend to the quartic correction as well. Basing on the existence of the infrared renormalon in perturbation theory, one would not expect that the quartic correction is calculable via the long perturbative series. Therefore, it is a challenge to explain the numerical observations on the perturbative series. [18,19,20].
The holographic approach [21] does suggest a mechanism for generating quartic corrections at short distances but much more is needed to be done to finally clarify the issue. In the holographic language the quadratic correction looks as a stringy correction 6 .
Taken at face values, these observations accumulate to a drastic change of expectations on behaviour of perturbative series at higher orders in pure gluonic sector. Instead of factorial divergences in the expansion coefficients and related power-like terms, there are coming from convergent and calculable series , or dual to the perturbative series ⊕ power-like terms. Moreover, the dual models provide both a gauge invariant parametrization of the mysterious quadratic correction and a mechanism for generating a calculable quartic correction. Adding to the credibility of the emerging picture, the infrared renormalons might still re-emerge at very high orders. However, the geometric character of the perturbative series for the gluon condensate indicates that so far the singularity due to the crossover from the strong to weak coupling dominates the series. Amusingly enough, an assumption on the geometric character of the perturbative series is not in an immediate contradiction with other known cases.