Renormalon-Free Gluon Condensate Scheme

• The paper introduces renormalon-free schemes that rearrange the perturbative series to cancel the O(Λ⁴) ambiguity in the operator-product expansion.
• These methods use dispersive techniques, minimal-term truncation, and gradient flow to isolate and subtract renormalon effects across observables like lattice plaquettes and the static potential.
• Practical implementations improve convergence in QCD observables, though choices like the renormalon norm introduce uncertainties that require careful calibration.

The renormalon-free gluon-condensate scheme is a class of prescriptions for defining the dimension-four gluon condensate in a way that removes the $O(\Lambda^4)$ ambiguity associated with the leading $u=2$ infrared renormalon of perturbative Wilson coefficients. In these schemes, the perturbative contribution and the condensate term are rearranged so that the operator-product expansion (OPE) becomes unambiguous term by term, and the condensate is promoted from a prescription-dependent remainder to a well-defined nonperturbative matrix element. Concrete realizations include large-$\beta_0$ dispersive constructions, minimal-term and principal-value prescriptions for the lattice plaquette, pNRQCD factorization for the static potential, $R$-dependent subtraction schemes for $\tau$ observables, and gradient-flow formulations in which the flow time $1/\sqrt{t}$ serves as a gauge-invariant factorization scale (Suzuki et al., 2018, Bali et al., 2015, Ayala et al., 2020, Takaura, 2017, Benitez-Rathgeb et al., 2022, Beneke et al., 2023).

1. Renormalon ambiguity at dimension four

In a standard single-scale OPE,

$$X(Q^2)=c_{1,X}(Q^2)\cdot \langle 1\rangle +c_{FF,X}(Q^2)\cdot \frac{\langle (\alpha/\pi)\{F^2\}\rangle}{Q^4} +O(1/Q^6),$$

the coefficient $c_{1,X}$ is computed in perturbation theory but is afflicted by the $u=2$ infrared renormalon, which induces an intrinsic uncertainty of order $(\Lambda^2/Q^2)^2\sim \Lambda^4/Q^4$, precisely the same order as the condensate term. This prevents a reliable determination of the gluon condensate, because any subtraction of $u=2$0 at finite order leaves an $u=2$1 remnant (Suzuki et al., 2018).

The same structural problem appears in lattice regularization. For the plaquette,

$u=2$2

the perturbative tail is asymptotic, and the identity operator mixes with the plaquette, producing a power-divergent contribution $u=2$3. In this setting, the separation of scales in the OPE does not correspond to a separation of perturbative and non-perturbative contributions, so the ambiguity attached to the perturbative series has to be absorbed into the definition of the condensate itself (Debbio et al., 2018).

For the Adler function in the conventional $u=2$4 OPE, the leading non-perturbative correction is the dimension-4 gluon condensate, while the perturbative coefficients contain the same $u=2$5 singularity that generates factorial growth and an $u=2$6 ambiguity in the standard condensate. This is the ambiguity targeted by renormalon-free gluon-condensate schemes (Benitez-Rathgeb et al., 2022).

2. General subtraction logic and scheme architectures

A common strategy is to introduce an auxiliary scale or prescription that isolates the renormalon-sensitive part of the perturbative coefficient and then absorb the matching ambiguity into a redefined condensate. In the large-$u=2$7 framework of Suzuki and Takaura, the perturbative Wilson coefficient is written in a dispersive form,

$u=2$8

with

$u=2$9

Introducing an infrared cutoff $\beta_0$0 yields $\beta_0$1, whose $\beta_0$2-dependent part can be matched against the ultraviolet-cutoff dependence of the condensate. This leads to

$\beta_0$3

which is $\beta_0$4-independent, free of $\beta_0$5 renormalon ambiguity, and universal in the sense that no observable label $\beta_0$6 appears in its definition (Suzuki et al., 2018).

Different implementations realize the same logic in different variables.

Framework	Observable	Characteristic subtraction
Large-$\beta_0$7 OPE (Suzuki et al., 2018)	$\beta_0$8	$\beta_0$9-dependent split of $R$0 and $R$1
Plaquette hyperasymptotics (Ayala et al., 2020)	Wilson plaquette	PV Borel sum $R$2 plus terminant $R$3
Minimal-term lattice schemes (Bali et al., 2015, Debbio et al., 2018)	Plaquette	truncation at the minimal term $R$4
pNRQCD static potential (Takaura, 2017)	$R$5	separation into $R$6, $R$7, and $R$8
$R$9-scheme for $\tau$0 moments (Benitez-Rathgeb et al., 2022)	Adler function and moments	subtraction $\tau$1
Gradient flow (Beneke et al., 2023, Beneke et al., 14 Oct 2025)	Adler function, $\tau$2 width	subtraction with flowed $\tau$3 coefficient $\tau$4

These formulations differ in the auxiliary quantity that carries the subtraction—$\tau$5, the truncation point, a principal-value prescription, the pNRQCD cutoffs $\tau$6, the infrared factorization scale $\tau$7, or the gradient-flow time $\tau$8—but each aims to cancel the same dimension-four ambiguity between perturbation theory and the condensate term.

3. Plaquette-based realizations on the lattice

The plaquette has supplied the most explicit numerical determinations of a renormalon-free gluon condensate. In four-dimensional SU(3) pure gauge theory with the Wilson action, the perturbative expansion

$\tau$9

was computed to 35 loops. The large-order coefficients exhibit the expected asymptotic behavior associated with the dimension-4 operator at $1/\sqrt{t}$0, and the natural prescription in early lattice work was truncation at the minimal term. Defining

$1/\sqrt{t}$1

with $1/\sqrt{t}$2 chosen so that $1/\sqrt{t}$3 is minimal, one extracts

$1/\sqrt{t}$4

Using high-precision Monte Carlo data for $1/\sqrt{t}$5 and perturbative coefficients up to $1/\sqrt{t}$6, Bali et al. obtained

$1/\sqrt{t}$7

with an intrinsic “renormalon-prescription” ambiguity

$1/\sqrt{t}$8

in pure gluodynamics (Bali et al., 2015).

A later NSPT study extended the same logic to $1/\sqrt{t}$9 massless QCD with massless staggered fermions and twisted boundary conditions. In that computation, the coefficients $$X(Q^2)=c_{1,X}(Q^2)\cdot \langle 1\rangle +c_{FF,X}(Q^2)\cdot \frac{\langle (\alpha/\pi)\{F^2\}\rangle}{Q^4} +O(1/Q^6),$$0 were measured up to $$X(Q^2)=c_{1,X}(Q^2)\cdot \langle 1\rangle +c_{FF,X}(Q^2)\cdot \frac{\langle (\alpha/\pi)\{F^2\}\rangle}{Q^4} +O(1/Q^6),$$1, the ratios $$X(Q^2)=c_{1,X}(Q^2)\cdot \langle 1\rangle +c_{FF,X}(Q^2)\cdot \frac{\langle (\alpha/\pi)\{F^2\}\rangle}{Q^4} +O(1/Q^6),$$2 approached the renormalon-predicted constant, and the minimal-term subtraction yielded a condensate that exhibited clean $$X(Q^2)=c_{1,X}(Q^2)\cdot \langle 1\rangle +c_{FF,X}(Q^2)\cdot \frac{\langle (\alpha/\pi)\{F^2\}\rangle}{Q^4} +O(1/Q^6),$$3 scaling and remained stable under modest variations of the prescription point $$X(Q^2)=c_{1,X}(Q^2)\cdot \langle 1\rangle +c_{FF,X}(Q^2)\cdot \frac{\langle (\alpha/\pi)\{F^2\}\rangle}{Q^4} +O(1/Q^6),$$4 or of the scale $$X(Q^2)=c_{1,X}(Q^2)\cdot \langle 1\rangle +c_{FF,X}(Q^2)\cdot \frac{\langle (\alpha/\pi)\{F^2\}\rangle}{Q^4} +O(1/Q^6),$$5. The quoted result was

$$X(Q^2)=c_{1,X}(Q^2)\cdot \langle 1\rangle +c_{FF,X}(Q^2)\cdot \frac{\langle (\alpha/\pi)\{F^2\}\rangle}{Q^4} +O(1/Q^6),$$6

for $$X(Q^2)=c_{1,X}(Q^2)\cdot \langle 1\rangle +c_{FF,X}(Q^2)\cdot \frac{\langle (\alpha/\pi)\{F^2\}\rangle}{Q^4} +O(1/Q^6),$$7 massless QCD (Debbio et al., 2018).

The hyperasymptotic formulation of Ayala et al. replaced optimal truncation by a principal-value Borel prescription and an explicit terminant associated with the leading renormalon. The perturbative sum is defined as

$$X(Q^2)=c_{1,X}(Q^2)\cdot \langle 1\rangle +c_{FF,X}(Q^2)\cdot \frac{\langle (\alpha/\pi)\{F^2\}\rangle}{Q^4} +O(1/Q^6),$$8

and the OPE becomes

$$X(Q^2)=c_{1,X}(Q^2)\cdot \langle 1\rangle +c_{FF,X}(Q^2)\cdot \frac{\langle (\alpha/\pi)\{F^2\}\rangle}{Q^4} +O(1/Q^6),$$9

Using the hyperasymptotic expansion

$c_{1,X}$0

they obtained

$c_{1,X}$1

The final result showed a very flat dependence on $c_{1,X}$2, repeating the analysis in the $c_{1,X}$3 scheme gave the same central value within the quoted error, and the scheme and scale dependence in $c_{1,X}$4, $c_{1,X}$5, and $c_{1,X}$6 canceled in the combination $c_{1,X}$7 (Ayala et al., 2020).

4. pNRQCD and the static QCD potential

In pNRQCD, the static QCD potential is organized as

$c_{1,X}$8

The soft singlet potential $c_{1,X}$9 contains renormalons at positive half-integers $u=2$0, and the leading $u=2$1-dependent ambiguity is the $u=2$2 renormalon $u=2$3. The next-to-leading ultrasoft term $u=2$4, which scales as $u=2$5 at small $u=2$6, contains the same $u=2$7 renormalon. With two cutoffs satisfying

$u=2$8

one writes

$u=2$9

$(\Lambda^2/Q^2)^2\sim \Lambda^4/Q^4$0

so that the $(\Lambda^2/Q^2)^2\sim \Lambda^4/Q^4$1 ambiguities cancel in $(\Lambda^2/Q^2)^2\sim \Lambda^4/Q^4$2 (Takaura, 2017).

At still longer wavelengths, the Wilson-line correlator in $(\Lambda^2/Q^2)^2\sim \Lambda^4/Q^4$3 admits a local expansion whose leading term is the local gluon condensate. In the same large-$(\Lambda^2/Q^2)^2\sim \Lambda^4/Q^4$4 framework, the condensate contribution carries an $(\Lambda^2/Q^2)^2\sim \Lambda^4/Q^4$5 ambiguity from the $(\Lambda^2/Q^2)^2\sim \Lambda^4/Q^4$6 renormalon, but this dependence cancels against the corresponding term in $(\Lambda^2/Q^2)^2\sim \Lambda^4/Q^4$7. The resulting renormalon-free condensate is

$(\Lambda^2/Q^2)^2\sim \Lambda^4/Q^4$8

and the final short-distance expansion reads

$(\Lambda^2/Q^2)^2\sim \Lambda^4/Q^4$9

This factorization is designed for very short distances $u=2$00. In that regime, $u=2$01 is negative, of order $u=2$02, decreases as $u=2$03, and at $u=2$04 is only a few MeV for $u=2$05; its effect on the force is at the $u=2$06 level (Takaura, 2017).

5. Infrared-subtracted condensates in $u=2$07 observables

The renormalon-free gluon-condensate scheme acquired particular prominence in analyses of the Adler function and hadronic $u=2$08 decay, where the discrepancy between contour-improved perturbation theory (CIPT) and fixed-order perturbation theory (FOPT) was traced to infrared sensitivity associated with the gluon-condensate renormalon. In the large-$u=2$09 test case, the Adler-function Borel transform has a simple pole at $u=2$10, and one isolates the pure gluon-condensate piece

$u=2$11

Subtracting this pole defines a subtracted perturbative Adler function and, at the same time, a renormalon-free condensate

$u=2$12

For the kinematic $u=2$13 weight $u=2$14, one has $u=2$15, so the explicit condensate term vanishes, yet the subtraction removes the bulk of the FOPT–CIPT discrepancy: a standard-scheme difference of about $u=2$16 around orders $u=2$17–$u=2$18 is reduced to $u=2$19 in the renormalon-free scheme (Benitez-Rathgeb et al., 2021).

The full-QCD $u=2$20-scheme introduced an infrared factorization scale $u=2$21 and defined the order-dependent standard condensate by

$u=2$22

with

$u=2$23

This subtraction removes the $u=2$24 renormalon from the perturbative Adler series and repackages it into the scale-dependent condensate. In this framework, FOPT contributes identically zero to gluon-condensate-suppressed moments, while the CIPT subtraction is nonzero and cancels the piece responsible for the earlier asymptotic separation. In the large-$u=2$25 toy model, in large-$u=2$26, and in the 5-loop plus renormalon model for full QCD, FOPT and CIPT in the renormalon-free scheme approach the same “true” Borel sum, and moments that were poorly convergent in the $u=2$27 scheme become well behaved as well (Benitez-Rathgeb et al., 2022).

The phenomenological implementation depends on the renormalon norm $u=2$28. A follow-up analysis used three methods to determine $u=2$29 for $u=2$30—a multi-renormalon Borel model, a conformal-mapping method, and an optimal-subtraction $u=2$31 method—and adopted

$u=2$32

Applying the scheme to state-of-the-art $u=2$33 determinations, the truncated-OPE analysis changed from $u=2$34 versus $u=2$35 in the standard scheme to $u=2$36 versus $u=2$37 in the renormalon-free scheme, with combined average

$u=2$38

In the DV-model analysis, the standard-scheme values $u=2$39 and $u=2$40 became $u=2$41 and $u=2$42, with combined average

$u=2$43

In both cases, the FOPT result was essentially unchanged and the CIPT result moved toward FOPT (Benitez-Rathgeb et al., 2022). A related presentation using a realistic high-order Borel model quoted $u=2$44 and emphasized that, for the kinematic and $u=2$45-suppressed moments, the renormalon-free scheme yields compatible FOPT and CIPT extractions near the input $u=2$46, while moments with unsuppressed gluon-condensate contributions also become more stable (Benitez-Rathgeb et al., 2022).

6. Gradient flow as a gauge-invariant renormalon subtraction

The gradient-flow approach replaces the auxiliary infrared cutoff by the flow time $u=2$47, which acts as a gauge-invariant factorization scale. One introduces flowed fields $u=2$48 satisfying

$u=2$49

and defines the flowed action density

$u=2$50

Its small-$u=2$51 OPE is

$u=2$52

or, in the notation of the later paper,

$u=2$53

Solving this relation for the condensate and substituting it into the Adler-function OPE produces a subtracted perturbative coefficient,

$u=2$54

so that the same infrared renormalon singularity in $u=2$55 and in $u=2$56 cancels exactly, order by order in $u=2$57, without introducing or determining the non-perturbative Stokes constant that would normalize the renormalon (Beneke et al., 2023).

Applied to hadronic $u=2$58 decay, this construction modifies only the contour-improved series in practice. Using $u=2$59, $u=2$60, and $u=2$61 chosen so $u=2$62, the quoted partial sums through $u=2$63 are

$u=2$64

so the subtraction leaves FO essentially unchanged but lifts CI into agreement with FO already by the fourth order (Beneke et al., 2023).

The later gradient-flowed OPE formulation used lattice data on the action density to estimate the flowed condensate and reported a marked improvement in convergence. At $u=2$65 with $u=2$66, the unsubtracted perturbative coefficients for $u=2$67 were

$u=2$68

while after subtraction, with $u=2$69,

$u=2$70

The partial sums

$u=2$71

stabilized already at $u=2$72 to $u=2$73. Repeating the analysis for $u=2$74, the paper reported significantly reduced theoretical uncertainty and extended reliable predictions down to $u=2$75; for $u=2$76 decay, the FO and CI prescriptions converged to the same result, in favour of the fixed-order treatment (Beneke et al., 14 Oct 2025).

7. Shared structural features, limitations, and scope

Across the existing literature, a renormalon-free gluon-condensate scheme has three recurring elements. First, the perturbative contribution is split into a renormalon-free part and a subtraction term tied to the $u=2$77 singularity. Second, the same subtraction is inserted into the condensate definition, producing a nonperturbative quantity that is independent of the auxiliary prescription variable. Third, the OPE is rewritten so that the perturbative coefficient and the condensate matrix element are separately free of the leading $u=2$78 ambiguity. This structure is explicit in the $u=2$79-dependent large-$u=2$80 OPE (Suzuki et al., 2018), in the PV and hyperasymptotic plaquette formulation (Ayala et al., 2020), in the $u=2$81-dependent $u=2$82 scheme (Benitez-Rathgeb et al., 2022), and in the gradient-flowed Adler-function expansion (Beneke et al., 14 Oct 2025).

The practical limitations differ by implementation. In the $u=2$83-scheme for $u=2$84 observables, the subtraction depends on the renormalon norm $u=2$85, which has to be supplemented independently and was assigned a $u=2$86 uncertainty in the dedicated determination (Benitez-Rathgeb et al., 2022). In the plaquette hyperasymptotic analysis, the error budget included a $u=2$87 error in the renormalon normalization $u=2$88 (Ayala et al., 2020). In the original large-$u=2$89 proposal based on gradient-flow energy-density data, the difference

$u=2$90

did not yet display a clean $u=2$91 behavior in the available $u=2$92-window, and higher-power or higher-log terms reflecting truncation errors spoiled a stable extraction; the authors therefore emphasized the need to go beyond the large-$u=2$93 approximation and beyond existing three-loop information (Suzuki et al., 2018).

These results suggest that “renormalon-free gluon-condensate scheme” is best understood not as a single unique prescription but as a family of OPE rearrangements adapted to different observables and regulators. What is common is the demand that the perturbative subtraction and the condensate definition be matched so that the leading renormalon ambiguity is canceled explicitly. Within that shared framework, the lattice plaquette, the static potential, $u=2$94 moments, and gradient-flowed operators provide complementary realizations of the same underlying principle.