Renormalon Resummation in QFT

• Renormalon-motivated resummation is a framework in quantum field theory that addresses series divergence by exploiting Borel-plane singularities.
• It reorganizes perturbative expansions via logarithmic derivative methods and a renormalon-inspired Borel ansatz to clearly separate perturbative and nonperturbative contributions.
• This approach enhances phenomenological predictions in QCD by refining lattice-continuum matching and reducing renormalization scale ambiguities in data fits.

Renormalon-motivated resummation refers to a suite of methodologies in quantum field theory (QFT) designed to handle divergent series arising in perturbative expansions, specifically those where the divergence is dominated by “renormalons”—singularities in the Borel plane stemming from certain classes of Feynman diagrams. Such divergences encode the intertwining of perturbative and nonperturbative effects. Renormalon-motivated resummation aims to systematically disentangle, resum, or otherwise subtract the perturbative ambiguities in these series, enabling a more robust extraction of the underlying nonperturbative physics, such as power corrections calculable in the framework of the operator product expansion (OPE).

1. Origin and Structure of Renormalon-induced Divergences

Perturbative expansions for QFT observables (e.g., current correlators, Wilson loops, decay rates) often yield asymptotic series whose coefficients grow factorially at high orders, typically as $n!\,\rho^n$ for some constant $\rho$. This universal divergence is attributed to renormalons—poles (or more generally, singularities) in the Borel transform of the series—classified as infrared (IR) or ultraviolet (UV) according to their position in the Borel $u$-plane.

Consider a generic observable $O$ with an asymptotic perturbative series in the coupling $a(Q^2) = \alpha_s(Q^2)/\pi$:

$O(Q^2) = \sum_{n=0}^\infty d_n\,a(Q^2)^{n+1}$

The Borel transform,

$$B[O](u) = \sum_{n=0}^\infty \frac{d_n}{n!\;\beta_0^n} u^n,$$

contains poles on the positive (IR renormalons) and negative (UV renormalons) real axis. For example, in QCD the leading IR renormalon often appears at $u=2$, governing the growth of $d_n$ as $n\to\infty$. The perturbative series is only Borel summable if the cut singularities are absent, which is almost never the case for physically relevant QFTs (Cvetic, 2019, Cvetic, 2018).

2. Renormalon-motivated Borel Resummation Methodologies

The central philosophy of renormalon-motivated resummation is to reconstruct QFT observables by exploiting their Borel-plane singularity structure, thereby resumming dominant large-order effects and extracting unambiguous, physically meaningful predictions. The general workflow may be summarized as:

1. Reorganization of the Expansion: Introduce an auxiliary expansion, often in logarithmic derivatives of the running coupling, to facilitate separation of renormalon-dominated contributions:

$$\mathcal{D}(Q^2) = \sum_{n=0}^\infty \tilde{d}_n(\kappa) \tilde{a}_{n+1}(\kappa Q^2)$$

with

$$\tilde{a}_{n+1}(Q^2) \equiv \frac{(-1)^n}{\beta_0^n n!} \left(\frac{d}{d\ln Q^2}\right)^n a(Q^2)$$

This expansion is exactly renormalization-scale independent at one-loop (Cvetic, 2018).

1. Motivated Borel Ansatz: Construct a Borel transform modeled after the expected renormalon structure, usually parameterizing both IR and UV singularities:

$$B[\mathcal{D}](u) = \exp(u)\,\pi\left[\sum_j \frac{r_j^{\mathrm{IR}}}{(p_j - u)^{k_j}} + \sum_k \frac{r_k^{\mathrm{UV}}}{(p_k + u)^{m_k}} + \cdots\right]$$

The residues $r_j$ are fixed by matching the known low-order coefficients.

1. Inverse Mellin (Characteristic) Function: Obtain the characteristic function (Neubert distribution) $F_\mathcal{D}(t)$ as the inverse Mellin transform of $B[\mathcal{D}](u)$:

$$F_\mathcal{D}(t) = \frac{1}{2\pi i} \int_{u_0-i\infty}^{u_0+i\infty} B[\mathcal{D}](u)\; t^u\,du$$

This function encodes the entire singularity information of the Borel plane in $t$-space (Cvetic, 2018, Cvetic, 2019).

1. Integral Representation of the Resummed Observable:

$$\mathcal{D}_{\mathrm{res}}(Q^2) = \int_0^\infty \frac{dt}{t}\; F_\mathcal{D}(t)\, a(te^{-C} Q^2)$$

(Here $C$ is a scheme-dependent constant.)

This construction ensures that all-order renormalon effects are incorporated into the prediction for $O(Q^2)$. Ambiguities associated with contour integration around Borel-plane singularities reflect the inescapable need for matching with nonperturbative OPE terms (Cvetic, 2018, Cvetic, 2019, Ayala et al., 2023).

3. Practical Implementation and Generalizations

The above methodology has been generalized along several axes:

• Non-integer Leading Powers: For observables where the expansion starts with a term $a(Q^2)^{\nu_0}$ ($0<\nu_0\leq1$), as encountered in cases with anomalous dimensions (e.g., chromomagnetic operator Wilson coefficients in HQET), a rescaling relation for the coefficients and a modified Borel representation allow extension of the method (Ayala et al., 10 Sep 2025):

$$\mathcal{D}(Q^2) = \sum_{n=0}^\infty d_n(\nu_0;\kappa)\,[a(\kappa Q^2)]^{\nu_0 + n}$$

The characteristic function is similarly generalized.

• Spacelike and Timelike Domains: The characteristic (Neubert) function is universal between spacelike and timelike quantities, with integration kernels and analytic continuation handled in the resummed formalism (Ayala et al., 10 Sep 2025).
• OPE Matching and Power Corrections: Integration of the resummed leading-twist contribution with OPE higher-twist terms ($D=2$, $D=4$) results in a controlled, ambiguity-aware evaluation of physical observables, enabling direct fits to experimental data sets (e.g., Adler function, Bjorken sum rule, $\tau$-decay ratios) (Ayala et al., 2023, Ayala et al., 2023, Cvetic et al., 2020).
• Fourier Transform-based Subtraction: The FTRS approach employs Fourier transformation to momentum space, leveraging parameter choices to simultaneously cancel multiple IR renormalons and to resum artificial and physical UV renormalons systematically. The method has demonstrated equivalence to principal value (PV) Borel resummation in test cases (Hayashi, 2021).
• Bilocal Expansion of Borel Transform: For lattice observables such as the average plaquette, the Borel transform is reconstructed via a bilocal expansion interpolating between the origin ($b=0$) and the leading renormalon singularity ($b=z_0$), allowing reliable subtraction of all perturbative ambiguities and extraction of nonperturbative remnants (e.g., the gluon condensate) (Lee, 2010).

4. Critical Assessment of Previous and Alternative Methods

Earlier renormalon subtraction schemes—especially those that attempted a matching of perturbative coefficients between lattice and continuum schemes—often failed due to the non-asymptotic nature of the available coefficients. This mismatch caused contamination of the extracted nonperturbative terms by residual perturbative components and led, for example, to spurious dimension-2 power corrections, at odds with OPE requirements (Lee, 2010). The bilocal Borel-based expansions and new resummation techniques obviate this issue by incorporating the full structure of the renormalon singularity, enforcing the correct scaling of the nonperturbative residue (e.g., $a^4$ for the gluon condensate).

In multi-field quantum field theories, renormalon analysis can be systematized via analytic solutions to coupled RGEs and the construction of multi-variable Borel transforms. Such studies show that as the number of active couplings increases, renormalon-induced ambiguities can become more severe and further constrain regions of perturbative resummability (Maiezza et al., 2018).

5. Numerical and Phenomenological Impact

Lattice and Continuum QCD Applications

• In the extraction of the gluon condensate from the lattice average plaquette, after bilocal Borel subtraction all extracted nonperturbative residues scale in accordance with OPE expectations as dimension-4 quantities (i.e., $\sim a^4$), validating the theoretical underpinning of the method (Lee, 2010).
• For the spacelike Adler function, applying the full machinery of the renormalon-motivated approach yields resummed results that match, and in some frameworks outperform, reference methods such as extended diagonal Padé (dPA) approximants in terms of renormalization scale independence and higher-order prediction (Cvetic, 2018, Cvetic, 2019).
• The method is also successfully applied to timelike observables (e.g., $\tau$ lepton decay) and in the evaluation of contributions to the muon $g-2$ (Cvetic et al., 2020).

Data Fitting, Uncertainties, and Coupling Extraction

For experimental sum rules such as the Bjorken polarized sum rule (BSR), the method is calibrated to high-$Q^2$ data regions ($Q^2_{\min}\gtrsim 1.7\,\mathrm{GeV}^2$ for pQCD couplings, extendable to $Q^2_{\min}\sim 0.6\,\mathrm{GeV}^2$ for analytic couplings), simultaneously fitting higher-twist parameters (e.g., $D=2$, $D=4$ OPE terms), and the strong coupling constant $\alpha_s(M_Z^2)$. The main limitations stem from experimental uncertainties (particularly correlated systematics) and Landau singularities in the running coupling, the latter addressed by holomorphic (AQCD) couplings (Ayala et al., 2023).

Significantly, in the OPE-resummed fits, residue parameters in the $D=2$ and $D=4$ corrections can be extracted in a manner consistent with the scaling dictated by renormalon structure, provided $\alpha_s$ is fixed to standard world-average values (Ayala et al., 2023, Ayala et al., 2023).

6. Outstanding Theoretical Issues and Future Directions

• Ambiguity Matching: The remaining renormalon ambiguity in the resummed perturbative contribution—manifest as integration contour dependence in the Borel plane—must be canceled by corresponding ambiguities in the definition of nonperturbative matrix elements. The proper matching and quantification of these combined ambiguities is central to any physically meaningful resummation (Cvetic, 2019, Lee, 2010).
• Analytic Couplings and IR Behavior: Using holomorphic QCD couplings eliminates Landau singularities and allows resummation integrals to be extended to lower $Q^2$ domains without regularization ambiguities. This simplifies phenomenological analysis and offers a technically clean arena for further QCD applications (Ayala et al., 2023, Cvetic et al., 2020).
• Extension to Multi-anomalous-dimension Cases: The generalization to noninteger leading powers and multi-field scenarios opens prospects for robust resummation in observables with nontrivial anomalous dimension structure or in complex theories beyond minimal QCD (Ayala et al., 10 Sep 2025, Maiezza et al., 2018).
• Resurgence and Transseries Structure: Nonperturbative corrections—exponentially small in the coupling—naturally emerge in the resurgent formalism. Borel non-summability is mirrored by the necessity of a transseries expansion encoding both perturbative and nonperturbative data, as exemplified in integrable field theory and large-$N$ studies (Bersini et al., 2019, Pietro et al., 2021).

7. Summary Table of Core Technical Ingredients

Component	Role in Resummation	Key References
Borel transform, $B(u)$	Encodes renormalon singularities; basis for integral resummation	(Cvetic, 2018, Cvetic, 2019)
Logarithmic derivative expansion	Yields scale-invariant auxiliary expansion with controlled RG properties	(Cvetic, 2018, Cvetic, 2019)
Characteristic function, $F(t)$	Inverse Mellin transform of $B(u)$; integral kernel for resummed observable	(Cvetic, 2018, Cvetic, 2019)
Bilocal expansion	Interpolates between low-order data and renormalon singularity in Borel transform	(Lee, 2010)
Fourier (FTRS) approach	Simultaneous subtraction/resummation of multiple renormalons using transform-space techniques	(Hayashi, 2021)
OPE parameter extraction	Fits higher-twist terms concurrently with resummed leading-twist contribution	(Ayala et al., 2023, Ayala et al., 2023)
Analytic couplings	Enables resummation down to IR without Landau singularities	(Ayala et al., 2023, Cvetic et al., 2020)

• (Lee, 2010)
• (Maiezza et al., 2018)
• (Cvetic, 2018)
• (Cvetic, 2019)
• (Pietro et al., 2021)
• (Hayashi, 2021)
• (Ayala et al., 2023)
• (Ayala et al., 2023)
• (Ayala et al., 10 Sep 2025)
• (Cvetic et al., 2020)
• (Bersini et al., 2019)

Renormalon-motivated resummation now stands as a powerful and systematically improvable framework in high-energy theory, enabling the extraction of nonperturbative content from divergent perturbative expansions across a wide range of phenomenological applications.