Heavy-Quark Current-Current Correlators

• Heavy-quark current-current correlators are two-point functions of quark bilinears used to probe QCD dynamics and heavy-quark properties.
• They enable precision determinations of heavy-quark masses and αₛ via low-energy moment expansions, asymptotic series, and lattice matching.
• Advanced multi-loop and HQET methods facilitate reliable reconstructions across different energy regimes and channel-specific renormalizations.

Searching arXiv for relevant papers on heavy-quark current-current correlators and closely related heavy-light/HQET extensions. Using arXiv search results to anchor the article in the relevant literature while restricting factual claims to the provided data. Heavy-quark current-current correlators are two-point functions of local quark bilinears built from heavy fields, most commonly of the form $\bar\psi \Gamma \psi$ for flavour-diagonal channels and $\bar\psi \Gamma \chi$ for flavour non-diagonal heavy-light channels. They are basic objects in perturbative QCD, lattice-QCD moment methods, relativistic quarkonium sum rules, HQET sum rules, and low-energy effective descriptions below heavy-quark thresholds. In practice, the subject is organized around Taylor coefficients at $q^2=0$, asymptotic coefficients for $|q^2|\gg m^2$, spectral densities, and Euclidean-time correlators, with the central use cases being determinations of $m_c$, $m_b$, and $\alpha_s$, lattice–continuum matching, and reconstruction of correlators away from directly accessible kinematic limits (Maier et al., 2011, McNeile et al., 2010).

1. Definitions and channel structure

For flavour-diagonal heavy-heavy channels, the correlators are defined by

$$\Pi^{\delta}(q^2)= i\int d^4x\, e^{iqx}\,\langle 0|T\,J^\delta(x)J^\delta(0)|0\rangle, \qquad J^\delta(x)=\bar\psi(x)\Gamma^\delta\psi(x),$$

with $\psi$ a heavy quark. For flavour non-diagonal heavy-light channels, the corresponding definition is

$$\Pi^{\delta}_{ND}(q^2)= i\int d^4x\, e^{iqx}\,\langle 0|T\,j^\delta(x)j^\delta(0)|0\rangle, \qquad j^\delta(x)=\bar\psi(x)\Gamma^\delta\chi(x),$$

where $\bar\psi \Gamma \chi$0 is a massless quark. The standard Dirac structures are $\bar\psi \Gamma \chi$1, corresponding to scalar, pseudo-scalar, vector, and axial-vector channels. In the non-diagonal massless-light-quark limit, $\bar\psi \Gamma \chi$2 and $\bar\psi \Gamma \chi$3, so only scalar and vector channels are independent (Maier et al., 2011).

Channel	$\bar\psi \Gamma \chi$4	Structural remark
Scalar	$\bar\psi \Gamma \chi$5	In the non-diagonal massless-light limit, coincides with pseudo-scalar
Pseudo-scalar	$\bar\psi \Gamma \chi$6	Related to longitudinal axial channel by Ward identities
Vector	$\bar\psi \Gamma \chi$7	Decomposes into transverse and longitudinal pieces
Axial-vector	$\bar\psi \Gamma \chi$8	In the non-diagonal massless-light limit, coincides with vector

For vector and axial-vector channels, the correlator is decomposed into transverse and longitudinal parts. In the non-diagonal vector case,

$\bar\psi \Gamma \chi$9

A standard normalization imposed on the renormalized correlators is $q^2=0$0. For two different nonzero quark masses, it is also convenient to introduce $q^2=0$1 and the mass ratio $q^2=0$2, and to write the renormalized low-$q^2=0$3 expansion as

$q^2=0$4

In this two-mass setting, $q^2=0$5 and $q^2=0$6, while the longitudinal moments obey

$q^2=0$7

These relations encode the usual Ward-identity structure and reduce redundant computation (Hoff et al., 2011).

At energies well below a heavy-quark threshold, the heavy-quark vector current itself admits a distinct effective description. Rather than matching onto a light conserved current of the same dimension, it is represented by total divergences of higher-dimensional operators, notably light-quark tensor currents and dimension-six gluonic operators. This low-energy representation is one of the reasons that sub-threshold heavy-current correlators are softer than ordinary light-current correlators (Meyer, 2023).

2. Moment expansions and analytic regimes

The central analytic regimes are the low-energy expansion around $q^2=0$8 and the high-energy asymptotic expansion for $q^2=0$9. In the low-energy region one writes

$|q^2|\gg m^2$0

so the correlator moments are the Taylor coefficients. In the high-energy region the generic structure is

$|q^2|\gg m^2$1

For the three-loop non-diagonal vector correlator the expansion contains logarithms up to cubic order at $|q^2|\gg m^2$2, i.e. at NNLO (Maier et al., 2011).

A major modern benchmark is the three-loop NNLO computation of low- and high-energy coefficients for heavy-quark correlators. In the non-diagonal scalar and vector channels, 30 coefficients were obtained in the low-energy expansion and 30 in the high-energy expansion. In the high-energy region, 30 coefficients were also obtained for the diagonal vector, axial-vector, scalar, and pseudo-scalar correlators. The practical significance of this extension is twofold: it enlarges the perturbative input for lattice matching and mass determinations, and it provides boundary data for Padé and Mellin–Barnes-inspired reconstructions of the full momentum dependence (Maier et al., 2011).

The same program extends to higher perturbative order in restricted kinematics. For flavour non-diagonal heavy-light correlators with one massless light quark, the first four physical low-energy moments $|q^2|\gg m^2$3 of the vector and scalar channels were computed at four loops in both on-shell and $|q^2|\gg m^2$4 mass schemes. These coefficients are the heavy-light analogue of the low-energy moment data long used in heavy-heavy precision analyses (Maier et al., 2015).

For genuinely two-mass non-diagonal correlators, the small-$|q^2|\gg m^2$5 moments are known through three loops for vector, axial-vector, scalar, and pseudo-scalar channels. The practical strategy is to compute complementary expansions around $|q^2|\gg m^2$6 and $|q^2|\gg m^2$7, with terms through $|q^2|\gg m^2$8 and $|q^2|\gg m^2$9 respectively for $m_c$0, and then interpolate in the narrow intermediate region. In the pseudo-scalar channel the quoted three-loop approximation is better than about $m_c$1 for $m_c$2, while outside this interval the direct asymptotic expansions are more accurate (Hoff et al., 2011).

3. Perturbative machinery and renormalization structure

The perturbative computation of heavy-quark correlators is built on highly automated multiloop workflows. In the NNLO three-loop low- and high-energy expansion program, diagrams are generated with QGRAF, mapped onto topologies with q2e and exp, reduced to master integrals by IBP identities using Crusher, and manipulated symbolically in FORM. Because full analytic master integrals as functions of $m_c$3 and $m_c$4 are not available in closed form, the coefficients are generated from differential equations, using

$m_c$5

with low-energy and high-energy ansätze adapted to the corresponding singularity structure in dimensional regularization (Maier et al., 2011).

At four loops in the low-energy heavy-light problem, the method is instead a momentum expansion followed by vacuum-tadpole reduction. The workflow consists of diagram generation with QGRAF, algebraic manipulation with TFORM/FORM, topology mapping onto 28 topologies, color algebra with the FORM package color, Taylor expansion in $m_c$6, tensor reduction, and Laporta reduction to known four-loop tadpole master integrals. A major check is the reproduction of the top-induced non-singlet four-loop correction to the electroweak $m_c$7 parameter, together with UV finiteness and gauge-parameter cancellation in representative coefficients (Maier et al., 2015).

Renormalization depends strongly on the current channel. In the standard correlator expansions, $m_c$8 is renormalized in $m_c$9, while heavy-quark masses are often treated in both $m_b$0 and on-shell schemes. For scalar and pseudo-scalar channels the current renormalizes like the quark mass, whereas vector currents are protected. For pseudo-scalar and axial-vector correlators, the treatment of $m_b$1 requires care: singlet diagrams containing exactly one $m_b$2 in a fermion trace are handled with the Larin prescription, while a naively anticommuting $m_b$3 is used otherwise (Maier et al., 2011).

This channel dependence persists in adjacent correlator problems. In the unequal-mass tensor-current correlator with full $m_b$4 dependence, the renormalized result requires both mass counterterms and tensor-current renormalization,

$m_b$5

with $m_b$6. In that case, omitting current renormalization changes the finite $m_b$7 part and leads to inconsistencies in moments and imaginary parts, a point that has become a useful cautionary example for current-dependent renormalization in correlator analyses (Generet et al., 2 Sep 2025).

4. Heavy-heavy correlators in precision determinations

The most developed phenomenology uses low moments of heavy-heavy correlators to determine heavy-quark masses and the QCD coupling. In lattice QCD, a standard choice is the Euclidean pseudoscalar correlator

$m_b$8

with reduced moments

$m_b$9

The continuum counterpart factorizes the perturbative coefficient $\alpha_s$0 from the hadron-to-quark mass ratio $\alpha_s$1, making the reduced moments a particularly clean bridge between lattice data and continuum perturbation theory (McNeile et al., 2010).

This framework produced early high-precision determinations from charm pseudoscalar, vector, and axial-vector correlators. Using new four-loop continuum input, the quoted results were

$\alpha_s$2

$\alpha_s$3

A later extension to finer lattices and heavier masses yielded

$\alpha_s$4

$\alpha_s$5

together with a nonperturbative HISQ mass-ratio check giving $\alpha_s$6 before combination (0805.2999, McNeile et al., 2010).

A more recent development is the replacement of fixed-order perturbation theory by renormalization-group summed perturbation theory. In that approach, the perturbative moments are reorganized as

$\alpha_s$7

which sums all RG-accessible logarithms. Applied to vector and pseudo-scalar low-energy moments, this substantially reduces renormalization-scale dependence, especially for the third and fourth moments. The quoted final values are

$\alpha_s$8

with the analysis also stressing that vector-channel condensate terms become pathological in fixed-order perturbation theory if the $\alpha_s$9 mass is used directly instead of the pole mass (Khan, 2023).

5. Heavy-light correlators, condensates, and HQET

Heavy-light current-current correlators preserve the moment-based logic of heavy-heavy analyses but introduce qualitatively new structure. In the lattice heavy-light pseudoscalar method, one studies

$$\Pi^{\delta}(q^2)= i\int d^4x\, e^{iqx}\,\langle 0|T\,J^\delta(x)J^\delta(0)|0\rangle, \qquad J^\delta(x)=\bar\psi(x)\Gamma^\delta\psi(x),$$0

and matches reduced moments to continuum moments depending on $$\Pi^{\delta}(q^2)= i\int d^4x\, e^{iqx}\,\langle 0|T\,J^\delta(x)J^\delta(0)|0\rangle, \qquad J^\delta(x)=\bar\psi(x)\Gamma^\delta\psi(x),$$1. In the HISQ-HISQ test case the current normalization is trivial, $$\Pi^{\delta}(q^2)= i\int d^4x\, e^{iqx}\,\langle 0|T\,J^\delta(x)J^\delta(0)|0\rangle, \qquad J^\delta(x)=\bar\psi(x)\Gamma^\delta\psi(x),$$2, and the extracted quantity is $$\Pi^{\delta}(q^2)= i\int d^4x\, e^{iqx}\,\langle 0|T\,J^\delta(x)J^\delta(0)|0\rangle, \qquad J^\delta(x)=\bar\psi(x)\Gamma^\delta\psi(x),$$3. The method is aimed ultimately at nonperturbative renormalization factors for NRQCD heavy-light currents, but the initial study emphasizes that heavy-light analyses are “currently a lot less accurate” than heavy-heavy ones (Koponen et al., 2010).

The main reason is the operator-product expansion. In heavy-light correlators the light-quark condensate appears already at tree level and is numerically large, contributing roughly $$\Pi^{\delta}(q^2)= i\int d^4x\, e^{iqx}\,\langle 0|T\,J^\delta(x)J^\delta(0)|0\rangle, \qquad J^\delta(x)=\bar\psi(x)\Gamma^\delta\psi(x),$$4 of the reduced moment for heavy masses between charm and bottom, with effective scaling $$\Pi^{\delta}(q^2)= i\int d^4x\, e^{iqx}\,\langle 0|T\,J^\delta(x)J^\delta(0)|0\rangle, \qquad J^\delta(x)=\bar\psi(x)\Gamma^\delta\psi(x),$$5. This term is absent in the heavy-heavy pseudoscalar moments used in precision $$\Pi^{\delta}(q^2)= i\int d^4x\, e^{iqx}\,\langle 0|T\,J^\delta(x)J^\delta(0)|0\rangle, \qquad J^\delta(x)=\bar\psi(x)\Gamma^\delta\psi(x),$$6, $$\Pi^{\delta}(q^2)= i\int d^4x\, e^{iqx}\,\langle 0|T\,J^\delta(x)J^\delta(0)|0\rangle, \qquad J^\delta(x)=\bar\psi(x)\Gamma^\delta\psi(x),$$7, and $$\Pi^{\delta}(q^2)= i\int d^4x\, e^{iqx}\,\langle 0|T\,J^\delta(x)J^\delta(0)|0\rangle, \qquad J^\delta(x)=\bar\psi(x)\Gamma^\delta\psi(x),$$8 work. In addition, the heavy-light perturbative coefficients depend sensitively on $$\Pi^{\delta}(q^2)= i\int d^4x\, e^{iqx}\,\langle 0|T\,J^\delta(x)J^\delta(0)|0\rangle, \qquad J^\delta(x)=\bar\psi(x)\Gamma^\delta\psi(x),$$9: the expansion works reasonably well for $\psi$0, but for $\psi$1, where $\psi$2, the mass-ratio dependence must be treated exactly at the available orders. The paper also stresses that continuum perturbation theory is known through $\psi$3 for heavy-heavy correlators but only through $\psi$4 for heavy-light correlators (Koponen et al., 2010).

That condensate sensitivity can be turned into a virtue. By deriving the heavy-light pseudoscalar OPE through $\psi$5 for the quark-condensate coefficient and fitting lattice heavy-strange moments $\psi$6, one obtains

$\psi$7

The analysis shows explicitly that the leading nonperturbative effect in heavy-light moments is $\psi$8, without suppression by the light-quark mass, in sharp contrast to heavy-heavy moments where the gluon condensate enters at order $\psi$9 and is numerically tiny (Davies et al., 2018).

In the static limit, heavy-light correlators are naturally reformulated in HQET. There the correlator is expanded as

$$\Pi^{\delta}_{ND}(q^2)= i\int d^4x\, e^{iqx}\,\langle 0|T\,j^\delta(x)j^\delta(0)|0\rangle, \qquad j^\delta(x)=\bar\psi(x)\Gamma^\delta\chi(x),$$0

or equivalently in momentum space and spectral density form,

$$\Pi^{\delta}_{ND}(q^2)= i\int d^4x\, e^{iqx}\,\langle 0|T\,j^\delta(x)j^\delta(0)|0\rangle, \qquad j^\delta(x)=\bar\psi(x)\Gamma^\delta\chi(x),$$1

The three-loop HQET OPE is known up to dimension 4, including perturbative terms, quark-condensate terms, and the gluon-condensate contribution with RG consistency. More recently, the perturbative coefficients for the operators $$\Pi^{\delta}_{ND}(q^2)= i\int d^4x\, e^{iqx}\,\langle 0|T\,j^\delta(x)j^\delta(0)|0\rangle, \qquad j^\delta(x)=\bar\psi(x)\Gamma^\delta\chi(x),$$2, $$\Pi^{\delta}_{ND}(q^2)= i\int d^4x\, e^{iqx}\,\langle 0|T\,j^\delta(x)j^\delta(0)|0\rangle, \qquad j^\delta(x)=\bar\psi(x)\Gamma^\delta\chi(x),$$3, $$\Pi^{\delta}_{ND}(q^2)= i\int d^4x\, e^{iqx}\,\langle 0|T\,j^\delta(x)j^\delta(0)|0\rangle, \qquad j^\delta(x)=\bar\psi(x)\Gamma^\delta\chi(x),$$4, and $$\Pi^{\delta}_{ND}(q^2)= i\int d^4x\, e^{iqx}\,\langle 0|T\,j^\delta(x)j^\delta(0)|0\rangle, \qquad j^\delta(x)=\bar\psi(x)\Gamma^\delta\chi(x),$$5 were extended to four loops in both coordinate-space and spectral-density representations. These HQET correlators are the natural short-distance input for static heavy-light sum rules and lattice HQET comparisons (Chetyrkin et al., 2021, Grozin, 2024).

6. Convergence limits, reconstructions, and adjacent generalizations

The moment expansions are powerful but not uniformly convergent across the full kinematic plane. In the NNLO three-loop study of heavy-quark correlators, the high-energy expansion was shown to converge only above

$$\Pi^{\delta}_{ND}(q^2)= i\int d^4x\, e^{iqx}\,\langle 0|T\,j^\delta(x)j^\delta(0)|0\rangle, \qquad j^\delta(x)=\bar\psi(x)\Gamma^\delta\chi(x),$$6

for non-diagonal correlators, because of a cut through three heavy-quark lines, and only above

$$\Pi^{\delta}_{ND}(q^2)= i\int d^4x\, e^{iqx}\,\langle 0|T\,j^\delta(x)j^\delta(0)|0\rangle, \qquad j^\delta(x)=\bar\psi(x)\Gamma^\delta\chi(x),$$7

for diagonal correlators, because the first four-particle cut appears at three loops. The newly available many-term asymptotic series make this breakdown near threshold more visible through rapidly growing coefficients. This is one reason the low- and high-energy expansions are best viewed as precise boundary data rather than complete solutions (Maier et al., 2011).

A complementary diagnosis comes from the large-$$\Pi^{\delta}_{ND}(q^2)= i\int d^4x\, e^{iqx}\,\langle 0|T\,j^\delta(x)j^\delta(0)|0\rangle, \qquad j^\delta(x)=\bar\psi(x)\Gamma^\delta\chi(x),$$8 limit. There the reduced moment coefficients admit exact Borel representations, and the ratios

$$\Pi^{\delta}_{ND}(q^2)= i\int d^4x\, e^{iqx}\,\langle 0|T\,j^\delta(x)j^\delta(0)|0\rangle, \qquad j^\delta(x)=\bar\psi(x)\Gamma^\delta\chi(x),$$9

are controlled by

$\bar\psi \Gamma \chi$00

These ratios exhibit a partial cancellation of the leading UV renormalon at $\bar\psi \Gamma \chi$01 and a reduced residue of the leading IR renormalon at $\bar\psi \Gamma \chi$02. The analysis also identifies improved combinations such as $\bar\psi \Gamma \chi$03, for which both leading UV and IR residues are reduced by about $\bar\psi \Gamma \chi$04 (Boito et al., 2021).

Below heavy-quark threshold, current correlators acquire a different effective interpretation. The heavy-quark vector current matches onto total divergences of light-quark tensor currents and dimension-six gluonic operators,

$\bar\psi \Gamma \chi$05

with

$\bar\psi \Gamma \chi$06

This formulation governs sub-threshold disconnected contributions and low-energy hadronic matrix elements of heavy currents (Meyer, 2023).

Several adjacent generalizations broaden the subject without changing its core logic. Full-$\bar\psi \Gamma \chi$07 unequal-mass tensor-current correlators are now known analytically at NLO, supplying dispersive input for unitarity bounds and QCD sum rules beyond the usual vector and scalar channels (Generet et al., 2 Sep 2025). In thermal QCD, the massive vector correlator separates into a transport peak and a quark–antiquark threshold contribution; the Euclidean correlator can agree well with NLO perturbation theory while still being relatively insensitive to detailed reshaping of the real-time spectral function, which complicates extractions of diffusion and quarkonium properties (Burnier et al., 2012). By contrast, heavy-quark “correlators” in the instanton liquid model often refer to static heavy-quark propagators and Wilson-loop correlators rather than local bilinear current-current correlators, and that distinction matters when comparing vacuum-polarization studies with static-potential models (Musakhanov et al., 2020).

Taken together, these developments define heavy-quark current-current correlators as a family of analytically structured and computationally precise observables whose utility depends on regime and channel: heavy-heavy low moments are precision short-distance observables for $\bar\psi \Gamma \chi$08, $\bar\psi \Gamma \chi$09, and $\bar\psi \Gamma \chi$10; heavy-light moments are sensitive probes of condensates and current renormalization; HQET correlators organize the static heavy-light limit; and low- and high-energy expansions remain the essential perturbative inputs whenever the full momentum dependence must still be reconstructed rather than computed directly (Maier et al., 2011).