---
title: Heavy-Quark Current-Current Correlators
url: https://www.emergentmind.com/topics/heavy-quark-current-current-correlators
type: topic
---

# Heavy-Quark Current-Current Correlators

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 [1110.5581][1004.4285].

## 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 \(\chi\) is a massless quark. The standard Dirac structures are \(\Gamma^\delta=1,\ i\gamma_5,\ \gamma_\mu,\ \gamma_\mu\gamma_5\), corresponding to scalar, pseudo-scalar, vector, and axial-vector channels. In the non-diagonal massless-light-quark limit, \(\Pi^s_{ND}=\Pi^p_{ND}\) and \(\Pi^v_{ND}=\Pi^a_{ND}\), so only scalar and vector channels are independent [1110.5581].

| Channel | \(\Gamma^\delta\) | Structural remark |
|---|---|---|
| Scalar | \(1\) | In the non-diagonal massless-light limit, coincides with pseudo-scalar |
| Pseudo-scalar | \(i\gamma_5\) | Related to longitudinal axial channel by Ward identities |
| Vector | \(\gamma_\mu\) | Decomposes into transverse and longitudinal pieces |
| Axial-vector | \(\gamma_\mu\gamma_5\) | 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,
\[
\Pi^{v,\mu\nu}_{ND}(q^2) = \left(-q^2 g^{\mu\nu}+q^\mu q^\nu\right)\Pi^{v}_{ND}(q^2) +q^\mu q^\nu \Pi^{v}_{ND,L}(q^2).
\]
A standard normalization imposed on the renormalized correlators is \(\Pi^\delta(0)=0\). For two different nonzero quark masses, it is also convenient to introduce \(z=q^2/m_1^2\) and the mass ratio \(x=m_2/m_1\), and to write the renormalized low-\(q^2\) expansion as
\[
\bar{\Pi}^\delta(q^2) = \frac{3}{16\pi^2} \sum_{n\ge -1} \bar{C}^\delta_n(x)\, z^n.
\]
In this two-mass setting, \(\bar{C}^s_n(x)=\bar{C}^p_n(-x)\) and \(\bar{C}^v_n(x)=\bar{C}^a_n(-x)\), while the longitudinal moments obey
\[
\bar{C}_{L,n}^v = (1-x)^2 \bar{C}_{n+1}^s,\qquad
\bar{C}_{L,n}^a = (1+x)^2 \bar{C}_{n+1}^p.
\]
These relations encode the usual Ward-identity structure and reduce redundant computation [1103.1481].

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 [2310.09085].

## 2. Moment expansions and analytic regimes

The central analytic regimes are the low-energy expansion around \(q^2=0\) and the high-energy asymptotic expansion for \(|q^2|\gg m^2\). In the low-energy region one writes
\[
\Pi^\delta(q^2)=\sum_{n\ge 0} C_n^\delta z^n,
\qquad z=\frac{q^2}{m^2},
\]
so the correlator moments are the Taylor coefficients. In the high-energy region the generic structure is
\[
\Pi^\delta(q^2)\sim \sum_{n,m} D_{n,m}^\delta\, z^{-n}\,\ln^m\!\left(\frac{-q^2}{m^2}\right).
\]
For the three-loop non-diagonal vector correlator the expansion contains logarithms up to cubic order at \(\mathcal O(\alpha_s^2)\), i.e. at NNLO [1110.5581].

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 [1110.5581].

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 \(n=1,2,3,4\) of the vector and scalar channels were computed at four loops in both on-shell and \(\overline{\mathrm{MS}}\) mass schemes. These coefficients are the heavy-light analogue of the low-energy moment data long used in heavy-heavy precision analyses [1506.00900].

For genuinely two-mass non-diagonal correlators, the small-\(q^2\) moments are known through three loops for vector, axial-vector, scalar, and pseudo-scalar channels. The practical strategy is to compute complementary expansions around \(x\to 0\) and \(x\to 1\), with terms through \(x^8\) and \((1-x)^9\) respectively for \(n\le 4\), and then interpolate in the narrow intermediate region. In the pseudo-scalar channel the quoted three-loop approximation is better than about \(5\%\) for \(x\in[0.1,0.5]\), while outside this interval the direct asymptotic expansions are more accurate [1103.1481].

## 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 \(q^2\) and \(m^2\) are not available in closed form, the coefficients are generated from differential equations, using
\[
\mathcal D_i\, M_i(q^2,m^2) = \left( 2q^2\frac{\partial}{\partial q^2} + 2m^2\frac{\partial}{\partial m^2} \right) M_i(q^2,m^2),
\]
with low-energy and high-energy ansätze adapted to the corresponding singularity structure in dimensional regularization [1110.5581].

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 \(q\), 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 \(\rho\) parameter, together with UV finiteness and gauge-parameter cancellation in representative coefficients [1506.00900].

Renormalization depends strongly on the current channel. In the standard correlator expansions, \(\alpha_s\) is renormalized in \(\overline{\mathrm{MS}}\), while heavy-quark masses are often treated in both \(\overline{\mathrm{MS}}\) 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 \(\gamma_5\) requires care: singlet diagrams containing exactly one \(\gamma_5\) in a fermion trace are handled with the Larin prescription, while a naively anticommuting \(\gamma_5\) is used otherwise [1110.5581].

This channel dependence persists in adjacent correlator problems. In the unequal-mass tensor-current correlator with full \(q^2\) dependence, the renormalized result requires both mass counterterms and tensor-current renormalization,
\[
\Pi_T = \Pi_T \big|_{\rm unren.} + \Pi_T \big|_{\rm CT-mass} + \Pi_T \big|_{\rm CT-curr},
\]
with \(Z_T=1+\frac{\alpha_s}{4\pi\epsilon}C_F+\mathcal O(\alpha_s^2)\). In that case, omitting current renormalization changes the finite \(\mathcal O(\epsilon^0)\) 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 [2509.02776].

## 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
\[
G(t) = a^6 \sum_{\mathbf{x}} (a m_{0h})^2 \langle 0 | j_5(\mathbf{x},t) j_5(0,0)|0\rangle,
\qquad
G_n=\sum_t (t/a)^n G(t),
\]
with reduced moments
\[
R_n =
\begin{cases}
G_4/G_4^{(0)}, & n=4,\\[4pt]
\dfrac{a m_{\eta_h}}{2 a m_{0h}}
\left(\dfrac{G_n}{G_n^{(0)}}\right)^{1/(n-4)}, & n\ge 6.
\end{cases}
\]
The continuum counterpart factorizes the perturbative coefficient \(r_n\) from the hadron-to-quark mass ratio \(z(\mu/m_h,m_{\eta_h})=m_{\eta_h}/(2m_h(\mu))\), making the reduced moments a particularly clean bridge between lattice data and continuum perturbation theory [1004.4285].

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
\[
m_c(3\,\mathrm{GeV},n_f=4)=0.986(10)\,\mathrm{GeV},\qquad
m_c(m_c,n_f=4)=1.268(9)\,\mathrm{GeV},
\]
\[
\alpha_{\overline{\mathrm{MS}}}(3\,\mathrm{GeV},n_f=4)=0.251(6),\qquad
\alpha_{\overline{\mathrm{MS}}}(M_Z,n_f=5)=0.1174(12).
\]
A later extension to finer lattices and heavier masses yielded
\[
m_c(3\,\mathrm{GeV},n_f=4)=0.986(6)\,\mathrm{GeV},\qquad
m_b(10\,\mathrm{GeV},n_f=5)=3.617(25)\,\mathrm{GeV},
\]
\[
\alpha_{\overline{\mathrm{MS}}}(M_Z,n_f=5)=0.1183(7),
\qquad
\frac{m_b}{m_c}=4.51(4),
\]
together with a nonperturbative HISQ mass-ratio check giving \(m_b/m_c=4.49(4)\) before combination [0805.2999][1004.4285].

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
\[
\mathcal M_n^{X,\Sigma}=m_q^{-2n}\sum_{i=0} x^i\,S_i(xL),
\]
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
\[
\alpha_s^{(5)}(M_Z)=0.1171(7),\qquad
\overline m_c = 1281.1(3.8)\ {\rm MeV},\qquad
\overline m_b = 4174.3(9.5)\ {\rm MeV},
\]
with the analysis also stressing that vector-channel condensate terms become pathological in fixed-order perturbation theory if the \(\overline{\rm MS}\) mass is used directly instead of the pole mass [2306.10323].

## 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
\[
G(t)=a^6\sum_{\vec{x}}(am_q)^2\langle 0|j_5(\vec{x},t)j_5(0,0) |0 \rangle,
\qquad
G_n=\sum_t\Big(\frac{t}{a}\Big)^nG(t),
\]
and matches reduced moments to continuum moments depending on \(m_l/m_h\). In the HISQ-HISQ test case the current normalization is trivial, \(Z=1\), and the extracted quantity is \(m_{\eta_h}/(2m_h(\mu))\). 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 [1011.1208].

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 \(10\%-30\%\) of the reduced moment for heavy masses between charm and bottom, with effective scaling \(1/m_h^3\). This term is absent in the heavy-heavy pseudoscalar moments used in precision \(m_c\), \(m_b\), and \(\alpha_s\) work. In addition, the heavy-light perturbative coefficients depend sensitively on \(m_l/m_h\): the expansion works reasonably well for \(B_s\), but for \(B_c\), where \(m_c(\mu)/m_b(\mu)\approx 0.22\), the mass-ratio dependence must be treated exactly at the available orders. The paper also stresses that continuum perturbation theory is known through \(\alpha_s^3\) for heavy-heavy correlators but only through \(\alpha_s^2\) for heavy-light correlators [1011.1208].

That condensate sensitivity can be turned into a virtue. By deriving the heavy-light pseudoscalar OPE through \(\alpha_s^2\) for the quark-condensate coefficient and fitting lattice heavy-strange moments \(n=4,6,8,10\), one obtains
\[
\langle \bar s s \rangle^{\overline{\mathrm{MS}}(2\,\mathrm{GeV})}=-(296(11)\,\mathrm{MeV})^3.
\]
The analysis shows explicitly that the leading nonperturbative effect in heavy-light moments is \(\langle\bar\psi\psi\rangle/M^3\), without suppression by the light-quark mass, in sharp contrast to heavy-heavy moments where the gluon condensate enters at order \(1/M^4\) and is numerically tiny [1811.04305].

In the static limit, heavy-light correlators are naturally reformulated in HQET. There the correlator is expanded as
\[
\Pi_P(\tau)=\sum_{\mathcal O} C_{\mathcal O}(\tau)\,\langle \mathcal O\rangle,
\]
or equivalently in momentum space and spectral density form,
\[
\Pi_P(\omega)=\sum_{\mathcal O} C_{\mathcal O}(\omega)\,\langle \mathcal O\rangle,
\qquad
\rho_P(\omega)=\sum_{\mathcal O} R_{\mathcal O}(\omega)\,\langle \mathcal O\rangle.
\]
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 \(1\), \(m\), \(m^2\), and \(\sum_i m_i^2\) 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 [2111.14571][2411.11080].

## 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
\[
z=9,\qquad v=\frac{z-1}{z+1}=0.8
\]
for non-diagonal correlators, because of a cut through three heavy-quark lines, and only above
\[
z=16,\qquad v=\sqrt{1-\frac{1}{z}}\approx 0.97
\]
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 [1110.5581].

A complementary diagnosis comes from the large-\(\beta_0\) limit. There the reduced moment coefficients admit exact Borel representations, and the ratios
\[
R_{q,n}^{\delta}=\frac{(M_{q,n}^{\delta})^{1/n}}{(M_{q,n+1}^{\delta})^{1/(n+1)}}
\]
are controlled by
\[
B_n^\delta(u)=\frac{S_n^\delta(u)}{n}-\frac{S_{n+1}^\delta(u)}{n+1}.
\]
These ratios exhibit a partial cancellation of the leading UV renormalon at \(u=-1\) and a reduced residue of the leading IR renormalon at \(u=2\). The analysis also identifies improved combinations such as \(\widehat{R}_c^V(-1/3,1,-1/3)\), for which both leading UV and IR residues are reduced by about \(70\%\) [2106.05660].

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,
\[
\bar Q\gamma^\mu Q \longrightarrow \sum_f d_2\,\partial_\nu(\bar q_f[\gamma^\mu,\gamma^\nu]q_f)
+ h_1\,\partial_\nu {\rm Tr}(G^{\mu\nu}G_{\alpha\beta}G^{\alpha\beta})
+ h_2\,\partial_\nu {\rm Tr}\!\left(\frac12\{G^{\mu\alpha},G^{\nu\beta}\}G_{\alpha\beta}\right),
\]
with
\[
d_2 \sim \frac{m_f}{M^2}\left(\frac{\alpha_s}{\pi}\right)^3,
\qquad
h_{1,2}\sim \frac{g_s^3}{16\pi^2 M^4}.
\]
This formulation governs sub-threshold disconnected contributions and low-energy hadronic matrix elements of heavy currents [2310.09085].

Several adjacent generalizations broaden the subject without changing its core logic. Full-\(q^2\) 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 [2509.02776]. 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 [1210.1064]. 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 [2006.01545].

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 \(m_c\), \(m_b\), and \(\alpha_s\); 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 [1110.5581].

Source: https://www.emergentmind.com/topics/heavy-quark-current-current-correlators