Laplace Sum Rules in QCD

• Laplace Sum Rules are exponentially weighted dispersion relations built on two-point correlators in QCD to extract hadronic observables.
• They utilize the Laplace (or Borel) transform to suppress high-energy states and enhance the low-lying resonance signals.
• The method relies on optimization of the Laplace parameter, continuum threshold, and renormalization scale to improve extraction of masses and couplings.

Searching arXiv for recent and foundational papers on Laplace sum rules to ground the article. Using arXiv search to confirm relevant sources and recent treatments. Laplace Sum Rules (LSR), also called Borel or inverse Laplace sum rules in the QCD spectral sum-rule literature, are exponentially weighted dispersion relations constructed from two-point correlation functions in order to relate hadronic observables to the operator product expansion (OPE) of Quantum Chromodynamics. In the sources considered here, LSR are defined through moments of a spectral density weighted by $e^{-t\tau}$, with $\tau$ the Laplace or Borel parameter and $t_c$ the continuum threshold, and are used to extract hadron masses, couplings, mass ratios, and QCD parameters from optimized regions of $\tau$, $t_c$, and, when radiative corrections are included, the renormalization scale $\mu$ (Narison, 2023). Across applications to hybrids, tetraquarks, heavy-light mesons, glueballs, heavy quarkonia, and light scalars, the formal structure is stable: one starts from a current with the desired quantum numbers, computes the corresponding correlator in QCD through perturbation theory plus condensates, models the hadronic spectral function by a pole plus continuum ansatz, and uses ratios of Laplace moments to isolate the ground-state mass (Li et al., 27 Jun 2025).

1. Definition and formal framework

The common starting point is a two-point correlator built from an interpolating current carrying the quantum numbers of the hadronic channel under study. In the hybrid example with $J^{PC}=1^{--}$, the current is

$$J^\mu(x) = \overline{\Psi}(x)\,\gamma_\rho\gamma^5\,T^n\,\widetilde{G}^{n\,\rho\mu}(x)\,\Psi(x),$$

and the corresponding correlator is

$$\Pi^{\mu\nu}(q^2) = i\int d^4x\, e^{-iqx}\,\langle 0 | T\{ J^\mu(x) J^{\dagger\,\nu}(0)\} | 0 \rangle,$$

with the transverse invariant amplitude $\Pi_v(q^2)$ carrying the $\tau$0 information (Li et al., 27 Jun 2025). In heavy-light vector channels, the same logic appears with $\tau$1 and its transverse part $\tau$2 (Narison, 2014). For scalar channels, the invariant correlator is written directly as $\tau$3 (Albuquerque et al., 2023). For tensor glueballs, the correlator is projected onto a spin-2 Lorentz structure $\tau$4, leaving a scalar function $\tau$5 to which the sum rule is applied (Li et al., 2023).

Analyticity yields a dispersion relation in terms of the spectral density. In the notation used repeatedly across the sources, the spectral function is

$\tau$6

or its channel-specific analogue. The Laplace sum rule is then defined as a finite-energy exponential moment,

$\tau$7

with variants such as $\tau$8 in the hybrid analysis and $\tau$9 in heavy-light mesons [(Albuquerque et al., 2022); (Narison, 2014); (Li et al., 27 Jun 2025)]. The ratio of successive moments,

$t_c$0

or equivalently $t_c$1, is interpreted as the squared ground-state mass in the single-pole approximation [(Narison, 2023); (Narison, 2014)].

The hadronic side is usually modeled by the minimal duality ansatz or pole plus continuum ansatz,

$t_c$2

with current-dependent powers of $t_c$3 multiplying the residue according to the operator dimension and normalization convention [(Li et al., 27 Jun 2025); (Narison, 2014); (Li et al., 2023)]. After continuum subtraction, the lowest-state contribution takes the form $t_c$4, which makes the mass accessible through the moment ratio while the coupling follows from the normalization of the lowest moment (Albuquerque et al., 2022).

2. Role of the Laplace transform

The defining feature of LSR is the exponential kernel $t_c$5. In the reviewed literature, three effects are emphasized. First, the transform removes subtraction polynomials in the dispersion relation. Second, it suppresses higher-energy states and the continuum, enhancing sensitivity to the lowest resonance. Third, it improves the convergence properties of the OPE by damping high-energy contributions and converting inverse powers of $t_c$6 into powers of $t_c$7 [(Li et al., 27 Jun 2025); (Narison, 2023); (Narison, 2014)].

This role is explicit in general reviews. Narison writes the core LSR as

$t_c$8

with $t_c$9 acting as the resolution scale and $\tau$0 the corresponding Borel mass (Narison, 2023). The exponential kernel is described as strongly suppressing large-$\tau$1 contributions and emphasizing the low-lying spectral region, a property repeatedly invoked in glueball, hybrid, and tetraquark analyses (Li et al., 2024, Li et al., 27 Jun 2025, Albuquerque et al., 2022).

The method is often described as “inverse Laplace” or “Borel” transform. Several sources note that there is no separate practical object beyond the same exponentially weighted moment integral. In the tetraquark and heavy-light applications, “inverse Laplace transform” simply refers to the standard Borel-Laplace transform used in QCD sum rules [(Albuquerque et al., 2022); (Narison, 2014)]. This terminological variation is common rather than substantive.

3. QCD side: perturbation theory and condensates

On the QCD side, the correlator is expanded through the OPE as a perturbative contribution plus vacuum condensates. The specific content depends on the channel, but the reviewed sources consistently include a perturbative term and gauge-invariant condensates of increasing dimension. In the light hybrid $\tau$2 channel, the OPE includes perturbative contributions at NLO together with condensates up to dimension 8: $\tau$3, $\tau$4, $\tau$5, $\tau$6, and $\tau$7 (Li et al., 27 Jun 2025). In the heavy-light decay-constant analysis, the vector-channel LSR contains perturbation theory through $\tau$8 and condensates up to $\tau$9, including $t_c$0, mixed condensates, $t_c$1, and four-quark structures parameterized by $t_c$2 (Narison, 2014).

For the $t_c$3 tensor di-gluonium, the OPE is especially transparent. The perturbative correlator is given at LO by

$t_c$4

with NLO logarithmic and constant corrections. The gluon-condensate term $t_c$5 contributes through a logarithm at NLO, while the $t_c$6 gluonic condensate enters as a $t_c$7 term parameterized by a factorization-violation parameter $t_c$8 (Li et al., 2023, Li et al., 2024). The same study shows that the $t_c$9 $\mu$0 contribution vanishes at LO in that tensor channel (Li et al., 2023).

In the light scalar analysis, the perturbative spectral function for $\mu$1 currents is written explicitly through $\mu$2, with an estimated $\mu$3 contribution used only for truncation errors, while condensates up to $\mu$4 are retained (Albuquerque et al., 2023). The authors also model missing higher-dimension effects by

$\mu$5

with $\mu$6 for three flavors in that analysis (Albuquerque et al., 2023).

A recurring theme is that higher-order perturbative corrections matter numerically and conceptually. In the hybrid case, NLO terms lower the predicted mass by roughly $\mu$7 GeV relative to LO (Li et al., 27 Jun 2025). In the tensor glueball case, NLO corrections raise the mass substantially compared with LO (Li et al., 2023). In heavy-heavy and heavy-light systems, NLO or N2LO corrections are stressed as necessary to justify the use of the running $\mu$8 heavy-quark mass rather than an ad hoc mass choice at LO [(Albuquerque et al., 2022); (Narison et al., 2021); (Narison, 2014)].

4. Hadronic modeling, mass extraction, and couplings

The phenomenological side is almost universally represented by a narrow pole plus continuum. For the hybrid channel, this is written as

$\mu$9

leading, after continuum subtraction, to

$J^{PC}=1^{--}$0

and hence

$J^{PC}=1^{--}$1

in the one-pole approximation (Li et al., 27 Jun 2025). The same structure appears in heavy-light mesons, where

$J^{PC}=1^{--}$2

and in tensor glueballs, where

$J^{PC}=1^{--}$3

under the minimal duality ansatz [(Narison, 2014); (Li et al., 2023)].

This ratio method eliminates the unknown coupling from the mass determination. The coupling is then obtained from the lowest moment after inserting the extracted mass. In the heavy-light review, this yields

$J^{PC}=1^{--}$4

(Narison, 2014). In tensor glueballs, the coupling is similarly determined through $J^{PC}=1^{--}$5 once $J^{PC}=1^{--}$6 is fixed, and then converted into a renormalization-group-invariant quantity $J^{PC}=1^{--}$7 (Li et al., 2023).

Some applications extend this framework. Double ratios of sum rules (DRSR) compare two channels sharing approximately the same $J^{PC}=1^{--}$8 and $J^{PC}=1^{--}$9,

$$J^\mu(x) = \overline{\Psi}(x)\,\gamma_\rho\gamma^5\,T^n\,\widetilde{G}^{n\,\rho\mu}(x)\,\Psi(x),$$0

leading to more precise mass ratios because heavy-quark masses, condensates, continuum thresholds, and scale dependences partially cancel (Albuquerque et al., 2022). This strategy is used to sharpen mass predictions for $$J^\mu(x) = \overline{\Psi}(x)\,\gamma_\rho\gamma^5\,T^n\,\widetilde{G}^{n\,\rho\mu}(x)\,\Psi(x),$$1-type states (Albuquerque et al., 2022, Albuquerque et al., 2022).

5. Stability criteria and optimization

LSR are not evaluated at arbitrary $$J^\mu(x) = \overline{\Psi}(x)\,\gamma_\rho\gamma^5\,T^n\,\widetilde{G}^{n\,\rho\mu}(x)\,\Psi(x),$$2, $$J^\mu(x) = \overline{\Psi}(x)\,\gamma_\rho\gamma^5\,T^n\,\widetilde{G}^{n\,\rho\mu}(x)\,\Psi(x),$$3, or $$J^\mu(x) = \overline{\Psi}(x)\,\gamma_\rho\gamma^5\,T^n\,\widetilde{G}^{n\,\rho\mu}(x)\,\Psi(x),$$4. The predictive content comes from optimized regions where the extracted observables are stable. Across the sources, three criteria recur.

The first is $$J^\mu(x) = \overline{\Psi}(x)\,\gamma_\rho\gamma^5\,T^n\,\widetilde{G}^{n\,\rho\mu}(x)\,\Psi(x),$$5-stability. One seeks a plateau, minimum, or inflection point in the $$J^\mu(x) = \overline{\Psi}(x)\,\gamma_\rho\gamma^5\,T^n\,\widetilde{G}^{n\,\rho\mu}(x)\,\Psi(x),$$6-dependence of the mass or coupling. In the heavy-light decay-constant study, pseudoscalar channels often exhibit minima while vector channels show inflection points (Narison, 2014). In the tetraquark and exotic analyses, $$J^\mu(x) = \overline{\Psi}(x)\,\gamma_\rho\gamma^5\,T^n\,\widetilde{G}^{n\,\rho\mu}(x)\,\Psi(x),$$7-stability defines the working window together with OPE convergence and pole dominance (Albuquerque et al., 2022, Narison et al., 2021). In the scalar review, the optimal values are described as those satisfying minimum sensitivity under $$J^\mu(x) = \overline{\Psi}(x)\,\gamma_\rho\gamma^5\,T^n\,\widetilde{G}^{n\,\rho\mu}(x)\,\Psi(x),$$8 and $$J^\mu(x) = \overline{\Psi}(x)\,\gamma_\rho\gamma^5\,T^n\,\widetilde{G}^{n\,\rho\mu}(x)\,\Psi(x),$$9 variation (Albuquerque et al., 11 Oct 2025).

The second is $$\Pi^{\mu\nu}(q^2) = i\int d^4x\, e^{-iqx}\,\langle 0 | T\{ J^\mu(x) J^{\dagger\,\nu}(0)\} | 0 \rangle,$$0-stability. The continuum threshold is scanned over a conservative range, and an acceptable extraction requires weak dependence of the observable on $$\Pi^{\mu\nu}(q^2) = i\int d^4x\, e^{-iqx}\,\langle 0 | T\{ J^\mu(x) J^{\dagger\,\nu}(0)\} | 0 \rangle,$$1 within the chosen $$\Pi^{\mu\nu}(q^2) = i\int d^4x\, e^{-iqx}\,\langle 0 | T\{ J^\mu(x) J^{\dagger\,\nu}(0)\} | 0 \rangle,$$2-window [(Albuquerque et al., 2022); (Narison, 2014)]. In tensor glueballs, $$\Pi^{\mu\nu}(q^2) = i\int d^4x\, e^{-iqx}\,\langle 0 | T\{ J^\mu(x) J^{\dagger\,\nu}(0)\} | 0 \rangle,$$3 is selected from the onset of $$\Pi^{\mu\nu}(q^2) = i\int d^4x\, e^{-iqx}\,\langle 0 | T\{ J^\mu(x) J^{\dagger\,\nu}(0)\} | 0 \rangle,$$4-stability to the beginning of $$\Pi^{\mu\nu}(q^2) = i\int d^4x\, e^{-iqx}\,\langle 0 | T\{ J^\mu(x) J^{\dagger\,\nu}(0)\} | 0 \rangle,$$5-stability (Li et al., 2023). The scalar analyses emphasize that $$\Pi^{\mu\nu}(q^2) = i\int d^4x\, e^{-iqx}\,\langle 0 | T\{ J^\mu(x) J^{\dagger\,\nu}(0)\} | 0 \rangle,$$6 need not coincide exactly with the first radial excitation mass, even though it is often of comparable scale (Albuquerque et al., 2023).

The third is $$\Pi^{\mu\nu}(q^2) = i\int d^4x\, e^{-iqx}\,\langle 0 | T\{ J^\mu(x) J^{\dagger\,\nu}(0)\} | 0 \rangle,$$7-stability when radiative corrections are included. This is strongly emphasized in modern LSR treatments. In heavy-light mesons, optimal values are chosen at minima or inflection points in the $$\Pi^{\mu\nu}(q^2) = i\int d^4x\, e^{-iqx}\,\langle 0 | T\{ J^\mu(x) J^{\dagger\,\nu}(0)\} | 0 \rangle,$$8-dependence, such as $$\Pi^{\mu\nu}(q^2) = i\int d^4x\, e^{-iqx}\,\langle 0 | T\{ J^\mu(x) J^{\dagger\,\nu}(0)\} | 0 \rangle,$$9 for $\Pi_v(q^2)$0 and $\Pi_v(q^2)$1 for $\Pi_v(q^2)$2 (Narison, 2014). In heavy-light tetraquarks, the authors use nearly universal values $\Pi_v(q^2)$3 and $\Pi_v(q^2)$4 derived from previous stability studies (Albuquerque et al., 2022). In heavy quarkonia, the extraction of $\Pi_v(q^2)$5, quark masses, and the gluon condensate is explicitly tied to the “$\Pi_v(q^2)$6-subtraction stability point” (Narison, 2020).

A stricter version of pole dominance is used in the scalar studies, where the ratio

$\Pi_v(q^2)$7

is required to satisfy $\Pi_v(q^2)$8 (Albuquerque et al., 2023, Albuquerque et al., 11 Oct 2025). This criterion excludes low-$\Pi_v(q^2)$9 solutions leading to on-shell scalar masses around $\tau$00 MeV in those analyses (Albuquerque et al., 2023).

6. Applications across channels and relation to other sum rules

The method is used in a wide range of channels. In light hybrids, LSR with NLO perturbation theory and condensates up to dimension 8 lead to a conservative mass range $\tau$01 for the light $\tau$02 hybrid, with the LSR analysis broadly consistent with sharper Gaussian sum-rule determinations (Li et al., 27 Jun 2025). In heavy-light decay constants, LSR give $\tau$03, $\tau$04, $\tau$05, $\tau$06, and $\tau$07 in one representative study (Narison, 2014). In tensor glueballs, LSR at NLO give $\tau$08 and $\tau$09 for $\tau$10, and $\tau$11, $\tau$12 for $\tau$13 (Li et al., 2023, Li et al., 2024).

LSR also serve as precision tools for QCD parameters. In heavy quarkonia, optimized ratios of relativistic LSR yield $\tau$14, $\tau$15, $\tau$16, and $\tau$17 in a representative global analysis (Narison, 2020).

The relationship of LSR to other QCD spectral sum rules is explicit in the literature. Finite-energy sum rules arise as polynomial-weighted moments and are often viewed as complementary constraints on $\tau$18 [(Narison, 2023); (Narison, 2014)]. Gaussian sum rules apply a Gaussian kernel rather than an exponential one and can be more stable in channels where the LSR window is delicate. In the $\tau$19 hybrid case, GSR give narrower mass predictions and coincide with the $\tau$20 region where LSR are most stable (Li et al., 27 Jun 2025). This suggests complementarity rather than competition between kernels.

A recurring limitation is that simple pole-plus-continuum models may not resolve nearby states or strong mixing. This is especially important in light scalar and exotic multiquark channels, where molecule, tetraquark, and $\tau$21 assignments often produce overlapping mass ranges (Albuquerque et al., 2023, Albuquerque et al., 2022). The scalar analyses conclude that the assignment of scalar mesons is “not crystal clear,” while the heavy-exotic studies often interpret observed states as mixed “tetramoles,” meaning superpositions of quasi-degenerate molecule and tetraquark configurations with almost equal couplings to the currents (Albuquerque et al., 11 Oct 2025, Albuquerque et al., 2022).

7. Conceptual significance and methodological caveats

LSR occupy a distinctive place in nonperturbative QCD because they combine analytic control over short-distance dynamics with an explicit parametrization of long-distance physics through condensates and a controlled spectral ansatz. Their strength lies in the exponential suppression of the continuum, the direct access to ground-state masses through moment ratios, and the existence of internal optimization criteria in $\tau$22, $\tau$23, and $\tau$24 [(Narison, 2023); (Narison, 2014)].

At the same time, the method is only as reliable as its working window and truncations. OPE convergence can be marginal in delicate channels, as illustrated by the $\tau$25 hybrid where dimension-8 terms spoil $\tau$26-stability (Li et al., 27 Jun 2025). Higher-dimension condensates and factorization assumptions can materially affect glueball and scalar analyses (Li et al., 2023, Albuquerque et al., 2023). The continuum threshold remains a model parameter rather than an observable, and the single-pole ansatz is an approximation to what may be a dense or mixed spectrum [(Narison, 2014); (Narison, 2023)].

Within those caveats, the reviewed literature presents LSR as a mature and adaptable framework. It is used to extract hadron masses and couplings, determine quark masses and condensates, quantify SU(3) breakings, sharpen predictions through ratios and double ratios, and cross-check more channel-specific kernels such as Gaussian sum rules. The method’s continued use across recent hybrid, glueball, tetraquark, heavy-light, and scalar studies indicates not only formal continuity with the SVZ program but also an active technical evolution driven by higher-order perturbation theory, improved stability criteria, and more systematic uncertainty accounting (Li et al., 27 Jun 2025, Li et al., 2024, Albuquerque et al., 2022).