Time-Dependent HAL QCD Method

• Time-dependent HAL QCD method is a lattice QCD framework that bypasses ground-state saturation by using the full elastic-state spectrum to extract non-local interaction potentials.
• The method employs a Schrödinger-like equation and a derivative expansion to approximate the energy-independent kernel while mitigating excited-state contamination.
• It enables calculations of scattering phase shifts, bound-state searches, and finite-volume spectra, demonstrating quantitative consistency with Lüscher’s finite-volume approach.

Searching arXiv for recent and foundational papers on the time-dependent HAL QCD method. The time-dependent HAL QCD method is a lattice QCD framework to extract hadron–hadron interaction potentials directly from correlation functions, without needing to isolate the ground state of the two-hadron system (Iritani et al., 2015). In this formulation, the central object is the normalized Nambu–Bethe–Salpeter (NBS) correlation function $R(\vec r,t)$, whose Euclidean-time evolution satisfies a Schrödinger-like equation with an energy-independent, generally non-local interaction kernel $U(\vec r,\vec r\,')$ below the inelastic threshold (Iritani et al., 2015). The method was developed to address the difficulty that, in multi-baryon systems and other hadronic channels on large volumes, elastic scattering levels are closely spaced, so genuine ground-state saturation requires very large Euclidean times where statistical noise becomes prohibitive (Iritani et al., 2018). By using the full superposition of elastic states rather than isolating a single eigenstate, the time-dependent HAL QCD method provides a potential-based route to scattering phase shifts, bound-state searches, and finite-volume spectra across baryon–baryon, meson–meson, meson–baryon, and heavy-hadron systems (Iritani et al., 2018).

1. Historical placement and conceptual motivation

The method emerged within the HAL QCD program as an extension of the original, time-independent potential method. In the original formulation, one reconstructs an interaction kernel from the spatial dependence of the equal-time NBS wave function for an eigenstate with energy $W_n$, typically requiring ground-state saturation (Iritani et al., 2015). The time-dependent formulation removes this requirement by deriving a Schrödinger-like equation directly for the correlation function $R(\vec r,t)$, so that all elastic states below the inelastic threshold contribute simultaneously to the determination of the same energy-independent potential (Iritani et al., 2015).

This development was motivated by a specific spectral problem. In multi-baryon systems, especially on large volumes, the energy spacing between elastic levels is $\mathcal{O}(1/L^2)$, much smaller than the inelastic gap $\mathcal{O}(\Lambda_{\mathrm{QCD}})$ (Iritani et al., 2018). In the $\Xi\Xi(^1S_0)$ example at $m_\pi = 0.51$ GeV, the first elastic excited state on the $64^3$ lattice is only $\sim 30$ MeV above the ground state, so suppression of elastic excited states would require Euclidean times of order $U(\vec r,\vec r\,')$0 fm, which are effectively inaccessible because of noise (Iritani et al., 2015). This leads to pseudo plateaux or fake plateaux in temporal effective energies, a phenomenon described as the “mirage problem” in two-baryon systems (Iritani et al., 2017).

Against this background, the methodological contrast with Lüscher’s finite-volume method is sharp. The direct method infers finite-volume eigenenergies from plateaux in temporal correlators and then converts them to phase shifts through Lüscher’s formula, whereas HAL QCD extracts observables from the non-local potential associated with a tempo-spatial correlator (Iritani et al., 2018). The former requires reliable identification of plateaux dominated by the finite-volume ground state; the latter requires only suppression of inelastic contamination and treats elastic excited states as signal rather than contamination (Iritani et al., 2018).

2. Formal structure of the time-dependent equation

For equal-mass baryons $U(\vec r,\vec r\,')$1, the normalized NBS correlation function is defined as

$U(\vec r,\vec r\,')$2

where $U(\vec r,\vec r\,')$3 is a single-baryon sink operator, $U(\vec r,\vec r\,')$4 is a two-baryon source operator, and $U(\vec r,\vec r\,')$5 is the single-baryon two-point function (Iritani et al., 2015). In the elastic region, its spectral decomposition is

$U(\vec r,\vec r\,')$6

with $U(\vec r,\vec r\,')$7 the equal-time NBS wave function, $U(\vec r,\vec r\,')$8, and $U(\vec r,\vec r\,')$9 the energy gap to the lowest inelastic threshold (Iritani et al., 2015).

The central time-dependent HAL QCD equation is

$W_n$0

with

$W_n$1

and $W_n$2 for equal-mass baryons (Iritani et al., 2015). An equivalent form, written with reordered operators, is used in later presentations of the method (Iritani et al., 2018). The same structure appears in mesonic systems such as $W_n$3 scattering (1711.01883), in hyperon and hyperonic channels (Nemura et al., 2016), and in heavy-hadron systems such as charmonium–nucleon interactions, where the relativistic correction term may be neglected when it is much smaller than the leading terms (Sugiura et al., 2017).

The defining feature is that this equation is valid for the full elastic sum. Since each elastic component shares the same energy-independent kernel $W_n$4, the mixture of elastic states does not obstruct potential extraction (Iritani et al., 2015). This suggests a change in the relevant saturation criterion: not ground-state saturation, but elastic-state saturation, meaning that inelastic contributions $W_n$5 are negligible (Iritani et al., 2018).

For channels with unequal masses, the time-dependent equation is generalized. In kaon–nucleon systems, for example, the coefficient of the second time derivative depends on the mass asymmetry parameter $W_n$6, and the potential is extracted from the corresponding unequal-mass version of the equation (Murakami et al., 2020). In laboratory-frame formulations with non-zero total momentum, the time-dependent equation acquires explicit $W_n$7-dependent longitudinal and transverse derivative structures, while still defining a CM-frame HAL QCD potential (Aoki et al., 2021).

3. Derivative expansion, scheme dependence, and Hermiticity

The HAL QCD interaction kernel is intrinsically non-local but energy-independent. In practice, it is approximated through a derivative expansion,

$W_n$8

or, in a truncated form,

$W_n$9

depending on the analysis (Iritani et al., 2018). At leading order for spin-singlet or central channels, one writes

$R(\vec r,t)$0

and obtains the local potential

$R(\vec r,t)$1

This expression is the standard LO time-dependent HAL QCD potential (Iritani et al., 2015).

The convergence of the derivative expansion is a central systematic issue. In the $R(\vec r,t)$2 system at $R(\vec r,t)$3 MeV, comparison of LO and N$R(\vec r,t)$4LO analyses showed that phase shifts agree very well at low energies, with differences appearing only at higher energies (Iritani et al., 2018). In $R(\vec r,t)$5, the point-sink scheme exhibits negligible NLO contribution over a wide range of energies, while the LapH smeared-sink scheme has non-negligible NLO terms; however, the resulting phase shifts agree across schemes once NLO is included (1711.01883). This is the operational content of the method’s “scheme independence”: exact observables derived from the exact non-local potential do not depend on the sink operator, but truncated derivative expansions can show scheme dependence that must be monitored and reduced (1711.01883).

A related formal issue is that the derivative-expanded HAL QCD potential is generically non-Hermitian beyond the LO local term, because NBS wave functions for different energies are not orthogonal to each other (Aoki et al., 2019). A formalism to hermitize the HAL QCD potential order by order in the derivative expansion was developed and shown to work to all orders; in particular, the NLO potential can be exactly hermitized without approximation (Aoki et al., 2019). In the $R(\vec r,t)$6 case, the NLO term gives relatively small corrections to the scattering phase shift, and the LO analysis remains justified at low energies (Aoki et al., 2019). A plausible implication is that hermitized versions are particularly useful when comparing HAL QCD interactions with phenomenological potentials or embedding them into many-body frameworks.

The derivative expansion can also be replaced or supplemented by other parametrizations of non-locality. In the SU(3)-symmetric study of $R(\vec r,t)$7-related meson–baryon channels, a separable potential was introduced in the time-dependent HAL QCD method to avoid the singular behavior of leading-order derivative-expanded potentials caused by zeros in the R-correlator (Murakami et al., 29 Jan 2025). This suggests that alternative non-local ansätze may be practically valuable in channels where the local LO extraction is numerically unstable.

4. Comparison with temporal plateau methods and Lüscher analyses

The most persistent controversy around the time-dependent HAL QCD method concerns its relation to the direct, plateau-based finite-volume method. In the direct method, one forms an effective energy shift such as

$R(\vec r,t)$8

or equivalently through a ratio of two-baryon and single-baryon correlators, and interprets a plateau at large $R(\vec r,t)$9 as the finite-volume energy shift (Iritani et al., 2015). In two-baryon systems, however, the close spacing of elastic levels causes moderate-time plateau-like behavior that can originate from cancellations among several low-lying states rather than genuine ground-state dominance (Iritani et al., 2018).

This was demonstrated concretely for $\mathcal{O}(1/L^2)$0 at $\mathcal{O}(1/L^2)$1 GeV. With both smeared and wall sources, $\mathcal{O}(1/L^2)$2 exhibits plateau-like behavior in the range $\mathcal{O}(1/L^2)$3–16, but the plateau values differ significantly between the two sources, outside mutual error bars (Iritani et al., 2015). Since the physical energy shift cannot depend on the source operator, at least one plateau, and likely both, must be fake (Iritani et al., 2015). The subsequent consistency study decomposed the two-baryon correlator into finite-volume eigenmodes obtained from the HAL QCD potential and showed that the pseudo-plateau at early times from the smeared source indeed originates from contamination of excited states, while the true plateau with ground-state saturation is realized only at $\mathcal{O}(1/L^2)$4–15 fm, depending on the volume (Iritani et al., 2018).

This eigenmode analysis provides the bridge between HAL QCD and Lüscher. Starting from the HAL QCD potential, one constructs the finite-volume Hamiltonian and obtains eigenvalues $\mathcal{O}(1/L^2)$5 and eigenfunctions $\mathcal{O}(1/L^2)$6 (Iritani et al., 2018). These same eigenmodes can be used to reconstruct the temporal correlator and its effective energy shift, including the pseudo-plateaux (Iritani et al., 2018). Moreover, they can be used to construct optimized two-baryon operators,

$\mathcal{O}(1/L^2)$7

such that the direct method, when supplied with highly optimized operators, reproduces the same finite-volume spectrum as the HAL QCD potential (Iritani et al., 2018). The resulting conclusion is explicit: Lüscher’s finite-volume method and the HAL QCD method are quantitatively consistent once excited-state contamination is properly removed in the former (Iritani et al., 2018).

This resolves the apparent contradiction between potential-based and energy-level approaches. The time-dependent HAL QCD method does not bypass finite-volume physics; rather, it reorganizes the same elastic information into an energy-independent non-local kernel that can then be used to generate both infinite-volume observables and finite-volume eigenmodes (Iritani et al., 2017).

5. Empirical applications across hadronic systems

The method has been applied across a wide range of channels. In the $\mathcal{O}(1/L^2)$8 system at $\mathcal{O}(1/L^2)$9 GeV, the extracted potential indicates an attractive interaction that is not strong enough to produce a bound state (Iritani et al., 2015). Finite-volume eigenvalues obtained from the HAL QCD Hamiltonian show a ground-state energy shift that is slightly negative and approaches zero as $\mathcal{O}(\Lambda_{\mathrm{QCD}})$0, while the phase shifts show no bound-state pole (Iritani et al., 2015). This case became the standard illustration of source independence in HAL QCD and pseudo-plateaux in the direct method (Iritani et al., 2016).

At nearly physical quark masses, the time-dependent HAL QCD method was used to study the $\mathcal{O}(\Lambda_{\mathrm{QCD}})$1 system on a $\mathcal{O}(\Lambda_{\mathrm{QCD}})$2 lattice with $\mathcal{O}(\Lambda_{\mathrm{QCD}})$3 MeV (Iritani et al., 2018). The central potential is attractive at all distances and is stable across $\mathcal{O}(\Lambda_{\mathrm{QCD}})$4–14 (Iritani et al., 2018). Solving the Schrödinger equation with the fitted potential gives a scattering length $\mathcal{O}(\Lambda_{\mathrm{QCD}})$5 fm, effective range $\mathcal{O}(\Lambda_{\mathrm{QCD}})$6 fm, and a bound state with binding energy $\mathcal{O}(\Lambda_{\mathrm{QCD}})$7 MeV (Iritani et al., 2018). Including the Coulomb interaction for $\mathcal{O}(\Lambda_{\mathrm{QCD}})$8 increases the binding energy to $\mathcal{O}(\Lambda_{\mathrm{QCD}})$9 MeV (Iritani et al., 2018).

In mesonic channels, the $\Xi\Xi(^1S_0)$0 system has served as a controlled testbed for scheme dependence and moving-frame extensions. The comparison of point-sink and LapH smeared-sink schemes showed that although the potentials differ, the scattering phase shifts agree once NLO is included (1711.01883). Later, the method was generalized to non-zero total momentum, with a time-dependent laboratory-frame formulation applied to $\Xi\Xi(^1S_0)$1 on a $\Xi\Xi(^1S_0)$2 lattice at $\Xi\Xi(^1S_0)$3 MeV (Aoki et al., 2021). The effective LO potentials and corresponding phase shifts agree with those obtained in the CM frame, and consistency with finite-volume methods was demonstrated across several frames (Aoki et al., 2021). This suggests a route toward studying resonance channels, including $\Xi\Xi(^1S_0)$4 and the $\Xi\Xi(^1S_0)$5, in moving frames.

In resonance physics, the $\Xi\Xi(^1S_0)$6 system at $\Xi\Xi(^1S_0)$7 MeV provided the first N$\Xi\Xi(^1S_0)$8LO extraction of a HAL QCD potential in a resonant channel requiring all-to-all propagators (Akahoshi et al., 2021). The resulting $\Xi\Xi(^1S_0)$9-meson mass is consistent with values in the literature, while the coupling $m_\pi = 0.51$0 is somewhat larger, most likely because the CM-frame setup lacks low-energy information near threshold (Akahoshi et al., 2021). This case makes clear that the time-dependent HAL QCD method can identify resonance structure and support pole analyses, but that the quantitative control of widths and couplings depends strongly on the energy coverage encoded in the correlators.

Heavy-hadron and meson–baryon applications further broaden the scope. The charmonium–nucleon effective central interactions were computed with the time-dependent HAL QCD method, yielding short-ranged attractive potentials but no bound state in the analyzed single-channel systems (Sugiura et al., 2017). In kaon–nucleon S-wave interactions at $m_\pi = 0.51$1 MeV, the method combined with all-to-all propagators and the one-end trick found stronger repulsion in the $m_\pi = 0.51$2 channel than in $m_\pi = 0.51$3, with phase shifts that qualitatively reproduce the energy dependence of experiment (Murakami et al., 2020). In the study of decuplet baryons as meson–baryon bound states, P-wave $m_\pi = 0.51$4 and $m_\pi = 0.51$5 interactions were analyzed at $m_\pi = 0.51$6 MeV and $m_\pi = 0.51$7 MeV, with the binding energies from HAL QCD agreeing with those from temporal 2-point functions within large systematic errors dominated by short-distance lattice artifacts (Murakami et al., 2022).

6. Systematic uncertainties, algorithmic infrastructure, and current directions

The principal systematic uncertainties are well established. First is the truncation of the derivative expansion: LO is often sufficient at low energies, but NLO or N$m_\pi = 0.51$8LO terms can matter in smeared schemes, at higher momenta, or in resonant channels (1711.01883). Second is the need to suppress inelastic contamination by working at moderate but sufficiently large Euclidean times; this requirement is typically far less demanding than ground-state saturation, but it remains nontrivial (Iritani et al., 2015). Third are numerical derivative errors, especially for second time derivatives and short-distance spatial Laplacians (Iritani et al., 2015). Fourth are short-distance lattice artifacts, which become particularly important for compact bound states and can dominate the systematic uncertainty in binding energies and root-mean-square radii (Murakami et al., 2022).

A large computational infrastructure has developed around these issues. Efficient evaluation of multi-baryon four-point correlators led to the effective baryon block algorithm and the unified contraction algorithm, which reduce the cost of computing many baryon–baryon channels and support large-volume, near-physical calculations (Nemura et al., 2016). For channels requiring quark annihilation diagrams or momentum-projected multi-hadron sources, all-to-all methods based on the one-end trick, sequential propagators, jackknife or AMA/CAA error reduction, and LapH or related smearing schemes have become integral to practical time-dependent HAL QCD analyses (Murakami et al., 2020, Akahoshi et al., 2021, Murakami et al., 29 Jan 2025).

Recent work has increasingly focused on convergence diagnostics based on finite-volume eigenmodes. In $m_\pi = 0.51$9 and $64^3$0 at nearly physical light quark masses and physical charm, the HAL QCD potentials determined at LO were used to obtain finite-volume eigenmodes; eigenmode-projected temporal correlators then reproduced the same ground and first excited spectra as the HAL QCD Hamiltonian (Doi et al., 2021). In the $64^3$1 case, the unprojected correlator is dominated by the first excited state rather than the ground state, yet the HAL QCD potential still reliably reproduces the ground-state binding energy (Doi et al., 2021). This provides explicit evidence that the method can extract ground-state information even from correlators dominated by excited scattering states.

Current directions extend along several axes. One is the refinement of non-local parameterizations beyond simple local LO forms, including separable potentials for channels where the LO derivative expansion becomes singular or numerically unstable (Murakami et al., 29 Jan 2025). Another is the extension to moving frames, which enlarges the accessible kinematic region and may improve studies of resonances that require low-energy information not present in the CM frame alone (Aoki et al., 2021, Akahoshi et al., 2021). A third is the use of hermitized derivative-expanded potentials for comparison with phenomenological interactions and for many-body calculations (Aoki et al., 2019). Taken together, these developments suggest that the time-dependent HAL QCD method has evolved from a workaround for ground-state saturation into a broader non-local effective-interaction framework for lattice QCD scattering theory.