Deuteron Longitudinal Response Function

• The deuteron longitudinal response function is defined as the squared matrix element of the nuclear charge operator between the deuteron ground state and continuum nucleon states, crucial for understanding nuclear structure and DIS observables.
• It is computed using methodologies such as chiral EFT, Lippmann–Schwinger solutions, and Chebyshev-expansion techniques, which ensure RG invariance, order-by-order convergence, and systematic uncertainty quantification.
• High-precision measurements and theoretical reconstructions of the response function enable refined corrections for nuclear modifications, thereby improving the accuracy of nucleon structure extractions in scattering experiments.

The longitudinal response function of the deuteron quantifies the probability that the deuteron absorbs energy and momentum through the time component of the nuclear current in an external probe, such as in electron or neutrino scattering. This response governs crucial observables in deuteron electrodisintegration and deep inelastic scattering (DIS) off the deuteron, central both for nuclear structure theory and for the extraction of nucleon structure from nuclear data. The precise calculation and measurement of the deuteron longitudinal response serve as benchmarks for theoretical frameworks such as chiral effective field theory (EFT), ab initio many-body methods, and advanced reconstruction techniques.

1. Formal Definition and Kinematic Structure

The deuteron longitudinal response function, often denoted $f_L(q^2, E_{np})$ or $R_L(q, \omega)$, is defined as the matrix element squared of the nuclear charge operator $J^0$ (or charge-density operator $\rho(q)$) between the deuteron ground state and all possible final-state nucleon-nucleon configurations, summed over spins and isospins: $$f_L(q^2, E_{np}) = \sum_f \left| \langle \psi_f | J^0(q) | \psi_d\rangle \right|^2 \delta(E_f - E_d - \omega)$$ where $\psi_d$ is the deuteron bound-state wave function, $q^\mu = (\omega, \mathbf{q})$ is the virtual-photon four-momentum, and $\psi_f$ represents continuum final states with relative energy $E_{np}$ in the $np$ center-of-mass (c.m.). An analogous formalism is used for weak probes, with appropriate substitutions for the current operator (Yang et al., 2013, Golak et al., 2019).

For DIS, the deuteron longitudinal structure function $F_L^D(x, Q^2)$ is projected out by contracting the hadronic tensor with the virtual boson's polarization, and is related directly to the "00" response.

2. Dynamical Content and Operator Expansion

Chiral EFT and Nuclear Currents

Within the chiral EFT framework, the deuteron response is calculated by expanding both the nucleon-nucleon (NN) potential and the nuclear charge operator in powers of $P/\Lambda_\chi$, where $P \sim p, m_\pi$, and $\Lambda_\chi \sim 1$ GeV denotes the chiral breakdown scale. At leading order (LO), only one-body operators contribute. Up to next-to-next-to-leading order (NNLO), the relevant operator structure is purely one-body with relativistic and nucleon structure corrections at $\mathcal{O}(eP^2)$; genuine two-body charge operators do not enter before N$^3$LO (Yang et al., 2013, Andis et al., 14 Dec 2025).

Explicitly, the charge operator in momentum space is structured as: $$\langle \mathbf{p}' | J^0(\mathbf{q}) | \mathbf{p} \rangle = \left[ ... \right] G_E^{(s)}(Q^2) + \left[ ... \right] G_E^{(v)}(Q^2)$$ where $G_E^{(s,v)}$ are isoscalar/isovector nucleon electric form factors, and the bracketed terms encode the isospin structure.

At NNLO in RG-improved chiral EFT, additional corrections appear: relativistic Foldy-Wouthuysen terms, nucleon finite-size corrections, and effective boost operators. No two-body current appears at this order (Andis et al., 14 Dec 2025).

3. Computational Approaches

Momentum-Space Solutions and Regularization

One computational strategy is to solve the Lippmann–Schwinger equation for the chiral EFT potential up to a given order, implementing a regulator—typically

$$\exp\left[-(p/\Lambda)^{2n}\right], \quad \Lambda \in [0.6, 1.0]\,\mathrm{GeV}$$

Cutoff dependence quantifies the residual theoretical uncertainty arising from higher-order corrections (Yang et al., 2013).

Perturbative and RG-Invariant Schemes

Perturbatively renormalized EFT schemes treat subleading potentials and operators as perturbations atop a nonperturbative LO ground state. Lorentz Integral Transform (LIT)-based frameworks allow for a clean perturbative expansion of the response, facilitating error control and RG invariance at each order. Observable convergence is assessed via cutoff variation and order-by-order stability (Andis et al., 14 Dec 2025).

Chebyshev-Expansion Reconstructions

The Chebyshev expansion bound-state method—recently developed for optimizing the reconstruction of response functions—employs stochastic regularization of the density of states to define adaptive, equal-area bins. This enables high-precision, histogram-based extraction of $R_L(q,\omega)$ from harmonic-oscillator representations of the Hamiltonian coupled with chiral interactions (Reis et al., 1 Jul 2025).

Method	Principle	Scale/Approximations
Chiral EFT	Systematic $P/\Lambda_\chi$ expansion	Cutoff $\Lambda$ varied, NNLO/impulse approx.
LIT	Integral transform + inversion	Perturbative expansion in potential and operator
Chebyshev Histogram	Polynomial expansion, stochastic binning	Harmonic-oscillator basis, chiral N$^3$LO

4. Phenomenology and Comparison to Data

Near Quasi-Free Ridge

In the region $|\mathbf{q}^2 - \mathbf{q}_{\rm qf}^2| \leq 4$ fm$^{-2}$, $E_{np} \leq 60$ MeV, final-state interactions (FSI) are suppressed and the impulse approximation is accurate. Here, both chiral EFT and high-precision phenomenological potentials (e.g., Bonn, AV18) yield predictions that agree within 10% theoretical uncertainty with each other and with experimental data (Yang et al., 2013).

Example (chiral EFT at NNLO vs. Bonn potential):

$\mathbf{q}^2\,(\mathrm{fm}^{-2})$	$f_L^{\chi\rm EFT}$	$f_L^{\rm Bonn}$	Relative Deviation
0.5	5.02 ± 0.10	4.98	2%
1.0	4.11 ± 0.05	4.08	1%
4.0	1.23 ± 0.12	1.30	10%
10.0	0.21 ± 0.02	0.25	15%

Agreement with experimental measurements is observed within combined statistical and theoretical uncertainties, notably on or near the quasi-free ridge. For larger momentum transfers ($\mathbf{q}^2 \gtrsim 3\,\mathrm{fm}^{-2}$), both chiral EFT and potential-model results underpredict the data by $1$–$3\sigma$, suggesting missing contributions beyond impulse approximation (Yang et al., 2013).

Convergence and Accuracy

Renormalization group invariance is demonstrated by the stability of the response under cutoff variation, and order-by-order convergence is observed in both fully nonperturbative and perturbative chiral approaches (Andis et al., 14 Dec 2025). Chebyshev-based reconstruction methods confirm convergence to within a few percent with respect to exact solutions and experimental data for both low and quasi-elastic regions (Reis et al., 1 Jul 2025).

5. Nuclear Modifications and Structure Function Ratios

The assumption of negligible nuclear corrections in the longitudinal-transverse structure function ratio, $R_N = F_L^N/(2xF_1^N)$, in extractions from deuteron data, has been challenged. Nuclear modifications due to the transverse motion of nucleons introduce mixing between longitudinal and transverse nucleon structure functions, with admixtures proportional to $\langle p_T^2 \rangle/Q^2$ (Kumano, 23 Jun 2025). The modified ratio in the deuteron,

$$R_D(x,Q^2) \approx R_N(x,Q^2) + \frac{\langle p_T^2 \rangle}{Q^2} \times (\text{admixture term})$$

exceeds the nucleon ratio $R_N(x,Q^2)$ by a few percent ($\sim3\%$ at $x \sim 0.6$, $Q^2=5\,\mathrm{GeV}^2$), growing as $Q^2$ decreases and toward larger $x$. This admixture vanishes in the Bjorken limit as $p_T^2/Q^2 \to 0$ but becomes non-negligible at moderate kinematics (Kumano, 23 Jun 2025). The underlying mechanism is the Fermi motion of bound nucleons, which projects transverse structure into the "longitudinal" channel.

Implications for precise nucleon structure extraction from deuteron measurements include the necessity to account for these nuclear modifications, particularly for future high-precision DIS and Rosenbluth-separation experiments at electron-ion colliders.

6. Numerical Methods and Uncertainty Quantification

Reconstruction and inversion of response functions from theoretical calculations or experimental data introduce technical uncertainties. In Chebyshev-expansion approaches, resolution, kernel width, number of moments, and binning strategy collectively determine reconstruction accuracy. Adaptive histogram binning—defined by optimizing the density of states via stochastic sampling—serves to balance discretization effects and statistical fluctuations, achieving sub-percent uncertainties with appropriate parameter choices (Reis et al., 1 Jul 2025).

Sources of overall theoretical uncertainty include:

• Truncation of the EFT expansion (estimated via cutoff variation and order-by-order differences; typical scale is $\leq10\%$ in the quasi-free domain (Yang et al., 2013, Andis et al., 14 Dec 2025)).
• Limitations of the one-body (impulse) approximation; two-body charge operators contribute starting at N$^4$LO ($\mathcal{O}(eP^4)$), with an estimated size at the few-percent level in $f_L$ (Yang et al., 2013).
• Basis truncation and model-space errors in numerical implementations.
• Inversion and reconstruction errors in integral-transform and histogram methods.

7. Open Problems and Future Directions

Several challenges and research avenues remain in the analysis of the deuteron longitudinal response:

• The persistent deficit of both chiral EFT and phenomenological potential predictions with respect to data at high momentum transfer ($\mathbf{q} \gtrsim 400$ MeV) points toward missing two-body charge operators and higher-order relativistic corrections (Yang et al., 2013).
• Quantitative uncertainty estimation requires explicit inclusion of two-body currents at N$^4$LO, and a systematic refitting of low-energy constants (LECs) in the NN potential to account for these effects (Yang et al., 2013).
• Extension of these techniques to transverse and interference response functions ($f_T,f_{LT},f_{TT}$) remains necessary for comprehensive tests of current operators and the underlying chiral dynamics.
• Ab initio generalization to heavier nuclei, leveraging perturbative RG-invariant EFT and Chebyshev-expansion techniques, promises systematic access to lepton-nucleus response functions in the medium-mass regime (Andis et al., 14 Dec 2025, Reis et al., 1 Jul 2025).
• Experimental advances, especially at future electron-ion colliders, are anticipated to test nuclear modifications in the longitudinal-transverse structure function ratios at small $x$, illuminating gluon dynamics in light nuclei (Kumano, 23 Jun 2025).

A plausible implication is that high-precision measurements of the deuteron longitudinal response, combined with systematic theoretical treatments of two-body currents and nuclear modifications, will critically inform both nuclear structure theory and the extraction of fundamental nucleon observables from nuclear targets.