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Toward extracting scattering phase shift from integrated correlation functions IV: Coulomb corrections

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Pith's one-line read This paper extends integrated-correlation-function phase-shift extraction to charged particles: subtracting the pure-Coulomb correlation function instead of the free one yields a divergence-free route to the short-range phase shift.

desk verdict Useful Coulomb extension of the integrated-correlation-function method, with a real gap in the derivation of the central formula; send to review and request the missing argument. read the letter →

arxiv 2506.01768 v2 pith:Z2KNAYZL submitted 2025-06-02 hep-lat cond-mat.othercond-mat.quant-gashep-phnucl-th

Peng Guo , Frank X. Lee This is my paper
classification hep-latcond-mat.othercond-mat.quant-gashep-phnucl-th
keywords scatteringphaseshiftintegratedcorrelationfunctionsCoulombcorrectionsFriedelformulaKreintracetrappedparticleslatticeQCD
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper extends a trap-based method for extracting scattering phase shifts to particles that also feel the long-range Coulomb force. With a Coulomb potential present, the original subtraction — interacting minus non-interacting correlation function — diverges, because the Coulomb force distorts the asymptotic wavefunction. The fix is to subtract, instead, the correlation function of the pure-Coulomb system: the two systems share the same divergent asymptotic pieces, so their difference stays finite, approaches a well-defined infinite-trap limit, and equals a weighted integral of the short-range part of the phase shift. The paper verifies this formula to numerical precision with an exactly solvable contact-interaction model in a spherical hard-wall trap. Because the relation converges fastest at short Euclidean times, it is aimed at lattice QCD determinations of charged-hadron scattering, where the Coulomb baseline would otherwise spoil the extraction.

What carries the argument

Two pieces carry the argument. On the wavefunction side, the normalization identity of Eq. (8), applied to the asymptotic Coulomb-distorted wavefunctions, shows that the short-range-plus-Coulomb and pure-Coulomb systems share the identical divergent structure, $\Lambda - \frac{1}{2}\frac{d}{dk}\big[\gamma\ln(2k\Lambda)\big] + \frac{\sin(2k\Lambda-\pi l-2\gamma\ln(2k\Lambda))}{4k}$, so the difference of their normalization integrals reduces to exactly $\frac{1}{2}\,d\delta_l^{(S)}/dk$. On the operator side, the central identity is the Coulomb-modified Friedel–Krein trace formula, Eq. (B18): $\mathrm{Tr}\big[G_l^{(\infty)}(\epsilon)-G_l^{(C,\infty)}(\epsilon)\big] = -\frac{1}{\pi}\int_0^\infty d\lambda\,\frac{\delta_l^{(S)}(\lambda)}{(\lambda-\epsilon)^2}$, which converts the diverging Green's-function trace into a dispersion integral over the short-range phase shift. These two mechanisms are what turn a cutoff-dependent object into the finite weighted integral of Eq. (27).

What would settle it

Evaluate the trace difference $\mathrm{Tr}\big[G_l^{(\infty)}(\epsilon)-G_l^{(C,\infty)}(\epsilon)\big]$ by brute-force numerical integration of the full Green's functions for a finite-range short-range potential (for example a square well or a Yukawa tail) on a Coulomb background, and compare it with $-\frac{1}{\pi}\int_0^\infty d\lambda\,\delta_l^{(S)}(\lambda)/(\lambda-\epsilon)^2$; any systematic mismatch beyond numerical precision would falsify the central identity. As a complementary check, rerun the paper's convergence test for a partial wave with $l\ge 1$ or for a finite-range potential in the hard-wall trap, since the exact verification in the paper uses only an S-wave contact interaction, where the oscillatory boundary terms of Eqs. (23)–(26) are at their simplest.

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Extended reading notes

Core claim

The central result is Eq. (27) and its Euclidean counterpart Eq. (28): for two nonrelativistic particles in a trap interacting through a short-range potential plus a Coulomb potential, the difference between the integrated correlation function $C_l(t)$ and the pure-Coulomb correlation function $C_l^{(C)}(t)$ converges, as the trap size grows, to $\frac{(2l+1)it}{\pi}\int_0^\infty d\epsilon\,\delta_l^{(S)}(\epsilon)e^{-i\epsilon t}$ in Minkowski time and to $\frac{(2l+1)\tau}{\pi}\int_0^\infty d\epsilon\,\delta_l^{(S)}(\epsilon)e^{-\epsilon\tau}$ in Euclidean time, where $\delta_l^{(S)}(\epsilon)$ is the infinite-volume phase shift of the short-range potential alone. The substance of the claim is that the Coulomb-distorted asymptotic wavefunctions with and without the short-range potential contain exactly the same divergent terms — the logarithmic phase $\gamma\ln(2kr)$, the cutoff-linear piece, and the oscillatory sine term — so their difference is finite and cutoff-independent, whereas the original subtraction against the free-particle wavefunction is cutoff-dependent and diverges. Once $\delta_l^{(S)}$ is inverted out of the weighted integral, the total phase shift is obtained simply by adding the analytic Coulomb phase shift $\delta_l^{(C)}(\epsilon)=\arg\Gamma(l+1+i\gamma)$. The paper demonstrates the relation numerically in an exactly solvable model — a contact interaction in a spherical hard-wall trap — and finds rapid approach to the infinite-volume limit as the trap radius grows.

Load-bearing premise

The result stands on the assumption that the short-range-plus-Coulomb system and the pure-Coulomb system differ, in their asymptotic behavior, by exactly the short-range phase shift — so that every divergent and cutoff-dependent piece cancels in the correlation-function difference and the asserted Coulomb-modified trace formula picks up no extra corrections from the long-range logarithmic phase.

Editorial extensions

If this is right

  • Short-range phase shifts for charged pairs become extractable from the difference of two trapped correlation functions, with no need to evaluate the diverging Coulomb baseline.
  • The relation converges rapidly with trap size and fastest at short Euclidean times, so modestly sized traps suffice for a reliable infinite-volume limit.
  • Once $\delta_l^{(S)}(\epsilon)$ is inverted out of Eq. (28), the total phase shift follows by adding the closed-form Coulomb phase $\arg\Gamma(l+1+i\gamma)$, so the Coulomb content of the answer adds no further work.
  • The formula applies to any central short-range potential and any rotationally symmetric trap (hard wall, harmonic oscillator); periodic boxes would require projection onto irreducible representations of the cubic group, as in the earlier formalism.
  • The authors see no conceptual obstacle to lifting the same subtraction to the relativistic version of the integrated-correlation-function formalism, whose structure is the same apart from the kinematic factor.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same subtraction tactic should transfer to lattice QCD+QED simulations in periodic boxes: compute the QED-only baseline in the same box as the full QCD+QED correlation function, and the remaining finite difference isolates the strong-interaction phase shift.
  • The paper's exact check uses only an S-wave contact interaction, where the oscillatory boundary terms in Eqs. (23)–(26) are simplest; testing $l\ge 1$ partial waves or a finite-range potential would reveal whether the pointwise cancellation of those terms, rather than some weaker integrated cancellation, is what makes Eq. (27) hold.
  • Because the Euclidean right-hand side of Eq. (28) weights the phase shift by $e^{-\epsilon\tau}$, the method is intrinsically most sensitive to low-energy quantities such as scattering lengths and effective ranges, and the short-time limit formally requires knowledge of the phase shift at all energies.
  • The construction suggests a general recipe: any long-range force whose asymptotic distortion of the wavefunction is known, such as dipole or magnetic $1/r^3$ interactions, could be bundled into the baseline correlation function, with the short-range phase shift recovered from a finite difference.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper extends the integrated correlation function formalism of Refs. [1–3] to systems with Coulomb plus short-range interactions. The central result, Eq. (27), states that in the infinite trap limit the difference between the trapped integrated correlation function for short-range plus Coulomb and for pure Coulomb is proportional to a weighted integral of the short-range scattering phase shift δ_l^(S). The authors motivate the formula via asymptotic wavefunction analysis in Sec. III and via a Friedel–Krein trace relation in Appendix B, and test it on an exactly solvable contact interaction in a spherical hard-wall trap in Sec. IV, finding rapid convergence with trap radius.

Significance. If the result is established rigorously, it provides a practical route to extract short-range phase shifts in the presence of Coulomb interactions from integrated correlation functions, avoiding the divergent Coulomb baseline that prevents direct use of the earlier formalism. The paper is commendable for having no fitted parameters: the trap radius, Coulomb strength, reduced mass, and contact coupling are all inputs, and the numerical validation is clean. The exactly solvable model and the explicit closed-form right-hand side in Eq. (31) are valuable. However, two technical gaps in the derivation—the unproved pointwise cancellation of oscillatory boundary terms in Sec. III and the unproved Coulomb extension of the trace formula in Eq. (B18)—mean the central claim is not yet fully established for general short-range potentials.

major comments (2)
  1. [Sec. III, Eqs. (23)–(26)] The pointwise cancellation used to obtain Eq. (26) is not correct as written. With the asymptotic form of Eq. (20), the radial function u_l^(∞)(r;k) = sin(kr − πl/2 + δ_l(k) − γ ln(2kr)), where δ_l = δ_l^(S) + δ_l^(C). Applying the identity in Eq. (8) gives an oscillatory boundary term whose phase contains the total phase δ_l, i.e., sin(2kΛ − πl + 2δ_l − 2γ ln(2kΛ))/(4k). Equation (23) as printed omits δ_l, and Eq. (25) omits δ_l^(C). The difference of the two oscillatory terms is therefore of order (δ_l^(S)/k) times cos(2kΛ − πl + 2δ_l^(C) + δ_l^(S) − 2γ ln(2kΛ)), which does not vanish pointwise as Λ → ∞. Consequently Eq. (26) does not follow as a pointwise limit. To rescue the derivation, you need to show that the oscillatory remainder vanishes in a distributional sense after integration against e^{−ik²t/(2μ)} (or e^{−k²τ/(2μ)}), for example via the Riemann–Lebesgue lemma or a stationary-phase estimate. Please supply that argument or replace the wavefunction derivation with one that avoids the pointwise step.
  2. [Appendix B, Eq. (B18)] The Coulomb-modified Friedel–Krein trace formula is asserted without derivation or citation. It is not evident that for an arbitrary short-range perturbation V^S on top of the Coulomb Hamiltonian, the trace of the difference of the Green's functions equals the second energy derivative of δ_l^(S)/π, with no additional term coming from the logarithmic Coulomb phase. The numerical test in Sec. IV uses a contact interaction and therefore cannot validate Eq. (B18) for general short-range potentials. Please either prove Eq. (B18) from scattering theory (e.g., via the spectral shift function and stationary scattering wavefunctions) or provide a precise reference where this result is established.
minor comments (5)
  1. [Sec. I] There are two typos in the introduction: "Precisie" should be "Precise" and "inifinite" should be "infinite".
  2. [Sec. IV, Eq. (38)] The phrase "to rival the phase shift" is unclear; consider replacing it with "to parallel the phase shift" or "to serve as the trap analog of the phase shift".
  3. [Eq. (31)] The typesetting of Eq. (31) is ambiguous; please insert explicit parentheses around the argument of the complementary error function and around the exponent of the final exponential factor.
  4. [Fig. 3 caption] The caption inconsistently uses "dashed red" both before and after the description of the R=π curve ("dashed purple"); please correct the color labels so that each curve is assigned a unique linestyle/color.
  5. [Sec. II and Sec. III] Please standardize the superscript notation for the noninteracting and pure-Coulomb correlation functions (C_l^(0) vs. C_l^(C)); the current text uses both C_l^(0) and C_l^{(0)} without a clear distinction.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the Coulomb-modified relation is derived analytically from asymptotic wavefunctions; heavy self-citation and the internal numerical check are not load-bearing circular steps.

full rationale

The central relation Eq.(27) is obtained by comparing the cutoff-dependent integrated wavefunction norm for the combined Coulomb-plus-short-range Hamiltonian, Eq.(23), with the same quantity for pure Coulomb, Eq.(25), and taking the difference. The difference eliminates the common divergent leading terms and leaves (1/2)dδ^(S)/dk in Eq.(26); this is an analytic derivation with V_R, Z, R, and μ as model inputs and no fitted parameter. The numerical test in Sec. IV is an internal consistency check rather than an independent prediction: the trapped energies are generated from the quantization condition Eq.(35), which already contains the same δ^(S) that appears on the right-hand side of Eq.(36). The agreement in Fig. 3 thus verifies the trace-formula identity for this solvable model, but it is not a circular fit because neither side is obtained by fitting δ^(S) to the other. The paper cites the authors' earlier formalism [1–3,35,54,65] heavily, but the new Coulomb-specific step does not reduce to those citations: the asymptotic forms (20) and (24) and the analytic Coulomb phase (22) are standard, and the non-Coulomb Friedel/Krein relation (17) is independently attributed to Friedel and Krein [66–68]. Appendix B asserts the Coulomb-modified trace formula Eq.(B18) without derivation, and the wavefunction route to Eq.(26) suppresses the question of whether leftover oscillatory boundary terms vanish pointwise or only distributionally after energy integration; these are mathematical-rigor and correctness concerns, not self-referential reductions. No claim in the paper is equivalent by construction to its own input.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The central identity is parameter-free; all listed parameters are inputs to the numerical demonstration. The main axioms are the asserted Coulomb-modified Krein trace formula and the applicability of the scattering-state identity to Coulomb wavefunctions.

free parameters (4)
  • V_R (renormalized contact coupling) = 0.5 (chosen in numerical check)
    Input of the exactly solvable model; not fitted to data. The central identity is independent of this value.
  • Z (Coulomb strength) = 0.1 (chosen for Figs. 1-3)
    Coulomb coupling in the numerical demonstration; physical input, not a fit.
  • R (hard-wall trap radius) = pi and 4pi (two values used)
    Trap size in the convergence study; not fitted.
  • mu (reduced mass) = 1 (set in figures)
    Reduced mass in the numerical demonstration; physical input, not a fit.
assumptions (4)
  • domain assumption For a short-range perturbation of a Coulomb Hamiltonian, the resolvent trace formula holds with spectral shift function delta^S/pi (Eq. B18).
    This is the key unproved mathematical input that makes Eq. (27) rigorous.
  • standard math Poliatzky's identity Eq. (8) applies to Coulomb-distorted scattering states, yielding the asymptotic expansions in Eqs. (23) and (25).
    The identity is cited from Ref. [64]; its extension to wavefunctions with logarithmic phase is assumed.
  • domain assumption Contact-interaction renormalization: the renormalized coupling V_R is the same in infinite volume and in the hard-wall trap, so the quantization condition Eq. (35) applies.
    Standard assumption for contact EFT in finite volume; the paper uses it without discussion.
  • domain assumption The trap preserves rotational symmetry so partial waves decouple.
    Explicitly stated in Sec. II; limits the present derivation to rotationally symmetric traps.

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Pith. "Pith review of Toward extracting scattering phase shift from integrated correlation functions IV: Coulomb corrections." pith.science (2026). https://pith.science/paper/Z2KNAYZL

@misc{pith2026250601768,
  author       = {Pith},
  title        = {Pith review of: Toward extracting scattering phase shift from integrated correlation functions IV: Coulomb corrections},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Z2KNAYZL}},
  note         = {Machine review of arXiv:2506.01768}
}
read the original abstract

The formalism developed in Refs.~\cite{Guo:2023ecc,Guo:2024zal,Guo:2024pvt} that relates the integrated correlation functions for a trapped system to the infinite volume scattering phase shifts through a weighted integral is further extended to include Coulomb interaction between charged particles. The original formalism cannot be applied due to different divergent asymptotic behavior resulting from the long-range nature of the Coulomb force. We show that a modified formula in which the difference of integrated correlation functions between particles interacting with Coulomb plus short-range interaction and with Coulomb interaction alone is free of divergence, and has rapid approach to its infinite volume limit. Using an exactly solvable model, we demonstrate that the short-range potential scattering phase shifts can be reliably extracted from the formula in the presence of Coulomb interaction.

Figures

Figures reproduced from arXiv: 2506.01768 by the authors.

Figure 1
Figure 1. FIG. 1: Coulomb effects in the generalized zeta function [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Quantization condition plot of Eq.( [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Convergence of Eq.( [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

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