Aharonov-Bohm effect and nucleon-nucleon phase shifts on the lattice

Paulo F. Bedaque

REVIEW

Aharonov-Bohm effect and nucleon-nucleon phase shifts on the lattice

Paulo F. Bedaque

Nuclear Theory · Submitted 14 Feb 2004 · 18 references

Pith's verdict: A clean, cheap idea — twist the boundary conditions to scan NN phase shifts — but the L = 5 fm headline rests on an illustrative estimate whose own error term is ~25% at the advertised momentum.

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arXiv IDnucl-th/0402051 (version 1)

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CitationPaulo F. Bedaque. "Aharonov-Bohm effect and nucleon-nucleon phase shifts on the lattice." arXiv:nucl-th/0402051 (2004). https://pith.science/paper/QFYJIBT2

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Editor's note

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Bedaque's idea is simple and, as far as I can tell, correct: put the two-nucleon system in a box with a U(1) vector potential of zero field strength — equivalently, twist the quark boundary conditions in z — and use the twist as a knob to slide the ground-state energy to whatever small momentum you want. That turns Lüscher's one-kinematic-point-per-lattice-size into a continuous scan, and it lets you work at L ≈ 5 fm instead of 8–10 fm. The finite-volume formula (Eq. 14) is a clean generalization of Lüscher's, the positivity argument for the two-flavor Wilson determinant is right, and the paper is upfront about its window (k < mπ/2, e^{−mπL} ≈ 0.03 at 5 fm). It also ships a C code for S(η,φ), which is more than most proposal papers bother with. Credit for that. What it is not is a demonstration. Fig. 4, which carries the L = 5 fm claim in the abstract, is experimental effective-range parameters fed into the leading-order formula. That is a plausibility argument, not evidence that lattice QCD at 5 fm will produce a usable signal. Two things I would press on. First, the paper's own error term in Eq. (12) is O((Q/mπ)³(φ−π)²). At the φ you need for k = mπ/2 in a 5 fm box — φ ≈ 0.56π — that factor is roughly 0.24, not small. The symmetry argument that kills p-waves only works at φ = 0 and φ = π, so the headline momentum reach sits exactly where the suppression is weakest. The paper states the error term but never evaluates it at the kinematic point it advertises. Second, the ERE truncation at r0 is shaky at k = mπ/2 in the singlet (r0k²/2 ≈ 0.17 fm⁻¹ against 1/|a| ≈ 0.04), though this degrades the illustrative figure rather than the inversion of Eq. 14 itself. I would push back on one reader complaint: the tensor-force cancellation is not merely asserted — the paper gives the q_i q_j − q²δ_ij/3 argument explicitly. The p-wave suppression is the softer of the two. Worth a referee. The method is real and cheap to test; the abstract should say 'may be sufficient' and the paper should show the error term evaluated at φ ≈ 0.56π.

Referee report

Major objections (4)

[§B, Eq. (12) and Fig. 4] The p-wave contamination is stated as O((Q/mπ)³(φ−π)²), but this estimate is only controlled near φ = π. To reach k ~ mπ/2 with L = 5 fm one needs φ ≈ kL ≈ 0.56π, where (φ−π)² ≈ 1.9 and (Q/mπ)³ ≈ 0.125, giving a ~20-25% effect. Fig. 4 does not include this contamination, and the symmetry arguments that forbid p-waves apply only at φ = 0 and φ = π. Since the abstract's claim (abstract:s3) is precisely about L = 5 fm and k ~ mπ/2, please either quantify the p-wave contribution at the relevant φ or restrict the quantitative claim to the region where it is suppressed.
[§B, Eq. (20)] The assertion that the linear tensor-force contribution vanishes after spin averaging is load-bearing for the triplet analysis in Fig. 4(b), but the trace is not displayed. The text says 'it is easy to see by an explicit calculation'; please show the spin sum explicitly, and state whether the cancellation holds at the φ values used, where the torus and the background field break rotational symmetry.
[§B, Eqs. (16)-(19)] The expansion of the twisted Lüscher formula is internally inconsistent as printed. Expanding Eq. (16) near the lowest pole gives E = φ²/(ML²) − (4π/(MkL³)) tan δ [1 + c0 tan δ/(πkL) + ...], whereas Eq. (18) has 4πk/(ML³) and c0kL tanδ; the two differ by factors of k². Moreover, at φ = 0 Eq. (19) reduces to E = (4πa/(ML³))[1 − c0a/L], which with c0 = −8.914 from Eq. (17) gives [1 + 8.914a/L], whereas the standard Lüscher result is [1 − 2.8373a/L] = [1 + c0a/(πL)]. Please check the algebra; the numerical results in Fig. 4 use the full S function and are unaffected, but Eqs. (18)-(19) are the formulas a reader would use.
[§B, Fig. 4] The estimates insert the effective-range expansion truncated at r0 into Eq. (14). At k = mπ/2 ≈ 0.35 fm⁻¹ the singlet ERE is dominated by the range term (r0s k²/2 ≈ 0.17 fm⁻¹ versus 1/|a_s| ≈ 0.04 fm⁻¹), so the truncation is marginal and the shape parameter could shift the levels. Please state the sensitivity of Fig. 4 to the truncation, or restrict the quantitative claim to lower k.

Minor objections (8)

[§A, Eq. (1)] The factor 3 in A = φ/(3L) ẑ is the quark baryon number; please state this explicitly, since it is what makes the nucleon pick up phase φ in Eq. (7).
[§B, Eq. (17)] With n defined as (nx, ny, nz + φ/2π) in Eq. (15), the summand 1/(n² + 2nzφ) is not consistent; the denominator should be n² − φ²/4π², or, with unshifted integer n, nx² + ny² + nz² + nzφ/π. Please correct.
[§A] The statement 'the determinant of M is positive' is stronger than what is shown: the argument establishes that det M is real, and positivity requires the product of the up and down determinants. Please rephrase.
[Fig. 4 caption] Please state whether the plotted energy includes the free φ²/(ML²) term and whether it is measured relative to the two-nucleon threshold; as drawn, the curves decrease monotonically with φ, whereas the free term increases.
[References] Ref. [7] is listed as arXiv:hep-lat/0312004, a preprint at the time of writing; the 8-10 fm benchmark should be accompanied by a statement of its assumptions, or by a published reference if available.
[§B] The sentence 'the phase shifts at arbitrary kinematical points are probed' should clarify that this requires tuning φ and that the reach is limited by the EFT validity, as stated later in the same paragraph.
[§B] The sentence 'Similarly, it might be of use in ameliorating some of the issues involved in the k → ππ lattice extraction' is too vague to be useful; please specify or remove it.
[Throughout] Typographical errors to correct: 'only only' (introduction), 'this contributions' and 'neccesarily' and 'By the other hand' (§B), 'decays constants' (introduction), and 'One inconvenient' (§B).

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The paper

The paper claims

A zero-field vector potential lets a small lattice box yield nucleon-nucleon phase shifts at many momenta.

The reading

Lattice QCD is the only known way to derive the nucleon-nucleon force from the underlying theory, but the standard finite-volume route ties each box size to essentially one scattering momentum, and the large nucleon scattering lengths force boxes of 8 to 10 fm before the ground state is low enough to be usable. This paper proposes simulating QCD in the presence of a U(1) vector potential whose field strength vanishes everywhere on the lattice, the Aharonov-Bohm configuration of a thin solenoid threaded through the torus, which is equivalent to giving the quarks twisted boundary conditions in one spatial direction. Because the phase of that twist shifts the allowed momenta continuously, the two-nucleon ground-state energy can be tuned at will, and each choice of twist supplies an independent determination of the phase shift. The author argues that with the twist set near $q = 2e$ (i.e. $q = 2e$ replaced by the flux $q = 2e$) the relevant energy levels stay small enough in boxes as small as 5 fm, and that phase shifts are then obtainable at any momentum below $m_1/2$.

What carries the argument

The carrying object is the finite-volume function $S(eta, phi) = lim sum_{|n|<N} 1/(n^2 - eta) - 4 pi N$, summed over the twisted momentum lattice, which replaces the ordinary Luescher zeta function. The twist enters only through the shifted third component of the integer vector, so the whole effect of the Aharonov-Bohm potential is captured by the allowed momenta $q_{x,y} = 2 pi n_{x,y}/L$, $q_z = 2 pi (n_z + phi/2 pi)/L$. Matching the poles of the finite-volume correlator to the effective-field-theory amplitude, which resums contact interactions with an increasing number of derivatives, gives the quantisation condition that converts a measured energy level into $k cot d$. The same change of variables that removes the potential from the quark action turns it into twisted boundary conditions on the nucleon fields, which is what makes the argument transparent in the effective theory.

Extended reading notes

Core claim

The claim is that a background gauge potential with zero field strength, combined with the standard relation between finite-volume energy shifts and scattering phase shifts, converts a lattice simulation into a tunable probe of nucleon-nucleon scattering. Concretely, the author derives the modified quantisation condition $k L cot d = (1/pi) S(k^2 L^2 / 4 pi^2, phi)$, where $S$ is a sum over momenta shifted by the twist, $n = (n_x, n_y, n_z + phi/2 pi)$, and shows that the pole structure of $S$ moves continuously with the flux. Tuning the twist near $phi = pi$ pushes the free two-nucleon ground state down to energies of a few MeV even for $L = 5$ fm, which the author illustrates for both the spin-singlet channel, using $a_s = 23.7$ fm and $r_{0s} = 2.73$ fm, and the spin-triplet channel, using $a_t = 5.425$ fm and $r_{0t} = 1.75$ fm. The payoff claimed is twofold: lattices roughly half the linear size previously thought necessary, and phase shifts at a continuum of momenta rather than at a single kinematic point per box.

Load-bearing premise

The feasibility estimate assumes that the leading contact interaction alone, evaluated with experimental scattering lengths and effective ranges, correctly predicts the finite-volume energy levels, and that higher partial waves and tensor forces stay suppressed at the twist values used.

Editorial extensions

If this is right

Lattice collaborations could extract nucleon-nucleon phase shifts across a range of momenta from a single small box by scanning the twist parameter, instead of enlarging the box to reach new kinematic points.
Boxes of about 5 fm become viable for the two-nucleon sector, which is close to the floor set by exponentially small finite-pion-mass corrections of order $e^{-m_pi L}$, about 0.03 at that size.
The same trick applies to any hadronic interaction where a conserved charge can be coupled to a zero-strength potential, though positivity of the fermion determinant restricts it to flavour-neutral or baryon-number couplings unless the calculation is quenched.
Because the method determines phase shifts rather than a potential, the output can be matched directly onto the low-energy constants of the contact effective field theory.

Reading between the lines

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

If the higher-partial-wave contamination really scales as $(Q/m_pi)^3$ and the tensor-force contribution as $(Q/m_pi)^4$, the method could be pushed somewhat above $m_pi/2$; the paper does not test where those suppressions actually break down numerically.
A practical test would be to run the same twisted-boundary simulation at two box sizes and check that the extracted $k cot d$ agrees, since a systematic error from neglected partial waves would show up as a residual volume dependence.
Extending the twist to more than one spatial direction would give a multi-parameter tuning knob, potentially allowing several energy levels to be brought into the useful window simultaneously rather than one at a time.

Original abstract

We propose a method for the lattice QCD computation of nucleon-nucleon low-energy interactions. It consists in simulating QCD in the background of a ''electromagnetic" field whose potential is non-vanishing, but whose field strength is zero. By tuning the background field, phase-shifts at any (but small) momenta can be determined by measuring the shift of the ground state energy. Lattice sizes as small as 5 Fermi can be sufficient for the calculation of phase shifts up to momenta of order of $m_{\pi}/2$.

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What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.73 · human participants 0.01 · controlled comparison 0.08 · randomized assignment 0.01 · states quantitative result 0.72 · compares with prior results 0.06 · mentions code or data availability 0.02.

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We propose a method for the lattice QCD computation of nucleon-nucleon low-energy interactions. It consists in simulating QCD in the background of a ''electromagnetic" field whose potential is non-vanishing, but whose field strength is zero. By tuning the background field, phase-shifts at any (but small) momenta can be determined by measuring the shift of the ground state energy. Lattice sizes as small as 5 Fermi can be sufficient for the calculation of phase shifts up to momenta of order of $m_{\pi}/2$.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We propose a method for the lattice QCD computation of nucleon-nucleon low-energy interactions.	contribution 0.99	0.56	0.24	asserted 0.77
s1	It consists in simulating QCD in the background of a ''electromagnetic" field whose potential is non-vanishing, but whose field strength is zero.	method 0.99	0.34	0.07	asserted 1.00
s2	By tuning the background field, phase-shifts at any (but small) momenta can be determined by measuring the shift of the ground state energy.	method 0.97	0.41	0.40	asserted 0.83
s3	Lattice sizes as small as 5 Fermi can be sufficient for the calculation of phase shifts up to momenta of order of $m_{\pi}/2$.	result 0.89	0.62	0.21	hedged possibility 0.89

What kind of paper the classifier reports this is: method or tool 1.00 · evidence from simulation or computation 0.47.

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References

Works cited (18)

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93 later papers cite it

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5 later papers by other authors are listed below: 5 used its method or tool.

Uses its method or tool 5 of 50 papers with this label

Used the so-called twisted boundary condition, which this paper has proven to be advantageous, to achieve its goal.

Scalar mesons in a finite volume (2010) · 0.93 · paraphrased: the licence does not allow quoting
We keep the same value of $p⃗_$ on the other lattices by introducing flavour-twisted boundary conditions [ref] for the strange quark (cf. Ref. [ref] ).

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'Bs --> Ds l nu' near zero recoil in and beyond the Standard Model (2013) · 1.00 · paraphrased: the licence does not allow quoting
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92 arXiv papers cite this in 113 sentences

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Uses its method or tool 50 papers

This spectrum can be used to constrain the infinite-volume scattering amplitudes via the Lüscher method [ref] – for a recent review see Ref. [ref] .

The quark-mass dependence of elastic $\pi K$ scattering from QCD · Introduction · 0.90
We obtain results at non zero momentum transfer with good accuracy by enforcing special boundary conditions on the quark fields, called flavor twisted [ref] , that shift by an arbitrary amount the discretized set of lattice momenta (see also [ref] ).

Quenched lattice calculation of the vector channel B --> D* l nu decay rate · Introduction · 0.91
In order to reach small momentum transfers for the pion form factor and $q^2=0$ for the $K→π$ form factors, we use partially twisted boundary conditions [ref] , combining gauge field configurations generated with sea quarks obeying periodic boundary conditions with valence quarks with twisted boundary conditions [ref] .

Kl3 and pion form factors using partially twisted boundary conditions · The Form Factors · 0.99

Cites it as background 8 papers

The precise relationship is encoded in the Lüscher quantization condition [ref] , which can be written as [This is valid for any two-hadron to two-hadron scattering process – significant recent steps towards a general quantization condition relevant for three-hadron channels have been made [ref] .]

Efficient solution of the multi-channel L\"uscher determinant condition through eigenvalue decomposition · Introduction · 0.83
In traditional lattice calculations [ref] , the on shell transition factor is determined by the extrapolation of the off shell transition factors with nonzero photon virtualities, or by so-called twisted boundary conditions [ref] where an appropriate twisted angle is tuned to put the transition factor on shell directly.

Lattice study of $J/\psi \rightarrow \gamma\eta_c$ using a method without momentum extrapolation · Introduction · 0.85
The progress made in the field is in large part due to the extensive and growing literature [ref] dedicated to deriving the relation between finite- and infinite-volume observables.

Inclusive reactions from finite Minkowski spacetime correlation functions · Introduction · 0.95

Discussion

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