Measurement of the TMD soft function on the lattice using the auxiliary field representation of the Wilson line

Anthony Francis; C.-J. David Lin; Issaku Kanamori; Wayne Morris; Yong Zhao

REVIEW 4 major objections 6 minor 19 references

Measurement of the TMD soft function on the lattice using the auxiliary field representation of the Wilson line

T0 review · 4 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper argues that the TMD soft function can be computed on a Euclidean lattice by rewriting Wilson lines as auxiliary one-dimensional fermion fields with a complex direction, matching the Collins soft function at one loop and using a…

desk verdict A plausible new route to the TMD soft function with a clean one-loop core, but the lattice result is orders of magnitude off and the central ratio is asserted rather than derived—an exploratory proposal, not a measurement. read the letter →

arxiv 2412.12645 v1 pith:ENZD3EOF submitted 2024-12-17 hep-lat

Anthony Francis , Issaku Kanamori , C.-J. David Lin , Wayne Morris , Yong Zhao This is my paper · ORCID

classification hep-lat

keywords transversemomentumdependentsoftfunctionCollins-SoperkernelWilsonlineauxiliaryfieldlatticeQCDrapiditydivergenceperturbationtheory

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 attempts to show that the soft function of transverse-momentum-dependent (TMD) factorization can be computed directly on the lattice rather than extracted from experiment or modeled. Its move is to rewrite the Wilson lines that define the soft function as one-dimensional auxiliary fermion fields and to take the Wilson-line direction in Euclidean space to be complex, with a purely imaginary time component, so that the direction parameter maps to a Minkowski-space rapidity. At one loop the Euclidean calculation reproduces the Collins soft function, and the paper proves that a ratio of finite-length Wilson lines removes the linear length divergence and cancels power corrections up to $b_\perp^2/L^2$. If this holds beyond perturbation theory, it gives a first-principles route to the soft function, the intrinsic soft function, and the Collins-Soper kernel needed for TMD physics.

What carries the argument

The central object is the auxiliary-field representation of the Wilson line, in which $P\exp(-ig\int ds\, n\cdot A)$ is rewritten as a path integral over one-dimensional fermion fields whose propagator $H_{\tilde n}$ satisfies $i\tilde n\cdot D_E H_{\tilde n}(x_E-y_E)=\delta^{(4)}(x_E-y_E)$. The lattice implementation constructs this propagator recursively, following the same recursive construction used for heavy-quark effective theory. The complex direction $\tilde n=(i n_0,\mathbf 0_\perp,n_3)$ with $|n_3/n_0|>1$ carries the argument: it converts the Euclidean computation into the Minkowski space-like soft function and fixes the mapping to rapidity. The second piece is the ratio in Eq. (13), which divides the finite-length soft function by $\sqrt{S_A S_B}$ so that the linear divergence in $L$ and the leading power corrections cancel; the oscillatory cutoff effects of the auxiliary propagator also cancel in the product.

What would settle it

Compute the ratio in Eq. (13) on two ensembles with different lattice spacings but the same physical $b_\perp$ and Wilson-line length $L$; if the matched results disagree, the auxiliary-field propagator does not have the assumed continuum limit. Also compute the one-loop linear divergence with a second UV regulator: if the cancellation in Eq. (13) is regulator-dependent, the ratio does not define the Collins soft function. Inspecting the pole structure of the lattice auxiliary-field propagator and checking that the analytic continuation contour avoids those poles would settle whether the Euclidean measurement is the Minkowski soft function.

Watch

Extended reading notes

Core claim

The central claim is that the lattice soft function defined through auxiliary-field Wilson lines with complex Euclidean directions is the Collins soft function in the space-like region. Concretely, choosing $\tilde n_A=(i n_0^A,\mathbf 0_\perp,n_3^A)$ and $\tilde n_B=(i n_0^B,\mathbf 0_\perp,-n_3^B)$ with $|n_3/n_0|>1$, and identifying $r_a=n_3^A/n_0^A$, $r_b=n_3^B/n_0^B$ with the rapidities of space-like Wilson lines, the one-loop result of Eq. (11) matches the Minkowski-space soft function exactly. Time-like directions are shown to fail: the relevant integral diverges. For finite Wilson-line length $L$, the ratio in Eq. (13) cancels the linear divergence and power corrections of order $b_\perp^2/L^2$ at one loop, and the authors expect the remaining corrections to be of order $b_\perp^4/L^4$. Exploratory numerical results with $r_a=r_b=1.01$ show the expected real, time-independent signal, with statistical errors that shrink when the number of source points per configuration is doubled.

Load-bearing premise

The method assumes that the discretized auxiliary-field propagator for the complex direction $\tilde n=(i n_0,\mathbf 0_\perp,n_3)$ has a well-defined continuum limit and that the Euclidean correlator can be analytically continued to the Minkowski-space Collins soft function; the paper itself notes that the pole structure of this propagator could complicate that continuation.

Editorial extensions

If this is right

A lattice computation of the ratio in Eq. (13) yields the Collins soft function as a function of transverse separation $b_\perp$ and rapidity parameters $r_a,r_b$.
Varying $r_a$ and $r_b$ on a single ensemble produces the soft function at many rapidity separations, and taking the large-rapidity limit extracts the intrinsic soft function $S_I(b_\perp,\mu)$ and the Collins-Soper kernel $K(\mu,b_\perp)$ through Eq. (12).
The ratio construction suppresses both the linear length divergence and the unphysical oscillatory factors of the Euclidean auxiliary propagator, leaving an observable that becomes time-independent for large Wilson-line length.
Because the soft function is time-independent, the calculation does not require a long Euclidean time direction, only Wilson lines long enough that power corrections of order $b_\perp^2/L^2$ are negligible.

Reading between the lines

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

Beyond the one-loop check reported here, the power-correction cancellation in Eq. (13) is only demonstrated to order $b_\perp^2/L^2$; if the ratio turns out not to cancel the linear divergence at higher orders, the lattice result would not match the Collins soft function without an additional subtraction.
A natural testable extension would be to compute Eq. (13) at two different lattice spacings at fixed physical $b_\perp$ and $L$; if the ratio is not independent of the lattice spacing after matching, the assumed continuum limit of the auxiliary-field propagator is not the one the one-loop calculation implies.
The numerical bottleneck is the denominator loops $S_A,S_B$, whose oscillatory cutoff effects produce much larger relative errors than the numerator; variance-reduction techniques beyond increased source count may be needed to make the method practical.
The same complex-direction auxiliary-field construction could in principle be applied to other Wilson-line observables with rapidity divergences, such as quasi-TMD matrix elements, where the need for a Euclidean realization of rapidity is equally acute.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague, here is the bottom line. The genuinely new piece is the use of complex Euclidean directional vectors—purely imaginary time component—to reach space-like Minkowski Wilson lines. The one-loop computation in Sec. 2.1 is coherent: the convergence condition |r_a|,|r_b|>1 follows from positivity of the quadratic form, and the mapping r=coth(y) recovers the Collins soft function. The observation that time-like mHQET directions fail because the integral diverges is real and worth knowing. That part is solid. The lattice strategy via auxiliary fields is a legitimate extension of the mHQET literature, and the authors are upfront that the analytic continuation may be complicated by pole structure.

The soft spots are where the stress-test note lands. Eq. (13), the ratio that is supposed to cancel the linear L divergence and b^2/L^2 power corrections, is asserted with 'we have determined' but no derivation is shown. For a proceedings this is thin; for a claim that the ratio 'enables a lattice computation' it is a load-bearing gap. More worrying, the only numerical result quoted is S_num/sqrt(S_A S_B) ≈ 2873.5/0.47 ≈ 6.1e3, about three orders of magnitude from the O(1) expectation from Eq. (17). The authors attribute this to an uncomputed matching between lattice and continuum schemes. A matching coefficient of O(10^3) is not a small correction; without it, the lattice ratio does not yet represent the soft function. The title promises a 'Measurement' that the paper does not deliver—the abstract and conclusions are more careful, but the mismatch should be fixed. On the citation side, the paper borrows the ratio from Collins and Ji-Liu-Liu and is transparent about it; no hidden fitting. The numerical setup is standard PACS-CS configurations, and the source-statistics scaling is a reasonable sanity check, but it does not address the normalization problem.

Overall: the idea is worth pursuing, the one-loop core is correct as far as I can see, and the authors are honest about what is missing. But this is an exploratory methodological study, not a measurement. It deserves referee time—the method is novel and the matching can in principle be computed—but it needs a derivation or numerical verification of the ratio, a matching calculation, and a title and framing that match the content. I would not carry the number forward, but I would cite the complex-direction mapping if I worked on TMD lattice.

Referee Report

4 major / 6 minor

Summary. The paper proposes a lattice method for the TMD soft function using an auxiliary-field representation of Wilson lines whose Euclidean directional vector has a purely imaginary time component, n~ = (i n0, 0, n3). Section 2.1 presents a one-loop Euclidean computation, derives the convergence condition |r_a|, |r_b| > 1, maps the ratios r = coth(y) to space-like Minkowski rapidities, and recovers the known Collins soft function at one loop. Section 2.2 introduces a finite-length ratio, Eq. (13), claimed to cancel the linear L-divergence and power corrections of order b^2/L^2. Section 3 outlines the lattice implementation of the auxiliary-field propagator, and Section 4 reports exploratory numerical results on a 32^3 x 64 PACS-CS ensemble. The reported ratio is approximately 2873.5/0.47 ~ 6.1 x 10^3 at tau/a = 3, b_perp/a = 3, r_a = r_b = 1.01, far above the O(1) value expected from tree-level estimates. Section 5 concludes with a caveat about the analytic continuation of the lattice results to Minkowski space.

Significance. If the method is established, it would provide a new lattice route to the TMD soft function and the Collins-Soper kernel, complementing existing approaches based on quasi-TMDs and mHQET. The one-loop derivation in Section 2.1 is a genuine strength: it is parameter-free, explicitly exhibits the convergence condition, and is checked against the known Minkowski result. The finite-length ratio idea is attractive and could be useful if properly derived. However, the current manuscript does not provide enough evidence that the lattice quantity equals the Collins soft function: the central ratio is asserted rather than derived, and the single numerical value is three orders of magnitude away from the expected result. The paper is therefore best viewed as an exploratory proceedings contribution whose central computational claim remains to be substantiated.

major comments (4)

[Section 2.2, Eq. (13)] The central ratio in Eq. (13) is asserted without derivation. The sentence 'We have determined that Eq. (13) holds to one loop in perturbation theory, using a Polyakov regulator' reports a check but does not show the calculation or state the result. Since this ratio is the entire lattice observable, its claimed cancellation of the linear L-divergence and the b_perp^2/L^2 power corrections is load-bearing. Please provide the one-loop derivation (or a self-contained reference with the explicit application to this complex-direction case) and state what is known beyond one loop, including the conditions under which the cancellation persists.
[Section 4, numerical results] The only reported value of the ratio, S_num/sqrt(S_A S_B) ~ 2873.5/0.47 ~ 6.1 x 10^3, is about three orders of magnitude larger than the O(1) value expected from the tree-level estimates in Eq. (17) and from the soft-function normalization. Attributing this to 'the need for the perturbative matching between the lattice and continuum regularization scheme' is not sufficient: a matching coefficient of O(10^3) would itself require a calculation or at least a bound before the lattice ratio can be called a measurement of the soft function. The abstract and conclusions should be softened to reflect that the numerical evidence is exploratory and that the matching is an uncomputed, essential step.
[Section 5, conclusions] The paper acknowledges that 'the direct analytic continuation of our lattice calculations to Minkowski space could be complicated by the pole structure of the auxiliary field propagator.' This is a central correctness risk for the proposed method, because the Euclidean complex-direction object must be connected to the Minkowski Collins soft function. The reference to [14] is not by itself an argument; please explain concretely how the mHQET consistency analysis applies to the direction n~ = (i n0, 0, n3), or state the conditions under which the continuation is expected to work.
[Equation (17) and Eq. (18)] The expected large-tau behaviour in Eq. (17) contains oscillatory factors in S_A and S_B that are said to cancel in sqrt(S_A S_B), but the cancellation is not demonstrated. Since the numerical calculation is done at tau/a = 3, which is not asymptotically large, it would be helpful to show quantitatively how the oscillatory terms cancel in the product and how sensitive the ratio in Eq. (18) is to this cancellation.

minor comments (6)

[Section 1 and Footnote 1] The footnote discusses an O(4) symmetry equivalence with n_3 < n_0, but the main text uses r_a = r_b = 1.01, which corresponds to n_3 > n_0. Please clarify whether this affects the claimed equivalence and whether the complex direction used here lies in the same region.
[Equation (3)] In Eq. (3), the expression after evaluation at s = -infinity is written for k_4 < 0, but the divergence condition is stated only in passing. The limit notation is unclear; please define the sign conventions for the Fourier transform and the integration contour more carefully.
[Notation in Eq. (13)] The symbol S is used for the finite-length Wilson-loop value, the infinite-length soft function, and the ratio in Eq. (13). Please introduce distinct notation or explicitly state which quantity each S denotes in the equation and in the surrounding text.
[Equation (12)] Equation (12) has a missing closing parenthesis in the logarithm argument; the formula should read log( S(b_perp, y_n, y_B, mu) / S(b_perp, y_A, y_n, mu) ).
[Figures 3-6] The axis labels 'R (S_num)', 'I (S_num)', and similar abbreviations are not defined in the captions. Please state that R and I denote real and imaginary parts, and define the plotted quantities in the captions.
[Conclusions] The word 'Wilon' in the final paragraph should be 'Wilson'.

Circularity Check

0 steps flagged · score 0.0 of 10

No load-bearing circularity: the one-loop derivation is self-contained, and the unproven lattice-ratio matching is a correctness gap rather than a circular reduction.

full rationale

The central derivation is a parameter-free one-loop computation. The Euclidean Wilson-line integrals in Eq. (2) are evaluated directly, giving Eq. (6); substituting the spacelike rapidity parameterization in Eqs. (8)-(10) turns this into the known Minkowski-space Collins soft function in Eq. (11). The mapping r_a = coth(y_A), r_b = coth(y_B) is an algebraic definition of the Euclidean directional vector components, not a fitted parameter, and the one-loop result is checked against an external benchmark rather than assumed. Equation (13), the finite-length ratio, is borrowed from the external references [2,10] and asserted to cancel the linear L divergence and b^2/L^2 power corrections at one loop; the paper gives no derivation, and the large numerical quotient in Sec. 4 (2873.5/0.47) is honestly reported as requiring a perturbative matching between lattice and continuum schemes. That is a missing-support or correctness concern, not a circular reduction: the ratio is not defined to be the Collins soft function, and the paper does not present a fitted parameter as a prediction. The only self-citation is reference [1], used for the starting complex-direction ansatz and in the priority footnote; the present derivation independently re-derives the one-loop mapping, so that self-citation is not load-bearing. Section 5 explicitly flags the pole-structure issue in the analytic continuation, which is a limitation, not circularity. Overall, no prediction in the paper reduces by construction to its inputs.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

No new physical entities are introduced. The complex directional vector n~ is a mathematical construction, not a postulated particle or force. The free-parameter count is low: the exploratory run fixes r_a = r_b = 1.01, and all lattice parameters (a = 0.0907 fm, m_ud, m_s) are taken from the PACS-CS ensemble. The real burden sits in the five domain assumptions, especially the analytic continuation (assumption 2) and the divergence cancellation of the ratio (assumption 3), both of which are central to the method and only partially supported.

free parameters (1)

r_a = r_b = 1.01 = 1.01
Kinematic parameters chosen for the exploratory lattice run; correspond to large rapidity y_A ≈ -y_B ≈ 2.65. They are not fitted to data, but they are hand-chosen inputs that set the Wilson line angles and must lie in the convergent region |r|>1.

assumptions (5)

domain assumption The auxiliary field representation of the Wilson line (Eq. 14) is exact, including for complex directions n~ with imaginary time component.
Used as the starting point for the lattice operator; cited to [11,12]. The extension to complex n is assumed without a dedicated proof.
domain assumption The Euclidean soft function with complex directions is time-independent and can be analytically continued to the Minkowski space-like soft function via the mapping r = coth(y).
Section 1 states 'the soft function remains independent in time'; Section 5 flags that the analytic continuation 'could be complicated by the pole structure of the auxiliary field propagator'. This is the load-bearing assumption for the whole matching.
domain assumption The ratio of finite-length Wilson lines in Eq. (13) cancels the linear L-divergence and all power corrections up to O(b^2/L^2).
Section 2.2: 'We have determined that Eq. (13) holds to one loop in perturbation theory, using a Polyakov regulator', but no derivation is shown; relies on Collins [10] and mHQET [2].
domain assumption The discretized auxiliary field propagator, obtained from Eq. (31) of [16], solves Eq. (16) and has a well-defined continuum limit.
Section 3: 'the central idea is to express the soft function in terms of lattice-regulated auxiliary field propagators... which can be shown to possess a well-defined continuum limit'; the proof is delegated to mHQET references [13,16].
domain assumption The one-loop perturbative results are representative of the all-order behavior of the r-to-y mapping and of the ratio's divergence cancellation.
The lattice method assumes the one-loop matching persists beyond leading order; a standard but unproven assumption in this proposal.

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Cite this review

Pith. "Pith review of Measurement of the TMD soft function on the lattice using the auxiliary field representation of the Wilson line." pith.science (2026). https://pith.science/paper/ENZD3EOF

@misc{pith2026241212645,
  author       = {Pith},
  title        = {Pith review of: Measurement of the TMD soft function on the lattice using the auxiliary field representation of the Wilson line},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ENZD3EOF}},
  note         = {Machine review of arXiv:2412.12645}
}

read the original abstract

The transverse momentum dependent (TMD) soft function and Collins-Soper (CS) kernel may be obtained by formulating the Wilson line in terms of auxiliary one-dimensional fermion fields on the lattice. Our computation takes place in the region of the lattice that corresponds to the "spacelike" region in Minkowski space in order to obtain the Collins soft function. The matching of our result to the Collins soft function is achieved through the mapping of the auxiliary field directional vector that to the Wilson line rapidity. In Euclidean space, this directional vector is complex, having a purely imaginary time component. We present some exploratory numerical results of our lattice calculation, and discuss the methodology employed.

Figures

Figures reproduced from arXiv: 2412.12645 by the authors.

**Figure 2.** Diagrams contributing at one-loop to the soft function, up to mirror diagrams. 2. Theoretical considerations We conduct a perturbative investigation for both infinite and finite-length Wilson lines. For the infinite case, we establish an equivalence between computations performed in Euclidean and Minkowski spaces at one loop. Additionally, we demonstrate that the results for finite-length Wilson lines converge towar… view at source ↗

**Figure 3.** Real and imaginary parts of the numerator loop, 𝑆num. The mean, 𝜇, is taken directly from the lattice measurement, and 𝜎 is the standard deviation of the bootstrap distribution. −0.25 0.00 0.25 0.50 0.75 1.00 1.25 R (SA) 0 20 40 60 80 100 120 140 160 180 ra = 1.01 rb = 1.01 τ /a = 3 b⊥/a = 3 16384 sources µ = 4.5 · 10−1 σ = 2.1 · 10−1 32768 sources µ = 4.7 · 10−1 σ = 1.6 · 10−1 −0.8 −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 0.… view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: Real and imaginary parts of the denominator A loop, 𝑆𝐴. The mean, 𝜇, is taken directly from the lattice measurement, and 𝜎 is the standard deviation of the bootstrap distribution. As discussed in [13, 14], meaningful solutions to Eq. (16) can only be obtained with a UV…

**Figure 5.** Figure 5: Real and imaginary parts of the denominator B loop, 𝑆𝐵. The mean, 𝜇, is taken directly from the lattice measurement, and 𝜎 is the standard deviation of the bootstrap distribution. 0.0 0.2 0.4 0.6 0.8 1.0 1.2 R √ SASB 0 20 40 60 80 100 120 140 ra = 1.01 rb = 1.01 τ /a…

**Figure 6.** Figure 6: Real and imaginary parts of full denominator term, √ 𝑆𝐴𝑆𝐵. The mean, 𝜇, is taken directly from the lattice measurement, and 𝜎 is the standard deviation of the bootstrap distribution. configurations using a non-perturbatively O (𝛼)-improved Wilson quark action and Iwasa…

Discussion (0). Continue with ORCID to comment.

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Pith tools

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