{"id":"eddbe02b-e83a-4c1e-990c-0f0512c82502","arxiv_id":"2412.12645","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Complex Euclidean Wilson line directions map to space-like rapidity, giving a one-loop match to the Collins soft function and a promising lattice ratio, though the numerical ratio is not yet physical.","lead":"This paper proposes a new lattice QCD method for the TMD soft function using Wilson lines with complex Euclidean directions that map to space-like rapidity. The authors show one-loop matching to the Collins soft function but report only exploratory numerics; the final ratio is not yet a physical measurement.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central ratio Eq. (13) is asserted without derivation and the reported lattice value is ~10^3 away from the expected O(1); without the missing matching, the claimed 'measurement' of the soft function is unsupported.","rationale":"The reader's weakest-assumption focuses on the continuum limit and analytic continuation of the auxiliary-field propagator, which is indeed a serious issue. My stress-test identifies a more immediately concrete and falsifiable weakness: the ratio Eq. (13), the central numerical construct, is asserted without derivation, and the one numerical result reported is three orders of magnitude from expectation. The paper itself acknowledges the need for an uncomputed matching, which means that the claimed 'measurement' is not yet established. This does not overturn the reader's CONDITIONAL verdict; it reinforces it. The condition should explicitly include (1) a derivation or numerical verification of the divergence cancellation in Eq. (13), and (2) a lattice-to-continuum matching calculation, as the reader already requests in their rationale. I mark agreement as 'partial' because my primary concern differs from the reader's stated weakest assumption, though both point to the same conclusion: the method is plausible but unverified.","tokens_in":8456,"tokens_out":6076,"duration_ms":55365,"concrete_test":"Compute the one-loop matching coefficient between the lattice auxiliary-field regularized ratio in Eq. (13) (using the same discretized auxiliary-field propagator) and the continuum Collins soft function, then apply that coefficient to the reported τ/a = 3, b⊥/a = 3, r_a = r_b = 1.01 result. If the corrected value is not O(1) and does not remain stable under increasing τ (e.g., τ/a = 4, 5), Eq. (13) fails to cancel the linear/oscillatory terms as claimed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central lattice observable is the ratio in Eq. (13), claimed to cancel the linear L-divergence and the b^2⊥/L^2 power corrections. The paper states that this was verified to one loop with a Polyakov regulator, but gives no derivation and no argument beyond one loop. More tellingly, the only reported numerical value of this ratio is S_num/√(S_A S_B) ≈ 2873.5/0.47 ≈ 6.1×10^3 (Sec. 4, r_a = r_b = 1.01, τ/a = 3, b⊥/a = 3), about three orders of magnitude larger than the O(1) value expected from the tree-level estimates in Eq. (17) and from soft-function normalization. The paper attributes this to an uncomputed 'perturbative matching between the lattice and continuum regularization scheme.' But if the matching coefficient must be O(10^3), the lattice ratio is not a controlled estimate of the Collins soft function, and the statement that Eq. (13) 'enables a lattice computation of the soft function' is not supported by the presented evidence. The analytic-continuation/pole-structure issue flagged in Sec. 5 is a genuine further concern, but the immediate load-bearing gap is the missing derivation and matching of the ratio, exactly where the numerical result fails.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":8676,"tokens_out":3745,"duration_ms":35141,"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":[{"comment":"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":"Section 2.2, Eq. (13)"},{"comment":"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":"Section 4, numerical results"},{"comment":"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.","section":"Section 5, conclusions"},{"comment":"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.","section":"Equation (17) and Eq. (18)"}],"minor_comments":[{"comment":"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.","section":"Section 1 and Footnote 1"},{"comment":"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.","section":"Equation (3)"},{"comment":"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.","section":"Notation in Eq. (13)"},{"comment":"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) ).","section":"Equation (12)"},{"comment":"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.","section":"Figures 3-6"},{"comment":"The word 'Wilon' in the final paragraph should be 'Wilson'.","section":"Conclusions"}],"recommendation":"major_revision","confidential_remarks":"This is a proceedings contribution, and the missing derivation of Eq. (13) together with the uncontrolled numerical matching would normally warrant rejection if the central claim were a completed measurement. However, the one-loop derivation in Section 2.1 is sound and the method may be viable, so I believe the appropriate action is major revision rather than rejection. The authors should be asked to either supply the derivation of the ratio, provide a quantitative matching estimate, or explicitly reframe the paper as an exploratory methodology study with no claim of having measured the soft function."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nOverall: 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.","headline":"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.","tokens_in":9284,"tokens_out":1845,"would_cite":true,"duration_ms":16474,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["transverse momentum dependent","soft function","Collins-Soper kernel","Wilson line","auxiliary field","lattice QCD","rapidity divergence","lattice perturbation theory"],"falsifier":"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.","tokens_in":8191,"feed_emoji":"⚛️","tokens_out":11924,"duration_ms":93336,"temperature":0.7,"pith_summary":"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.","feed_headline":"One-loop match opens lattice route to the TMD soft function","feed_subtitle":"A one-loop match plus a divergence-cancelling ratio makes Wilson-line data yield the Collins-Soper kernel.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"supplies the heavy-quark-effective-theory-inspired proposal that the soft function can be obtained from Wilson-line form factors and is the source of the finite-length ratio in Eq. (13).","marker":"[2]"},{"why":"defines the Collins soft function and its rapidity regularization scheme, the target of the lattice computation.","marker":"[3]"},{"why":"provides the alternative Collins definition that, together with [2], motivates the ratio in Eq. (13).","marker":"[10]"},{"why":"introduces the path-integral representation of a Wilson line in terms of auxiliary fields used in Eq. (14).","marker":"[11]"},{"why":"independently establishes the auxiliary-field representation of Wilson lines that underlies the method.","marker":"[12]"},{"why":"shows that the Euclidean auxiliary-field propagator requires a UV cutoff, motivating the lattice regulator and the recursive construction.","marker":"[13]"},{"why":"analyzes the consistency, renormalization, and pole structure of the auxiliary-field propagator, which the paper cites for the continuum limit and analytic continuation.","marker":"[14]"},{"why":"provides the recursive relation used to construct the lattice auxiliary-field propagator in the numerical computation.","marker":"[16]"},{"why":"supplies the dynamical gauge configurations used in the exploratory numerical study.","marker":"[17]"}],"fun_headline_variants":["Auxiliary-field Wilson lines yield TMD soft function on lattice","Lattice QCD: complex directions yield Collins-Soper kernel","One-loop match: complex Wilson lines produce Collins soft function","Complex Euclidean Wilson lines give TMD soft function on lattice"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Auxiliary-field Wilson lines yield TMD soft function on lattice","Lattice QCD: complex directions yield Collins-Soper kernel","One-loop match: complex Wilson lines produce Collins soft function","Complex Euclidean Wilson lines give TMD soft function on lattice"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001182,"raw_usage":{"total_tokens":4873,"prompt_tokens":924,"completion_tokens":3949,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":540,"completion_tokens_details":{"reasoning_tokens":3879}},"tokens_in":540,"tokens_out":3949,"duration_ms":23715,"temperature":1.0,"reasoning_tokens":3879,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T13:52:55.787517+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Collins, Foundations of perturbative QCD, Cambridge monographs on particle physics, nuclear physics, and cosmology","cited_arxiv_id":null,"evidence_quote":"defines the Collins soft function and its rapidity regularization scheme, the target of the lattice computation."},{"cited_title":"New definition of TMD parton densities","cited_arxiv_id":"1107.4123","evidence_quote":"provides the alternative Collins definition that, together with [2], motivates the ratio in Eq. (13)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"introduces the path-integral representation of a Wilson line in terms of auxiliary fields used in Eq. (14)."},{"cited_title":"Gervais and A","cited_arxiv_id":null,"evidence_quote":"independently establishes the auxiliary-field representation of Wilson lines that underlies the method."},{"cited_title":"Aglietti, M","cited_arxiv_id":null,"evidence_quote":"shows that the Euclidean auxiliary-field propagator requires a UV cutoff, motivating the lattice regulator and the recursive construction."},{"cited_title":"Aglietti,Consistency and lattice renormalization of the effective theory for heavy quarks, Nucl","cited_arxiv_id":null,"evidence_quote":"analyzes the consistency, renormalization, and pole structure of the auxiliary-field propagator, which the paper cites for the continuum limit and analytic continuation."},{"cited_title":"Mandula and M","cited_arxiv_id":null,"evidence_quote":"provides the recursive relation used to construct the lattice auxiliary-field propagator in the numerical computation."}],"review_version":1}