Saddle-point method for resummed form factors in QCD

Giancarlo Ferrera; Ugo Giuseppe Aglietti; Wan-Li Ju

REVIEW 3 major objections 5 minor 24 references

Saddle-point method for resummed form factors in QCD

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The paper claims that analytic inversion of resummed QCD form factors succeeds when the expansion is made around the true saddle point of the interacting theory, matching exact numerical inversion within perturbative uncertainty where the…

desk verdict A genuinely new saddle-point inversion for event-shape resummed form factors that agrees with exact numerics far better than CTTW, but the printed central formula has a sign error that needs a fix before the claim is fully assessable. read the letter →

arxiv 2506.18707 v1 pith:E43MRGQD submitted 2025-06-23 hep-ph hep-th

Ugo Giuseppe Aglietti , Giancarlo Ferrera , Wan-Li Ju This is my paper · ORCID

classification hep-phhep-th

keywords saddle-pointmethodSudakovresummationeventshapedistributionsthrustformfactorinversionLandausingularityanalyticQCDcouplingtruesaddlepoint

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

This paper takes on a long-standing technical problem in perturbative QCD: converting an all-orders resummed form factor from the moment space where it factorizes into the physical space where event-shape data are collected. The authors present a fully analytic inversion formula obtained by expanding the exponent of the Laplace integral around the saddle point of the full interacting theory instead of the free-theory saddle point used by the classical Taylor-based inversion. For the thrust distribution at leading, next-to-leading, and next-to-next-to-leading logarithmic accuracy, the formula agrees with the exact numerical inverse transform at the percent level down to $\tau \sim 0.002$, while the standard analytic inversion deviates outside its perturbative uncertainty band. If correct, the result supplies an accurate analytic route to momentum-space Sudakov resummation and confirms, in a fully analytic framework, the earlier numerical finding that the standard momentum-space formula biases the extraction of $\alpha_S(m_Z)$ from thrust.

What carries the argument

The load-bearing mechanism is the saddle-point method applied to the Laplace-inversion exponent $g_y(N)$. Its characteristic move is to expand around the 'true' saddle point $\bar N(y)$ — the stationary point of the full interacting theory — rather than around the free-theory point $N = 1/y$ used by the classical inversion, and the paper reduces the resulting transcendental equation to the nested-logarithm fixed-point problem $u = \Phi(u)$ for $u = y\bar N$, solved analytically by recursion $u_{n+1} = \Phi(u_n)$ with rapid convergence. The Gaussian evaluation of the integral then produces the closed form-factor formula, with the Landau singularity handled beforehand by replacing the running coupling with a Landau-free analytic coupling, and with the frozen-coupling limit supplying the formal control parameter $\ell = \ln(1/y)$ for the expansion.

What would settle it

Evaluate the inverse-transform integral in Eq. (15) numerically to high precision for the regularized thrust form factor at a fixed perturbative value such as $\tau = 0.01$, and compare it with the closed Gaussian formula in Eq. (19) while checking that the anharmonic coefficients $c_k$ of Eq. (23) form a convergent, subleading series: the central claim requires percent-level agreement and a small convergent anharmonic correction, so a deviation of several percent, or a non-convergent anharmonic series, would settle the claim negatively.

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Extended reading notes

Core claim

On the paper's own terms, the central claim is that the inverse transform $\Sigma(y,\alpha_S)$ of the resummed form factor is dominated by the saddle point $\bar N(y)$ of the interacting theory — the solution of $g'_y(\bar N)=0$ for the full exponent $g_y(N) = yN - \ln N + \ln N\, f_1(\lambda) + \sum_{n\ge 0}(\alpha_S/\pi)^n f_{n+2}(\lambda)$ — and that a quadratic expansion around it yields the closed analytic form $\Sigma(y,\alpha_S) \simeq \exp[h_y(0)]/\sqrt{2\pi h''_y(0)}$. The saddle point is obtained either numerically or by iterating the fixed-point equation $u = \Phi(u)$ with $u = y\bar N$, starting from the free-theory value $u_0 = 1$; the recursion converges to better than $0.01\%$ within a handful of steps. The paper shows that this Gaussian formula tracks the exact numerical inversion of the Landau-regularized form factor within perturbative uncertainty for $\tau$ down to about $0.002$, at LL, NLL, and NNLL accuracy, whereas the classical Taylor expansion of the exponent around the free-theory point $N = 1/y$ leaves the uncertainty band. The same analytic construction reproduces the authors' earlier numerical conclusion that the standard momentum-space formulation yields an $\alpha_S(m_Z)$ from thrust inconsistent with the world average.

Load-bearing premise

The load-bearing premise is that the inversion integrand decays fast enough around the true saddle point for the Gaussian term to dominate: in the running-coupling case the authors point out that no large control parameter can be extracted from the integral, so the accuracy of the closed formula rests on the unproven supposition that the anharmonic corrections remain small.

Editorial extensions

If this is right

Momentum-space Sudakov resummation for event shapes becomes analytically reliable without numerical integration: the closed saddle-point formula reproduces exact inversions at the percent level throughout the perturbative two-jet region.
Because the exponent structure of the form factor is universal, the same true-saddle-point construction carries over to other event-shape variables, such as heavy-jet mass and the C-parameter, and to threshold resummation, with only the coefficient functions changing.
The discrepancy between the standard Taylor-based momentum-space resummation and the exact inversion is reproduced in a fully analytic setting, tying the thrust $\alpha_S(m_Z)$ bias seen in the earlier numerical study to the choice of the free-theory expansion point.
The method sets its own validity boundary: no saddle point exists below roughly $\tau \sim 0.004$–$0.01$, so the disappearance of the saddle point marks the scale $\tau \sim \Lambda_{\mathrm{QCD}}/Q$ at which non-perturbative corrections take over.

Reading between the lines

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

A refit of existing thrust data with the saddle-point-inverted form factor — which this paper does not perform — is the direct test of the implied claim that the extracted $\alpha_S(m_Z)$ moves from the low value produced by the standard momentum-space formula toward the world average.
The convergence of the fixed-point iteration $u_{n+1} = \Phi(u_n)$ is demonstrated numerically, not proven; establishing a contraction bound on $\Phi$ would turn the recursion into a rigorous existence proof for the true saddle point in the running-coupling case.
Repeating the saddle-point-versus-exact comparison under alternative Landau-singularity regularizations, beyond the minimal prescription, would separate regularization dependence from inversion-method error in the final form factor.
The critical $\tau$ where the saddle point disappears is effectively a parameter-free estimate of the non-perturbative boundary, and comparing that boundary across event shapes and collision energies would test whether it scales as $\Lambda_{\mathrm{QCD}}/Q$.

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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

Dear colleague,

You should know two things about arXiv:2506.18707. First, it does something genuinely new: it inverts the resummed event-shape form factor by expanding around the true saddle point of the interacting theory, rather than around the free-theory expansion point N = 1/y used by CTTW and successors. Second, the numerical evidence is strong: with an analytic coupling that removes the Landau pole, their saddle-point inversion reproduces the exact numerical Laplace inversion at the percent level from tau ~ 0.05 down to ~0.002, while the CTTW Taylor inversion misses by more than the perturbative uncertainty band over most of that range. That difference matters, because it is the same approximation that feeds alpha_s determinations from thrust.

What the paper does well: it sets up the saddle-point equation, solves it by a simple recursion that converges quickly, regularizes the Landau singularity with an analytic coupling, and checks the approximation at LL, NLL, and NNLL, plus a frozen-coupling limit where a large parameter (ell = ln 1/tau) formally justifies the expansion. The authors are also honest that no such large parameter can be extracted for the running-coupling case, and they give a plausibility argument rather than a proof.

The soft spot is real and in the central formula. Eq (15) defines the form factor as ∫ dnu exp[h_y(nu)], and for that integral to converge h_y must have a maximum at nu=0, i.e. h_y''(0) < 0. But Eq (20) states h_y''(0) = -g_y''(N) > 0, and the Gaussian denominator in Eq (19) uses that positive value. With h_y''>0, the Gaussian integrand grows and the integral is not defined; the change of variables in Eq (21) then produces e^{-x^2} only if the second derivative is negative. This is almost certainly a sign typo — the correct denominator is sqrt(2 pi g_y''(N)) — but it is load-bearing as written: without correcting it, the central formula cannot be checked. The authors should re-derive Eqs (19)-(23) with explicit signs.

Beyond that, the lack of code or data is a minor weakness; and the convergence of the recursion is only shown for one point, but that is easy to extend. The paper is for resummation practitioners, especially anyone using analytic momentum-space formulas for event shapes or threshold resummation, and for those studying analytic inversion of Laplace integrals in QCD.

My recommendation: send it to peer review. The method is credible, the numerics are convincing, and the sign issue appears fixable. A referee should ask for the corrected formula, a clarification of the sign convention, and ideally the code or numbers behind the figures. After those revisions, I would take the result seriously.

Referee Report

3 major / 5 minor

Summary. The paper presents a fully analytic saddle-point inversion formula for the resummed QCD form factor of event-shape distributions, expanding the exponent of the inverse-transform integrand around the solution of the full (interacting-theory) saddle-point equation rather than around the free-theory point N = 1/y. The saddle point is obtained numerically and by an analytic recursion, and the resulting form factor is compared with the exact numerical inverse transform for thrust at LL, NLL, and NNLL accuracy, finding percent-level agreement while the standard CTTW Taylor expansion deviates by more than the estimated perturbative uncertainty. A frozen-coupling limit is used to exhibit a formal large expansion parameter.

Significance. If the central formula is corrected, the paper would provide a useful analytic route from N-space resummed form factors to momentum-space event-shape predictions, and it would independently confirm the authors' earlier numerical finding that standard Taylor-based analytic inversions can differ substantially from exact numerical inversion. The recursive saddle-point solution and the explicit regularized form-factor expressions are concrete and testable. However, the printed derivation of the central formula contains a load-bearing sign inconsistency that prevents the claim from being assessed as written.

major comments (3)

[Sec. 3, Eqs. (19)-(23)] The printed saddle-point formula is internally inconsistent. Eq. (20) asserts h''_y(0) = -g''_y(N) > 0, but for the integral in Eq. (15) to converge the exponent must have a maximum at ν = 0, which requires h''_y(0) < 0; indeed Eq. (21) uses e^{-x^2}, which corresponds to the opposite sign. With h''_y(0) > 0 the Gaussian integrand grows away from ν = 0, and the denominator sqrt(2π h''_y(0)) in Eq. (19) is not real. The correct expression is Σ ≃ exp[h_y(0)]/sqrt(-2π h''_y(0)) = exp[g_y(N)]/sqrt(2π g''_y(N)) when g''_y(N) > 0. The same sign error propagates into the change of variable x = ν sqrt(h''_y(0)/2) and into the coefficients c_k in Eq. (23), so the stated expansion is not well-defined as written. Because Eq. (19) is the paper's central result, the authors must correct the sign, re-derive Eqs. (21)-(23), and confirm that the numerical curves in Figs. 2-4 were obtained with the corrected formula.
[Sec. 3, Eqs. (21)-(23)] The treatment of anharmonic corrections is also not well-defined as printed. After the sign of h''_y is fixed, x must be defined with |h''_y(0)|, and the statement that 'anharmonic terms can be included only if c_k i^k < 0' is at most a necessary condition for an individual term; it does not guarantee convergence of the integral in Eq. (21) when several coefficients are present, because the large-x asymptotic behavior is controlled by the highest-order term retained. The paper should specify exactly which terms of K(x) are kept in the 'sextic' approximation, verify the convergence of the resulting integral, and state the numerical impact of the quartic and sextic terms separately.
[Sec. 4, Figs. 2-4] The 'exact numerical inversion' used as the benchmark is not sufficiently specified. The contour, integration method, numerical precision, and the value of the regularization parameter r in Eq. (42) (in particular the scale at which α_S is evaluated) are not given. Since the central evidence is the agreement between the saddle-point approximation and this exact inversion, the numerical procedure must be described in enough detail for the comparison to be reproducible and independently checkable.

minor comments (5)

[Sec. 3, after Eq. (20)] The sentence stating that 'the function h_y(ν) has its minimum at ν = 0' should read 'maximum' once the sign of h''_y(0) is corrected; the maximum is required for the Laplace-type approximation used here.
[Sec. 4, final paragraph] The statement that the saddle-point method is 'fully consistent with the numerical inversion within the Minimal Prescription' for the unregularized form factor is not accompanied by any plot, table, or quantitative estimate; adding this comparison would strengthen the claim.
[Sec. 2, Eq. (8)] The notation F(k)(α_S, ℓ) for the k-th derivative with respect to ℓ should use a superscript F^{(k)} to avoid confusion with an argument k, and ℓ = ln(1/y) should be explicitly distinguished from L = ln N throughout.
[Sec. 3.1] The convergence of the recursion u_{n+1} = Φ(u_n) is demonstrated for a single value y = 0.01 at LL accuracy; the authors should state the conditions under which this fixed-point iteration is expected to converge and comment on how the number of iterations varies with y and with the logarithmic order.
[App. A, final paragraph] The claim that the frozen-coupling limit provides a good approximation to the running-coupling case because the analytic coupling saturates at π/β_0 is heuristic; the relation between the value π/β_0 ≃ 1.6 and the artificially chosen A = 0.2 in Eq. (46) is not explained.

Circularity Check

0 steps flagged · score 2.0 of 10

Derivation is self-contained; no fitted-input circularity — but the printed saddle-point formula (Eq. 19) has a sign inconsistency that blocks verification of the agreement claim.

full rationale

The claimed derivation is not circular in the input-output sense. The paper starts from the N-space resummed form factor (Eqs. 6 and 10) and approximates the inverse Laplace transform (Eq. 9) by expanding exp[h_y(nu)] around the saddle point N(y) determined by g'_y(N)=0 (Eqs. 17-18). The saddle point is solved numerically and by a fixed-point recursion (Sec. 3.1); no parameter is fitted to the benchmark. The comparison with exact numerical inversion in Figs. 2-4 is a self-contained accuracy test of the same integral, not an input to the formula. The anharmonic coefficients c_k in Eq. (23) are derived from derivatives of the input exponent, not from the target Sigma. Self-citations are present but not load-bearing: Ref. [10] supplies higher-order resummation functions and is the earlier numerical result being confirmed; Ref. [17] supplies the analytic-coupling regularization. Neither is used as the premise that saddle-point inversion is accurate; the paper recomputes that here. However, there is a non-circular but serious internal inconsistency: Eq. (19) places h_y''(0)>0 (Eq. 20) in the denominator of a Gaussian result for the integral d(nu/(2pi)) exp[+h_y(nu)]; with h_y''>0 the integrand grows away from nu=0 and the Gaussian approximation is not defined. The opposite sign is needed for a maximum. This sign issue must be fixed before the agreement in Figs. 2-4 can be checked against the printed formula, and the paper gives no code or data for that check. The paper itself states the validity limitation: 'there is no proof that our (saddle-point inspired) approximation of the integral in Eq. (15) is valid for any value of y' (Sec. 3). This is a convergence/validity risk, not a circularity. Overall: minor self-citation, no circular reduction.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the standard saddle-point approximation, the CTTW exponentiation structure, a specific analytic-coupling regularization from Ref [17], and the numerical convergence of the recursion. None of these are new entities. The only hand-chosen numbers are the illustrative frozen-coupling constants A and B; no fitted parameters are used in the main comparison.

free parameters (1)

A and B (frozen-coupling model) = A=0.2, B=0 (illustrative)
Chosen in App A to ensure convergence of the frozen-coupling integral; illustrative only, not used in the main running-coupling results.

assumptions (4)

domain assumption The saddle-point (Laplace) method formula Eq (11) applies to the integral in Eq (15) with no extracted large parameter s.
Invoked in Sec 3; the authors state there is no proof of validity for all y and only frozen-coupling asymptotics justifies the expansion.
domain assumption The N-space resummed form factor exponentiation structure Eq (6) from CTTW is valid for event-shape distributions.
Starting point of Sec 2; taken from Ref [1] and assumed throughout.
ad hoc to paper Replacing the standard QCD coupling with the analytic coupling of Ref [17] yields a form factor whose inversion accuracy represents the physical form factor.
Sec 3.2; the authors assert the regularization method is not relevant to the accuracy question, but this is not proven.
domain assumption The recursive sequence Eq (28) converges to the unique real saddle point for the τ range studied.
Sec 3.1; convergence demonstrated numerically for one value τ=0.01, not proven in general.

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Cite this review

Pith. "Pith review of Saddle-point method for resummed form factors in QCD." pith.science (2026). https://pith.science/paper/E43MRGQD

@misc{pith2026250618707,
  author       = {Pith},
  title        = {Pith review of: Saddle-point method for resummed form factors in QCD},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/E43MRGQD}},
  note         = {Machine review of arXiv:2506.18707}
}

read the original abstract

We consider the form factor appearing in QCD resummation formalism for event shape distributions in the two-jet (or Sudakov) region. We present an analytic formula for the inverse transform of the form factor, namely from the conjugate moment space to the (physical) momentum space, based on the saddle-point method. The saddle-point itself is determined by means of an analytic recursion method as well as by standard numerical methods. The results we have found are in very good agreement with the exact (numerical) evaluation of the inverse transform, while they significantly differ from classical analytical formulations of resummation in momentum space. The latter are based on a Taylor expansion of the form factor around the free-theory saddle point.

Figures

Figures reproduced from arXiv: 2506.18707 by the authors.

**Figure 1.** Saddle-point solutions at leading logarithmic accuracy (Eq. (27)) for the thrust at y = τ = 0.01. Numerical comparison of the first ten recursions N = u1/y, u2/y, u3/y, · · · (blue dots) with the exact numerical solutionN = u/y (red line). Therefore, in order to obtain a non-trivial saddle point, at least one recursion has to be made, even at LL. In fact, u = 1 (i.e. N = 1/y) is the saddle-point of the “free” theory… view at source ↗

**Figure 2.** Resummed form factor at leading logarithmic accuracy obtained through: exact numerical inversion (red solid line); analytic saddle-point inversion (dashed blue line); also including anharmonic sextic corrections (dot-dashed blue line) and analytic Taylor inversion (black dotted line). The lower panel shows the ratios with respect to the exact numerical inversion and the band between the two red curves is the estimat… view at source ↗

**Figure 3.** Resummed form factor at next-to-leading logarithmic accuracy obtained through: exact numerical inversion (red solid line); analytic saddle-point inversion (dashed blue line); including also anharmonic sextic corrections (dot-dashed blue line) and analytic Taylor inversion (black dotted line). The lower panel shows the ratios with respect to the exact numerical inversion and the band between the two red curves is t… view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: Resummed form factor at next-to-next-to-leading logarithmic accuracy obtained through: exact numerical inversion (red solid line), analytic saddle-point inversion (dashed blue line) including anharmonic sextic corrections (dot-dashed blue line) and analytic Taylor inve…

**Figure 5.** Figure 5: Exact (numeric) frozen-coupling resummed form factor, Eq. (48), (red solid line) compared with the saddle-point result, Eq. (53), (blue dashed line). References [1] S. Catani, L. Trentadue, G. Turnock and B. R. Webber, Nucl. Phys. B 407 (1993), 3-42 doi:10.1016/0550-32…

Review history (2 revisions) →

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Works this paper leans on

24 extracted references · 5 canonical work pages

[1]

Catani, L

S. Catani, L. Trentadue, G. Turnock and B. R. Webber, Nucl. Phys. B 407 (1993), 3-42 doi:10.1016/0550-3213(93)90271-P

doi:10.1016/0550-3213(93)90271-p 1993
[2]

C. W. Bauer, S. P. Fleming, C. Lee and G. F. Sterman, Phys. Rev. D 78 (2008), 034027 doi:10.1103/PhysRevD.78.034027 [arXiv:0801.4569 [hep-ph]]

arXiv 2008
[3]

L. G. Almeida, S. D. Ellis, C. Lee, G. Sterman, I. Sung and J. R. Walsh, JHEP 04 (2014), 174 doi:10.1007/JHEP04(2014)174 [arXiv:1401.4460 [hep-ph]]

arXiv 2014
[4]

Gardi and J

E. Gardi and J. Rathsman, Nucl. Phys. B 609 (2001), 123-182 doi:10.1016/S0550- 3213(01)00284-X [arXiv:hep-ph/0103217 [hep-ph]]

arXiv 2001
[5]

Banfi, G

A. Banfi, G. P. Salam and G. Zanderighi, JHEP 01 (2002), 018 doi:10.1088/1126- 6708/2002/01/018 [arXiv:hep-ph/0112156 [hep-ph]]

arXiv 2002
[6]

M. D. Schwartz, Phys. Rev. D 77 (2008), 014026 doi:10.1103/PhysRevD.77.014026 [arXiv:0709.2709 [hep-ph]]. 18

arXiv 2008
[7]

Becher and M

T. Becher and M. D. Schwartz, JHEP 07 (2008), 034 doi:10.1088/1126- 6708/2008/07/034 [arXiv:0803.0342 [hep-ph]]

arXiv 2008
[8]

Abbate, M

R. Abbate, M. Fickinger, A. H. Hoang, V. Mateu and I. W. Stewart, Phys. Rev. D 83 (2011), 074021 doi:10.1103/PhysRevD.83.074021 [arXiv:1006.3080 [hep-ph]]

arXiv 2011

Show all 24 references

[9]

P. F. Monni, T. Gehrmann and G. Luisoni, JHEP 08 (2011), 010 doi:10.1007/JHEP08(2011)010 [arXiv:1105.4560 [hep-ph]]

2011 arXiv
[10]

U. G. Aglietti, G. Ferrera, W. L. Ju and J. Miao, [arXiv:2502.01570 [hep-ph]]

arXiv
[11]

Farhi, Phys

E. Farhi, Phys. Rev. Lett. 39 (1977), 1587-1588 doi:10.1103/PhysRevLett.39.1587

1977 doi
[12]

Catani, M

S. Catani, M. L. Mangano, P. Nason and L. Trentadue, Nucl. Phys. B 478 (1996), 273-310 doi:10.1016/0550-3213(96)00399-9 [arXiv:hep-ph/9604351 [hep-ph]]

1996 arXiv
[13]

Bonvini, S

M. Bonvini, S. Forte and G. Ridolfi, Phys. Rev. Lett. 109 (2012), 102002 doi:10.1103/PhysRevLett.109.102002 [arXiv:1204.5473 [hep-ph]]

2012 arXiv
[14]

Bonvini, S

M. Bonvini, S. Forte, G. Ridolfi and L. Rottoli, JHEP 01 (2015), 046 doi:10.1007/JHEP01(2015)046 [arXiv:1409.0864 [hep-ph]]

2015 arXiv
[15]

D. Kang, C. Lee and V. Vaidya, JHEP 04 (2018), 149 doi:10.1007/JHEP04(2018)149 [arXiv:1710.00078 [hep-ph]]

2018 arXiv
[16]

Grewal, Z

M. Grewal, Z. B. Kang, J. W. Qiu and A. Signori, Phys. Rev. D 101 (2020) no.11, 114023 doi:10.1103/PhysRevD.101.114023 [arXiv:2003.07453 [hep-ph]]

2020 arXiv
[17]

Aglietti and G

U. Aglietti and G. Ricciardi, Phys. Rev. D 70 (2004), 114008 doi:10.1103/PhysRevD.70.114008 [arXiv:hep-ph/0407225 [hep-ph]]

2004 arXiv
[18]

Forte, G

S. Forte, G. Ridolfi, J. Rojo and M. Ubiali, Phys. Lett. B 635 (2006), 313-319 doi:10.1016/j.physletb.2006.02.058 [arXiv:hep-ph/0601048 [hep-ph]]

2006 arXiv
[19]

Aglietti, G

U. Aglietti, G. Ferrera and G. Ricciardi, Nucl. Phys. B 768 (2007), 85-115 doi:10.1016/j.nuclphysb.2007.01.014 [arXiv:hep-ph/0608047 [hep-ph]]

2007 arXiv
[20]

Aglietti, G

U. Aglietti, G. Corcella and G. Ferrera, Nucl. Phys. B 775 (2007), 162-201 doi:10.1016/j.nuclphysb.2007.04.014 [arXiv:hep-ph/0610035 [hep-ph]]

2007 arXiv
[21]

P.-S. Laplace. Essai philosophique sur les probabilit´ es. Courcier, Paris, 1814
[22]

Catani, G

S. Catani, G. Turnock, B. R. Webber and L. Trentadue, Phys. Lett. B 263 (1991), 491-497 doi:10.1016/0370-2693(91)90494-B

1991 doi
[23]

D. V. Shirkov and I. L. Solovtsov, JINR Rapid Comm., No. 2[76]-96, 5, [arXiv:hep- ph/9604363 [hep-ph]]
[24]

D. V. Shirkov and I. L. Solovtsov, Phys. Rev. Lett. 79 (1997), 1209-1212 doi:10.1103/PhysRevLett.79.1209 [arXiv:hep-ph/9704333 [hep-ph]]. 19

1997 arXiv

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

Catani, L

S. Catani, L. Trentadue, G. Turnock and B. R. Webber, Nucl. Phys. B 407 (1993), 3-42 doi:10.1016/0550-3213(93)90271-P

doi:10.1016/0550-3213(93)90271-p 1993

[2] [2]

C. W. Bauer, S. P. Fleming, C. Lee and G. F. Sterman, Phys. Rev. D 78 (2008), 034027 doi:10.1103/PhysRevD.78.034027 [arXiv:0801.4569 [hep-ph]]

arXiv 2008

[3] [3]

L. G. Almeida, S. D. Ellis, C. Lee, G. Sterman, I. Sung and J. R. Walsh, JHEP 04 (2014), 174 doi:10.1007/JHEP04(2014)174 [arXiv:1401.4460 [hep-ph]]

arXiv 2014

[4] [4]

Gardi and J

E. Gardi and J. Rathsman, Nucl. Phys. B 609 (2001), 123-182 doi:10.1016/S0550- 3213(01)00284-X [arXiv:hep-ph/0103217 [hep-ph]]

arXiv 2001

[5] [5]

Banfi, G

A. Banfi, G. P. Salam and G. Zanderighi, JHEP 01 (2002), 018 doi:10.1088/1126- 6708/2002/01/018 [arXiv:hep-ph/0112156 [hep-ph]]

arXiv 2002

[6] [6]

M. D. Schwartz, Phys. Rev. D 77 (2008), 014026 doi:10.1103/PhysRevD.77.014026 [arXiv:0709.2709 [hep-ph]]. 18

arXiv 2008

[7] [7]

Becher and M

T. Becher and M. D. Schwartz, JHEP 07 (2008), 034 doi:10.1088/1126- 6708/2008/07/034 [arXiv:0803.0342 [hep-ph]]

arXiv 2008

[8] [8]

Abbate, M

R. Abbate, M. Fickinger, A. H. Hoang, V. Mateu and I. W. Stewart, Phys. Rev. D 83 (2011), 074021 doi:10.1103/PhysRevD.83.074021 [arXiv:1006.3080 [hep-ph]]

arXiv 2011