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Higher-Order Corrections in Threshold Resummation

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S. Moch (DESY, Zeuthen), J.A.M. Vermaseren (NIKHEF, Amsterdam), A. Vogt (Durham Univ., IPPP)

High Energy Physics - Phenomenology · Submitted 28 Jun 2005 · 38 references

Pith's verdict: Solid N3LL extension for DIS with a trustworthy B_q,3, but the gluon jet coefficients — especially B_g,3 — rest on an unpublished calculation the reader cannot check.

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arXiv IDhep-ph/0506288 (version 2)
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CitationAmsterdam), A. Vogt (Durham Univ., Ippp), J.A.M. Vermaseren (NIKHEF, S. Moch (DESY, Zeuthen). "Higher-Order Corrections in Threshold Resummation." arXiv:hep-ph/0506288 (2005). https://pith.science/paper/A3W4JWUB
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The paper

The paper claims

The authors extend threshold resummation for deep-inelastic scattering to fourth logarithmic order and extract the quark and gluon jet-function coefficients through third order in the strong coupling.

The reading

The paper asks how far the large logarithms that dominate deep-inelastic scattering near the threshold $x$ approaching 1 can be resummed, and what those resummed logs reveal about the still-uncomputed four-loop coefficient function. Working in Mellin-$N$ space, the authors extend the threshold exponent $G^N$ to fourth logarithmic order, determining the function $g_4$ that multiplies $a_s^2$ and collects all powers $(a_s L)^n$ with $L = 1/(b_0 a_s)$, equivalently all powers $(a_s L)^n$ with $L = 1/(b_0 a_s)$ in the paper's notation. They then match the expanded exponent to their three-loop DIS coefficient functions and read off the universal jet-function coefficients $B_q$ through $a_s^3$ and $B_g$ through $a_s^3$, extending the quark result by one order and the gluon result by two. The payoff is a complete N$^3$LL resummation for DIS, a prediction of the first seven towers of the four-loop coefficient function, and a numerical check that the new corrections are small at large scales and still modest at $a_s$ around 0.3.

What carries the argument

The load-bearing device is the Mellin-space exponentiation $C^N/C^N_{LO}=g_0 exp(G^N)$, with $G^N=L g_1(1/(b_0 a_s))+g_2(1/(b_0 a_s))+a_s g_3(1/(b_0 a_s))+a_s^2 g_4(1/(b_0 a_s))+...$ and $L=1/(b_0 a_s)$ in the paper's shorthand. Each $g_i$ is an integral over the running coupling of the universal cusp and jet functions $A_p$, $B_p$, plus a process-dependent large-angle soft term $D$; for DIS the paper takes $D_{DIS}=0$ from Refs. [23,24]. The extraction works by expanding this exponent in powers of $a_s$ and comparing the coefficients of $L^k$ with the fixed-order DIS coefficient functions, so that the highest $L$ term at each order isolates the new $B$ coefficient. The same expansion, truncated to its leading towers, generates the four-loop predictions and the numerical comparison between exponentiation and tower summation.

Extended reading notes

Core claim

The central claim is that threshold resummation for inclusive DIS can be carried to N$^3$LL accuracy, with the fourth logarithmic function $g_4(1/(b_0 a_s))$ computed from the running coupling up to the four-loop $\beta$ function and from the cusp anomalous dimension $A_q$ up to a Padé estimate of $A_{q,4}$. Matching the $L$ terms of the known one-, two-, and three-loop DIS coefficient functions to the expansion of the exponent fixes $B_{q,1}$, $B_{q,2}$, and the new $B_{q,3}$ in Eq. (4.12); the analogous scalar-exchange calculation fixes $B_{g,2}$ and $B_{g,3}$. The authors also report a second-order relation, Eq. (4.19), linking the Drell-Yan large-angle soft coefficients $D_2$, the jet coefficients $B_{p,2}$, and the diagonal two-loop splitting-function constants $P^{(1)}_{p,delta}$, with all $zeta$-function terms cancelling. The practical consequence they emphasize is that the N$^3$LL corrections are much smaller than the N$^2$LL ones and that the tower expansion reproduces the exponentiated result except very close to threshold.

Load-bearing premise

The extraction assumes the process-dependent large-angle soft term for DIS vanishes to all orders, $D_{DIS}=0$, as proved in Refs. [23,24]; if that term is nonzero at three loops, the extracted $B_{q,3}$ would absorb it and no longer be a pure jet-function coefficient.

Editorial extensions

If this is right

  • Inclusive DIS threshold resummation can be carried to N$^3$LL accuracy, with the new $g_4$ correction small at large $Q^2$ and below about 10 percent even for $a_s$ around 0.3.
  • The first seven towers of the four-loop DIS coefficient function are fixed, giving predictions for the $D_7$ through $D_2$ terms and a $D_1$ result up to the unknown $A_{q,4}$.
  • The jet-function coefficients $B_q$ and $B_g$ are now known through $a_s^3$, with $B_{g,2}$ sufficient to extend NNLL resummation to processes with final-state gluons once their process-dependent soft terms are available.
  • The second-order relation Eq. (4.19) connects splitting functions, jet functions, and Drell-Yan large-angle soft emissions, and its right-hand sides vanish as $b_0$ goes to zero.
  • The tower expansion agrees with the exponentiated result away from $x$ approaching 1, supporting the minimal-prescription treatment of the Mellin inversion.

Reading between the lines

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

  • Editorial inference: the second-order relation in Eq. (4.19) looks like a residual of a common colour-weight structure; a future four-loop calculation of the diagonal splitting functions or the DIS jet function could turn it into a sharper check of the resummation framework.
  • Editorial inference: the large third-order $B_{q,3}$ and $B_{g,3}$ suggest the series may grow factorially rather than converge smoothly; a four-loop extraction would show whether N$^4$LL remains numerically under control.
  • Editorial inference: because $B_{g,2}$ and $B_{g,3}$ are now known, the same matching strategy can be applied to any process with a final-state gluon once its process-dependent soft term is computed, extending NNLL resummation beyond DIS and Drell-Yan.
  • Editorial inference: the agreement between tower expansion and exponentiation away from $x$ approaching 1 suggests a practical split, using fixed towers for most of phase space and exponentiation only in the extreme threshold region.
Original abstract

We extend the threshold resummation exponents G^N in Mellin-N space to the fourth logarithmic (N^3LL) order collecting the terms alpha_s^2 (alpha_s ln N)^n to all orders in the strong coupling constant as. Comparing the results to our previous three-loop calculations for deep-inelastic scattering (DIS), we derive the universal coefficients B_q and B_g governing the final-state jet functions to order alpha_s^3, extending the previous quark and gluon results by one and two orders. A curious relation is found at second order between these quantities, the splitting functions and the large-angle soft emissions in Drell-Yan type processes. We study the numerical effect of the N^3LL corrections using both the fully exponentiated form and the expansion of the coefficient function in towers of logarithms.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,758 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.09 · states limitation 0.08 · human participants 0.01 · controlled comparison 0.13 · randomized assignment 0.01 · states quantitative result 0.22 · compares with prior results 0.96 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

We extend the threshold resummation exponents G^N in Mellin-N space to the fourth logarithmic (N^3LL) order collecting the terms alpha_s^2 (alpha_s ln N)^n to all orders in the strong coupling constant as. Comparing the results to our previous three-loop calculations for deep-inelastic scattering (DIS), we derive the universal coefficients B_q and B_g governing the final-state jet functions to order alpha_s^3, extending the previous quark and gluon results by one and two orders. A curious relation is found at second order between these quantities, the splitting functions and the large-angle soft emissions in Drell-Yan type processes. We study the numerical effect of the N^3LL corrections using both the fully exponentiated form and the expansion of the coefficient function in towers of logarithms.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We extend the threshold resummation exponents G^N in Mellin-N space to the fourth logarithmic (N^3LL) order collecting the terms alpha_s^2 (alpha_s ln N)^n to all orders in the strong coupling constant as. contribution 1.00 0.94 0.31 asserted 1.00
s1 Comparing the results to our previous three-loop calculations for deep-inelastic scattering (DIS), we derive the universal coefficients B_q and B_g governing the final-state jet functions to order alpha_s^3, extending the previous quark and gluon results by one and two orders. result 0.82 0.96 0.29 asserted 1.00
s2 A curious relation is found at second order between these quantities, the splitting functions and the large-angle soft emissions in Drell-Yan type processes. result 0.97 0.93 0.10 asserted 1.00
s3 We study the numerical effect of the N^3LL corrections using both the fully exponentiated form and the expansion of the coefficient function in towers of logarithms. contribution 0.73 0.95 0.12 asserted 0.97

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.98.

What was examined, and what was not

sources read
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not examined
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  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. Left: the LL, NLL, N2LL and N3LL approximations for the resummation exponent for standard DIS. Right: the convolutions of the exponentiated results with a typical input shape. 0 2 4 6 8 0.6 0.7 0.8 0.9 1 x (exp GDIS ⊗ f ) / f LL NLL N 2 LL N 3 LL αS = 0.2, Nf = 4 x (exp GDIS ⊗ f ) / f xf = x0.5 (1−x)3 αS = 0.3, Nf = 3 0 2 4 6 8 0.5 0.6 0.7 0.8 0.9
Figure 2. As the right part of the previous figure, but for a diff
Figure 3. As Fig. 1, but for inclusive DIS by exchange of a scal
Figure 4. Convolution of the two-loop (left) and three-loop
Figure 5. As the previous figure, but for large-N expansion in terms of decreasing powers of ln N. 15
Figure 6. Left: the successive approximations of the four-l

Figures reproduced from arXiv:hep-ph/0506288 by the authors.

References
How later papers used it

61 later papers cite it

Evidence: substantial. 56 of the citing papers share no author with it, and 15 of those carry a confident label from Pith's citation classifier.

Pith read 23 of the 141 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

Builds on it 1 paper
Uses its method or tool 1 of 6 papers with this label · 1 by the same authors

By the same authors

How the literature uses it

59 arXiv papers cite this in 141 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 95 sentences without a confident label

Builds on it 1 paper
Uses its method or tool 6 papers
Uses its data or values 6 papers
  • The coefficient $g_5^b$ needed for N4LL resummation can be found from the same for DIS [ref] with the relation $g_5^b() = g_5^DIS(2)$.

    Next-to-soft threshold effects on Higgs boson production via bottom quark annihilation · Inclusive production · 0.86

  • At order $α_s^2$, comparing to Eq. ([ref]) we reproduce Eq. ([ref]), and we find [equation] which fully determine the NNLL resummed result, together with the NNLO cusp anomalous dimension given in Refs. [ref] ).

    Threshold resummation of Semi-Inclusive Deep-Inelastic Scattering · Double soft · 0.83

  • Here, the universal process-independent coefficients are given by [ref] A^(1) = 2 C_F, A^(2) = 2 C_F [C_A (67/18 - ζ_2 ) - 5/9 n_f], A^(3) = 1/2 C_F [C_A^2 (245/24 - 67/9ζ_2 + 11/6ζ_3 + 11/5ζ_2^2 ) +C_F n_f (2ζ_3 - 55/24 ) + C_A n_f (10/9ζ_2 - 7/3 ζ_3 - 209/108 ) - n_f^2/27 ] and [ref] D^(2)=2C_F[C_A( -10127+113ζ_2+72ζ_3) +n_f(1427-23ζ_2)].

    Slepton pair production with aNNLO+NNLL precision · Threshold resummation at NNLL accuracy · 0.92

Compares with it 9 papers
Cites it as background 20 papers

Discussion

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