Threshold resummation for electroweak annihilation from DIS data

Eric Laenen; Lorenzo Magnea

REVIEW

Threshold resummation for electroweak annihilation from DIS data

Eric Laenen, Lorenzo Magnea

High Energy Physics - Phenomenology · Submitted 28 Aug 2005 · 34 references

Pith's verdict: A clean, well-checked derivation of the three-loop Drell-Yan resummation coefficient D(3), independently confirmed by Moch-Vogt; the only real soft spot is that the all-order relation is an ansatz validated to three loops, not a theorem.

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arXiv IDhep-ph/0508284 (version 1)

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CitationEric Laenen, Lorenzo Magnea. "Threshold resummation for electroweak annihilation from DIS data." arXiv:hep-ph/0508284 (2005). https://pith.science/paper/OQSMUWHS

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Editor's note

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The thing to know: this paper gets D(3) for Drell-Yan threshold resummation out of three-loop DIS data, and it is right. Moch and Vogt computed the same coefficient by a different route and the authors say so in the Note added. That kind of independent agreement is worth more than any amount of internal consistency checking, and it settles the main result. If you do N3LL Drell-Yan or Higgs phenomenology, this is the paper that gives you the last coefficient you were missing, up to the four-loop cusp anomalous dimension, which everyone already treats as negligible numerically. I would cite it without hesitation. The all-order relation Eq. (4.4), D = 4 B_delta - 2 Gtilde + beta-hat d/dalpha F_MS, is the part with lasting value; it turns a Drell-Yan calculation into a DIS calculation, which is a genuinely useful reduction. The checks are the right ones: D(1) = 0 and D(2) against Hamberg-van Neerven-Matsuura, and the nf^2 term of D(3) against the Beneke-Braun and Gardi renormalon results. Nothing is fitted to the target. Where I would push back is on the framing of Eq. (4.4). The paper calls it an all-order ansatz, and that is honest, but the reparametrization argument below it (define a modified quark density including eikonal effects, poles still cancel) is a plausibility argument for the finite remainder, not a proof. The phi_V/phi_R split inherited from Ref. [5] is more constrained than a casual reader might think, since Gtilde is pinned by pole cancellation plus the pure-counterterm condition, but the finite N-independent pieces are where a misassignment would bite, and that is checked only at one and two loops. The gluon extension is flagged by the authors as an argument checked to three loops, not a proof, and they are right to flag it; the CF -> CA replacement is exactly the kind of thing that tends to fail once d-tensor Casimirs appear. Neither of these lands on D(3). One small thing: the erratum note on Eq. (4.6) of Ref. [5] is buried in a footnote and matters to anyone reusing G(2)_0. This is a letter, so the pole-cancellation chain is compressed; a referee should be willing to redo a few of the intermediate steps. Send it out. It is short, the claim is sharp, and the confirmation is already in the manuscript.

Referee report

Significance

If correct, this extends threshold resummation for Drell-Yan and Higgs production to N3LL accuracy, short of only the four-loop cusp anomalous dimension, which is plausibly negligible. The result is a genuine parameter-free derivation: no coefficient is fitted to the target, all inputs are published DIS and Drell-Yan quantities, and the output is checked three independent ways — the one- and two-loop coefficients reproduce known direct calculations, the n_f^2 term of D^(3) matches the renormalon analyses of Beneke–Braun and Gardi, and the full D^(3) agrees with the independent Moch–Vogt computation reported in the Note added. The all-order relation Eq. (4.4) is a compact and reusable structural statement, and the paper is commendably explicit about which step is an ansatz rather than a theorem.

Major objections (2)

[Section 4, Eq. (4.4)] The relation Eq. (4.4) is presented as an 'all-order ansatz' and is verified at one, two, and three loops, but the step from the three-loop constraints of Eqs. (4.1)–(4.3) to the all-order form is justified by the reparametrization argument that 'one could define a modified quark density including eikonal effects, and poles would still cancel.' That argument establishes that the pole structure is insensitive to the φ_V/φ_R split, but it does not by itself prove that the finite N-independent remainder organizes into the β̂ dF_MS/dα_s term at every order. The authors are honest about this, and the three-loop coefficient itself is independently confirmed, so this is not a correctness problem for D^(3); I would ask, however, that the abstract and the opening of Section 4 state the scope explicitly — Eq. (4.4) is established through three loops and conjectured beyond — so that a reader does not take the all-order form as proved. A sentence identifying what a four-loop check would require would also help.
[Section 4, Eq. (4.6)] The two-loop constant F_MS^(2) entering Eq. (4.3) is obtained 'by comparing our exponentiated expression with the two-loop calculation of Ref. [10],' but the comparison is not spelled out: the reader is not told which coefficients of the two-loop Drell-Yan cross section (which powers of D_k(N) and which N-independent terms) were matched, nor in which scheme the matching was performed. Since F_MS^(2) enters the determination of D^(3) linearly through Eq. (4.3), a short explicit statement of the matching — even one line giving the relation between F_MS^(2) and the corresponding coefficient of Ref. [10] — would make the derivation reproducible without redoing the comparison. The final agreement with Moch–Vogt mitigates the risk, but the intermediate step should be transparent.

Minor objections (6)

[Abstract and Eq. (4.7)] The abstract writes the three-loop coefficient as D_3 while the body consistently uses D^(3); the notation should be unified, and it would help to distinguish it clearly from the distributions D_k(N) defined in Eq. (1.1), which use the same letter.
[Section 2, Eq. (2.11)] The constant F_MS(α_s) is scheme dependent, as is D(α_s) itself. The paper states the MSbar scheme repeatedly but a one-sentence reminder at Eq. (2.11) that both F_MS and D carry scheme dependence, and that Eq. (4.4) is a scheme-specific relation, would prevent misapplication.
[Section 4, footnote 1] The correction of the misprint in Eq. (4.6) of Ref. [5] is useful. It would be worth stating explicitly whether the corrected coefficient of C_A C_F in G_0^(2) changes any numerical results previously published using the erroneous value.
[Section 5] The speculation that the C_F → C_A replacement for D_g may fail at higher orders once symmetric-tensor Casimirs d^{abc} contribute is physically reasonable but unsupported; since the Note added confirms the three-loop gluon result independently, the paragraph could be shortened to state the three-loop agreement and flag the higher-order question in a single sentence.
[Section 5] The letter argues that the new coefficient will be useful for LHC phenomenology but gives no indication of its numerical size or of the size of the N3LL correction it induces. A brief numerical remark, even at the level of the coefficient's magnitude relative to D^(2), would make the phenomenological motivation concrete.
[Whole manuscript] There are a number of typographical and typesetting errors in the arXiv version that should be corrected in proof, among them 'th reshold' and 'comput e' in the abstract, 'arising' and 'arbitariness' in Section 2, 'iself' and 'aswer' in Section 3, 'or result' in Section 4, and several broken words in the running text.

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The paper

The paper claims

The last unknown coefficient in Drell-Yan threshold resummation can be read off from deep inelastic scattering, and evaluated at three loops.

The reading

Threshold resummation reorganises the QCD prediction for Drell-Yan-like cross sections into an exponentiated tower of logarithms of $N$, the Mellin moment variable. Each tower is fixed by a small set of anomalous dimensions and coefficient functions, and the single-logarithmic tower is governed by a function $D(α_s)$ that had been known only to two loops. This paper argues that $D(α_s)$ is not an independent quantity at all: it is fixed by the virtual (delta-function) part of the nonsinglet splitting function, by the infrared pole coefficients of the quark form factor, and by the $N$-independent constants of the Drell-Yan cross section at one lower order. Using the three-loop splitting functions and the three-loop quark form factor obtained from DIS, the authors evaluate $D^{(3)}$ explicitly, which is what is needed to extend Drell-Yan resummation to N$^3$LL accuracy apart from the still-unknown four-loop cusp anomalous dimension.

What carries the argument

The load-bearing device is a split of the MS-bar mass-factorization counterterm into a virtual piece and a real piece, $φ_{MS}(N,ε) = φ_V(ε) φ_R(N,ε)$, where $φ_V$ is deliberately constructed to mimic the quark form factor and to cancel all of its infrared and collinear poles exactly. That construction uniquely fixes the auxiliary function $G̃(α_s)$, and with it the finite remainder. The unsubtracted cross section is then written as a product of a virtual factor and a real factor, each finite on its own, and the requirement that the real factor be finite order by order in $α_s$ generates the constraints that relate $D$ to $B_δ$, $G̃$, and $F_{MS}$. The authors stress that the detailed split between the soft eikonal function and the quark distribution drops out of the final relation, so the identity is robust against how one apportions real and virtual contributions.

Extended reading notes

Core claim

The paper's central claim is the all-order identity $D(α_s) = 4 B_δ(α_s) - 2 G̃(α_s) + β̂(α_s) dF_{MS}(α_s)/dα_s$ (Eq. 4.4), where $B_δ$ is the coefficient of $δ(1-x)$ in the nonsinglet splitting function, $G̃$ is the finite part of the quark form factor's scale-dependent anomalous dimension after the virtual subtraction is fixed, $F_{MS}$ collects the exponentiated $N$-independent Drell-Yan constants, and $β̂$ is the QCD $\beta$ function. The identity says that the single-logarithmic resummation coefficient for electroweak annihilation is not new physics but a bookkeeping of DIS quantities plus lower-order Drell-Yan information. Feeding in the three-loop splitting functions and the three-loop quark form factor, the authors obtain $D^{(3)}$ in closed form (Eq. 4.7), a result whose highest power of $n_f$ independently matches existing renormalon calculations. A note added at the end records that a parallel calculation by other authors, using a different line of argument, agrees with Eq. (4.7) for both quark- and gluon-initiated scattering.

Load-bearing premise

The derivation depends on splitting the MS-bar counterterm into a virtual factor and a real factor, with the virtual factor chosen to cancel the quark form factor's infrared poles exactly; that choice fixes the auxiliary function $G̃$ and hence $D$, so if finite $N$-independent pieces were misassigned between the two factors, the value of $D^{(3)}$ would move.

Editorial extensions

If this is right

Threshold resummation for Drell-Yan in the MS-bar scheme can be carried to N$^3$LL accuracy, with the only remaining unknown being the four-loop cusp anomalous dimension.
The same identity, with the quark replaced by a gluon and $C_F$ replaced by $C_A$, gives N$^3$LL resummation for Higgs production via gluon fusion in the large-top-mass effective theory.
Because the highest power of $n_f$ in $D^{(3)}$ matches renormalon expectations, the result supplies an independent cross-check between the resummation and renormalon pictures of the same series.
The result makes the resummed prediction for $Z^0$ and Higgs production at the LHC more precise, and allows the convergence of the tower expansion itself to be tested rather than assumed.

Reading between the lines

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

If the identity holds to all orders, the single-logarithmic tower of any electroweak annihilation process is a derived quantity, so future resummation work could focus on the cusp anomalous dimension and the $N$-independent constants rather than on new process-specific coefficients.
The pattern suggests a practical strategy for other processes at hadron colliders: identify the analogous virtual subtraction, then read the single-log coefficient off from DIS splitting functions and the relevant form factor, rather than computing it diagram by diagram.
The argument that the $C_F \to C_A$ replacement survives at three loops rests on isolating the maximally nonabelian part of the form factor; whether this survives at four loops, where symmetric higher-rank Casimir structures can enter, is an open question the authors flag but do not settle.
A numerical study comparing the N$^3$LL resummed Drell-Yan cross section with the fixed-order NNLO result, and with the residual scale variation, would show whether the new tower actually shrinks the theoretical uncertainty at LHC energies.

Original abstract

We show that higher-order coefficients required to perform threshold resummation for electroweak annihilation processes, such as Drell-Yan or Higgs production via gluon fusion, can be computed using perturbative results derived in Deep Inelastic Scattering. As an example, we compute the three-loop coefficient D_3, generating most of the fourth tower of threshold logarithms for the Drell-Yan cross section in the MSbar scheme, using the recent three-loop results for splitting functions and for the quark form factor, as well as a class of exponentiating two-loop contributions to the Drell-Yan process.

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We show that higher-order coefficients required to perform threshold resummation for electroweak annihilation processes, such as Drell-Yan or Higgs production via gluon fusion, can be computed using perturbative results derived in Deep Inelastic Scattering. As an example, we compute the three-loop coefficient D_3, generating most of the fourth tower of threshold logarithms for the Drell-Yan cross section in the MSbar scheme, using the recent three-loop results for splitting functions and for the quark form factor, as well as a class of exponentiating two-loop contributions to the Drell-Yan process.

background contribution method result interpretation main claim

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We show that higher-order coefficients required to perform threshold resummation for electroweak annihilation processes, such as Drell-Yan or Higgs production via gluon fusion, can be computed using perturbative results derived in Deep Inelastic Scattering.	contribution 0.91	0.78	0.87	asserted 1.00
s1	As an example, we compute the three-loop coefficient D_3, generating most of the fourth tower of threshold logarithms for the Drell-Yan cross section in the MSbar scheme, using the recent three-loop results for splitting functions and for the quark form factor, as well as a class of exponentiating two-loop contributions to the Drell-Yan process.	contribution 0.60	0.94	0.18	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.90 · evidence from proof or derivation 0.50.

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References

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Soft gluon resummation for gluon-induced Higgs Strahlung
hep-ph 2014-10 accept novelty 5.0 of 10

Threshold resummation at next-to-leading-log accuracy reduces the scale uncertainty of gluon-induced ZH production at the LHC by about a factor of two.
Measurements of fiducial and differential cross sections for Higgs boson production in the diphoton decay channel at $\sqrt{s}=8$ TeV with ATLAS
hep-ex 2014-07 accept novelty 5.0 of 10

ATLAS measures the pp to H to two-photon fiducial cross section as 43.2 +/- 9.4 (stat) fb and twelve differential distributions, all broadly consistent with Standard Model predictions within the statistical precision.
N$^3$LO Higgs and Drell-Yan production at threshold: the one-loop two-emission contribution
hep-ph 2014-04 conditional novelty 5.0 of 10

The authors compute the two-emission, one-loop real-virtual contribution to the three-loop soft function for threshold Higgs and Drell-Yan production, giving all nine master integrals to all orders in the dimensional ...
A next-to-next-to-leading order calculation of soft-virtual cross sections
hep-ph 2012-09 accept novelty 5.0 of 10

A universal, process-independent expression for the NNLO soft-virtual QCD corrections to colourless-final-state production, valid to all orders in the dimensional regulator, together with the hard coefficient for NNLL...
HiggsBounds 2.0.0: Confronting Neutral and Charged Higgs Sector Predictions with Exclusion Bounds from LEP and the Tevatron
hep-ph 2011-02 conditional novelty 5.0 of 10

Version 2.0.0 of the HiggsBounds code extends model-independent Higgs exclusion testing to charged Higgs bosons and 81 LEP and Tevatron analyses, then applies it to invisible Higgs decays and the Randall-Sundrum scala...
Renormalization-Group Improved Prediction for Higgs Production at Hadron Colliders
hep-ph 2008-09 conditional novelty 5.0 of 10

Resumming pi-squared-enhanced corrections via a time-like matching scale raises the NNLO Higgs production cross section by 8% at the LHC and 13% at the Tevatron for m_H = 120 GeV.
Threshold Resummation for Higgs Production in Effective Field Theory
hep-ph 2005-09 conditional novelty 5.0 of 10

An effective-field-theory route reproduces the known next-to-next-to-next-to-leading-log threshold resummation for Higgs production, confirming the standard QCD result by an independent derivation.
NLO Inspired Effective Lagrangians for Higgs Physics
hep-ph 2012-09 conditional novelty 4.0 of 10

A dimension-six effective-Lagrangian strategy for Higgs couplings is set out and audited, with an admissibility rule that keeps operator insertions from spoiling ultraviolet finiteness.
Handbook of LHC Higgs Cross Sections: 1. Inclusive Observables
hep-ph 2011-01 accept novelty 4.0 of 10

The report provides a harmonized set of inclusive LHC Higgs cross sections and uncertainty estimates for Standard Model and MSSM production modes at 7 and 14 TeV.
Handbook of LHC Higgs Cross Sections: 4. Deciphering the Nature of the Higgs Sector
hep-ph 2016-10 accept novelty 3.0 of 10

The report fixes the community's baseline numbers for Higgs boson production and decay at the 13 TeV LHC, including a 48.6 pb gluon-fusion cross section at N3LO with an explicit uncertainty budget.
Parton Distributions: Summary Report
hep-ph 2005-11 accept novelty 3.0 of 10

The report concludes that PDF uncertainties on LHC W and Z rapidity spectra sit near 8% at central rapidity, dominated by the low-x gluon, and that cross-section ratios cut this to roughly 4%.
QCD resummation for hadronic final states
hep-ph 2015-05 accept novelty 2.0 of 10

The review consolidates the direct-QCD resummation program: soft and collinear factorization, NLL event-shape resummation via Caesar, non-global logarithms, and groomed jet substructure, with the factorization-breakin...

How the literature uses it

48 arXiv papers cite this in 70 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 · 47 sentences without a confident label

Uses its method or tool 3 papers

For this, we use the approach of well-established SV threshold resummation [ref] and recently introduced NSV resummation in [ref] for resumming the large logarithms for the total cross-section and rapidity distribution, respectively.

Next-to-soft threshold effects on Higgs boson production via bottom quark annihilation · Introduction · 0.97
We extract all the process-dependent and universal coefficients needed for N$^3$LL resummation following the formalism developed in [ref] .

Threshold enhanced cross sections for colorless productions · Introduction · 0.97
The calculation of $σ(gg →h)_SM$ is based on [ref] , which combines the next-to-next-to-leading fixed-order corrections [ref] with resummation of both threshold logarithms from soft-gluon emission [ref] and terms of the form $(N_c πα_s )^n$ [ref] .

The Custodial Randall-Sundrum Model: From Precision Tests to Higgs Physics · Higgs-Boson Production · 0.87

Uses its data or values 1 paper

The ingredients for N$^3$LL$^ $ resummation are all available in the literature~ [ref] , including the $ (1-z)$ term at N$^3$LO which is taken from Ref.~ [ref] and the recent four-loop cusp anomalous dimension computed in Ref.~ [ref] .

On the approaches to threshold resummation of rapidity distributions for the Drell-Yan process · All-order resummed results for BNX/BFR · 0.82

Compares with it 2 papers

Then eq.([ref]) becomes [equation] which should be equivalent to eq.(4.4) in [ref] , and reproduces eq.(36) in [ref] .

Constant terms in threshold resummation and the quark form factor · Threshold resummation of the physical anomalous dimension (DY case) · 0.96
Using this result, we reproduce the perturbative expression for the radiation function $D(α_s)$ given in [ref] up to third order in $α_s$.

Renormalization-Group Improved Prediction for Higgs Production at Hadron Colliders · Renormalization-group analysis and resummation · 0.84

Cites it as background 15 papers

A complete calculation at NNLO [ref] and leading logarithms at N$^3$LO in the threshold limit [ref] and NNLL soft gluon resummation [ref] for the scalar Higgs boson production are known for more than a decade.

Pseudo-scalar Higgs Boson Production at Threshold N$^3$LO and N$^3$LL QCD · Introduction · 0.87
The fixed order cross-section is further improved by performing threshold resummation at the next-to-next-to-leading logarithmic (NNLL) accuracy [ref] and to the third logarithmic accuracy (N3LL) [ref] as well as resummation of $π^2$ terms [ref] arising from time-like Sudakov form factor.

Higgs boson rapidity distribution in bottom annihilation at NNLL and beyond · Introduction · 0.87
In the case of the inclusive GF cross section, one of the most studied and well-developed all-order techniques is threshold (or large $x$) resummation [ref] , which has been pushed to next-to-next-to-next-to-leading logarithmic (N$^3$LL) accuracy [ref] ,

Double resummation for Higgs production · Introduction · 0.81

Discussion

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