Decoupling Relations to O(alpha_s^3) and their Connection to Low-Energy Theorems

B.A. Kniehl; K.G. Chetyrkin; M. Steinhauser (MPI Munich)

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Decoupling Relations to O(alpha_s^3) and their Connection to Low-Energy Theorems

K.G. Chetyrkin, B.A. Kniehl, M. Steinhauser (MPI Munich)

High Energy Physics - Phenomenology · Submitted 5 Aug 1997 · 39 references

Pith's verdict: The standard reference for three-loop decoupling and four-loop C1; the vacuum-integral reduction is the real advance, and the one soft spot is a compressed step in the LET derivation.

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CitationB.A. Kniehl, K.G. Chetyrkin, M. Steinhauser (MPI Munich). "Decoupling Relations to O(alpha_s^3) and their Connection to Low-Energy Theorems." arXiv:hep-ph/9708255 (1997). https://pith.science/paper/KWQUBDMI

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Bottom line: this is the paper people cite for three-loop decoupling constants and the four-loop Higgs-gluon coefficient, and it earns that. The technical move — reducing ζg and ζm to massive vacuum integrals via Eqs. (11)–(19) — is what made the three-loop calculation tractable at all, and it is the part I would hand a student. The all-orders LETs (39) and (42) are clean and do real work: they turn four-loop C1 and C2 into derivatives of the three-loop ζ's plus the published four-loop β and γm. The photon extension in Section 5 is a sensible bonus, not padding. The cross-checks are thorough and I take them seriously: ξ-independence of the ζm combination, agreement with [19] and the revised [5] at two loops, agreement of C1 at O(αs³) with the direct diagrammatic [12], and the [26]/[29]/[31] comparisons. The paper is also honest about what comes from the companion [8] and [12] versus what is new here. The one place I would push: the step from Eqs. (35)–(36) to Eq. (37) is asserted, not shown. Combining (35), (36c), (36d) directly and using Eq. (18) gives −2C1 = −a/4 − b/2 + c/2 with a = ∂ ln ζ3, b = ∂ ln ζ̃3, c = ∂ ln ζ̃1, whereas ∂ ln(ζg²) = a + 2b + 2c. These agree only with an extra identity among a, b, c — presumably a Slavnov-Taylor relation at zero momentum — that the text never states. At O(αs) the naive combination even gives the wrong sign relative to the known C1, so the printed equations cannot be combined the way the sentence suggests. The final result is right (it matches [12]), so this is a transparency problem, not a correctness one, but it is exactly the step a referee should ask them to spell out. Minor: the operator-basis completeness in Eq. (27) is assumed rather than argued, which is standard but worth a sentence; and the remark about being unable to locate a definition of βt in [36] is fair but reads a little pointed. Who should read it: anyone doing threshold matching, αs running, or Higgs production at NNLO and beyond. It deserves a serious referee and I would cite it without hesitation.

Referee report

Significance

The three-loop decoupling constants are a genuine and needed ingredient for consistent four-loop running across flavour thresholds, and the paper demonstrates this with the bottom-threshold evolution study of Figs. 3 and 4. The all-orders low-energy theorems are conceptually valuable: they reduce the heavy-quark contributions to the ggH and qq̄H effective couplings to derivatives of quantities already computed, and the resulting four-loop C_1 and C_2 are directly usable in Higgs production and decay phenomenology. The results are cross-checked against independent calculations where available (the O(α_s^2) ζ_m against Ref. [5], ζ_g against Ref. [19] and the revised Ref. [5], and the O(α_s^3) C_1 against the direct diagrammatic result of Ref. [12]), which materially strengthens confidence in the numbers. The reduction of the problem to vacuum integrals is a real technical advance over earlier methods that would have required diagrams with nonvanishing external momenta or even four loops.

Major objections (3)

[§4, Eqs. (35)–(39)] The derivation of the central all-orders low-energy theorem, Eq. (39), is compressed to the point that the reader cannot verify it. The text states that solving Eqs. (35) and (36) for C_1^0 yields Eq. (37), −2C_1^0 = ∂_{0h} ln(ζ_g^0)^2, but the algebra connecting the five relations in Eqs. (35)–(36) to this single relation is not shown. A direct combination of the displayed equations does not obviously reproduce Eq. (37) without an additional relation among the gluon, ghost, and gluon-ghost-vertex decoupling constants (the analogue of a Ward/Slavnov-Taylor identity). Since Eq. (39) is the load-bearing result of the paper, the authors should either display the intermediate steps or state explicitly the identity that makes the combination collapse. The agreement of the final C_1 with Ref. [12] at O(α_s^3) is reassuring for the number, but it does not by itself establish the all-orders statement that the abstract advertises.
[§4, Eq. (27) and the paragraph preceding Eq. (33)] The derivation of the low-energy theorems rests on the claim that, in the m_h → ∞ limit, the heavy-quark operator O_h = m_h h̄h can be written as a linear combination of the operators O'_1 through O'_5 of Eq. (27). The completeness of this basis of scalar dimension-four operators is asserted rather than argued. Because the all-orders validity of Eqs. (39) and (42) depends on no operator being omitted, the authors should add a sentence or two justifying completeness (or citing where it is established), and should state explicitly which operators can and cannot mix under the symmetries preserved by the effective theory.
[§6, Eq. (60) and the discussion of Ref. [36]] The comparison with Ref. [36] is left in a state that is hard for the reader to adjudicate. The authors state that the O(α_s^3) formula for κ_{h,H} in the revised Ref. [36] agrees with their Eq. (43) to this order, but then note that they cannot locate a proper general definition of β_t(α_s) in the literature and that it remains unclear whether Eq. (8) of Ref. [36] agrees with Eq. (41) in higher orders. Since this bears on whether the two formalisms are equivalent beyond the order checked, the authors should sharpen the statement: either exhibit the agreement explicitly at O(α_s^3) with the scale choices of Eq. (60), or state plainly that the equivalence is established only through that order and identify what would be needed to extend it.

Minor objections (6)

[§3, Eq. (20) and following] The text notes that the constant term in O(α_s^2) of ζ_g slightly differs from the original Ref. [5] and that the authors of Ref. [5] have since revised their analysis to agree with Ref. [19]. It would help the reader to state in one place which of the earlier results are superseded and which are confirmed, rather than distributing this information between §3 and the references.
[§3, Eq. (23) and Eq. (24)] The numerical coefficients in the approximate expressions are quoted at μ = μ_h and μ = M_h respectively; the text should state explicitly, at each equation, the numerical value of α_s and the number of flavours used, so that the approximate forms can be reproduced without reverse-engineering.
[§2, Eq. (12)] The simplification ζ_m^0 = 1 − Σ_V^{0h}(0) − Σ_S^{0h}(0) is stated to hold through three loops; it would be useful to say briefly why the denominator expansion truncates at this order, since the exact expression is a ratio.
[§5, Eq. (50)] The O(ᾱ) term of ζ_{gγ}^2 carries a factor Q_h^2 while the O(ᾱα_s) term also carries Q_h^2; the text should clarify that the leading QED term is retained at O(ᾱ) only and that higher QED powers are dropped, so that the counting in ᾱ is unambiguous.
[Appendix B, Eq. (B.2)] The gauge-dependent constants ζ_2 and ζ_3 are given in the covariant gauge with the parameter ξ defined by Eq. (5); it would be helpful to repeat that convention in the appendix caption so the expressions are self-contained.
[§6, Figs. 3 and 4] The figures would be easier to read if the matching scale at which the curves are anchored were marked, and if the input values (α_s^{(4)}(M_τ) = 0.36, M_b = 4.7 GeV, μ_c = 1.2 GeV, α_s^{(5)}(M_Z) = 0.118) were repeated in the captions rather than only in the body text.

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The paper

The paper claims

The paper claims that the heavy-quark corrections to the Higgs boson's couplings to gluons and light quarks are exactly the logarithmic derivatives of the QCD decoupling constants with respect to the heavy-quark mass, and uses that identity

The reading

In QCD renormalized by minimal subtraction, the strong coupling and the light-quark masses jump discontinuously when a heavy-flavour threshold is crossed, and the size of the jump is carried by two matching constants, $Z_g$ and $Z_m$ (written $Z$ here as the decoupling constants $Z_g$, $Z_m$). The paper's first aim is to compute these constants through three loops, which needs a real three-loop calculation rather than the two-loop bookkeeping available before. Its second and larger aim is to show that the same constants already contain everything about how a heavy quark alters the Higgs boson's couplings to gluons and to light quarks: those effective couplings are the logarithmic derivatives of the decoupling constants with respect to the heavy-quark mass. If that identity holds to all orders, then a three-loop matching calculation combined with the known four-loop anomalous dimensions of QCD yields the Higgs-gluon and Higgs-quark coefficients through four loops without any new four-loop Feynman diagrams. The payoff is a consistent four-loop description of threshold crossing, which the authors illustrate by showing that the predicted $Z$-scheme values of $Z$ and $Z$ become nearly independent of where the bottom threshold is placed.

What carries the argument

The load-bearing device is the low-energy theorem that converts a zero-momentum Higgs insertion into a mass derivative. Because the Higgs couples proportionally to the heavy-quark mass, differentiating the heavy-quark propagator with respect to $m_h^2$ generates exactly the Green function with one zero-momentum operator insertion. Combined with a complete basis of scalar dimension-four operators built from light fields only, and with the statement that every scaleless tadpole vanishes in dimensional regularization, this collapses the matching equation to $-2C_1 = partial_h ln Z_g^2$ and its quark-mass analogue. The supporting machinery is the reduction of the decoupling constants themselves to massive tadpole integrals at zero external momentum, which is what makes the three-loop evaluation feasible with the available tools.

Extended reading notes

Core claim

The paper makes two linked claims. The technical one is that the bare decoupling constants of QCD can be extracted from the hard parts of light-particle self-energies and vertices evaluated at vanishing external momentum, for example $Z_2^0 = 1+Sigma_V^{0h}(0)$ and $Z_3^0 = 1+Pi_G^{0h}(0)$, with the coupling constant assembled from the ghost and gluon pieces. Because the external momenta are set to zero, only massive tadpole integrals survive, and that is what makes three loops tractable at all. The physical one is that the coefficient functions $C_1$ and $C_2$ of the heavy-quark effective Lagrangian obey $-2C_1 = partial_h ln Z_g^2$ and $C_2 = 1 + 2 partial_h ln Z_m$ to all orders in the strong coupling, where $partial_h$ is the logarithmic derivative with respect to the heavy-quark mass. The authors give the explicit three-loop decoupling constants and, from them, $C_1$ and $C_2$ through order $alpha_s^4$, plus the analogous photon case. The claim is that the entire virtual heavy-quark content of the Higgs-gluon and Higgs-quark couplings sits in the two matching constants and can be released by one derivative.

Load-bearing premise

The derivation assumes that in dimensional regularization every integral without a mass scale vanishes, so that in the infinite-heavy-mass limit only the heavy-quark part of each Green function survives and all higher-order terms on the right-hand side of the matching equation drop out.

Editorial extensions

If this is right

The gluon-fusion Higgs production coefficient is obtained through order $alpha_s^4$ in the heavy-top effective theory, extending the previously known three-loop result by one order.
The Higgs-to-light-quark coefficient is obtained through the same order, giving the four-loop Yukawa matching needed for Higgs decays and for light-quark mass terms in the effective Lagrangian.
Four-loop running combined with three-loop matching makes the relation between low- and high-scale couplings nearly independent of the matching scale, which the authors demonstrate numerically for the bottom threshold.
The same construction carries over to the electromagnetic case, producing the three-loop decoupling constant for the fine-structure constant and the four-loop coefficient for the Higgs-photon-photon coupling.
Decoupling constants for the quark and gluon fields in the covariant gauge are supplied as a by-product for calculations with external quarks and gluons.

Reading between the lines

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

Because $C_1$ is a pure mass derivative of the matching constant, any future four-loop tadpole computation of the decoupling constant would immediately upgrade the Higgs-gluon coefficient to five loops; the diagrammatic cost of the Higgs coupling is shifted entirely onto the matching constant.
The identity hints that the heavy-quark contribution to the QCD beta function at vanishing momentum transfer, which the authors discuss and say they could not find a proper general definition for in the literature, might be definable as a difference between the effective and full beta functions; that remains an open point rather than a result here.
The derivative structure relies only on the Higgs coupling being proportional to the fermion mass, so it should extend to any additional CP-even scalar with the same Yukawa structure; a useful cross-check would be to run the construction for a second heavy quark flavour.
The residual matching-scale dependence at four loops is a ready-made diagnostic for the size of the neglected five-loop terms, and could be quoted as a theory uncertainty in threshold-crossing analyses.

Original abstract

If quantum chromodynamics (QCD) is renormalized by minimal subtraction (MS), at higher orders, the strong coupling constant alpha_s and the quark masses m_q exhibit discontinuities at the flavour thresholds, which are controlled by so-called decoupling constants, zeta_g and zeta_m, respectively. Adopting the modified MS (MS-bar) scheme, we derive simple formulae which reduce the calculation of zeta_g and zeta_m to the solution of vacuum integrals. This allows us to evaluate zeta_g and zeta_m through three loops. We also establish low-energy theorems, valid to all orders, which relate the effective couplings of the Higgs boson to gluons and light quarks, due to the virtual presence of a heavy quark h, to the logarithmic derivatives w.r.t. m_h of zeta_g and zeta_m, respectively. Fully exploiting present knowledge of the anomalous dimensions of alpha_s and m_q, we thus calculate these effective couplings through four loops. Finally, we perform a similar analysis for the coupling of the Higgs boson to photons.

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The abstract, sentence by sentence

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

If quantum chromodynamics (QCD) is renormalized by minimal subtraction (MS), at higher orders, the strong coupling constant alpha_s and the quark masses m_q exhibit discontinuities at the flavour thresholds, which are controlled by so-called decoupling constants, zeta_g and zeta_m, respectively. Adopting the modified MS (MS-bar) scheme, we derive simple formulae which reduce the calculation of zeta_g and zeta_m to the solution of vacuum integrals. This allows us to evaluate zeta_g and zeta_m through three loops. We also establish low-energy theorems, valid to all orders, which relate the effective couplings of the Higgs boson to gluons and light quarks, due to the virtual presence of a heavy quark h, to the logarithmic derivatives w.r.t. m_h of zeta_g and zeta_m, respectively. Fully exploiting present knowledge of the anomalous dimensions of alpha_s and m_q, we thus calculate these effective couplings through four loops. Finally, we perform a similar analysis for the coupling of the Higgs boson to photons.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	If quantum chromodynamics (QCD) is renormalized by minimal subtraction (MS), at higher orders, the strong coupling constant alpha_s and the quark masses m_q exhibit discontinuities at the flavour thresholds, which are controlled by so-called decoupling constants, zeta_g and zeta_m, respectively.	background 1.00	0.07	0.06	asserted 1.00
s1	Adopting the modified MS (MS-bar) scheme, we derive simple formulae which reduce the calculation of zeta_g and zeta_m to the solution of vacuum integrals.	method 0.78	0.96	0.12	asserted 1.00
s2	This allows us to evaluate zeta_g and zeta_m through three loops.	result 0.45	0.93	0.11	asserted 0.97
s3	We also establish low-energy theorems, valid to all orders, which relate the effective couplings of the Higgs boson to gluons and light quarks, due to the virtual presence of a heavy quark h, to the logarithmic derivatives w.r.t. m_h of zeta_g and zeta_m, respectively.	contribution 0.67	0.92	0.31	asserted 1.00
s4	Fully exploiting present knowledge of the anomalous dimensions of alpha_s and m_q, we thus calculate these effective couplings through four loops.	result 0.74	0.95	0.24	asserted 1.00
s5	Finally, we perform a similar analysis for the coupling of the Higgs boson to photons.	contribution 0.98	0.93	0.06	asserted 0.97

What kind of paper the classifier reports this is: theoretical result 0.99 · evidence from proof or derivation 0.99.

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arXiv listing abstract v2

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full text
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P abstract

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T body sentences
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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.** Actually, through three loops, Eq. (12) simplifies to

**Figure 2.** Typical three-loop diagrams pertinent to Π

**Figure 3.** µ (5) dependence of α (5) s (MZ) calculated from α (4) s (Mτ ) = 0.36 and Mb = 4.7 GeV using Eq. (1) at one (dotted), two (dashed), three (dot-dashed), and four (solid) loops in connection with Eqs. (7) and (24) at the respective orders. 0.36 [34], then obtain α (5) s (µ (5)) from Eqs. (7) and (24) with Mb = 4.7 GeV, and finally compute α (5) s (MZ) with Eq. (1). For consistency, N-loop evolution must be accompanied by (N − 1)-loop matching, i.e. if we omit terms of O(α N+2 s ) on the right-hand side of Eq. (1), we need to discard those of O(α N s ) in Eq. (24) at the same time. In

**Figure 4.** µ (5) dependence of m(5) c (MZ) calculated from µc = m(4) c (µc) = 1.2 GeV, Mb = 4.7 GeV, and α (5) s (MZ) = 0.118 using Eq. (59) at one (dotted), two (dashed), three (dotdashed), and four (solid) loops in connection with Eqs. (8) and (22) at the respective orders. Similar analyses may be performed for the light-quark masses as well. For illustration, let us investigate how the µ (5) dependence of the relation between µc = m(4) c (µc) and m(5) c (MZ) changes under the inclusion of higher orders in evolution and matching. As typical input parameters, we choose µc = 1.2 GeV, Mb = 4.7…

Figures reproduced from arXiv:hep-ph/9708255 by the authors.

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Precise predictions for $t \bar t H$ production at the LHC: inclusive cross section and differential distributions
hep-ph 2024-11 accept novelty 8.0 of 10

First fully differential NNLO QCD predictions for ttH production at the LHC, with the missing two-loop virtual part estimated by two complementary approximations and merged with NLO electroweak corrections.
Higgs Boson Production at Hadron Colliders at N3LO in QCD
hep-ph 2018-02 accept novelty 8.0 of 10

The paper computes the exact next-to-next-to-next-to-leading-order QCD correction to Higgs boson production via gluon fusion, the first exact hadron collider cross section at this order and the first to require ellipt...
Five-Loop Running of the QCD coupling constant
hep-ph 2016-06 accept novelty 8.0 of 10

The five-loop coefficient of the QCD beta function is computed analytically, completing the running of the strong coupling to fifth order in perturbation theory.
Top Quark Mediated Higgs Decay into Hadrons to Order alpha(s)**5
hep-ph 2006-04 conditional novelty 8.0 of 10

At MH=120 GeV the N3LO QCD correction factor for H to gg is 1 + 0.650 + 0.201 + 0.018, with the new alpha_s^3 term only one-tenth the size of the previous correction.
Analytic two-loop corrections to top-quark pair production at hadron colliders
hep-ph 2026-10 conditional novelty 7.0 of 10

The full two-loop QCD amplitudes for quark-antiquark and gluon-fusion top-pair production are obtained in epsilon-factorised analytic form, including the first genus-two iterated integrals in a four-dimensional finite...

Show 55 more citing papers

Dyeing form factors as amplitudes
hep-th 2026-06 unverdicted novelty 7.0 of 10

A dyeing procedure maps form factors to colored amplitudes, recovering the originals by bleaching and explaining double-copy features through standard BCJ relations.
Two-loop leading-color QCD corrections for Higgs plus two-jet production in the heavy-top limit
hep-ph 2026-05 unverdicted novelty 7.0 of 10

Two-loop leading-color helicity amplitudes for H+2 jets in the heavy-top limit are computed analytically and validated, enabling NNLO phenomenology and revealing an anomalous threshold.
Higgs boson decay to massive bottom quarks at order $\alpha_s^4$ induced by top-quark Yukawa couplings
hep-ph 2026-03 accept novelty 7.0 of 10

The O(α_s^4) top-Yukawa-induced correction to H→bb is computed analytically, increasing the width by 0.4% and lowering scale uncertainty to 0.4%.
Fully differential Higgs boson pair production at N$^3$LO with top quark mass effects
hep-ph 2026-01 conditional novelty 7.0 of 10

First fully differential N3LO QCD predictions for gg->hh in the heavy-top limit, with NLO top-mass effects added; heavy-top scale uncertainty shrinks about 3x, to roughly 1-3%.
Beyond Scale Variations: Perturbative Theory Uncertainties from Nuisance Parameters
hep-ph 2024-11 conditional novelty 7.0 of 10

A framework that turns missing higher-order perturbative coefficients into fit-able theory nuisance parameters, giving correlated and statistically meaningful theory uncertainties.
Fully Differential Higgs Boson Production to Third Order in QCD
hep-ph 2021-02 accept novelty 7.0 of 10

The first fully differential N3LO QCD prediction for gluon-fusion Higgs production is obtained by combining an NNLO Higgs-plus-jet calculation with the inclusive N3LO rapidity distribution through the Projection-to-Bo...
Transverse momentum dependent PDFs at N$^3$LO
hep-ph 2020-06 conditional novelty 7.0 of 10

Quark and gluon TMD beam-function matching kernels are computed at N3LO, completing the ingredient set for qT subtraction and N3LL-prime resummation at the LHC.
Higgs boson pair production via gluon fusion at N$^3$LO in QCD
hep-ph 2019-09 accept novelty 7.0 of 10

The N3LO QCD correction to gluon-fusion Higgs pair production raises the NNLO cross section by 3.0% at 13 TeV and cuts the scale uncertainty below 3%, in the infinite top-quark mass limit.
On Higgs decays to hadrons and the R-ratio at N^4LO
hep-ph 2017-07 accept novelty 7.0 of 10

The five-loop absorptive part of the Higgs-to-gluons self-energy is computed, completing the N^4LO QCD correction to Higgs decay into hadrons in the heavy-top limit.
Bootstrapping rapidity anomalous dimension for transverse-momentum resummation
hep-ph 2016-04 conditional novelty 7.0 of 10

The three-loop rapidity anomalous dimension for transverse-momentum resummation is computed, and it is found to equal the threshold soft anomalous dimension minus beta-function and constant terms.
The Three-Loop Splitting Functions in QCD: The Helicity-Dependent Case
hep-ph 2014-09 conditional novelty 7.0 of 10

The paper presents the first complete set of next-to-next-to-leading-order splitting functions for longitudinally polarized parton densities in perturbative QCD.
The $\Upsilon$ and $\Upsilon^{\prime}$ Leptonic Widths, $a_{\mu}^b$ and $m_b$ from full lattice QCD
hep-lat 2014-08 conditional novelty 7.0 of 10

Full lattice QCD with radiatively improved NRQCD bottom quarks yields Γ(Υ→e+e−) = 1.19(11) keV and Γ(Υ'→e+e−) = 0.69(9) keV, plus m_b(m_b, nf=5) = 4.196(23) GeV and a_μ^b = 0.271(37)×10^-10.
The singular behavior of massive QCD amplitudes
hep-ph 2006-12 accept novelty 7.0 of 10

Massive QCD amplitudes in the small-mass limit factorize into a universal process-independent factor times the corresponding massless amplitude, enabling three-loop predictions for the heavy-quark form factor.
QCD Decoupling at Four Loops
hep-ph 2005-12 accept novelty 7.0 of 10

A four-loop calculation fixes the matching condition for the strong coupling at heavy-quark thresholds in the MS-bar scheme, and from it a partial five-loop Higgs-gluon effective coupling follows.
Four-Loop Decoupling Relations for the Strong Coupling
hep-ph 2005-12 accept novelty 7.0 of 10

The four-loop decoupling relation for the strong coupling is computed analytically, enabling consistent five-loop running of alpha_s across quark thresholds and giving the five-loop Higgs-gluon effective coupling.
Higher-Order Soft Corrections to Lepton Pair and Higgs Boson Production
hep-ph 2005-08 accept novelty 7.0 of 10

The complete threshold-enhanced N^3LO QCD coefficient functions for Drell-Yan and Higgs production are derived, and the D3 resummation coefficient is extracted, giving about 5% (LHC) and 10% (Tevatron) corrections to ...
Bounds on hadronic contributions to the anomalous magnetic moment of the $\tau$ lepton from QCD Sum Rules
hep-ph 2026-10 conditional novelty 6.0 of 10

A QCD sum-rule analysis using combined Laplace and finite-energy sum rules yields a determination of the tau lepton's (g-2) hadronic vacuum polarization contribution.
Radiative corrections in neutral-current (anti)neutrino elastic scattering at $\text{GeV}$ energies I: Nucleon targets
hep-ph 2026-07 conditional novelty 6.0 of 10

Radiative corrections to neutral-current (anti)neutrino-nucleon elastic scattering are computed within low-energy EFT and reach a few percent, comparable to the strange-quark effects they must be disentangled from.
Double copy of form factors with multiple operator insertions
hep-th 2026-06 unverdicted novelty 6.0 of 10

A dyeing procedure generalizes the double copy to form factors with multiple operator insertions by mapping spurious poles to physical propagators and revealing a new scalar-ordering structure in gravity.
Threshold Resummation for Semi-Inclusive Single-Hadron Production with Effective Field Theory
hep-ph 2024-11 accept novelty 6.0 of 10

Momentum-space SCET threshold resummation for SIA is extended to N4LL for quark and gluon channels, with new large-x coefficients through N3LO and partial N4LO.
Inclusive Production Cross Sections at N3LO
hep-ph 2022-09 conditional novelty 6.0 of 10

Associated W-Higgs and Z-Higgs production is computed to third order in QCD for the first time, via a public code that reuses existing Drell-Yan corrections, with the residual uncertainty dominated by parton distribut...
Three-loop form factors for Higgs boson pair production in the large top mass limit
hep-ph 2019-09 accept novelty 6.0 of 10

New 1/m_t^2 expansion terms, up to 1/m_t^8 for the box form factors and 1/m_t^14 for the triangle form factor, are computed for the three-loop gg to HH amplitude.
Precision Predictions at N$^3$LO for the Higgs Boson Rapidity Distribution at the LHC
hep-ph 2018-10 conditional novelty 6.0 of 10

A threshold-expansion calculation of the gluon-fusion Higgs rapidity distribution at N3LO finds a nearly uniform correction over NNLO and a scale uncertainty reduced to about [-3.4%, +0.9%] for |Y| < 3.
Fiducial distributions in Higgs and Drell-Yan production at N$^3$LL+NNLO
hep-ph 2018-05 accept novelty 6.0 of 10

Matched N3LL+NNLO predictions for Higgs and Drell-Yan transverse-momentum and phi*_eta distributions, fully exclusive in the Born phase space so that arbitrary fiducial cuts can be applied, reach few-percent scale unc...
Precise QCD Description of the Higgs Boson Transverse Momentum Spectrum
hep-ph 2018-05 accept novelty 6.0 of 10

Matching next-to-next-to-leading-order QCD with third-order logarithmic resummation yields a Higgs transverse-momentum spectrum with about six percent perturbative uncertainty below 35 GeV.
New physics searches in nuclear and neutron $\beta$ decay
hep-ph 2018-03 conditional novelty 6.0 of 10

A combined fit of superallowed Fermi, neutron, and nuclear beta-decay data yields updated bounds on exotic scalar and tensor couplings and CA/CV = -1.27510(66) under standard-model assumptions.
Weak Mixing Angle in the Thomson Limit
hep-ph 2017-12 conditional novelty 6.0 of 10

Evolving the MS-bar weak mixing angle from the Z pole to zero momentum transfer with five-loop QCD and a data-plus-lattice flavor separation gives sin^2 theta_W(0) = 0.23868(5)(2).
FlexibleSUSY 2.0: Extensions to investigate the phenomenology of SUSY and non-SUSY models
hep-ph 2017-10 accept novelty 6.0 of 10

The paper documents FlexibleSUSY 2.0, which extends an automated spectrum generator to non-supersymmetric models, complex parameters, W-mass prediction from muon decay, muon g-2, a semi-analytic boundary value solver,...
Compilation of low-energy constraints on 4-fermion operators in the SMEFT
hep-ph 2017-06 accept novelty 6.0 of 10

A global fit of low-energy precision data yields a 61-parameter Gaussian likelihood for flavor-generic 4-fermion operators in the SMEFT.
Renormalization-group evolution of new physics contributions to (semi)leptonic meson decays
hep-ph 2017-06 conditional novelty 6.0 of 10

Running new-physics tensor operators from 1 TeV to the bottom-quark scale generates scalar and pseudoscalar contributions large enough to change B-decay predictions by tens of percent.
A Subleading Operator Basis and Matching for $gg \to H$
hep-ph 2017-03 accept novelty 6.0 of 10

A complete, helicity-organized basis of power-suppressed SCET operators for gluon-fusion Higgs production, with tree-level Wilson coefficients computed for every operator that contributes to the cross section at order...
A Second Higgs Doublet in the Early Universe: Baryogenesis and Gravitational Waves
hep-ph 2016-11 conditional novelty 6.0 of 10

A hierarchical Type II two-Higgs-doublet benchmark survives the ACME electron EDM bound for baryogenesis and simultaneously predicts an acoustic gravitational wave signal within LISA reach, but only in a narrow tuned window.
$\overline{\rm MS}$-on-shell quark mass relation up to four loops in QCD and a general SU$(N)$ gauge group
hep-ph 2016-06 accept novelty 6.0 of 10

The four-loop MS-bar to on-shell quark mass conversion is computed for general SU(N) colour factors, reaching 0.2% numerical uncertainty in QCD and improving on the authors' earlier 3% result.
High precision determination of the gluon fusion Higgs boson cross-section at the LHC
hep-ph 2016-02 accept novelty 6.0 of 10

The gluon-fusion Higgs production cross-section at 13 TeV is predicted to be 48.58 pb, with theory and PDF uncertainties broken down, using QCD corrections through N3LO in the heavy-top limit.
Automated event generation for loop-induced processes
hep-ph 2015-06 accept novelty 6.0 of 10

The authors extend MadGraph5_aMC@NLO to compute cross-sections and generate events automatically for loop-induced processes, and tabulate inclusive rates for essentially all Standard Model cases with up to three final...
Higgs boson gluon-fusion production beyond threshold in N3LO QCD
hep-ph 2014-11 conditional novelty 6.0 of 10

The complete next-to-soft term and the exact coefficients of log^5,4,3(1-z) are computed for Higgs production at N3LO, and the threshold approximation is shown not to converge at that order.
Approximate N^3LO Higgs-boson production cross section using physical-kernel constraints
hep-ph 2014-08 conditional novelty 6.0 of 10

Using single-log constraints from the physical evolution kernel, the authors derive the ln^5,4,3(1-z) N3LO coefficients for Higgs production and estimate that N3LO increases the 14 TeV cross section by (10±2)% at mu_R...
Virtual corrections to Higgs boson pair production in the large top quark mass limit
hep-ph 2014-08 accept novelty 6.0 of 10

The three-loop matching coefficient C_HH for Higgs pair production in gluon fusion is computed, completing the ingredients for a consistent NNLO QCD prediction in the heavy-top limit.
Threshold resummation at N$^3$LL accuracy and soft-virtual cross sections at N$^3$LO
hep-ph 2014-05 accept novelty 6.0 of 10

The authors extract the last unknown universal coefficient needed for third-order threshold resummation of colourless hadroproduction, and their Drell-Yan prediction agrees with an independent calculation.
On Higgs-exchange DIS, physical evolution kernels and fourth-order splitting functions at large x
hep-ph 2009-12 conditional novelty 6.0 of 10

Third-order Higgs-exchange DIS coefficient functions are computed and, assuming physical evolution kernels remain single-logarithmic at large x, used to predict the leading ln^k(1-x) terms of the four-loop singlet spl...
Finite top mass effects for hadronic Higgs production at next-to-next-to-leading order
hep-ph 2009-09 conditional novelty 6.0 of 10

Top mass corrections to the NNLO gluon fusion Higgs cross section are below 0.5 percent for Higgs masses of 100 to 300 GeV, confirming the heavy-top limit.
Top mass effects in Higgs production at next-to-next-to-leading order QCD: virtual corrections
hep-ph 2009-07 accept novelty 6.0 of 10

The top-mass-suppressed terms of the three-loop virtual amplitude for gluon-fusion Higgs production are computed through the first three non-vanishing orders in M_H^2/M_t^2.
Virtual three-loop corrections to Higgs boson production in gluon fusion for finite top quark mass
hep-ph 2009-07 conditional novelty 6.0 of 10

Five terms of the expansion of the three-loop virtual corrections to gluon-fusion Higgs production in M_H^2/M_t^2 are computed, and the truncated series appears to converge up to near the top-antitop threshold.
Analytic one-loop amplitudes for a Higgs boson plus four partons
hep-ph 2009-05 accept novelty 6.0 of 10

Explicit analytic one-loop amplitudes are given for a Higgs boson plus four quarks, and plus two quarks and two opposite-helicity gluons, in the large-top-mass effective theory.
Mixed QCD-electroweak corrections to Higgs boson production in gluon fusion
hep-ph 2008-11 conditional novelty 6.0 of 10

The three-loop mixed QCD-electroweak correction to light-quark Higgs production gives a Wilson coefficient of 7/6 rather than the factorized 11/4, and lowers Tevatron Higgs cross sections by 4 to 6 percent.
On Third-Order Timelike Splitting Functions and Top-Mediated Higgs Decay into Hadrons
hep-ph 2007-09 conditional novelty 6.0 of 10

The authors derive the next-to-next-to-leading-order timelike quark-quark and gluon-gluon splitting functions for flavour-singlet fragmentation by analytic continuation from deep-inelastic scattering, and use the same...
Reconstruction of heavy quark current correlators at O(\alpha_s^3)
hep-ph 2009-07 conditional novelty 5.5 of 10

Padé approximants built from known low-energy, threshold, and high-energy expansions reconstruct heavy-quark correlators at O(alpha_s^3) and yield estimates for unknown expansion coefficients.
Soft Contributions Stabilize NNLO QCD Corrections to Quarkonium Production and Decay
hep-ph 2026-06 unverdicted novelty 5.0 of 10

Soft contributions stabilize NNLO QCD corrections for S-wave color-singlet quarkonium processes, yielding better perturbative convergence and experimental agreement.
Perturbative QCD fitting of KEDR and BESIII $e^+e^-$ data for R(s) and $\alpha_s$ determination
hep-ph 2026-03 unverdicted novelty 5.0 of 10

Fits of combined KEDR and truncated BESIII data yield α_s(M_Z)=0.1179 (NLO), 0.1221 (NNLO), 0.1313 (N3LO), with the rise attributed to π² analytic-continuation effects.
New high-precision $b$, $c$, and $s$ masses from pseudoscalar-pseudoscalar correlators in $n_f=4$ lattice QCD
hep-lat 2025-08 accept novelty 5.0 of 10

New high-precision MS-bar masses m_b(m_b, n_f=5)=4.1923(63) GeV, m_c(3 GeV, n_f=4)=0.9813(34) GeV, and m_s(3 GeV, n_f=4)=83.39(26) MeV from HISQ lattice QCD correlators including quenched QED.
Global SMEFT Fits at Future Colliders
hep-ph 2022-06 conditional novelty 5.0 of 10

Global fits to projected future-collider data indicate that e+e- and muon colliders would tighten SMEFT Wilson-coefficient constraints by one to three orders of magnitude beyond HL-LHC, with linear colliders gaining m...
The gluon-fusion production of Higgs boson pair: N$^3$LO QCD corrections and top-quark mass effects
hep-ph 2019-12 accept novelty 5.0 of 10

Third-order (N3LO) QCD corrections to gluon-fusion Higgs pair production, combined with top-quark mass effects via three approximation schemes, cut scale uncertainties fourfold relative to NNLO.
iHixs 2 - Inclusive Higgs Cross Sections
hep-ph 2018-02 conditional novelty 5.0 of 10

The authors release iHixs 2, a public code combining third-order QCD, electroweak, and finite quark-mass corrections to give 48.67 pb for gluon-fusion Higgs production at 13 TeV with an automated uncertainty budget.
The MSR Mass and the ${\cal O}(\Lambda_{\rm QCD})$ Renormalon Sum Rule
hep-ph 2017-04 accept novelty 5.0 of 10

An analytic sum rule extracts the O(Lambda_QCD) renormalon normalization of any perturbative series from its known coefficients, giving N_1/2 = 0.526, 0.492, 0.446 for 3, 4, 5 massless flavors.
NNLO QCD corrections to Higgs boson production at large transverse momentum
hep-ph 2016-07 conditional novelty 5.0 of 10

The NNLO QCD corrections to Higgs-plus-jet and Higgs transverse momentum spectra are moderate, positive, and reduce the theory scale uncertainty, with top-mass effects handled by a leading-order reweighting that becom...

See all 75 Pith citations

How later papers used it

150 later papers cite it

Evidence: substantial. 127 of the citing papers share no author with it, and 43 of those carry a confident label from Pith's citation classifier.

Pith read 13 of the 229 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

2 later papers by other authors are listed below: 1 built on it and 1 used its data or values. 1 more paper by the same authors is listed: 1 used its method or tool.

Builds on it 1 of 4 papers with this label

Extended this paper's result by one order in perturbation theory.

Higher order QCD results for the fermionic contributions of the Higgs-boson decay into two photons and the decoupling function for the MSbar renormalized fine-structure constant (2014) · 1.00 · paraphrased: the licence does not allow quoting

Uses its method or tool 1 of 28 papers with this label · 1 by the same authors

By the same authors

Performed the transition from alpha_s^(6) to alpha_s^(5) with the help of the formulae derived in this paper.

Virtual three-loop corrections to Higgs boson production in gluon fusion for finite top quark mass (2009) (same authors) · 0.91 · paraphrased: the licence does not allow quoting

Uses its data or values 1 of 20 papers with this label

Combined the matching coefficients of the quark masses given in this paper with the two-loop matching coefficient of C_7 given in another work to find its own values of xi_T,1 and xi_T,2.

Renormalization-group evolution of new physics contributions to (semi)leptonic meson decays (2017) · 0.95 · paraphrased: the licence does not allow quoting

How the literature uses it

150 arXiv papers cite this in 227 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 whose main result relates to this one's

Found among the papers closest to this one by title and main claim. Pith's classifier compared each pair's main results and a second model family confirmed the relation; checked against two further families, shown conflicts agreed 118 times in 120 and shown same results 218 times in 239. The quoted sentence is the other paper's main claim.

States the same result 1 paper

We also establish low-energy theorems, valid to all orders, which relate the effective couplings of the Higgs boson to gluons and light quarks, due to the virtual presence of a heavy quark h, to the logarithmic derivatives w.r.t. m_h of zeta_g and zeta_m, respectively.

Decoupling relations, effective Lagrangians and low-energy theorems · 0.99

Papers citing it · 145 sentences without a confident label

Builds on it 4 papers

This extends the result of Ref. [ref] by one order in perturbation theory.

Higher order QCD results for the fermionic contributions of the Higgs-boson decay into two photons and the decoupling function for the MSbar renormalized fine-structure constant · Introduction · 1.00
In order to extend it to $α_e$ we apply the procedure described in Ref. [ref] and build an effective Lagrangian in which the heavy quark flavors have been integrated out.

Regularization-scheme dependence of QCD amplitudes in the massive case · Decoupling transformations · 0.83
The decoupling functions in Eqs. ([ref]) and ([ref]) are known in QCD to high order in perturbation theory [ref] and can be used for the QED calculation considered here after inserting the proper values for the color factors.

Leptonic contributions to the effective electromagnetic coupling at four-loop order in QED · Decoupling functions · 0.80

Uses its method or tool 29 papers

For this purpose we used the matching formula near mass thresholds for the $MS$ scheme at three-loop order in Refs. [ref] .

Precise determination of the strong coupling constant in Nf=2+1 lattice QCD with the Schr\"odinger functional scheme · Our strategy · 0.99
In order to compute the decoupling effects of heavy particles from the running of $α_s$ one can use the well-established formalism derived in Ref. [ref] .

Decoupling constant for $\alpha_s$ and the effective gluon-Higgs coupling to three loops in supersymmetric QCD · Decoupling of heavy supersymmetric particles · 0.99
The bare strong coupling in the full theory is related to the bare strong coupling in the effective theory $α_s^'0$ by the decoupling constants $ζ_g^0$ [ref] , [equation] Similiarly, [equation] Using these relations, we obtain the bare Wilson coefficient as a function of the bare parameters in the effective theory and of the bare mass of the heavy quarks in the full theory, $C_1^0 =C_1^0( α_s^'0, m_q^0)$.

The NNLO gluon fusion Higgs production cross-section with many heavy quarks · Decoupling and renormalization · 0.82

Uses its data or values 18 papers

The expansion coefficients up to order $^2$ are known from the results of Refs. [ref] and are given in Appendix [ref].

Higher order QCD results for the fermionic contributions of the Higgs-boson decay into two photons and the decoupling function for the MSbar renormalized fine-structure constant · Generalities and notation · 0.98
The coefficients $β_L-1^(n_f)$ for $L≤4$ have been obtained in [ref] (see [ref] for the five-loop result); the two-loop decoupling coefficient $d_2 = (- 15 C_F + 32/3 C_A) T_F$ is from [ref] .

Anomalous dimension of the heavy-light quark current in HQET up to four loops · The ratio $f_B/f_D$ · 0.90
They have been computed in Ref. [ref] and can be found in explicit form in Eq. (12) of Ref. [ref] .

Completely automated computation of the heavy-fermion corrections to the three-loop matching coefficient of the vector current · Matching coefficient · 0.85

Compares with it 6 papers

We have also used the logarithmic parts of Eqs. ([ref]) and ([ref]) in order to derive the contributions which are proportional to the charge factor $Q_q_i$ of the light quarks and find agreement with the result of Ref. [ref] .

Higher order QCD results for the fermionic contributions of the Higgs-boson decay into two photons and the decoupling function for the MSbar renormalized fine-structure constant · Calculation and results · 0.99
Using Eq. [ref], or equivalently the relation underneath, we find [equation] independent of $M_pole$ and in agreement with Ref. [ref] .

Secondary massive quarks with the Mellin-Barnes expansion · Secondary massive quark · 0.98
At the NNLO the leading term in the inverse top quark mass expansion agrees with the results of Refs. [ref] .

Virtual three-loop corrections to Higgs boson production in gluon fusion for finite top quark mass · Results · 1.00

Cites it as background 15 papers

However, it is well known [ref] , that the QCD scale has to be modified (“matching”) when crossing flavour thresholds and switching the number of active flavours.

Determination of the strong coupling constant from the CLEO measurement of the total hadronic cross section in $e^+e^-$ annihilation below 10.56 GeV · The strong coupling at the scale of $M_Z$ · 0.90
Within the framework of QCD decoupling constants for the strong coupling $α_s$ are known at two- [ref] , three [ref] and even four-loop order [ref] .

Decoupling constant for $\alpha_s$ and the effective gluon-Higgs coupling to three loops in supersymmetric QCD · Introduction · 0.93
While the NLO results are exact in the top quark and Higgs boson masses (in this context, see also Ref. [ref] ), the NNLO results rely on the effective theory built in the limit of the large top quark mass (see, e.g., Refs. [ref] for the three-loop corrections to the effective $ggH$ coupling).

Finite top quark mass effects in NNLO Higgs boson production at LHC · Introduction · 0.86

Discussion

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