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Transverse Parton Distribution and Fragmentation Functions at NNLO: the Quark Case

T0 review · 4 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper establishes that the exponential regulator treats quark transverse-momentum-dependent PDFs and fragmentation functions consistently at NNLO, and extracts from them the two-loop jet function that completes N3LL resummation of…

desk verdict A careful NNLO calculation that delivers the last two-loop ingredient for EEC resummation, with an honest discrepancy in the TMDFF sector that needs referee scrutiny. read the letter →

arxiv 1908.03831 v1 pith:J5FZUNIL submitted 2019-08-11 hep-ph

Ming-Xing Luo , Xing Wang , Xiaofeng Xu , Li Lin Yang , Tong-Zhi Yang , Hua Xing Zhu This is my paper · ORCID
classification hep-ph
keywords transversemomentumdependentPDFfragmentationfunctionrapiditydivergencesexponentialregulatorenergy-energycorrelatorN3LLresummationsoft-collineareffectivetheoryNNLOmatchingcoefficients
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 establishes that the exponential regulator, an alternative scheme for taming rapidity divergences in transverse-momentum-dependent factorization, works consistently for the quark transverse-momentum-dependent parton distribution function (TMDPDF) and fragmentation function (TMDFF) at next-to-next-to-leading order (NNLO). Using it, the authors compute the full matching coefficients for quark TMDPDFs and TMDFFs to order $\alpha_s^2$, including the $\mathcal{O}(\epsilon^2)$ terms in dimensional regularization that a future N3LO calculation needs. The calculation also produces the two-loop quark jet function for the energy-energy correlator (EEC) in the back-to-back limit, which had been the only missing ingredient for resumming that observable at N3LL accuracy. If correct, the framework makes those N3LL predictions possible and settles a small discrepancy with an earlier NNLO fragmentation-function result.

What carries the argument

The carrying mechanism is the exponential regulator, defined in momentum space by inserting $\exp(-b_0\tau k^0)$ in every phase-space integral and taking $\tau\to 0$ after integration; at the operator level it shifts the field points in the TMD definitions by $(-ib_0\tau,-ib_0\tau,b_\perp)$. Because the regulator acts on the total momentum of extra emissions, it factorizes multi-emission phase space and preserves the structure of cut propagators, so the double-real contributions can be reduced with integration-by-parts identities and solved with canonical differential equations. The remaining rapidity singularities are expanded, via identities such as Eq. (3.30), directly into delta functions and plus-distributions. The zero-bin subtraction, Eq. (2.27), which subtracts exactly the TMD soft function from the collinear and fragmentation sectors, removes the double counting, and the final two-loop jet function emerges as the Mellin moment $\int_0^1 dx\,x\,C_{iq}(x,b_\perp/x,\mu,\nu)$ of the TMDFF matching coefficients.

What would settle it

A direct, independent calculation of the two-loop coefficient $c_2^J$, for example by computing the EEC in the back-to-back limit from full matrix elements without invoking TMD factorization, would settle the claim: if the extracted $\delta(1-z)$ constant disagrees with Eq. (4.19), then the TMDFF result and the exponential-regulator framework are wrong.

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Extended reading notes

Core claim

The paper's central claim is that the exponential regulator is a consistent, simplifying regulator for rapidity divergences in both the soft and collinear sectors, and that it delivers the first $\mathcal{O}(\epsilon^2)$ bare NNLO results for the quark TMDPDF and TMDFF matching coefficients. The regulator multiplies each real-emission phase-space measure by $\exp(-b_0\tau k^0)$ and takes $\tau\to 0$ after integration, turning rapidity singularities into plus-distributions and delta functions at the integrand level. With a zero-bin subtraction that identifies the overlap with the TMD soft function, the TMDPDF coefficients fully agree with earlier NNLO calculations, but the TMDFF coefficients differ from the earlier fragmentation calculation in the coefficient of a $C_AC_F\pi^4\delta(1-z)$ term. The authors validate their TMDFF result by using it to compute the two-loop quark jet function for the EEC in the back-to-back limit; the new constant $c_2^J$ is the last missing ingredient for N3LL resummation of that observable, and the resulting endpoint distribution passes the leading-transcendentality check and the momentum-conservation sum rule.

Load-bearing premise

The load-bearing assumption is that the exponential regulator and the zero-bin subtraction—which removes the overlap between the collinear and fragmentation sectors and the soft sector by subtracting exactly the TMD soft function—are together a consistent scheme in those sectors at two loops.

Editorial extensions

If this is right

  • The two-loop jet function constant $c_2^J$ completes the perturbative input for N3LL resummation of the energy-energy correlator in the back-to-back limit.
  • The bare NNLO TMDPDF and TMDFF expressions through $\mathcal{O}(\epsilon^2)$ provide the missing perturbative input for extending the matching to N3LO.
  • The exponential regulator is a consistent rapidity regulator in the collinear and fragmentation sectors, not just for soft functions, so the same simplified machinery can be reused for other transverse-momentum-dependent observables.
  • The disputed $C_AC_F\pi^4\delta(1-z)$ term in the quark TMDFF is resolved: the EEC jet function built from these TMDFFs passes the leading-transcendentality and sum-rule checks, supporting the new fragmentation result.
  • The factorization setup is universal and applies to processes such as Drell-Yan, SIDIS, and $e^+e^-$ jet production at small transverse momentum.

Reading between the lines

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

  • The same regulator plus the $\mathcal{O}(\epsilon^2)$ expressions should carry directly to the gluon TMDPDF and TMDFF case, where other regulators are known to be harder; the paper leaves this to future work.
  • Because the regulator is defined at the operator level, its path-shifted TMD definitions could be connected to non-perturbative determinations of the same objects, for instance through Euclidean or lattice methods, a step the paper does not take.
  • A practical test of the claim is to compute a full N3LL+NNLO back-to-back EEC spectrum using $c_2^J$ and compare it to $e^+e^-$ event-shape data; agreement would independently confirm the disputed TMDFF term.
  • The resolution implies that $\delta(1-z)$ constants in TMDFFs are fixed by universal soft physics; recomputing the same Mellin moment with an independent rapidity regulator would make that point explicit.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper revisits the NNLO calculation of quark transverse momentum dependent parton distribution functions (TMDPDFs) and fragmentation functions (TMDFFs) using the exponential regulator for rapidity divergences. The authors show that the regulator can be applied at the operator level, simplifies the phase-space integrals, and enables the use of IBP and differential-equation techniques. They present the scale-independent NNLO matching coefficients, report agreement with existing TMDPDF results, and find a discrepancy with Ref. [40] in the q-to-q TMDFF channel. As a by-product, they extract the two-loop quark jet function for the energy-energy correlator (EEC) in the back-to-back limit, including the new constant cJ2, and claim this is the last missing ingredient for N3LL resummation.

Significance. If the results are correct, the paper provides a valuable technical step: it demonstrates that the exponential regulator works for collinear TMD sectors, supplies O(epsilon^2) bare results needed for future N3LO calculations, and delivers a previously missing two-loop ingredient for the EEC. The calculation uses modern and reliable tools (IBP reduction, canonical differential equations, HPLs), and the paper includes several internal consistency checks: reproduction of RGE and rapidity-evolution structures, agreement with prior TMDPDF calculations, a leading-transcendentality check against N=4 SYM, and a momentum-conservation sum rule. These strengths are substantial. However, the checks do not fully pin down the disputed constant term, and one load-bearing assumption is stated rather than derived.

major comments (4)
  1. [Sec. 2.2, Eq. (2.27)] The identification of the zero-bin subtraction with the full TMD soft function S_qbarq is stated without derivation. This equality is load-bearing because the disputed q-to-q TMDFF term C_A C_F pi^4 delta(1-z) in Sec. 4.1 is attributed precisely to the universal soft function. The exponential regulator is implemented differently in Eqs. (2.24)-(2.26): a coordinate shift for the collinear matrix elements versus a Wilson-loop definition for the soft function, and the zero-bin is the soft limit of the collinear graph rather than an independently defined object. Please provide a derivation or an explicit diagram-by-diagram check that the soft limit of each collinear cut diagram equals the corresponding soft-function contribution with the same tau prescription, including subleading-in-tau regions. If this identification fails in any channel, the O(epsilon^2) bare results and the constant cJ2 in Eq. (4.16) are scheme-dependent.
  2. [Sec. 4.2, Eq. (4.17)] The statement that the two-loop plus-distribution terms D_n(z) are in full agreement with the 'analytical NLO calculation' in Ref. [82] is difficult to interpret, because Ref. [82] is a one-loop calculation and cannot determine O(alpha_s^2) D_n coefficients. Please clarify what comparison was actually performed, for example whether the two-loop leading singular terms were checked against an independent factorization-based calculation rather than against the NLO result. As written, this statement appears to claim a check that the cited calculation cannot provide.
  3. [Sec. 4.2, Eq. (4.20)] The claimed verification of the momentum-conservation sum rule does not test the two-loop endpoint contribution. The analytical NLO formula from Ref. [82] supplies only the one-loop cross section, while Eq. (4.17) contains the two-loop endpoint term dsigma^(2)/dz. A two-loop verification of the sum rule would require the full two-loop EEC away from the endpoint or an independent derivation of the endpoint integral. As presented, this check constrains only lower orders and leaves cJ2 in Eq. (4.16) and c_{z=1}^{(2)} in Eq. (4.19) without an independent numerical or analytical test beyond the partial N=4 leading-transcendentality check.
  4. [Sec. 3.2.1, Eqs. (3.15)] The claim that the loop integral in the real-virtual contribution 'does not need to be regularized' is not demonstrated. The master integrals IRV_1 and IRV_2 in Eq. (3.15) contain factors [nbar dot l]^{-a4}, and the cancellation of rapidity singularities between the real and virtual contributions is a nontrivial consistency requirement when the exponential regulator acts only on the real-emission phase space. Please show explicitly, or cite a derivation, that the combined real-virtual plus double-real contributions are finite in the tau -> 0 limit before zero-bin subtraction.
minor comments (4)
  1. [Sec. 3.1, Eq. (3.9)] The quantity L_nu = ln(nu^2/mu^2) is used in Eq. (3.9) but is not defined until the appendix; please define it near Eq. (2.38) for readability.
  2. [Sec. 2.1] There is a typo in the sentence 'probability distribution of find a quark'; it should read 'of finding a quark'.
  3. [Sec. 2.2, Eqs. (2.22) and (2.24)] The phase-space regulator in Eq. (2.22) uses exp(-b0 tau k0) with a real exponent, while the operator definitions in Eqs. (2.24)-(2.26) use a shift by -i b0 tau in spacetime arguments. Please clarify the relationship between these two representations and the Euclidean-signature convention for the transverse components.
  4. [Sec. 3.2.3] The O(epsilon^2) bare results are said to be provided in an electronic file attached to the arXiv submission, but the file format and the precise definitions of the plotted or listed quantities are not described in the text. A short summary of the file contents would help readers use these results.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the NNLO constants are diagrammatically computed, and the EEC checks are external constraints rather than inputs.

full rationale

The central results of the paper are the scale-independent NNLO matching coefficients I^(2) and C^(2). These are obtained by direct calculation of cut Feynman diagrams using IBP reduction and differential equations, not by fitting to the EEC or to any quantity they later claim to predict. The scale- and rapidity-dependent parts are reconstructed from known RGEs and anomalous dimensions, which is a standard consistency requirement, not a circular reduction. The zero-bin subtraction identity S0b = S_qbarq (Eq. 2.27) is an input assumption about the exponential regulator; even if it were wrong, that would be a correctness risk, not a circularity, because the paper does not derive the soft function from the TMDFF or vice versa. The EEC jet function is defined as a Mellin moment of the computed TMDFFs (Eq. 4.13), so the subsequent checks — the momentum sum rule (Eq. 4.20) and the N=4 leading-transcendental comparison — are external constraints applied to the resulting expression, and the paper explicitly verifies rather than assumes them. The use of the exponential regulator and the two-loop soft function from prior work by the same group is normal reliance on established results; the disputed C_A C_F pi^4 delta(1-z) term is checked against the EEC sum rule and SYM limit rather than being imposed. No output quantity is identical by construction to an input, and no fitted parameter is renamed as a prediction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The calculation introduces no new fitted parameters and no new entities. It relies on standard SCET machinery and on previously computed inputs (splitting functions, anomalous dimensions, TMD soft function). The one genuinely new postulate is the consistency of the exponential regulator in the collinear sectors, which the paper supports with agreement checks and the EEC application.

assumptions (5)
  • domain assumption Dimensional regularization (d = 4 - 2*epsilon) and light-cone gauge n.A = 0 are used for all loop and phase-space integrals; UV divergences are renormalized in the MS scheme.
    Invoked throughout Secs. 2 and 3; it is the standard framework for SCET TMD calculations and determines the form of all poles.
  • ad hoc to paper The exponential rapidity regulator, Eq. (2.22), replaces phase-space measures with exp(-b0 tau k0) and requires taking tau -> 0 after integration; it is assumed to regularize rapidity divergences consistently in the collinear sectors and to preserve non-Abelian exponentiation.
    This is the paper's central methodological postulate, introduced in Sec. 2.2 and used for all NNLO results; if false, the O(epsilon^2) results and the TMDFF claim do not follow.
  • domain assumption Zero-bin subtraction with the TMD soft function, Eq. (2.27), removes the overlap between collinear and soft sectors without additional terms.
    Used to define the genuinely collinear TMDPDF and TMDFF after Eq. (2.26); any mismatch would shift the final matching coefficients.
  • standard math The partonic collinear PDFs, FFs, splitting functions, and anomalous dimensions at NNLO (Appendix A and Eqs. (2.31), (2.41)) are taken from the cited literature.
    These inputs are needed for the RGE checks and subtraction procedure; they are not derived in this paper.
  • domain assumption The EEC back-to-back factorization, Eq. (4.12), and the jet function identity, Eq. (4.13), from Ref. [35] are valid.
    Used to derive the two-loop jet function from the TMDFFs and to validate the endpoint contribution; if the factorization were different, the check would not apply.

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Cite this review

Pith. "Pith review of Transverse Parton Distribution and Fragmentation Functions at NNLO: the Quark Case." pith.science (2026). https://pith.science/paper/J5FZUNIL

@misc{pith2026190803831,
  author       = {Pith},
  title        = {Pith review of: Transverse Parton Distribution and Fragmentation Functions at NNLO: the Quark Case},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/J5FZUNIL}},
  note         = {Machine review of arXiv:1908.03831}
}
abstract

We revisit the calculation of perturbative quark transverse momentum dependent parton distribution functions and fragmentation functions using the exponential regulator for rapidity divergences. We show that the exponential regulator provides a consistent framework for the calculation of various ingredients in transverse momentum dependent factorization. Compared to existing regulators in the literature, the exponential regulator has a couple of advantages which we explain in detail. As a result, the calculation is greatly simplified and we are able to obtain the next-to-next-to-leading order results up to $\mathcal{O}(\epsilon^2)$ in dimensional regularization. These terms are necessary for a higher order calculation which is made possible with the simplification brought by the new regulator. As a by-product, we have obtained the two-loop quark jet function for the Energy-Energy Correlator in the back-to-back limit, which is the last missing ingredient for its N$^3$LL resummation.

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Forward citations

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Reference graph

Works this paper leans on

102 extracted references · 6 canonical work pages · cited by 8 Pith papers

  1. [40]

    M. G. Echevarria, I. Scimemi, and A. Vladimirov, Unpolarized Transverse Momentum Dependent Parton Distribution and Fragmentation Functions at next-to-next-to-leading order, JHEP 09 (2016) 004, [ arXiv:1604.07869]

    arXiv 2016

  2. [82]

    L. J. Dixon, M.-X. Luo, V. Shtabovenko, T.-Z. Yang, and H. X. Zhu, Analytical Computation of Energy-Energy Correlation at Next-to-Leading Order in QCD , Phys. Rev. Lett. 120 (2018), no. 10 102001, [ arXiv:1801.03219]

    arXiv 2018

  3. [1]

    Collins, Foundations of perturbative QCD , Camb

    J. Collins, Foundations of perturbative QCD , Camb. Monogr. Part. Phys. Nucl. Phys. Cosmol. 32 (2011) 1–624

    2011

  4. [2]

    Angeles-Martinez et al., Transverse Momentum Dependent (TMD) parton distribution functions: status and prospects , Acta Phys

    R. Angeles-Martinez et al., Transverse Momentum Dependent (TMD) parton distribution functions: status and prospects , Acta Phys. Polon. B46 (2015), no. 12 2501–2534, [arXiv:1507.05267]

    arXiv 2015

  5. [3]

    Y. L. Dokshitzer, D. Diakonov, and S. I. Troian, On the Transverse Momentum Distribution of Massive Lepton Pairs , Phys. Lett. 79B (1978) 269–272

    1978

  6. [4]

    Parisi and R

    G. Parisi and R. Petronzio, Small Transverse Momentum Distributions in Hard Processes , Nucl. Phys. B154 (1979) 427–440

    1979

  7. [5]

    J. C. Collins, D. E. Soper, and G. F. Sterman, Transverse Momentum Distribution in Drell-Yan Pair and W and Z Boson Production , Nucl. Phys. B250 (1985) 199–224. – 32 –

    1985

  8. [6]

    P. B. Arnold and R. P. Kauffman, W and Z production at next-to-leading order: From large q(t) to small , Nucl. Phys. B349 (1991) 381–413

    1991

Show all 102 references
  1. [7]

    G. A. Ladinsky and C. P. Yuan, The Nonperturbative regime in QCD resummation for gauge boson production at hadron colliders , Phys. Rev. D50 (1994) R4239, [hep-ph/9311341]

    1994 arXiv

  2. [8]

    Bozzi, S

    G. Bozzi, S. Catani, G. Ferrera, D. de Florian, and M. Grazzini, Production of Drell-Yan lepton pairs in hadron collisions: Transverse-momentum resummation at next-to-next-to-leading logarithmic accuracy, Phys. Lett. B696 (2011) 207–213, [arXiv:1007.2351]

    2011 arXiv

  3. [9]

    Becher, M

    T. Becher, M. Neubert, and D. Wilhelm, Electroweak Gauge-Boson Production at Small qT : Infrared Safety from the Collinear Anomaly , JHEP 02 (2012) 124, [ arXiv:1109.6027]

    2012 arXiv

  4. [10]

    Bizon, A

    W. Bizon, A. Gehrmann-De Ridder, T. Gehrmann, N. Glover, A. Huss, P. F. Monni, E. Re, L. Rottoli, and D. M. Walker, The transverse momentum spectrum of weak gauge bosons at N3LL+NNLO, arXiv:1905.05171

    1905 arXiv

  5. [11]

    Bertone, I

    V. Bertone, I. Scimemi, and A. Vladimirov, Extraction of unpolarized quark transverse momentum dependent parton distributions from Drell-Yan/Z-boson production , JHEP 06 (2019) 028, [ arXiv:1902.08474]

    2019 arXiv

  6. [12]

    Ji, J.-p

    X.-d. Ji, J.-p. Ma, and F. Yuan, QCD factorization for semi-inclusive deep-inelastic scattering at low transverse momentum , Phys. Rev. D71 (2005) 034005, [ hep-ph/0404183]

    2005 arXiv

  7. [13]

    Ji, J.-P

    X.-d. Ji, J.-P. Ma, and F. Yuan, QCD factorization for spin-dependent cross sections in DIS and Drell-Yan processes at low transverse momentum , Phys. Lett. B597 (2004) 299–308, [hep-ph/0405085]

    2004 arXiv

  8. [14]

    P. Sun, J. Isaacson, C. P. Yuan, and F. Yuan, Nonperturbative functions for SIDIS and Drell–Yan processes, Int. J. Mod. Phys. A33 (2018), no. 11 1841006, [ arXiv:1406.3073]

    2018 arXiv

  9. [15]

    Z.-B. Kang, A. Prokudin, P. Sun, and F. Yuan, Extraction of Quark Transversity Distribution and Collins Fragmentation Functions with QCD Evolution , Phys. Rev. D93 (2016), no. 1 014009, [ arXiv:1505.05589]

    2016 arXiv

  10. [16]

    X. Liu, F. Ringer, W. Vogelsang, and F. Yuan, Lepton-jet Correlations in Deep Inelastic Scattering at the Electron-Ion Collider , Phys. Rev. Lett. 122 (2019), no. 19 192003, [arXiv:1812.08077]

    2019 arXiv

  11. [17]

    J. C. Collins and D. E. Soper, Back-To-Back Jets in QCD , Nucl. Phys. B193 (1981) 381. [Erratum: Nucl. Phys.B213,545(1983)]

    1981

  12. [18]

    J. C. Collins and D. E. Soper, Back-To-Back Jets: Fourier Transform from B to K-Transverse, Nucl. Phys. B197 (1982) 446–476

    1982

  13. [19]

    Neill, I

    D. Neill, I. Scimemi, and W. J. Waalewijn, Jet axes and universal transverse-momentum-dependent fragmentation, JHEP 04 (2017) 020, [ arXiv:1612.04817]

    2017 arXiv

  14. [20]

    Gutierrez-Reyes, I

    D. Gutierrez-Reyes, I. Scimemi, W. J. Waalewijn, and L. Zoppi, Transverse momentum dependent distributions with jets , Phys. Rev. Lett. 121 (2018), no. 16 162001, [arXiv:1807.07573]

    2018 arXiv

  15. [21]

    Gutierrez-Reyes, I

    D. Gutierrez-Reyes, I. Scimemi, W. J. Waalewijn, and L. Zoppi, Transverse momentum dependent distributions in e+e− and semi-inclusive deep-inelastic scattering using jets , arXiv:1904.04259. – 33 –

    1904 arXiv

  16. [22]

    Gutierrez-Reyes, Y

    D. Gutierrez-Reyes, Y. Makris, V. Vaidya, I. Scimemi, and L. Zoppi, Probing Transverse-Momentum Distributions With Groomed Jets , arXiv:1907.05896

    1907 arXiv

  17. [23]

    E. L. Berger and J.-w. Qiu, Differential cross-section for Higgs boson production including all orders soft gluon resummation , Phys. Rev. D67 (2003) 034026, [ hep-ph/0210135]

    2003 arXiv

  18. [24]

    Bozzi, S

    G. Bozzi, S. Catani, D. de Florian, and M. Grazzini, Transverse-momentum resummation and the spectrum of the Higgs boson at the LHC , Nucl. Phys. B737 (2006) 73–120, [hep-ph/0508068]

    2006 arXiv

  19. [25]

    Y. Gao, C. S. Li, and J. J. Liu, Transverse momentum resummation for Higgs production in soft-collinear effective theory, Phys. Rev. D72 (2005) 114020, [ hep-ph/0501229]

    2005 arXiv

  20. [26]

    M. G. Echevarria, T. Kasemets, P. J. Mulders, and C. Pisano, QCD evolution of (un)polarized gluon TMDPDFs and the Higgs qT -distribution, JHEP 07 (2015) 158, [arXiv:1502.05354]. [Erratum: JHEP05,073(2017)]

    2015 arXiv

  21. [27]

    Neill, I

    D. Neill, I. Z. Rothstein, and V. Vaidya, The Higgs Transverse Momentum Distribution at NNLL and its Theoretical Errors , JHEP 12 (2015) 097, [ arXiv:1503.00005]

    2015 arXiv

  22. [28]

    Bizon, P

    W. Bizon, P. F. Monni, E. Re, L. Rottoli, and P. Torrielli, Momentum-space resummation for transverse observables and the Higgs p ⊥ at N 3LL+NNLO, JHEP 02 (2018) 108, [arXiv:1705.09127]

    2018 arXiv

  23. [29]

    X. Chen, T. Gehrmann, E. W. N. Glover, A. Huss, Y. Li, D. Neill, M. Schulze, I. W. Stewart, and H. X. Zhu, Precise QCD Description of the Higgs Boson Transverse Momentum Spectrum, Phys. Lett. B788 (2019) 425–430, [ arXiv:1805.00736]

    2019 arXiv

  24. [30]

    Bizo´ n, X

    W. Bizo´ n, X. Chen, A. Gehrmann-De Ridder, T. Gehrmann, N. Glover, A. Huss, P. F. Monni, E. Re, L. Rottoli, and P. Torrielli, Fiducial distributions in Higgs and Drell-Yan production at N 3LL+NNLO, JHEP 12 (2018) 132, [ arXiv:1805.05916]

    2018 arXiv

  25. [31]

    H. X. Zhu, C. S. Li, H. T. Li, D. Y. Shao, and L. L. Yang, Transverse-momentum resummation for top-quark pairs at hadron colliders , Phys. Rev. Lett. 110 (2013), no. 8 082001, [arXiv:1208.5774]

    2013 arXiv

  26. [32]

    H. T. Li, C. S. Li, D. Y. Shao, L. L. Yang, and H. X. Zhu, Top quark pair production at small transverse momentum in hadronic collisions , Phys. Rev. D88 (2013) 074004, [arXiv:1307.2464]

    2013 arXiv

  27. [33]

    Catani, M

    S. Catani, M. Grazzini, and A. Torre, Transverse-momentum resummation for heavy-quark hadroproduction, Nucl. Phys. B890 (2014) 518–538, [ arXiv:1408.4564]

    2014 arXiv

  28. [34]

    Catani, M

    S. Catani, M. Grazzini, and H. Sargsyan, Transverse-momentum resummation for top-quark pair production at the LHC , JHEP 11 (2018) 061, [ arXiv:1806.01601]

    2018 arXiv

  29. [35]

    Moult and H

    I. Moult and H. X. Zhu, Simplicity from Recoil: The Three-Loop Soft Function and Factorization for the Energy-Energy Correlation , JHEP 08 (2018) 160, [arXiv:1801.02627]

    2018 arXiv

  30. [36]

    A. Gao, H. T. Li, I. Moult, and H. X. Zhu, The Transverse Energy-Energy Correlator in the Back-to-Back Limit, arXiv:1901.04497

    1901 arXiv

  31. [37]

    Catani and M

    S. Catani and M. Grazzini, Higgs Boson Production at Hadron Colliders: Hard-Collinear Coefficients at the NNLO , Eur. Phys. J. C72 (2012) 2013, [ arXiv:1106.4652]. [Erratum: Eur. Phys. J.C72,2132(2012)]. – 34 –

    2012 arXiv

  32. [38]

    Gehrmann, T

    T. Gehrmann, T. Lubbert, and L. L. Yang, Transverse parton distribution functions at next-to-next-to-leading order: the quark-to-quark case , Phys. Rev. Lett. 109 (2012) 242003, [arXiv:1209.0682]

    2012 arXiv

  33. [39]

    Gehrmann, T

    T. Gehrmann, T. Luebbert, and L. L. Yang, Calculation of the transverse parton distribution functions at next-to-next-to-leading order , JHEP 06 (2014) 155, [arXiv:1403.6451]

    2014 arXiv

  34. [41]

    Catani, S

    S. Catani, S. Devoto, M. Grazzini, S. Kallweit, J. Mazzitelli, and H. Sargsyan, Top-quark pair hadroproduction at next-to-next-to-leading order in QCD , Phys. Rev. D99 (2019), no. 5 051501, [arXiv:1901.04005]

    2019 arXiv

  35. [42]

    Catani, S

    S. Catani, S. Devoto, M. Grazzini, S. Kallweit, and J. Mazzitelli, Top-quark pair production at the LHC: Fully differential QCD predictions at NNLO , JHEP 07 (2019) 100, [arXiv:1906.06535]

    2019 arXiv

  36. [43]

    Y. Li, D. Neill, and H. X. Zhu, An Exponential Regulator for Rapidity Divergences , Submitted to: Phys. Rev. D (2016) [arXiv:1604.00392]

    2016 arXiv

  37. [44]

    Becher and M

    T. Becher and M. Neubert, Drell-Yan Production at Small qT , Transverse Parton Distributions and the Collinear Anomaly , Eur. Phys. J. C71 (2011) 1665, [arXiv:1007.4005]

    2011 arXiv

  38. [45]

    Becher and G

    T. Becher and G. Bell, Analytic Regularization in Soft-Collinear Effective Theory , Phys. Lett. B713 (2012) 41–46, [ arXiv:1112.3907]

    2012 arXiv

  39. [46]

    J.-Y. Chiu, A. Jain, D. Neill, and I. Z. Rothstein, A Formalism for the Systematic Treatment of Rapidity Logarithms in Quantum Field Theory , JHEP 05 (2012) 084, [arXiv:1202.0814]

    2012 arXiv

  40. [47]

    J.-y. Chiu, A. Fuhrer, A. H. Hoang, R. Kelley, and A. V. Manohar, Soft-Collinear Factorization and Zero-Bin Subtractions , Phys. Rev. D79 (2009) 053007, [arXiv:0901.1332]

    2009 arXiv

  41. [48]

    M. G. Echevarria, I. Scimemi, and A. Vladimirov, Universal transverse momentum dependent soft function at NNLO , Phys. Rev. D93 (2016), no. 5 054004, [arXiv:1511.05590]

    2016 arXiv

  42. [49]

    M. A. Ebert, I. Moult, I. W. Stewart, F. J. Tackmann, G. Vita, and H. X. Zhu, Subleading power rapidity divergences and power corrections for q T , JHEP 04 (2019) 123, [arXiv:1812.08189]

    2019 arXiv

  43. [50]

    Li and H

    Y. Li and H. X. Zhu, Bootstrapping Rapidity Anomalous Dimensions for Transverse-Momentum Resummation, Phys. Rev. Lett. 118 (2017), no. 2 022004, [arXiv:1604.01404]

    2017 arXiv

  44. [51]

    K. G. Chetyrkin and F. V. Tkachov, Integration by Parts: The Algorithm to Calculate beta Functions in 4 Loops, Nucl. Phys. B192 (1981) 159–204

    1981

  45. [52]

    Laporta, High precision calculation of multiloop Feynman integrals by difference equations, Int

    S. Laporta, High precision calculation of multiloop Feynman integrals by difference equations, Int. J. Mod. Phys. A15 (2000) 5087–5159, [ hep-ph/0102033]

    2000 arXiv

  46. [53]

    Z. Bern, L. J. Dixon, and D. A. Kosower, Dimensionally regulated pentagon integrals, Nucl. Phys. B412 (1994) 751–816, [ hep-ph/9306240]. – 35 –

    1994 arXiv

  47. [54]

    Gehrmann and E

    T. Gehrmann and E. Remiddi, Differential equations for two loop four point functions , Nucl. Phys. B580 (2000) 485–518, [ hep-ph/9912329]

    2000 arXiv

  48. [55]

    J. M. Henn, Multiloop integrals in dimensional regularization made simple , Phys. Rev. Lett. 110 (2013) 251601, [ arXiv:1304.1806]

    2013 arXiv

  49. [56]

    C. W. Bauer, S. Fleming, and M. E. Luke, Summing Sudakov logarithms in B —¿ X(s gamma) in effective field theory , Phys. Rev. D63 (2000) 014006, [ hep-ph/0005275]

    2000 arXiv

  50. [57]

    C. W. Bauer, S. Fleming, D. Pirjol, and I. W. Stewart, An Effective field theory for collinear and soft gluons: Heavy to light decays , Phys. Rev. D63 (2001) 114020, [ hep-ph/0011336]

    2001 arXiv

  51. [58]

    C. W. Bauer, D. Pirjol, and I. W. Stewart, Soft collinear factorization in effective field theory, Phys. Rev. D65 (2002) 054022, [ hep-ph/0109045]

    2002 arXiv

  52. [59]

    C. W. Bauer, S. Fleming, D. Pirjol, I. Z. Rothstein, and I. W. Stewart, Hard scattering factorization from effective field theory , Phys. Rev. D66 (2002) 014017, [ hep-ph/0202088]

    2002 arXiv

  53. [60]

    Beneke, A

    M. Beneke, A. P. Chapovsky, M. Diehl, and T. Feldmann, Soft collinear effective theory and heavy to light currents beyond leading power , Nucl. Phys. B643 (2002) 431–476, [hep-ph/0206152]

    2002 arXiv

  54. [61]

    J. C. Collins and D. E. Soper, Parton Distribution and Decay Functions , Nucl. Phys. B194 (1982) 445–492

    1982

  55. [62]

    Beneke, Lectures on soft-collinear effective theory, in Helmholtz International Summer School on Heavy Quark Physics , (Dubna), 2005

    M. Beneke, Lectures on soft-collinear effective theory, in Helmholtz International Summer School on Heavy Quark Physics , (Dubna), 2005

    2005

  56. [63]

    A. A. Vladimirov, Correspondence between Soft and Rapidity Anomalous Dimensions , Phys. Rev. Lett. 118 (2017), no. 6 062001, [ arXiv:1610.05791]

    2017 arXiv

  57. [64]

    M. A. Ebert, I. W. Stewart, and Y. Zhao, Determining the Nonperturbative Collins-Soper Kernel From Lattice QCD, Phys. Rev. D99 (2019), no. 3 034505, [ arXiv:1811.00026]

    2019 arXiv

  58. [65]

    M. A. Ebert, I. W. Stewart, and Y. Zhao, Towards Quasi-Transverse Momentum Dependent PDFs Computable on the Lattice , arXiv:1901.03685

    1901 arXiv

  59. [66]

    L¨ ubbert, J

    T. L¨ ubbert, J. Oredsson, and M. Stahlhofen, Rapidity renormalized TMD soft and beam functions at two loops , JHEP 03 (2016) 168, [ arXiv:1602.01829]

    2016 arXiv

  60. [67]

    M. G. Echevarria, I. Scimemi, and A. Vladimirov, Transverse momentum dependent fragmentation function at next-to–next-to–leading order , Phys. Rev. D93 (2016), no. 1 011502, [arXiv:1509.06392]. [Erratum: Phys. Rev.D94,no.9,099904(2016)]

    2016 arXiv

  61. [68]

    Lustermans, W

    G. Lustermans, W. J. Waalewijn, and L. Zeune, Joint transverse momentum and threshold resummation beyond NLL, Phys. Lett. B762 (2016) 447–454, [ arXiv:1605.02740]

    2016 arXiv

  62. [69]

    Nogueira, Automatic Feynman graph generation, J

    P. Nogueira, Automatic Feynman graph generation, J. Comput. Phys. 105 (1993) 279–289

    1993

  63. [70]

    J. A. M. Vermaseren, New features of FORM , math-ph/0010025

    arXiv

  64. [71]

    H. X. Zhu, On the calculation of soft phase space integral , JHEP 02 (2015) 155, [arXiv:1501.00236]

    2015 arXiv

  65. [72]

    Anastasiou and K

    C. Anastasiou and K. Melnikov, Higgs boson production at hadron colliders in NNLO QCD , Nucl. Phys. B646 (2002) 220–256, [ hep-ph/0207004]

    2002 arXiv

  66. [73]

    A. V. Smirnov, FIRE5: a C++ implementation of Feynman Integral REduction , Comput. Phys. Commun. 189 (2015) 182–191, [ arXiv:1408.2372]. – 36 –

    2015 arXiv

  67. [74]

    R. N. Lee, Presenting LiteRed: a tool for the Loop InTEgrals REDuction , arXiv:1212.2685

    arXiv

  68. [75]

    Meyer, Algorithmic transformation of multi-loop master integrals to a canonical basis with CANONICA, Comput

    C. Meyer, Algorithmic transformation of multi-loop master integrals to a canonical basis with CANONICA, Comput. Phys. Commun. 222 (2018) 295–312, [ arXiv:1705.06252]

    2018 arXiv

  69. [76]

    Meyer, Transforming differential equations of multi-loop Feynman integrals into canonical form, JHEP 04 (2017) 006, [ arXiv:1611.01087]

    C. Meyer, Transforming differential equations of multi-loop Feynman integrals into canonical form, JHEP 04 (2017) 006, [ arXiv:1611.01087]

    2017 arXiv

  70. [77]

    Panzer, Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals, Comput

    E. Panzer, Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals, Comput. Phys. Commun. 188 (2015) 148–166, [ arXiv:1403.3385]

    2015 arXiv

  71. [78]

    Remiddi and J

    E. Remiddi and J. A. M. Vermaseren, Harmonic polylogarithms, Int. J. Mod. Phys. A15 (2000) 725–754, [ hep-ph/9905237]

    2000 arXiv

  72. [79]

    Maitre, HPL, a mathematica implementation of the harmonic polylogarithms , Comput

    D. Maitre, HPL, a mathematica implementation of the harmonic polylogarithms , Comput. Phys. Commun. 174 (2006) 222–240, [ hep-ph/0507152]

    2006 arXiv

  73. [80]

    Y. L. Dokshitzer, G. Marchesini, and B. R. Webber, Nonperturbative effects in the energy energy correlation, JHEP 07 (1999) 012, [ hep-ph/9905339]

    1999 arXiv

  74. [81]

    de Florian and M

    D. de Florian and M. Grazzini, The Back-to-back region in e+ e- energy-energy correlation , Nucl. Phys. B704 (2005) 387–403, [ hep-ph/0407241]

    2005 arXiv

  75. [83]

    L. J. Dixon, I. Moult, and H. X. Zhu, Collinear limit of the energy-energy correlator , Phys. Rev. D100 (2019), no. 1 014009, [ arXiv:1905.01310]

    2019 arXiv

  76. [84]

    G. P. Korchemsky, Energy correlations in the end-point region , arXiv:1905.01444

    1905 arXiv

  77. [85]

    J. M. Henn, E. Sokatchev, K. Yan, and A. Zhiboedov, Energy-energy correlations at next-to-next-to-leading order, arXiv:1903.05314

    1903 arXiv

  78. [86]

    A. V. Kotikov and L. N. Lipatov, DGLAP and BFKL equations in the N = 4 supersymmetric gauge theory, Nucl. Phys. B661 (2003) 19–61, [ hep-ph/0208220]. [Erratum: Nucl. Phys.B685,405(2004)]

    2003 arXiv

  79. [87]

    Kologlu, P

    M. Kologlu, P. Kravchuk, D. Simmons-Duffin, and A. Zhiboedov, The light-ray OPE and conformal colliders, arXiv:1905.01311

    1905 arXiv

  80. [88]

    G. P. Korchemsky and A. V. Radyushkin, Renormalization of the Wilson Loops Beyond the Leading Order, Nucl. Phys. B283 (1987) 342–364

    1987

  81. [89]

    I. A. Korchemskaya and G. P. Korchemsky, On lightlike Wilson loops , Phys. Lett. B287 (1992) 169–175

    1992

  82. [90]

    Gehrmann, T

    T. Gehrmann, T. Huber, and D. Maitre, Two-loop quark and gluon form-factors in dimensional regularisation, Phys. Lett. B622 (2005) 295–302, [ hep-ph/0507061]

    2005 arXiv

  83. [91]

    S. Moch, J. A. M. Vermaseren, and A. Vogt, The Quark form-factor at higher orders , JHEP 08 (2005) 049, [ hep-ph/0507039]

    2005 arXiv

  84. [92]

    Becher and M

    T. Becher and M. Neubert, On the Structure of Infrared Singularities of Gauge-Theory Amplitudes, JHEP 06 (2009) 081, [ arXiv:0903.1126]. [Erratum: JHEP11,024(2013)]

    2009 arXiv

  85. [93]

    Y. Li, A. von Manteuffel, R. M. Schabinger, and H. X. Zhu, Soft-virtual corrections to Higgs production at N 3LO, Phys. Rev. D91 (2015) 036008, [ arXiv:1412.2771]. – 37 –

    2015 arXiv

  86. [94]

    D. J. Gross and F. Wilczek, Ultraviolet Behavior of Nonabelian Gauge Theories , Phys. Rev. Lett. 30 (1973) 1343–1346. [,271(1973)]

    1973

  87. [95]

    H. D. Politzer, Reliable Perturbative Results for Strong Interactions? , Phys. Rev. Lett. 30 (1973) 1346–1349. [,274(1973)]

    1973

  88. [96]

    W. E. Caswell, Asymptotic Behavior of Nonabelian Gauge Theories to Two Loop Order , Phys. Rev. Lett. 33 (1974) 244

    1974

  89. [97]

    D. R. T. Jones, Two Loop Diagrams in Yang-Mills Theory , Nucl. Phys. B75 (1974) 531

    1974

  90. [98]

    Egorian and O

    E. Egorian and O. V. Tarasov, Two Loop Renormalization of the QCD in an Arbitrary Gauge, Teor. Mat. Fiz. 41 (1979) 26–32. [Theor. Math. Phys.41,863(1979)]

    1979

  91. [99]

    Altarelli and G

    G. Altarelli and G. Parisi, Asymptotic Freedom in Parton Language , Nucl. Phys. B126 (1977) 298–318

    1977

  92. [100]

    Furmanski and R

    W. Furmanski and R. Petronzio, Singlet Parton Densities Beyond Leading Order , Phys. Lett. 97B (1980) 437–442

    1980

  93. [101]

    Curci, W

    G. Curci, W. Furmanski, and R. Petronzio, Evolution of Parton Densities Beyond Leading Order: The Nonsinglet Case , Nucl. Phys. B175 (1980) 27–92

    1980

  94. [102]

    Ritzmann and W

    M. Ritzmann and W. J. Waalewijn, Fragmentation in Jets at NNLO , Phys. Rev. D90 (2014), no. 5 054029, [ arXiv:1407.3272]. – 38 –

    2014 arXiv

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