Spiral Cholesterics: Surface Effects

We have computed the fourth-order n 2 f contributions to all three non-singlet quark-quark splitting functions and their four n 3 f flavour-singlet counterparts for the evolution of the parton distributions of hadrons in perturbative QCD with n f effectively massless quark flavours. The analytic form of these functions is presented in both Mellin N-space and momentum-fraction x-space; the large-x and small-x limits are discussed. Our results agree with all available predictions derived from lower-order information. The large-x limit of the quark-quark cases provides the complete n 2 f part of the four-loop cusp anomalous dimension which agrees with two recent partial computations.

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“…Refs. [54][55][56], and invariant under the x-space transformation f (x) → x f (1/x). The (combinations of) harmonic sums in Eq.…”

Section: )mentioning

confidence: 99%

Large-n contributions to the four-loop splitting functions in QCD

et al. 2017

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“…Eq. (6.1) has been applied and verified for the NNLO non-singlet [6] and gluon-gluon [8] splitting functions. In the latter case its applicability is obvious for Quantum Gluodynamics, n f = 0, but it is found to hold for all terms in the limit C F = 0.…”

Section: Dms Relation and 'Non-singlet' Resummation Of P T Ggmentioning

confidence: 99%

“…The splitting functions (1.4) and coefficient functions (1.8) are known (with a minor caveat in the former case which is not relevant in the present context) to order α 3 s and α 2 s , respectively, see Refs. [4][5][6][7][8][9] and references therein. These results provide the next-to-next-to-leading order (NNLO) approximation of (renormalization-group improved) perturbative QCD except for F L , where the third-order contributions to Eq.…”

Section: Introductionmentioning

confidence: 99%

Resummed small-x and first-moment evolution of fragmentation functions in perturbative QCD

Kom

Vogt

Yeats

2012

J. High Energ. Phys.

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We study the splitting functions for the evolution of fragmentation distributions and the coefficient functions for single-hadron production in semi-inclusive e + e − annihilation in massless perturbative QCD for small values of the momentum fraction and scaling variable x, where their fixed-order approximations are completely destabilized by huge double logarithms of the form α n s x −1 ln 2n−a x. Complete analytic all-order expressions in Mellin-N space are presented for the resummation of these terms at the next-to-next-to-leading logarithmic accuracy. The poles for the first moments, related to the evolution of hadron multiplicities, and the small-x instabilities of the next-to-leading order splitting and coefficient functions are removed by this resummation, which leads to an oscillatory small-x behaviour and functions that can be used at N = 1 and down to extremely small values of x. First steps are presented towards extending these results to the higher accuracy required for an all-x combination with the state-of-the-art next-to-next-to-leading order large-x results.If the constants up to F (m) n,ℓ are known for all n and ℓ, then the splitting functions and coefficient functions can be determined at N m LL accuracy at all orders of the strong coupling. As observed in Ref. [19], the n th order small-x contributions to F T, φ are built up from n terms of the form(2.4)Since the terms with ε −2n+1 , . . . , ε −n−1 have to cancel in sum (2.1), there are n−1 relations between the LL coefficients A n,k which lead to the constants F (0) n,ℓ in Eq. (2.3), n−2 relations between the NLL coefficients B n,k etc. As discussed above, a N m LO calculation fixes the (nonvanishing) coefficients of ε −n , . . . , ε −n+m at all orders n, adding m + 1 more relations between the coefficients in Eq. (2.4). Consequently the highest m+1 double logarithms, i.e., the N m LL approximation, can be determined order by order from the N m LO results. Finally the resulting series, here calculated to order α 18 s using FORM and TFORM [24], can be employed to find their generating functions via over-constrained systems of linear equations. The whole procedure is analogous to, if computationally more involved than, the large-x resummation in Ref. [25].

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“…(1.2) are known to order α 2 s [7][8][9][10], see also Ref. [11], i.e., to the next-to-next-to-leading order (NNLO) for F T and F A and to the next-to-leading order (NLO) for F L which vanishes for α s = 0. Here and throughout this article we identify, without loss of information, the MS renormalization and mass factorization scales with the physical hard scale Q 2 .…”

Section: Introductionmentioning

confidence: 99%

Resummation of small-x double logarithms in QCD: semi-inclusive electron-positron annihilation

Vogt

2011

J. High Energ. Phys.

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We have derived the coefficients of the highest three 1/x -enhanced small-x logarithms of all timelike splitting functions and the coefficient functions for the transverse fragmentation function in one-particle inclusive e + e − annihilation at (in principle) all orders in massless perturbative QCD. For the longitudinal fragmentation function we present the respective two highest contributions. These results have been obtained from KLN-related decompositions of the unfactorized fragmentation functions in dimensional regularization and their structure imposed by the mass-factorization theorem. The resummation is found to completely remove the huge small-x spikes present in the fixed-order results, allowing for stable results down to very small values of the momentum fraction and scaling variable x. Our calculations can be extended to (at least) the corresponding α n s ln 2n−ℓ x contributions to the above quantities and their counterparts in deep-inelastic scattering.

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Spiral Cholesterics: Surface Effects

Cited by 15 publications

References 0 publications

Large-n contributions to the four-loop splitting functions in QCD

Large-n contributions to the four-loop splitting functions in QCD

Resummed small-x and first-moment evolution of fragmentation functions in perturbative QCD

Resummation of small-x double logarithms in QCD: semi-inclusive electron-positron annihilation

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