We have calculated the coefficient functions for the structure functions F_2, F_L and F_3 in nu-nubar charged-current deep-inelastic scattering (DIS) at the third order in the strong coupling alpha_s, thus completing the description of unpolarized inclusive W^(+-) exchange DIS to this order of massless perturbative QCD. In this brief note, our new results are presented in terms of compact approximate expressions that are sufficiently accurate for phenomenological analyses. For the benefit of such analyses we also collect, in a unified notation, the corresponding lower-order contributions and the flavour non-singlet coefficient functions for nu+nubar charged-current DIS. The behaviour of all six third-order coefficient functions at small Bjorken-x is briefly discussed

Davies, J

Moch, S

Vermaseren, JAM

Vogt, A

We have calculated the coefficient functions for the structure functions F_2,
F_L and F_3 in nu-nubar charged-current deep-inelastic scattering (DIS) at the
third order in the strong coupling alpha_s, thus completing the description of
unpolarized inclusive W^(+-) exchange DIS to this order of massless
perturbative QCD. In this brief note, our new results are presented in terms of
compact approximate expressions that are sufficiently accurate for
phenomenological analyses. For the benefit of such analyses we also collect, in
a unified notation, the corresponding lower-order contributions and the flavour
non-singlet coefficient functions for nu+nubar charged-current DIS. The
behaviour of all six third-order coefficient functions at small Bjorken-x is
briefly discussed

University of Liverpool Repository

arXiv:1606.08907v1  [hep-ph]  28 Jun 2016LTH 1089, DESY 16-110, Nikhef 2016-030 June 2016Non-singlet coefficient functions for charged-currentdeep-inelastic scattering to the third order in QCDJ. Davies∗Department of Mathematical Sciences, University of Liverpool, Liverpool L69 3BX, UKE-mail: Joshua.Davies@liv.ac.ukS. MochII. Institute for Theoretical Physics, University of Hamburg, D-22761 Hamburg, GermanyE-mail: Sven-Olaf.Moch@desy.deJ.A.M. VermaserenNikhef Theory Group, Science Park 105, 1098 XG Amsterdam, The NetherlandsE-mail: t68@nikhef.nlA. VogtDepartment of Mathematical Sciences, University of Liverpool, Liverpool L69 3BX, UKE-mail: Andreas.Vogt@liv.ac.ukWe have calculated the coefficient functions for the structure functions F2, FL and F3 in ν− ¯νcharged-current deep-inelastic scattering (DIS) at the third order in the strong coupling αs , thuscompleting the description of unpolarized inclusive W±-exchange DIS to this order of masslessperturbative QCD. In this brief note, our new results are presented in terms of compact approxi-mate expressions that are sufficiently accurate for phenomenological analyses. For the benefit ofsuch analyses we also collect, in a unified notation, the corresponding lower-order contributionsand the flavour non-singlet coefficient functions for ν+ ¯ν charged-current DIS. The behaviour ofall six third-order coefficient functions at small Bjorken-x is briefly discussed.XXIV International Workshop on Deep-Inelastic Scattering and Related Subjects11-15 April 2016, DESY Hamburg, Germany∗Speaker.c© Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). http://pos.sissa.it/Non-singlet coefficient functions for charged-current DIS J. Davies1. IntroductionWith six structure functions, F2, FL and F3 for W+ and W− exchange [1, 2], inclusive charged-current DIS is an important source of information on the parton structure of nucleons and nucleiand on Standard Model parameters such as the strong coupling αs and the weak mixing sin2 θW [3].Future facilities, e.g., the LHeC [4], will be required to fully realize its phenomenological potential.Charged-current structure functions are, at very small values of the Bjorken variable x, also ofinterest for the scattering of high-energy cosmic neutrinos, see Ref. [5] and references therein.Here we address the corresponding coefficient functions (mass-factorized partonic cross sec-tions) in massless perturbative QCD. These functions are relevant also beyond DIS, e.g., for Higgsproduction via vector boson fusion [6, 7]. For recent progress on heavy-quark effects see Ref. [8].2. First-order and ν + ¯ν non-singlet coefficient functionsThe first-order coefficient functions for unpolarized inclusive DIS were derived in the early days ofQCD, see Refs. [1,9]. The second- and third-order contributions for the ν + ¯ν charged-current casehave been calculated in Refs. [10–17]. Here we collect the corresponding flavour non-singlet resultsin the MS scheme for the standard choice µF = µR = Q of the renormalization and factorizationscales. The additional contributions at other values of µF and/or µR are determined by these resultsand the corresponding splitting functions [18, 19]; see, e.g., Eqs. (2.16) – (2.18) in Ref. [20].We denote the non-singlet quark coefficient functions for the charged-current structure func-tions F ν p± ¯ν p2,3,L (x,Q2) in neutrino-proton DIS by Ca,±, and write their perturbative expansion asCa,±(x,Q2) = ∑n=0ans c(n)a,±(x) with as ≡ αs(Q2)/(4pi) . (2.1)In this notation the zeroth- and first-order coefficient functions are given byc(0)2,±(x) = c(0)3,±(x) = δ (x1) , c(0)L,±(x) = 0 , c(1)L,±(x) = 4CF x , (2.2)c(1)2,±(x) = CF{4D1− 3D0− (9+ 4ζ2)δ (x1)− 2(1+ x)(L1−L0)− 4x−11 L0 + 6+ 4x} , (2.3)c(1)3,±(x) = c(1)2,±(x)− 2CF (1+ x) (2.4)with CF = (n2c −1)/(2nc) = 4/3 in QCD. Here and below we use the abbreviationsx1 = 1− x , L0 = ln x , L1 = ln x1 , Dk = [x−11 Lk1 ]+ , (2.5)where [a(x)]+ denotes +-distributions defined via∫ 10 dx [a(x)]+ f (x) ≡∫ 10 dx a(x){ f (x)− f (1)}.The coefficient functions c(n)a,+ and c(n)a,− differ at n > 1. The 2nd- and 3rd-order contributions tothe former quantities read, in an approximate but sufficiently accurate form given in Refs. [15–17],c(2)2,+(x)∼= 128/9D3− 184/3D2− 31.1052D1+ 188.641D0− 338.513δ (x1)− 17.74L31+ 72.24L21− 628.8L1− 181.0− 806.7x+L0L1(37.75L0− 147.1L1)+ 0.719xL40− 28.384L0− 20.70L20− 80/27L30 (2.6)+ nf{16/9D2− 232/27D1+ 6.34888D0+ 46.8531δ (x1)− 1.500L21+ 24.87L1− 7.8109− 17.82x−12.97x2+ 8.113L0L1− 0.185xL30+ 16/3L0+ 20/9L20},c(2)L,+(x)∼= − 37.338+ 89.53x+33.82x2+ 128/9xL21− 46.50xL1+ xL0(32.90+ 18.41L0)− 84.094L0L1− 128/9L0− 0.012δ (x1) + 16/27nf{6xL1− 12xL0− 25x+ 6}, (2.7)1Non-singlet coefficient functions for charged-current DIS J. Daviesc(2)3,+(x)∼= 128/9D3− 184/3D2− 31.1052D1+ 188.641D0− 338.572δ (x1)− 16.40L31+ 78.46L21− 470.6L1− 149.75− 693.2x+0.218xL40+L0L1(33.62L0− 117.8L1)− 49.30L0− 94/3L20− 104/27L30 (2.8)+ nf{16/9D2− 232/27D1+ 6.34888D0+ 46.8464δ (x1)+ 0.066L31− 0.663L21+ 24.86L1− 5.738− 5.845x−10.235x2− 0.190xL30+ 4.265L0L1 + 20/9L0(4+L0)}andc(3)2,+(x)∼= 512/27D5− 5440/27D4+ 501.099D3+ 1171.54D2− 7328.45D1+ 4442.76D0− 9170.38δ (x1)− 512/27L51+ 704/3L41− 3368L31− 2978L21+ 18832L1− 4926+ 7725x+ 57256x2+ 12898x3− 56000x1L21−L0L1(6158+ 1836L0)+ 4.719xL50− 775.8L0− 899.6L20− 309.1L30− 2932/81L40− 32/27L50+ nf{640/81D4− 6592/81D3+ 220.573D2+ 294.906D1− 729.359D0+ 2574.687δ (x1)− 640/81L41+ 153.5L31− 828.7L21− 501.1L1+ 831.6− 6752x− 2778x2+ 171.0x1L41 +L0L1 (4365+ 716.2L0− 5983L1)+ 4.102xL40+ 275.6L0+ 187.3L20+ 12224/243L30+ 728/243L40} (2.9)+ n2f{64/81D3− 464/81D2+ 7.67505D1+ 1.00830D0− 103.2366δ (x1)− 64/81L31+ 18.21L21− 19.09L1+ 129.2x+ 102.5x2+L0L1 (−96.07− 12.46L0+ 85.88L1)− 8.042L0− 1984/243L20− 368/243L30},c(3)L,+(x)∼= 512/27L41− 177.40L31+ 650.6L21− 2729L1− 2220.5− 7884x+4168x2− (844.7L0+ 517.3L1)L0L1 +(195.6L1− 125.3)x1L31 + 208.3xL30− 1355.7L0− 7456/27L20− 1280/81L30+ 0.113δ (x1)+ nf{1024/81L31− 112.35L21+ 344.1L1+ 408.4− 9.345x−919.3x2+(239.7+ 20.63L1)x1L21 +(887.3+ 294.5L0− 59.14L1)L0L1− 1792/81xL30+ 200.73L0+ 64/3L20 + 0.006δ (x1)} (2.10)+ n2f{3xL21 +(6− 25x)L1− 19+(317/6−12ζ2)x− 6xL0L1 + 6xLi2(x)+ 9xL20− (6− 50x)L0}64/81 ,c(3)3,+(x)∼= 512/27D5− 5440/27D4+ 501.099D3+ 1171.54D2− 7328.45D1+ 4442.76D0− 9172.68δ (x1)− 512/27L51+ 8896/27L41− 1396L31+ 3990L21+ 14363L1− 1853− 5709x+ xx1(5600− 1432x)−L0L1(4007+ 1312L0)− 0.463xL60− 293.3L0− 1488L20− 496.95L30− 4036/81L40− 536/405L50+ nf{640/81D4− 6592/81D3+ 220.573D2+ 294.906D1− 729.359D0+ 2575.46δ (x1)− 640/81L41+ 32576/243L31− 660.7L21+ 959.1L1+ 31.95x1L41+ 516.1− 465.2x+ xx1(635.3+ 310.4x)+L0L1 (1496+ 270.1L0− 1191L1)− 1.200xL40+ 366.9L0+ 305.32L20+ 48512/729L30+ 304/81L40} (2.11)+ n2f{64/81D3− 464/81D2+ 7.67505D1+ 1.00830D0− 103.2602δ (x1)− 64/81L31+ 992/81L21− 49.65L1+ 11.32− xx1(44.52+ 11.05x)+ 51.94x+0.0647xL40−L0L1 (39.99+ 5.103L0− 16.30L1)− 16.00L0− 2848/243L20− 368/243L30}+ f l02 nf{2.147L21− 24.57L1+ 48.79− x1(242.4− 150.7x)−L0L1 (81.70+ 9.412L1)+ xL0 (218.1+ 82.27L20)− 477.0L0− 113.4L20 + 17.26L30− 16/27L50}x1 .2Non-singlet coefficient functions for charged-current DIS J. Davies3. ν− ¯ν non-singlet coefficient functionsThe differences between corresponding ν + ¯ν and ν − ¯ν coefficient functions are, as conjecturedin Ref. [23], suppressed at large x by two powers of 1−x. Hence it is convenient to present thecoefficient functions for ν− ¯ν charged-current DIS in terms of differences which we define asδC2,L ≡ C ν p+ ¯ν p2,L −Cν p− ¯ν p2,L , δC3 ≡ Cν p− ¯ν p3 −Cν p+ ¯ν p3 . (3.1)The flavour class f l02 , see Fig. 1 of Ref. [16], does not contribute to the flavour asymmetriesprobed in the ν− ¯ν combinations, hence it is understood that the corresponding part of Eq. (2.11)is removed before the difference for F3 is formed. The δCa can be perturbatively expanded asδCa = ∑n=2ans δ c(n)a , (3.2)where as is defined in Eq. (2.1) above. The second-order results were already given in Ref. [21] inexact and parametrized form. The later results, written in terms of the abbreviations (2.5), readδc(2)2 (x) ∼= {−9.1587−57.70x+72.29x2 −5.689x3− xL0 (68.804+24.40L0+2.958L20 )+0.249L0 +8/9 L20 (2+L0)}x1 , (3.3)δc(2)L (x) ∼= {10.663−5.248x−7.500x2 +0.823x3 + xL0 (11.10+2.225L0−0.128L20 )+64/9 L0 }x21 , (3.4)δc(2)3 (x) ∼= {−29.65+116.05x−71.74x2 −16.18x3 + xL0 (14.60+69.90x−0.378L20 )−8.560L0 +8/9 L20 (4+L0)} x1 . (3.5)The corresponding third-order corrections are the new results of the present contribution. Theysupersede the previous approximate expressions in Eqs. (3.7) – (3.9) of Ref. [21], which werebased on the lowest five even- integer and odd-integer Mellin moments of C3,− and Ca,−, a = 2, L,respectively, computed in Ref. [22]. Our new exact results can parametrized asδc(3)2 ∼={273.59−44.95x−73.56x2 +40.68x3 +0.1356L50 +8.483L40 +55.90L30+120.67L20 +388.0L0−329.8L0L1− xL0(316.2+71.63L0)+46.30L1+5.447L21}x1−0.0008δ (x1) (3.6)+ nf{(−19.093 +12.97x+36.44x2 −29.256x3−0.76L40−5.317L30−19.82L20−38.958L0−13.395L0L1 + xL0(14.44+17.74L0)+1.395L1)x1 +0.0001δ (x1)},δc(3)L ∼={−620.53−394.5x+1609x2 −596.2x3 +0.217L30 +62.18L20 +208.47L0−482.5L0L1− xL0(1751−197.5L0)+105.5L1 +0.442L21}x21 (3.7)+ nf{−6.500−12.435x+23.66x2 +0.914x3 +0.015L30−6.627L20−31.91L0− xL0(5.711+28.635L0)}x21 ,δc(3)3 ∼={−553.5+1412.5x−990.3x2 +361.1x3 +0.1458L50 +9.688L40 +90.62L30+83.684L20−602.32L0−382.5L0L1− xL0(2.805+325.92L0)+133.5L1+10.135L21}x1−0.0029δ (x1) (3.8)+ nf{(−16.777+77.78x−24.81x2 −28.89x3−0.7714L40−7.701L30−21.522L20−7.897L0−16.17L0L1 + xL0(43.21+67.04L0)+1.519L1)x1 +0.00006δ (x1)}.3Non-singlet coefficient functions for charged-current DIS J. Davies4. DiscussionWith the exception of the nf part of Eq. (2.7) and the n2f part of Eq. (2.10) which are exact, thesecond- and third-order expressions in sections 2 and 3 have been obtained by fitting the coefficientsnot written as fractions in the non-distribution parts to the exact coefficient functions at x≥ 10−6.Where useful, the coefficients of δ (1−x) have been adjusted (even from zero) to fine-tune theaccuracy of Mellin moments and convolutions. The resulting accuracy of Eqs. (2.6) – (2.11) and(3.3) – (3.8) and their convolutions with typical quark distributions of hadrons is 0.1% or betterexcept where the functions are very small. Towards smaller x the accuracy deteriorates, but theresults are still accurate to about 1% and 3% at x = 10−8 and x = 10−10, respectively.FORTRAN subroutines of these functions can be obtained from the preprint server ARXIV.ORGby downloading the source of this note. They are also available from the authors upon request.Analogous parametrizations for the pure-singlet quark and gluon coefficient functions for F2and FL have been given in Ref. [15] and section 4 of Ref. [16]. The partly very lengthy exactexpression corresponding to Eqs. (2.6) – (2.11) and (3.3) – (3.5) can be found in Ref. [16] – wherethe f l11 contribution has to be disregarded for the present charged-current case – and Refs. [17,21];those for Eqs. (3.6) – (3.8) will be presented in Ref. [25]. Only the latter expressions allow for theanalytical calculation of all integer Mellin moments of the coefficient functions.The second moments of δc(3)2 and δc(3)L are of particular relevance, since they enter the QCDcorrections to the Paschos-Wolfenstein relation [24] for the determination of sin2 θW from charged-current DIS [3]. The truncated numerical values of these moments for nf light flavours areδ c(3)2 (N = 2) = −20.4001+0.72202nf , δ c(3)L (N = 2) = −24.7755+0.80134nf . (4.1)Within their error estimates, the previous approximate results for these moments [21] agree withEq. (4.1). The corresponding analytical expressions will also by presented in Ref. [25].-20-1001010 -6 10 -4 10 -2 1-20-1001010 -6 10 -4 10 -2 1a = +a = −xc2,a (x)(3)∗ 1/2000, nf = 4a = +a = −xc3,a (x)(3)∗ 1/2000, nf = 4with fl02fl02 = 0a = +a = −xcL,a (x)(3)∗ 1/2000, nf = 4-20246810 -6 10 -4 10 -2 1Figure 1: The third-order non-singlet coefficient functions for F2, FL and F3 in ν p+ ¯ν p (a=+) and ν p− ¯ν p(a = −) charged-current DIS for four light flavours. The function c(3)3,+(x) in Eq. (2.11) is shown with andwithout the f l02 contribution. The factor 1/2000 approximately converts the curves to an expansion in αs.4Non-singlet coefficient functions for charged-current DIS J. DaviesThe a3s coefficients c(3)a,± in Eq. (2.1) are illustrated in Fig. 1 over a wide range in x. All sixfunctions exhibit a sharp small-x rise, but only at x < 10−5 for F2 and F3 and at x <∼ 10−4 for FL .With the exception of the flavour structure f l02 that occurs at three loops for the first time but dom-inates c(3)a,+ at small x, at least four of five lnkx small-x terms are required for a good approximationfor c(3)2,± and c(3)3,±, and all three such terms for c(3)L,±, even below the x-range shown in the figure.Further discussions and illustrations of these coefficient functions will be presented in Ref. [25].AcknowledgementsThis work has been supported by the UK Science & Technology Facilities Council (STFC) grantsST/L000431/1 and ST/K502145/1, by the Deutsche Forschungsgemeinschaft (DFG) via contractMO 1801/1-1, and by the European Research Council (ERC) Advanced Grant 320651, HEPGAME.References[1] A.J. Buras, Rev. Mod. Phys. 52 (1980) 199, and references therein[2] K.A. Olive et al., Particle Data Group, Chin. Phys. C38 (2014) 090001[3] G. Zeller et al., NuTeV, Phys. Rev. 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Non-singlet coefficient functions for charged-current deep-inelastic scattering to the third order in QCD

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