Non-singlet Contribution Research Articles

Two-loop helicity amplitudes for V+jet production including axial vector couplings to higher orders in ϵ

We compute the two-loop Quantum Chromodynamics (QCD) corrections to all partonic channels relevant for the production of an electroweak boson V = Z, W±, γ* and a jet at hadron colliders. We consider the decay of a vector boson V to three partons V → qq¯g\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ q\\overline{q}g $$\\end{document}, V → ggg with a vector and axial vector coupling in both channels, including singlet and non-singlet contributions. For the quark channel, we use a recent tensor decomposition and extend the calculation to O\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{O} $$\\end{document}(ϵ2). For the gluonic channel, we define a new tensor decomposition which allows us to compute the vector and the axial vector amplitudes at once and to perform the computation of the amplitudes to O\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{O} $$\\end{document}(ϵ2). We provide finite remainders of the helicity amplitudes analytically continued to all relevant scattering regions qq¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ q\\overline{q} $$\\end{document} → Vg, qg → Vq and gg → Vg. The axial vector contribution to the gluon-induced channel completes the set of two-loop amplitudes for this process, while the extension to O\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{O} $$\\end{document}(ϵ2) represents the first step in the calculation of next-to-next-to-next-to-leading-order (N3LO) QCD corrections to Z+jet production at hadron colliders.

N-jettiness for muon jet pairs in electroweak high-energy processes

We study the N-jettiness in the electroweak high-energy process for the final muon jet pairs, e−e+ → μ+ jet + μ−jet. Compared to QCD, the main difference is that there exist additional gauge nonsinglet contributions in the weak interaction, which make the factorization more elaborate. Especially the nonsinglet contributions arise due to the Block-Nordsieck violation in electroweak processes, which yields the Sudakov logarithms and the rapidity divergence. They change the evolution of the factorized parts considerably in the N-jettiness. There are two possible channels, initiated from the gauge bosons WW → ℓμ overline{ell} μ, and from the electrons ℓe overline{ell} e → ℓμ overline{ell} μ, where ℓ denotes the weak doublet. The latter was discussed previously, and we complete the analysis by studying the first. The factorization for WW → ℓμ overline{ell} μ can be proceeded in a similar way as in the factorization for ℓe overline{ell} e → ℓμ overline{ell} μ, and the result exhibits a rich structure. The new ingredients in this study consist of the W beam functions, and the complex color structure of the soft functions and the hard functions. The resummation of the large logarithms is performed by solving the renormalization group equations with respect to the renormalization scale and the rapidity scale. In the numerical analysis, we confine to the SU(2) weak gauge interaction, and the numerical results are presented for both channels at next-to-leading-logarithmic accuracy including the singlet and the nonsinglet contributions. The nonsinglet contributions turn out to be appreciable in the 2-jettiness.

Singlet and nonsinglet three-loop massive form factors

We consider Quantum Chromodynamics with external vector, axial-vector, scalar and pseudo-scalar currents and compute three-loop corrections to the corresponding vertex function taking into account massive quarks. We consider all non-singlet contributions as well as those singlet contributions where the external current couples to a massive quark loop. We apply a semi-numerical method which is based on expansions around singular and regular kinematical points. They are matched at intermediate values of the squared partonic center-of-mass energy $s$ which allows to cover the whole kinematic range for negative and positive values of $s$. Our method permits a systematic increase of the precision by varying the expansion depth and the choice of the intermediate matching points. In our current set-up we have at least seven significant digits for the finite contribution of all form factors. We present our results as a combination of series expansions and interpolation functions which allows for a straightforward use in practical applications.

Three-loop nonsinglet matching coefficients for heavy quark currents

We compute the matching coefficients between QCD and non-relativistic QCD for external vector, axial-vector, scalar and pseudo-scalar currents up to three-loop order. We concentrate on the non-singlet contributions and present precise numerical results with an accuracy of about ten digits. For the vector current the results from arXiv:1401.3004 are confirmed, increasing the accuracy by several orders of magnitude.

The overline{mathrm{MS}} -scheme {alpha}_s^5 QCD contributions to the Adler function and Bjorken polarized sum rule in the Crewther-type two-fold {β}-expanded representation

We consider the two-fold expansion in powers of the conformal anomaly and of the strong coupling αs for the non-singlet contributions to Adler D-function and Bjorken polarized sum rule calculated previously in the overline{mathrm{MS}} -scheme at the four-loop level. This representation provides relations between definite terms of different loop orders appearing within the {β}-expansion of these quantities. Supposing the validity of this two-fold representation at the five-loop order and using these relations, we obtain some mathcal{O} ( {alpha}_s^5 ) corrections to the D-function, to the R-ratio of e+e−-annihilation into hadrons and to Bjorken polarized sum rule. These corrections are presented both analytically in the case of the generic simple gauge group and numerically for the SU(3) color group. The arguments in the favor of validity of the two-fold representation are given at least at the four-loop level. Within the {β}-expansion procedure the analytical Riemann ζ4-contributions to the five-loop expressions for the Adler function and Bjorken polarized sum rule are also fixed for the case of the generic simple gauge group.

The qT and \u2206\u03d5 spectra in W and Z production at the LHC at N3LL\u2032+N2LO

The production of weak gauge bosons, W± and Z, are at the core of the LHC precision measurement program. Their transverse momentum spectra as well as their pairwise ratios are key theoretical inputs to many high-precision analyses, ranging from the W mass measurement to the determination of parton distribution functions. Owing to the different properties of the W and Z boson and the different accessible fiducial regions for their measurement, a simple one-dimensional correlation is insufficient to capture the differing vector and axial-vector dynamics of the produced lepton pair. We propose to correlate them in two observables, the transverse momentum qT of the lepton pair and its azimuthal separation ∆ϕ. Both quantities are purely transverse and therefore accessible in all three processes, either directly or by utilising the missing transverse momentum of the event. We calculate all the single-differential qT and ∆ϕ as well as the double-differential (qT, ∆ϕ) spectra for all three processes at N3LL′+N2LO accuracy, resumming small transverse momentum logarithms in the soft-collinear effective theory approach and including all singlet and non-singlet contributions. Using the double-differential cross sections we build the pairwise ratios {mathrm{mathcal{R}}}_{W^{+}/Z} , {mathrm{mathcal{R}}}_{W^{-}/Z} , and {mathrm{mathcal{R}}}_{W^{+}/{W}^{-}} and determine their uncertainties assuming fully correlated, partially correlated, and uncorrelated uncertainties in the respective numerators and denominators. In the preferred partially correlated case we find uncertainties of less than 1% in most phase space regions and up to 3% in the lowest qT region.

Heavy quark form factors at three loops in the planar limit

We compute the color-planar and complete light quark non-singlet contributions to the heavy quark form factors in the case of the axialvector, scalar and pseudoscalar currents at three loops in perturbative QCD. We evaluate the master integrals applying a new method based on differential equations for general bases, which is applicable for all first order factorizing systems. The analytic results are expressed in terms of harmonic polylogarithms and real-valued cyclotomic harmonic polylogarithms.

Renormalization scheme and gauge (in)dependence of the generalized Crewther relation: what are the real grounds of the \u03b2-factorization property?

The problem of scheme and gauge dependence of the factorization property of the renormalization group β-function in the SU(Nc) QCD generalized Crewther relation (GCR), which connects the flavor non-singlet contributions to the Adler and Bjorken polarized sum rule functions, is investigated at the mathcal{O}left({a}_s^4right) level of perturbation theory. It is known that in the gauge-invariant renormalization overline{mathrm{MS}} -scheme this property holds in the QCD GCR at least at this order. To study whether this factorization property is true in all gauge-invariant schemes, we consider the MS-like schemes in QCD and the QED-limit of the GCR in the overline{mathrm{MS}} -scheme and in two other gauge-independent subtraction schemes, namely in the momentum MOM and the on-shell OS schemes. In these schemes we confirm the existence of the β-function factorization in the QCD and QED variants of the GCR. The problem of the possible β-factorization in the gauge-dependent renormalization schemes in QCD is studied. To investigate this problem we consider the gauge non-invariant mMOM and MOMgggg-schemes. We demonstrate that in the mMOM scheme at the mathcal{O}left({a}_s^3right) level the β-factorization is valid for three values of the gauge parameter ξ only, namely for ξ = −3, −1 and ξ = 0. In the mathcal{O}left({a}_s^4right) order of PT it remains valid only for case of the Landau gauge ξ = 0. The consideration of these two gauge-dependent schemes for the QCD GCR allows us to conclude that the factorization of RG β-function will always be implemented in any MOM-like renormalization schemes with linear covariant gauge at ξ = 0 and ξ = −3 at the mathcal{O}left({a}_s^3right) approximation. It is demonstrated that if factorization property for the MS-like schemes is true in all orders of PT, as theoretically indicated in the several works on the subject, then the factorization will also occur in the arbitrary MOM-like scheme in the Landau gauge in all orders of perturbation theory as well.

The analytical expression for the polarized Bjorken sum rule in the miniMOM scheme and the consequences for the generalized Crewther relation

The analytical perturbative QCD expression for the flavour non-singlet contribution to the Bjorken polarized sum rule in the rather applicable at present gauge–dependent miniMOM scheme is obtained. For the considered three values of the gauge parameter, namely ξ = 0 (Landau gauge), ξ = −1 (anti–Feynman gauge) and ξ = −3 (Stefanis–Mikhailov gauge), the scheme-dependent coefficients are considerably smaller than the gauge-independent results. It is found that the fundamental property of the factorization of the QCD renormalization group β-function in the generalized Crewther relation, which is valid in the gauge-invariant scheme up to -level at least, is unexpectedly valid at the same level in the miniMOM-scheme for ξ = 0, and for ξ = −1 and ξ = −3 in part.

Deviation pattern approach for optimizing perturbative terms of QCD renormalization group invariant observables

We first consider the idea of renormalization group-induced estimates, in the context of optimization procedures, for the Brodsky-Lepage-Mackenzie approach to generate higher-order contributions to QCD perturbative series. Secondly, we develop the deviation pattern approach (DPA) in which through a series of comparisons between lower-order RG-induced estimates and the corresponding analytical calculations, one could modify higher-order RG-induced estimates. Finally, using the normal estimation procedure and DPA, we get estimates of $\alpha_s^4$ corrections for the Bjorken sum rule of polarized deep-inelastic scattering and for the non-singlet contribution to the Adler function.

Polarized DIS in 𝒩 = 4 SYM: where is spin at strong coupling?

Using the AdS/CFT correspondence, we calculate the polarized structure functions in strongly coupled = 4 supersymmetric Yang-Mills theory deformed in the infrared. We find that the flavor singlet contribution to the g1 structure function is vanishingly small, while the flavor non-singlet contribution shows the Regge behavior at small?x with an intercept slightly less than 1. We explicitly check that the latter satisfies the moment sum rule. We discuss the `spin crisis' problem and suggest that at strong coupling the spin of a hadron entirely comes from the orbital angular momentum.

Power corrections to fragmentation functions in flavour-singlet deep-inelastic scattering

We investigate the power-suppressed corrections to fragmentation functions in flavour-singlet deep-inelastic lepton scattering, to complement the previous results for the non-singlet contribution. Our method is a dispersive approach based on an analysis of Feynman graphs containing massive gluons. As in non-singlet deep-inelastic scattering we find that the leading corrections are proportional to 1/ Q 2.

Power corrections to flavour-singlet structure functions

We investigate the power-suppressed corrections to structure functions in flavour-singlet deep inelastic lepton scattering, to complement the previous results for the non-singlet contribution. Our method is a dispersive approach based on an analysis of Feynman graphs containing massive gluons; and our results agree with those obtained from leading infrared renormalon contributions. As in non-singlet deep inelastic scattering we find that the leading corrections are proportional to 1 Q 2 . We find that the singlet contribution becomes important small x.

Renormalon model predictions for power corrections to flavor-singlet deep-inelastic structure functions

We analyze power corrections to flavor-singlet deep-inelastic scattering structure functions in the framework of the infrared renormalon model. Our calculations, together with previous results for the non-singlet contribution, allow us to model the x-dependence of higher twist corrections to F 2, F L and g 1 in the whole x domain.

On the resummation of the α In2 z terms for QED corrections to deep-inelasticep scattering ande + e− annihilation

The resummation of the α ln2 (z) non-singlet contributions is performed for initial state QED corrections. As examples, the effect of the resummation on neutral-current deep-inelastic scattering and thee + e − → μ+μ− scattering cross section near theZ o-peak is investigated.

Neutron spin-dependent structure function, Bjorken sum rule, and first evidence for singlet contribution at lowx

We perform the isospin decomposition of proton and neutron SLAC data for moderately low $x$ in the region $0.01 \leq x \leq 0.1$. The isovector part is well described by a power behaviour $x^{\alpha}$, where $\alpha$ does not correspond to the intercept of single $a_1$ Regge trajectory, as expected. However, the observed power leads to the validity of Bjorken sum rule and it is consistent with the power extracted from all previous data using NLO evolution. At the same time, the isoscalar part behaviour may be interpreted as a partial cancellation between a positive non-singlet contribution and a strongly negative singlet one. Further experimental consequences are mentioned.

SU(3) breaking effects in hyperon semi-leptonic decays and the extraction of F and D

The analysis of hyperon semi-leptonic decay data is re-examined in the light of a recent suggestion that SU (3) symmetry breaking effects may be taken into account by applying a correction to the F D ratio obtained via naïve linear extrapolation in the hyperon masses. Comparison is made with the physically better motivated approach of applying so-called centre-of-mass corrections. This study (including all available data) reveals certain shortcomings of the former of the above treatments, highlights some interesting aspects of this type of analysis and attempts to pinpoint the decay data that might usefully be improved. A tantalising result of the SU (3) breaking analysis performed here is that the magnitude of recoil correction required by the data corresponds closely to that required for the standard explanation of the reduction of g A from its SU (6) value of 5 3 . We also comment on other recent suggestions for taking into account the effects of SU (3) breaking. Finally a few remarks are made on the relevance for predicting the flavour nonsinglet contribution to the proton g 1 and the Ellis-Jaffe sum rule.

SU(3) breaking effects in hyperon semi-leptonic decays and the EMC/SLAC proton spin measurements

The problem of analysing hyperon semi-leptonic decay data is discussed with particular reference to SU(3) symmetry breaking effects and the resulting predictions for the spin structure of the proton. It turns out that the present decay data can be accomodated exceedingly well by a simple approach treating solely the so-called recoil effect or centre-of-mass correction. The extracted Fand Dparameters, of particular relevance to present discussions of the structure functions for polarized deep-inelastic scattering, are very stable and striking close to the results of earlier analyses. The implication is that the Fand D thus extracted may reliably be used in predicting the flavour non-singlet contribution to the proton g 1. This last being thus still in marked conflict with the EMC/SLAC results on polarized deep-inelastic scattering.

A QCD calculation of the transverse momentum distribution of Drell-Yan pairs including soft emission and non-abelian effects

The well-known Parisi and Petronzio LLA result for the QCD transverse momentum distribution of Drell-Yan pairs at p T⪡ Q takes into account the non-singlet contribution only and totally neglects non-abelian effects due to the singlet contribution. The latter becomes substantial and sometimes dominant at pp and pp collider energies, and thus any comparison with data from these machines will require computation of its effects. We present a QCD Monte Carlo procedure which allows one to calculate the transverse momentum distribution with both non-singlet and singlet contributions included. At the same time it computes the evolution of parton density functions and the correlations between transverse and longitudinal distributions. Phase-space effects are duly taken into account. It is found that, at collider energies, the singlet contribution completely alters QCD predictions for dimuon transverse momenta giving at times 〈 p T〉 ∼ 4 times larger than the non-singlet contribution. Quantitative results are presented and a comparison with existing data is made, showing satisfactory agreement with the experimentally observed behaviours.

Quantum chromodynamics corrections to polarised deep-inelastic electron-nucleon scattering

Quantum chromodynamics corrections to orderα s (the running coupling constant), using the quark-parton approach are calculated for the spin-dependent structure functions in deep-inelastic polarised electron-nucleon scattering. Consequences of these corrections for the Bjorken sum rule and the asymmetry in the case of longitudinally polarised (with respect to the beam) nucleons is discussed which could provide possible tests of quantum chromodynamics. Comparison of our results with the moments of the flavour non-singlet contribution to the structure functions obtained using operator product expansion is also given.