The QCD Coupling and Parton Distributions at High Precision

📝 Abstract

A survey is given on the present status of the nucleon parton distributions and related precision calculations and precision measurements of the strong coupling constant $\alpha_s(M_Z^2) $. We also discuss the impact of these quantities on precision observables at hadron colliders.

💡 Analysis

A survey is given on the present status of the nucleon parton distributions and related precision calculations and precision measurements of the strong coupling constant $\alpha_s(M_Z^2) $. We also discuss the impact of these quantities on precision observables at hadron colliders.

📄 Content

arXiv:1007.5202v2 [hep-ph] 10 Sep 2010 1 THE QCD COUPLING AND PARTON DISTRIBUTIONS AT HIGH PRECISION ∗ JOHANNES BL¨UMLEIN Deutsches Elektronen-Synchrotron, DESY, Platanenallee 6, D-15738 Zeuthen, Germany E-mail: Johannes.Bluemlein@desy.de A survey is given on the present status of the nucleon parton distributions and re- lated precision calculations and precision measurements of the strong coupling constant αs(M2 Z). We also discuss the impact of these quantities on precision observables at hadron colliders. Keywords: Deep-inelastic scattering, strong coupling constant, heavy flavors.

1. Inside Nucleons The physics of the strong interactions always has been tightly connected to the study of nucleons at shorter and shorter distances. The measurement of the anoma- lous magnetic moments of the proton1 and neutron2 in 1933 and 1939 made clear that nucleons are no elementary particles. During the 1950ies the Hofstadter exper- iments3 revealed the charge distributions inside nucleons4 at scales Q2 ≃0.5 · M 2 N. Yet it was unknown how these distributions came about. In 1964 Murray Gell– Mann5 proposed the quark model, to catalog the plethora of observed baryons and mesons. Independently G. Zweig suggested aces6 as the building blocks of hadrons. A direct connection to the lepton-nucleon scattering data was not made at that time. Back in 1954 C.N. Yang and R Mills7 proposed novel bosonic field theories based on gauge invariance with respect to non-abelian groups. This development went unrelated to strong interactions for a long time. With the advent of the Stan- ford Linear Accelerator in 1968 the nucleon structure could be resolved at much shorter distances by the MIT-SLAC experiments8–10 beyond the resonant region W ≥2GeV for values Q2 up to 30 GeV2. The remarkable finding by these experi- ments were that i) the structure function νW2(ν, Q2) which has been expected to depend on both kinematic variables ν and Q2 independently, turned out to take the same values for fixed values of x = Q2/(2MNν) irrespectively of ν and Q2 at high enough values. This phenomenon is called scaling. ii) The ratio of the longitudinal structure function WL and W2 turned out to be very small. Bjorken13 had predicted ∗Dedicated to M. Gell–Mann on the occasion of his 80th birthday. 2 scaling at asymptotic scales Q2, ν →∞in 1969. Learning about the SLAC-MIT results R. Feynman very quickly proposed the parton model14, which is equivalent to Bjorken’s description but based on the observed strict microscopic correlation between Q2 and ν = q.pi W(x, Q2) = X i e2 i Z 1 0 dxifi(xi)δ q.pi M 2 −Q2 M 2  , (1) where ei and fi denote the parton’s charge and distribution functions. Would the parton model be unique in describing the new data? This has been challenged by other popular formalisms like vector meson dominance16. However, they failed to describe the behaviour observed for WL, which corresponded to that of spin 1/2 partons, according to the calculations by Callan and Gross17. Yang–Mills theories7 became building blocks of the electro-weak Standard Model18, although there renormalizibility had not been proven yet, a conditio sine qua non for a physical theory. The proof was an urgent matter and in 1971 it was achieved both for massless and spontaneously broken Yang-Mills theories, along with designing practical loop computations in this sophisticated theory in an au- tomated way20,26. Quantum Chromodynamics (QCD) was proposed as the theory of the strong interactions in 1972 by Gell–Mann and Fritzsch and Leutwyler27,28 as a renormalized Yang-Mills field theory based on SU(3) gauge interactions29. D. Gross, F. Wilczek30 and D. Politzer31 studied the asymptotic behaviour of color octet gluon Yang-Mills theory, cf. also32, and found asymptotic freedom. This is the essential ingredient, which makes it possible to perform perturbative calculations at large scales in a theory with strong interactions at low scales. At short distances the nucleon structure functions Fi(x, Q2) obey the light-cone expansion34. At large scales Q2 the contributions of lowest twist dominate and the representation Fi(x, Q2) = X j Cj i (x, Q2/µ2) ⊗fj(x, µ2) (2) holds. Here Cj i (x, Q2/µ2) denote the Wilson coefficients and fj(x, µ2) are the parton densities. µ2 is an arbitrary factorization scale and ⊗denotes the Mellin convolution. The scale behavior of the nucleon structure functions Fi(x, Q2) obey renormal- ization group equations, an important aspect of renormalizable Quantum Field The- ories to which Murray Gell–Mann made very essential contributions very early37a. Transforming Eq. (2) to Mellin space one obtains the following Callan-Symanzik40 equations :  µ ∂ ∂µ + β(g) ∂ ∂g −2γψ(g)  Fi(N, Q2) = 0 (3) aIt is interesting to note that different approaches to renormalization result into different mathe- matical structures as shown in38.Thus the method by Gell–Mann and Low37 is in general related to a cocycle, while that by St¨uckelberg and Petermann39 relates to a group. I th

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