rpp2010-rev-frag-functions

rpp2010-rev-frag-functions - 17 Fragmentation functions in...

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17. Fragmentation functions in e + e - , ep and pp collisions 1 17. FRAGMENTATION FUNCTIONS IN e + e - , ep AND pp COLLISIONS Revised October 2009 by O. Biebel (Ludwig-Maximilians-Universit¨at, Munich, Germany), D. de Florian (Dep. de F´ ısica, FCEyN-UBA, Buenos Aires, Argentina), D. Milstead (Fysikum, Stockholms Universitet, Sweden), and A. Vogt (Dep. of Mathematical Sciences, University of Liverpool, UK). 17.1. Introduction to fragmentation The term ‘fragmentation functions’ is widely used for two related if conceptually different sets of functions describing final-state single particle energy distributions in hard scattering processes (see Refs. [1,2] for introductory reviews, and Refs. [3,4] for summaries of recent experimental and theoretical research in this field). The first are cross-section observables such as the functions F T,L,A ( x, s ) in semi- inclusive e + e annihilation at center-of-mass (CM) energy s via an intermediate photon or Z -boson, e + e γ/Z h + X , given by 1 σ 0 d 2 σ h dx d cos θ = 3 8 (1 + cos 2 θ ) F h T + 3 4 sin 2 θ F h L + 3 4 cos θ F h A . (17 . 1) Here x = 2 E h / s 1 is the scaled energy of the hadron h (in practice the approximation x x p = 2 p h / s is often used), and θ is its angle relative to the electron beam in the CM frame. Eq. (17 . 1) is the most general form for unpolarized inclusive single-particle production via vector bosons [5]. The transverse and longitudinal fragmentation functions F T and F L represent the contributions from γ/Z polarizations transverse or longitudinal with respect to the direction of motion of the hadron. The parity-violating term with the asymmetric fragmentation function F A arises from the interference between vector and axial-vector contributions. Normalization factors σ 0 used in the literature range from the total cross section σ tot for e + e hadrons, including all weak and QCD contributions, to σ 0 = 4 πα 2 N c / 3 s with N c = 3, the lowest-order QED cross section for e + e μ + μ times the number of colors N c . LEP1 measurements of all three fragmentation functions are shown in Fig. 17.1. K. Nakamura et al. , JPG 37 , 075021 (2010) (http://pdg.lbl.gov) July 30, 2010 14:36
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2 17. Fragmentation functions in e + e - , ep and pp collisions 0.0005 0.001 0.002 0.005 0.01 0.02 0.05 0.1 0.2 0.5 1 2 5 10 20 50 100 200 500 F T,L (x) s=91 GeV LEP F T LEP F L 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x F A (x) -0.8 -0.4 0 0.4 0.8 LEP F A Figure 17.1: LEP1 measurements of total transverse ( F T ), longitudinal ( F L ), and asymmetric ( F A ) fragmentation functions [6,7,8]. Data points with relative errors greater than 100% are omitted. Integration of Eq. (17 . 1) over θ yields the total fragmentation function F h = F h T + F h L , 1 σ 0 h dx = F h ( x, s ) = i 1 x dz z C i ( z, α s ( μ ) , s μ 2 ) D h i ( x z , μ 2 ) + O ( 1 s ) (17 . 2) with i = u, ¯ u, d, ¯ d, . . . , g . Here we have introduced the second set of functions mentioned in the first paragraph, the parton fragmentation functions (or fragmentation densities) D h i . These functions are the final-state analogue of the initial-state parton distributions addressed in Section 16 of this Review . Due to the different sign of the squared four-momentum q 2
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