Physics – High Energy Physics – High Energy Physics - Phenomenology
Scientific paper
2012-02-22
Physics
High Energy Physics
High Energy Physics - Phenomenology
25 pages, 5 figures
Scientific paper
We investigate the soft spectator scattering contribution for the FF $F_{1}$. We focus our attention on factorization of the hard-collinear scale $\sim Q\Lambda$ corresponding to transition from SCET-I to SCET-II. We compute the leading order jet functions and find that the convolution integrals over the soft fractions are logarithmically divergent. This divergency is the consequence of the boost invariance and does not depend on the model of the soft correlation function describing the soft spectator quarks. Using as example a two-loop diagram we demonstrated that such a divergency corresponds to the overlap of the soft and collinear regions. As a result one obtains large logarithm $\ln Q/\Lambda$ which must be included in the correct factorization formalism. We conclude that a consistent description of the factorization for $F_{1}$ implies the end-point collinear divergencies in the hard and soft spectator contributions, i.e. convolution integrals with respect to collinear fractions are not well-defined. Such scenario can only be realized when the twist-3 nucleon distribution amplitude has non-vanishing end-point behavior at a certain low normalization point. Such behavior leads to the violation of the collinear factorization for the hard spectator scattering contribution. We also discuss the physical subtraction scheme for SCET-I factorization which can be used for systematical analysis of the hadronic processes in the range of moderate values of $Q^{2}\sim 5-20$ GeV$^{2}$ where the hard collinear scale $\sim Q\Lambda$ is still not large.
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