Semi-inclusive DIS: Factorization Page: 3 of 4
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It is instructive to demonstrate this fac-
torization at one-loop order. For example,
the one-loop real corrections to the struc- ,
ture function F are shown in Fig. 1. There
is no contribution to the hard scattering ker- (a) (b) (c)
nel from any of these diagrams. In Fig. la,
the gluon radiation generates a transverse- Figure 1: One-loop real diagrams for SIDIS.
momentum for the struck quark. There is
no contribution from the fragmentation function because the contribution from the final
state with a gluon parallel to final state quark is power suppressed. Therefore, the diagram
must be factorizable into the parton distribution. Similarly for Fig. lb, which again can be
reproduced by the factorization formula with the one-loop fragmentation function and the
soft factor S, and the tree-level parton distribution and the hard part. For Fig. ic and its
hermitian conjugate, we find three distinct contributions: where the first term corresponds
to a gluon collinear to the initial quark, the second term a gluon collinear to the final state
quark, and the third term a soft gluon. All these terms are reproduced by the factorization
formula with one-loop parton distribution, fragmentation function, and the soft factor. The
virtual contribution can be analyzed accordingly .
For arguments toward a factorization to all orders, we follow the discussions in [2, 7].
The procedure for this argument is the following. First, for any high order Feynman dia-
grams, using the power counting rules identifies the leading region contributions . The
leading regions clearly separate the soft, collinear, and hard gluons' contributions to the
cross section (the cut diagram), where the soft gluons are only attached to the jet functions
(parton distributions and/or fragmentation functions); hard gluons are included in the hard
part; collinear gluons attached the jet functions to the hard part. On top of that, we can
further use the Grammer-Yennie approximation to factorize out the soft factor, which can be
expressed as matrix element of Wilson lines [2, 7]. The Ward Identity will be used to further
factorize the collinear gluons from the hard part, which results in a Wilson line (gauge link)
association in the definition of the jet functions. The variation of the gauge link gives the
Collins-Soper evolution equation for the jet functions . After these procedures, the hard
part only contains hard gluons, which can be calculated from perturbative QCD.
The above factorization argument is also applicable to the spin and azimuthal depen-
dent SIDIS  at leading order of 1/Q. In particular, the Sivers-type single transverse
spin asymmetry in SIDIS has been studied in , where the gluon radiation generates
large transverse momentum Sivers function and fragmentation. When combining both
contributions with the soft factor, the Sivers-type SSA in SIDIS can be factorized into
the similar factorization formula as the above. The calculations of  are based one
collinear factorization approach, and the SSA comes from the twist-three quark-gluon cor-
relation function as we show in the diagram of Fig. 2(a). At large transverse momen-
tum Ph_ ~ Q, the SSA in of higher-twist, and will be suppressed by 1/Phi. At small
PhI < Q, a factorization in terms of TMD parton distribution applies , involving
in case of the SSA the Sivers functions. If PhI is much larger than AQCD, the depen-
dence of these functions on transverse momentum can be computed using QCD perturba-
tion theory. At the same time, the result obtained from the diagram Fig. 2(a) can also
be extrapolated into the regime AQCD < Phi < Q, and it can be shown that the re-
sult of this extrapolation is identical to that obtained using the TMD approach . In
this sense, the two mechanisms widely held responsible for the observed SSAs are unified.
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Yuan, Feng. Semi-inclusive DIS: Factorization, article, December 10, 2008; Berkeley, California. (https://digital.library.unt.edu/ark:/67531/metadc897140/m1/3/: accessed March 20, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.