Abstract
We study the one-loop correction in Transverse-Momentum-Dependent(TMD) factorization for Drell-Yan processes at small transverse momentum of the lepton pair. We adopt the so-called subtractive approach, in which one can systematically construct contributions for subtracting long-distance effects represented by diagrams. The perturbative parts are obtained after the subtraction. We find that the perturbative coefficients of all structure functions in TMD factorization at leading twist are the same. The perturbative parts can also be studied with scattering of partons instead of hadrons. In this way, the factorization of many structure functions can only be examined by studying the scattering of multi-parton states, where there are many diagrams. These diagrams have no similarities to those treated in the subtractive approach. As an example, we use existing results of one structure function responsible for Single-Spin-Asymmetry, to show that these diagrams in the scattering of multi-parton states are equivalent to those treated in the subtractive approach after using Ward identity.
Highlights
In Drell-Yan processes, one-loop correction of some structure functions can be obtained by studying partonic scattering and TMD parton distributions of a single parton, where one replaces each hadron with a single parton, i.e., the scattering a + b → γ∗ + X with a or b as a single parton state
By using the subtractive approach we have studied TMD factorization for Drell-Yan processes beyond tree-level
In given diagrams there are nonperturbative contributions which need to be subtracted into TMD parton distributions
Summary
We consider the Drell-Yan process: hA(PA) + hB(PB) → γ∗(q) + X → − + + + X. In this work we will only give results for those structure functions which receive leading-twist contributions in TMD factorization. In Γwe neglect the ξ−-dependence of the hadronic matrix element Another approximation can be made is that the leading contributions are only given by the matrix elements containing the good component of quark fields. For q⊥ ΛQCD one can make a further approximation by neglecting or expanding the ξ⊥dependence in hadron matrix elements in Γ or Γ Using the above approximated results for Γ and Γ, one obtains the hadronic tensor at leading order of αs as. It should be noted that M and Mare diagonal in colour space With these decomposition one can work out the hadronic tensor at leading order. The results for the tensor can be represented with structure functions and each structure function is factorized with corresponding TMD parton distributions
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