$k_T$ factorization is violated in production of high-transverse-momentum particles in hadron-hadron collisions

We show that hard-scattering factorization is violated in the production of high-$p_T$ hadrons in hadron-hadron collisions, in the case that the hadrons are back-to-back, so that $k_T$ factorization is to be used. The explicit counterexample that we construct is for the single-spin asymmetry with one beam transversely polarized. The Sivers function needed here has particular sensitivity to the Wilson lines in the parton densities. We use a greatly simplified model theory to make the breakdown of factorization easy to check explicitly. But the counterexample implies that standard arguments for factorization fail not just for the single-spin asymmetry but for the unpolarized cross section for back-to-back hadron production in QCD in hadron-hadron collisions. This is unlike corresponding cases in $e^{+}{e}^{-}$ annihilation, Drell-Yan, and deeply inelastic scattering. Moreover, the result endangers factorization for more general hadroproduction processes.

Figure 1

Lowest-order graph for SIDIS in our model. The initial-state particle is a color-singlet Dirac particle. The spectator line is for a Dirac “quark” field, and the active parton is for a scalar “diquark” field, as can be enforced by a condition on the detected outgoing particle of momentum $q+k$. The arrows on the internal lines indicate the flow of color charge, and the vertical line cutting the graph denotes the final state.

Figure 2

Virtual one-gluon-exchange corrections to Fig. 1 that give a SSA.

Figure 3

Virtual one-gluon-exchange correction to parton density. The upper double line is the Wilson line, and the graph shown, together with its Hermitian conjugate gives the first contribution to the Sivers function.

Figure 4

Lowest-order graph for the Drell-Yan process.

Figure 5

Virtual one-gluon-exchange corrections to Fig. 4 relevant for a SSA when the lower hadron has transverse spin $s$. Graph (a) and its Hermitian conjugate (b) have imaginary parts (at the amplitude level) that give a nonzero SSA. Gluon exchange between the active partons, graph (c) and its not-shown conjugate, gives an imaginary part in the vertex correction that does not give a SSA. Spectator-spectator gluon exchange graphs, (d) and (e), do contribute individually to the SSA, but the two contributions cancel (at leading power).

Figure 6

Lowest-order graph in model for hadroproduction of hadrons of high transverse momentum. The initial-state particles are color-singlet Dirac particles. The spectator lines are for Dirac quark fields of charges $g_1$ and $g_2$, and the active partons are for scalar diquark fields. The exchanged gluon line is thickened to denote the hard scattering.

Figure 7

One-gluon exchange in model for hadroproduction of hadrons of high transverse momentum. The specification of the process is as in Fig. 6. Only graphs that contribute to the SSA are shown. Hermitian conjugates of these graphs also contribute, with an equal value.

Figure 8

The exchange of two extra gluons, as in this graph, will tend to give nonfactorization in unpolarized cross sections.

Figure 9

In a conventional perturbative QCD calculation for an unpolarized partonic cross section, nonfactorization by the mechanisms discussed in this paper would first appear in graphs of this order.