What are the low-$Q$ and large-$x$ boundaries of collinear QCD factorization theorems?

Familiar factorized descriptions of classic QCD processes such as deeply inelastic scattering (DIS) apply in the limit of very large hard scales, much larger than nonperturbative mass scales and other nonperturbative physical properties like intrinsic transverse momentum. Since many interesting DIS studies occur at kinematic regions where the hard scale, $Q\sim 1–2\mathrm{GeV}$, is not very much greater than the hadron masses involved, and the Bjorken scaling variable $x_{\mathrm{bj}}$ is large, $x_{\mathrm{bj}}\gtrsim 0.5$, it is important to examine the boundaries of the most basic factorization assumptions and assess whether improved starting points are needed. Using an idealized field-theoretic model that contains most of the essential elements that a factorization derivation must confront, we retrace the steps of factorization approximations and compare with calculations that keep all kinematics exact. We examine the relative importance of such quantities as the target mass, light quark masses, and intrinsic parton transverse momentum, and argue that a careful accounting of parton virtuality is essential for treating power corrections to collinear factorization. We use our observations to motivate searches for new or enhanced factorization theorems specifically designed to deal with moderately low-$Q$ and large-$x_{\mathrm{bj}}$ physics.

Figure 1

The sequence of approximations leading to the canonical parton model picture: (a) A physical picture of the complete QCD event. The symbols $\subset$ represent the final state hadronization process. (b) The leading-power topological region contributing to the inclusive cross section. (c) The kinematical approximation (represented by the green dotted horizontal line) that produces the parton model cross section. The line is an instruction to replace the parton momentum by its approximated values (see Sec. 4). The momentum labels are discussed in the text.

Figure 2

Contributions to the hadronic tensor from diagrams allowed by the interaction Lagrangian (2) to $O(\alpha {\lambda }^2)$ in the couplings. Graph (a) is a manifestation of the familiar handbag diagram and represents the topology of the leading region. Graphs (b) and (c) are suppressed by powers of $1/Q$ when $k_{\mathrm{T}}$ is small, but are needed for gauge invariance. The Hermitian conjugate for (c) is not shown. The momenta on the various legs are as indicated.

Figure 3

The steps in the usual factorization approximation applied to a handbag topology. (a) Unapproximated handbag topology, with $H$ denoting the hard scattering of a virtual photon from a quark with momentum $k$ to one with momentum $k^{\prime }=k+q$, $J(k^{\prime })$ is the jet function, and $L(k,P)$ is the soft target amplitude. (b) Handbag diagram with standard factorization, with the parton momentum approximated by $\widehat{k}$ in the hard function $H$, and by $\tilde{k}$ in the jet and soft functions. The hooks represent the point of application of kinematic approximations on parton momentum. (c) Application of the $O({\lambda }^2)$ contribution in the theory from Sec. 2.

Figure 4

The unintegrated structure function $k_{\mathrm{T}}{\mathcal{F}}_1$ for $x_{\mathrm{bj}}=0.6$ and $Q=2\mathrm{GeV}$ (top row) and $Q=20\mathrm{GeV}$ (bottom row), for different values of $m_q$ and $m_s$ calculated using both the exact expressions (solid red curves) and the canonical collinear factorization approximation (dashed blue curves). The choices of $m_s$ are to fix $k^2$ at the values discussed in Sec. 5a. At the higher $Q$ value the collinear calculation is almost indistinguishable from the exact, while at the lower $Q$ value the exact calculation diverges as it approaches the kinematical upper limit of $k_{\mathrm{T}}$.

Figure 5

The dependence of the parton virtuality $v\equiv \sqrt{-k^2}$ on $k_{\mathrm{T}}$ evaluated at exact (solid red curves) and approximate collinear (dashed blue curves) kinematics, for $x_{\mathrm{bj}}=0.6$ at fixed $Q=2\mathrm{GeV}$ (left panel) and $Q=20\mathrm{GeV}$ (right panel), for quark mass $m_q=0.3\mathrm{GeV}$ and spectator diquark mass $m_s$ corresponding to $v(k_{\mathrm{T}}=0)=0.5\mathrm{GeV}$ (see Table 1).

Figure 6

Unintegrated structure function $k_{\mathrm{T}}{\mathcal{F}}_1$ for $x_{\mathrm{bj}}=0.6$ and $Q=3\mathrm{GeV}$, with quark mass $m_q=0.3\mathrm{GeV}$ and virtuality $v=0.5\mathrm{GeV}$ for the exact result (solid red curves), approximate collinear approximation (dashed blue curves), and collinear result with the replacement $x_{\mathrm{bj}}\to x_{\mathrm{n}}$ (dot-dashed green curves). The right-hand panel shows the results when the nucleon mass increased by a factor of 2.