Fully unintegrated parton correlation functions and factorization in lowest-order hard scattering

Motivated by the need to correct the potentially large kinematic errors in approximations used in the standard formulation of perturbative QCD, we reformulate deeply inelastic lepton-proton scattering in terms of gauge-invariant, universal parton correlation functions which depend on all components of parton four-momentum. Currently, different hard QCD processes are described by very different perturbative formalisms, each relying on its own set of kinematical approximations. In this paper we show how to set up formalism that avoids approximations on final-state momenta, and thus has a very general domain of applicability. The use of exact kinematics introduces a number of significant conceptual shifts already at leading order, and tightly constrains the formalism. We show how to define parton correlation functions that generalize the concepts of parton density, fragmentation function, and soft factor. After setting up a general subtraction formalism, we obtain a factorization theorem. To avoid complications with Ward identities, the full derivation is restricted to Abelian gauge theories; even so, the resulting structure is highly suggestive of a similar treatment for non-Abelian gauge theories.

Figure 1

(a) Parton-model handbag diagram for DIS. (b) Handbag diagram with outgoing jet factor. (c) The general reduced diagram for full QCD with one jet. Note that in graphs like (c), the target collinear gluons may pass through the cut inside the target bubble.

Figure 2

The amplitude for ${\gamma }^{*}p$ scattering into two jets with fixed masses.

Figure 3

An event in which the collinear, on-shell matrix element—the photon-quark vertex in this example—is accompanied by initial- and final-state showers. (In the showers, solid lines denote both quarks and gluons.).

Figure 4

Target-collinear gluons explicit in the definition of the standard integrated PDF.

Figure 5

A schematic representation of the approximation performed in Eq. (36).

Figure 6

The approximation of Eq. (36).

Figure 7

(a) Range of rapidity for exchanged gluons in DIS. Rightmost is the target region, and leftmost is the outgoing jet from the struck quark. (b) Rapidity for gluons in PDF defined with a lightlike Wilson line (or for PDF defined in light-cone gauge without the Wilson line).

Figure 8

Structure of the Wilson loop that appears in the definition of the soft factor (43). Dot-dashed lines indicate exactly lightlike directions.

Figure 9

Path of the Wilson line (in a highly boosted frame) in the definition of the parton correlation function in the target. (The fully unintegrated PDF.)

Figure 10

Path of the Wilson line in the definition of the fragmentation function.

Figure 11

A graph that yields a contribution to the soft and collinear regions.

Figure 12

Pinch singularities for gluon exchange in massless theory. The singularities are labeled: “S” for soft, “T” for target-collinear, and “J” for jet-collinear. The solid lines indicate the positions of the actual pinch singularities, and the dots indicate their extrapolations to where there is no pinch at the collinear singularities.

Figure 13

Distribution of gluon rapidity. The solid line is the exact distribution, whereas the dotted line represents the distribution obtained from the soft approximator, $T_S$.

Figure 14

Corrections to current vertex, and kinematic projection for hard scattering.

Figure 15

Factorization of one soft gluon.

Figure 16

Illustrating the different detailed leading regions for an exchanged gluon in a spectator model. The double lines are for a color-singlet scalar field treated as an analog of a hadron. The dashed arrows in graphs (b) and (c) indicate the lines that are pushed far off shell by the inclusion of the extra gluon.

Figure 17

(i) Ranges of gluon transverse and rapidity where the different graphs in Fig. 16 dominate. We use a logarithmic scale for $l_2^T$. The upper diagonal lines correspond to where the energy of the gluon in the Breit frame is of order $Q$, and the vertical lines correspond to the rapidities of the jet and the target. (ii) The case that the gluon attaches to the target remnant and that $x_{\mathrm{Bj}}$ is close to unity.

Figure 18

The approximated graph of Eq. (77).

Figure 19

The target-collinear contribution for the gluon in Fig. 11 gives this factor that is a contribution to the fully unintegrated PCF. See Eq. (81).

Figure 20

Virtual correction graph.

Figure 21

The deformation of the contour for the integral over $l_2^{+}$ in Eq. (87).

Figure 22

Graphical representation of the target-collinear contribution for one virtual collinear gluon, as in (95). From this is to be subtracted a soft term.

Figure 23

One-loop correction to current vertex.

Figure 24

Conversion to factorized form: (a) All leading reduced graphs with generalized handbag structure. (b) After application of Ward identities to connection of gluons from soft subgraphs to collinear subgraphs. (c) After application of Ward identities to connection of collinear subgraphs to hard subgraphs.

Figure 25

Definition of momentum variables for a given region. The subgraphs are coded by color as follows: $\mathrm{\text{red}}=\mathrm{\text{soft}}$, $\mathrm{\text{blue}}=\mathrm{\text{jet-collinear}}$, $\mathrm{\text{green}}=\mathrm{\text{target-collinear}}$, and $\mathrm{\text{black}}=\mathrm{\text{hard}}$.

Figure 26

(a) Representation of subgraphs for region $R$. (b) Representation for smaller region $R^{\prime }$. (c) Soft subgraph of $R^{\prime }$ overlaid on subgraphs for $R$. The color coding is the same as in Fig. 25.

Figure 27

(a)–(c) A graph for DIS, and two of its regions. (d)–(f) Another graph, and two of its regions. The diagrams in the left-hand column, all in black, are particular Feynman graphs for the process, i.e., a candidate for $\Gamma$. The colored diagrams in the middle column represent a particular region (a candidate for $R$) for each of these graphs where one of the gluons is soft and one is jet-collinear. The diagrams in the right-hand column represent a smaller region, a candidate for $R^{\prime }$ where both gluons are soft. The color coding is the same as in Fig. 25.

Figure 28

Lowest-order connection to the hard subgraph of the target-collinear gluon. The hooked lines on the outermost fermion lines indicate where fermion momenta are replaced by massless on-shell approximations.

Figure 29

(a) This is not an allowed connection of a target-collinear gluon to the hard subgraph, since the gluon and the attached quark are both target-collinear: the intermediate quark is therefore collinear, not hard. (b) The corrected treatment of the graph: The hard subgraph is just the vertex for the electromagnetic current.

Figure 30

(a) and (b) The lowest-order possibilities for two target-collinear gluons attaching to the hard subgraph. (c) Subregion with $l_1$ collinear and $l_2$ soft is allowed only for graph (a) and not graph (b), but in the subtraction the intermediate quark is on shell.

Figure 31

(a) The application of the Ward identity when both gluons are collinear. (b) The application of the Ward identity when one of the gluons is soft.

Figure 32

Applying the Ward identity to factorize a single soft-gluon insertion.

Figure 33

Result of applying the Ward identity with multiple soft-gluon insertions.

Figure 34

Using Eq. (b7) to rewrite Fig. 33 for the case of two soft-gluon insertions.