Hard diffraction from parton rescattering in QCD

We analyze the QCD dynamics of diffractive deep inelastic scattering. The presence of a rapidity gap between the target and diffractive system requires that the target remnant emerges in a color-singlet state, which we show is made possible by the soft rescattering of the struck quark. This rescattering is described by the path-ordered exponential (Wilson line) in the expression for the parton distribution function of the target. The multiple scattering of the struck parton via instantaneous interactions in the target generates dominantly imaginary diffractive amplitudes, giving rise to an “effective Pomeron” exchange. The Pomeron is not an intrinsic part of the proton but a dynamical effect of the interaction. This picture also applies to diffraction in hadron-initiated processes. Because of the different color environment the rescattering is different in virtual photon- and hadron-induced processes, explaining the observed nonuniversality of diffractive parton distributions. This framework provides a theoretical basis for the phenomenologically successful soft color interaction (SCI) model which includes rescattering effects and thus generates a variety of final states with rapidity gaps. We discuss developments of the SCI model to account for the color coherence features of the underlying subprocesses.

Figure 1

Virtual photon scattering on the proton in (a) the parton model frame and (b) the dipole frame. The LF time increases from left to right and the rescattering effects are indicated.

Figure 2

Gluon-induced DIS at small $x$ with color flux tube, or string, configuration in (left) the conventional Lund string model connection of partons and (right) after a soft color-octet exchange (dashed gluon line) between the remnant and the hard scattering system resulting in a phase space region without a string leading to a rapidity gap after hadronization.

Figure 3

Distribution of the size $\Delta y_{\max }$ of the largest rapidity gap in DIS events at HERA simulated using Lepto (standard small-$x$ dominated DIS event sample with $Q^2\ge 4{\mathrm{GeV}}^2$ and $x\ge {10}^{-4}$). The dashed-dotted curve represents the parton level obtained from hard, perturbative processes (matrix element plus parton showers). The dashed curve is for the hadronic final state after standard Lund model hadronization, whereas adding the soft color interaction model results in the full curve. The dotted curve is when the SCI probability parameter $P$ has been lowered from its standard value 0.5 to 0.1.

Figure 4

Generic diagram for diffractive deep inelastic scattering.

Figure 5

Low-order rescattering correction to DIS in the parton model frame where the virtual photon momentum is along the negative $z$ axis with $q=(q^{+},q^{-},{\mathit{q}}_{\perp })\simeq (-m_p{x}_B,2\nu ,0)$ and the target is at rest, $p=(m_p,m_p,0)$. The struck parton absorbs nearly all the photon momentum, $p_1\simeq (0,2\nu ,{\mathit{p}}_{1\perp })$ (aligned jet configuration). In (a) the virtual photon strikes a quark and the diffractive system is formed by the $\mathrm{q}\overline{\mathrm{q}}$ pair $(p_1,p_2)$ which rescatters coherently from the target via instantaneous longitudinal ($A^{+}$) gluon exchange with momentum $k_2$. In (b) the $Q\overline{Q}$ quark pair which is produced in the ${\gamma }^{*}g\to Q\overline{Q}$ subprocess has a small transverse size $r_{\perp }\sim 1/Q$ and rescatters like a gluon. The diffractive system is then formed by the $(Q\overline{Q})g$ system. The possibility of hard gluon emission close to the photon vertex is indicated. Such radiation (labeled $k$) emerges at a short transverse distance from the struck parton and is not resolved in the rescattering.

Figure 6

The amplitude of Fig. 5a as viewed in the dipole frame. The photon splits into a $\mathrm{q}\overline{\mathrm{q}}$ pair of large transverse size which multiply scatters on the target such that the overall exchange is color neutral.

Figure 7

A scalar Abelian model for deep inelastic scattering with one-, two-, and three-gluon exchanges. At each order only one representative diagram is shown. In the $x_B\to 0$ limit and at the orders considered no other final states contribute to the total DIS cross section.

Figure 8

The two-gluon exchange amplitude $A_2$ becomes purely imaginary as $x_B\to 0$. The intermediate state indicated by the dashed line is thus on-shell and the full amplitude is given by the product of the two subamplitudes on either side of the cut. (Only one of the contributing Feynman diagrams is shown.)

Figure 9

Illustration of diffraction through rescattering in $NN\to 2\mathrm{\text{jets}}+X$ in analogy with, and using the same notation as, the DIS case in Fig. 5. The compact $\mathrm{q}\overline{\mathrm{q}}$ pair which forms the jets is assumed to be in a color-octet ($\mathbf{8}$) configuration. This pair rescatters coherently and thus retains its color.