Spacelike and timelike structure functions: a dispersive crossing relation

Abstract

Crossing symmetry suggests that deep inelastic scattering and semi inclusive electron-positron annihilation are governed by analytic continuations of a single forward amplitude. Drell, Levy, and Yan proposed that the hadronic tensor admits analytic continuation and demonstrated, in reasonable models, that connected contributions to the cross-section continue. They also identified obstructions to continuation of the current correlator. In this work we supplement their observation with a new dispersive proposal for analytic continuation of the correlator and, assuming polynomial boundedness, derive subtracted dispersion relations relating spacelike and timelike cross sections. We introduce a new factorized function that quantifies the obstruction to crossing and compute its hard kernel at lowest order. The resulting identity connects distribution functions in deep inelastic scattering to fragmentation functions in annihilation.

Figures (8)

• Figure 1: Analytic structure of $T_a(Q^2,x)$ in the complex $x$-plane at fixed $Q^2<0$. Branch cuts (red) lie on $x\in(-1,0)\cup(0,1)$. The contour $C_a$ (blue) encircles the cut, picking up $\mathrm{Disc}\,T_a = 2i\,W_a$. After deformation, $C_a$ collapses onto the residue contour $C_{\pm\tilde{x}}$ (teal) encircling the evaluation points $\tilde{x} = 1/x_F > 1$.
• Figure 2: Analytic structure of $T_a(q^2,\tilde{x})$ in the complex $q^2$-plane at fixed $\tilde{x} = 1/x_F > 1$. The branch cut (red) lies on the positive real axis for $q^2 \ge m_p^2$, where $\mathrm{Disc}\,T_a = 2i\,\bar{W}_a+2i\Gamma_a$. The Hankel contour $\mathcal{C}$ (blue) wraps the cut; the arc at infinity vanishes by polynomial boundedness, yielding Eq. \ref{['eq:dispersion-SIA']}. After deformation, $\mathcal{C}$ collapses onto the residue contours $C_{Q_1}$ and $C_{Q_2}$ (teal) encircling the Euclidean subtraction points $-|Q_1^2|$ and $-|Q_2^2|$.
• Figure 3: Analytic structure of $T_a(Q^2,\omega)$ in the complex $\omega = 1/x$ plane at fixed $Q^2<0$. Branch cuts (red) lie on the real axis for $|\omega|\ge 1$, i.e. $\omega\in(-\infty,-1]\cup[1,\infty)$, with the unit circle $|\omega|=1$ shown as a dashed reference. The contour $C$ (blue) wraps each semi-infinite cut, picking up $\mathrm{Disc}\,T_a = 2i\,W_a$. Inside the unit disk $T_a$ is analytic and admits the Taylor expansion $T_a = \sum_N c_N\,\omega^N$. The teal contour encircles $\omega=0$ and extracts the Taylor coefficients via the Cauchy formula; the point $x_F\in(0,1)$ is reached by evaluating the series there.
• Figure 4: Fully disconnected graphs that potentially contribute to the imaginary part of $T^{\mu\nu}$. Cuts of this graph have singularities in $q^2$ alone, independently of the value of $x$. However, they don't survive LSZ reduction.
• Figure 5: Graphs that are fully connected on either side of the cut. Such graphs, when crossed yield semi-inclusive annihilation cross-section.