Twist-4 contributions to semi-inclusive $e^{+}{e}^{-}$ annihilation process

We present the complete twist-4 results for the semi-inclusive annihilation process $e^{+}+e^{-}\to h+\overline{q}+X$ at the tree level of perturbative quantum chromodynamics. The calculations are carried out by using the formalism obtained by applying the collinear expansion to this process, where the multiple gluon scattering is taken into account and gauge links are obtained systematically and automatically. We present the results for structure functions in terms of gauge invariant fragmentation functions up to twist-4 and the corresponding results for the azimuthal asymmetries and polarizations of hadrons produced. The results obtained show in particular that, similar to that for semi-inclusive deeply inelastic scattering, for structure functions associated with the sine or cosine of an odd number of azimuthal angle(s), there are only twist-3 contributions, while for those of an even number of azimuthal angle(s), there are leading twist and twist-4 contributions. For all of those structure functions that have leading twist contributions, there are twist-4 addenda to them. Hence, twist-4 contributions may even have large influences on extracting leading twist fragmentation functions from the data. We also suggest a method for a rough estimation of twist-4 contributions based on the leading twist fragmentation functions.

Figure 1

Illustrating diagram for $e^{+}+e^{-}\to h+\overline{q}+X$.

Figure 2

The first few diagrams as examples of the considered diagram series with exchange of $j$-gluon(s) and different cuts. We see (a) $j=0$, (b1) $j=1$ and left cut, (b2) $j=1$ and right cut, (c1) $j=2$ and left cut, (c2) $j=2$ and middle cut, and (c3) $j=2$ and right cut, respectively.

Figure 3

A rough estimation of the twist-4 contribution factor $\alpha /k_{\perp M}^2$ as a function of $z$ at $y=0.5$ and $Q=M_Z$.

Figure 4

The first four of the four-quark diagrams where no multiple gluon scattering is involved. In (a), we have $k_1^{\prime }=k_1-k$ and $k_2^{\prime }=k_2-k$; in (b) we have the interchange of $k_1$ with $k_1^{\prime }$; in (c) we have the interchange of $k_2$ with $k_2^{\prime }$; in (d) we have both interchanges of $k_1$ with $k_1^{\prime }$ and $k_2$ with $k_2^{\prime }$.