Higher twist effects in $e^+e^-$ annihilation at high energies

In the two papers published recently, we apply collinear expansion to both inclusive ($e^+ e^- \to h+X$) and semi-inclusive ($e^+ e^- \to h + \bar q + X$) hadron production in $e^+ e^-$ annihilation to derive a formalism suitable for a systematic study of leading as well as higher twist contributions to fragmentation functions at the tree level. We carry out the calculations for hadrons with spin-0, spin-1/2 as well as spin-1. This proceeding is mainly a summary of these two papers.


Introduction
The e + e − annihilation process is most suitable to study fragmentation functions among all different high energy reactions, as there is no hadron involved in the initial state. The one dimensional fragmentation functions can be studied via inclusive process, while, the 3D information can only be extracted from semi-inclusive process.
Higher twist terms may have very important contributions to azimuthal asymmetries and spin asymmetries 3 , which are usually measured in experiments to study the properties of fragmentation functions and parton distribution functions. Collinear expansion, first developed in 1980s, seems to be the unique method to calculate leading twist and higher twist contributions in a systemic way.
This method was applied to inclusive DIS process 4 to get the cross section up to twist-4 level at first. It was summarized as four steps 5 , and was applied to SIDIS to get the form of azimuthal asymmetries up to twist-3 5 and twist-4 6 . Recently, we applied collinear expansion to inclusive e + e − annihilation process 1 and semiinclusive process 2 . This is an Open Access article published by World Scientific Publishing Company. It is distributed under the terms of the Creative Commons Attribution 3.0 (CC-BY) License. Further distribution of this work is permitted, provided the original work is properly cited.

Collinear Expansion
Collinear expansion was inspired by the collinear approximation, which is stated as follows.
• We only keep the collinear component of the quark momentum, k i ≈ p/z i . • We only keep the plus component of the gluon field, A µ ≈ A +nµ .
This approximation is reasonable, if we are only care about the leading twist contributions, since other components are power suppressed compared to the plus component. We apply this approximation to the following diagrams, and we have, where, Γ µ = γ µ (c q V − c q A γ 5 ) is the vertex for weak interaction. And the soft matrix Ξ approx is a little different for inclusive process and semi-inclusive process. For the inclusive process, this soft matrix does not dependent on k ′ ⊥ , While, for the semi-inclusive process, F is just the Fourier Transformation operator. We see, these soft matrices are automatically gauge invariant. If we are going to the twist-3 level, these transverse components can not be simply neglected any more. So, we need the collinear expansion to take them back into account by the following steps 1,2 .
• Make a Taylor expansion to the hard part, • Decompose the gluon fields in this way, Reducing them with Ward identities, we get the hadronic tensor twist by twist, where,W is the new hadronic tensor, whose leading contribution is at twist-3. The new quark gluon correlators are defined as,Ξ These Ξ's defined above are all 4 by 4 matrices which can alway be decomposed in terms of Gamma matrices. In this case, only γ α and γ 5 γ α that will contribute.
where, parity invariant constrains the possible structures of them, since Ξ (i) α is a vector andΞ (i) α is an axis vector. We will discuss the details of this decomposition in the following two sections for inclusive process and semi-inclusive process separately.
3. Inclusive Process e + e − → h + X These Ξ's are made up by p, n and S for the inclusive process. S here refers to the spin parameters of produced hadrons. For spin-1/2 hadrons, we only need the spin vector S µ , while for vector mesons, we need also a spin tensor T µν , which can be decomposed in terms of S LL , S µ LT and S µν T T .

Spin Independent Part and Spin Vector Dependent Part
LT S (z) .
For spin-0 hadrons, only the spin independent terms that will contribute, For spin-1/2 particles, both the spin independent part and spin vector dependent part contribute. The spin vector dependent part is given by, Here, the summation over quark flavor and color is not written out explicitly. We see, the first term is at leading twist and the last two terms are at twist-3. So, it is obvious that at leading twist, produced hadrons will be longitudinally polarized.
It is just proportional to the polarization of quark, P q (y) = T q 1 (y)/T q 0 (y), times a spin transfer, ∆D q→h 1L (z). There are also two transverse polarizations at twist-3. One is perpendicular to the leptonic plane, ǫ l ⊥ S ⊥ ⊥ , while the other one lies in the leptonic plane, l ⊥ · S ⊥ .
The first one is T-odd. Hence this is NO corresponding term in inclusive DIS process. The last one is P-odd, which will disappear in the electromagnetic process.

Spin Tensor Dependent Part
LT S (z). (18) For vector mesons, the cross section contains contributions from spin independent, vector polarization dependent and tensor polarization dependent parts.
The most interesting spin parameter of vector mesons is spin alignment (ρ 00 ). This leading twist effect is verified by LEP experiment. We predict that this tensor polarization can also be easily measured by BES experiment since it is P-even.
4. Semi-inclusive Process e + e − → h +q + X For the semi-inclusive process, these Ξ's are also functions of the transverse momentum k ′ ⊥ . So, we have much more new structures.

Spin Independent Part and Azimuthal Asymmetries
And the corresponding cross section is, We see immediately that there are two azimuthal asymmetries at twist-3, The second one is P-odd, so it will disappear in electromagnetic process.

Spin Vector Dependent Part
The cross section is then given by, We see, that there is a leading twist polarization in the longitudinal direction, The most suitable directions to study transverse polarization and corresponding fragmentation functions are those in and transverse to the production plane. At leading twist, Those twist-3 terms gives us corrections 2 which are suppressed by M/Q.

Spin Alignment
The Lorentz structures for vector mesons are much more complicated, since there are much more spin parameters. We would not show you all those structures in this proceeding, for the limitation of length. Please find the details in [2] if you are interested in this topic. Here we only show you results concerning spin alignment.
The corresponding cross section is given by, And then, we get the spin alignment up to twist-3,

Summary
In inclusive process, there is a leading twist longitudinal polarization for spin-1 2 hadrons and also spin alignment (ρ 00 = 1 3 ) for vector mesons. At twist-3, there are transverse polarizations for spin-1 2 hadrons in and transverse to the leptonic plane. In semi-inclusive process, for spin-0 hadrons, there are two azimuthal asymmetries at twist-3. For spin-1 2 hadrons, there is a longitudinal polarization and also transverse polarizations in and transverse to the production plane at leading twist.