The forward-backward asymmetry in the electron positron annihilation process at twist-4

A quantity of particular experimental interest is the forward-backward asymmetry in the angular distribution of positively and negatively charged fermions produced in $Z^0$ decays. Measurements of this asymmetry can enable independent determinations of the neutral-current couplings of these fermions, i.e. the $Z^0$ boson couplings for left- and right-handed fermions, respectively. Due to the quark confinement, however, it is difficult to determine the electroweak interactions of quarks, especially for light quarks. In the hadron production electron positron annihilation process, the parton model with factorization theorem gives a reliable approximate description. Quantities are thus expressed in terms of fragmentation functions in the annihilation process. In this paper, we consider the vector meson production in the inclusive electron positron annihilation process and calculate the forward-backward asymmetry in the hadronic level. Calculations are carried out by applying the collinear expansion in the parton model at leading order twist-4. We note here this process provides not only a tool for analyzing the hadronic weak interactions but also an opportunity for understanding the parton model of the strong interaction. In other words, the results can be used to test the electroweak and strong interactions simultaneously.


I. INTRODUCTION
The Standard Model (SM) of elementary particles and their interactions has achieved great success in the past decades. It has two basic components, the spontaneously broken electroweak (EW) theory and the color gauge theory or quantum chromodynamics (QCD). Since the leftand right-handed fermions in the EW theory live in different representations of the fundamental gauge group, they have different couplings for the gauge bosons, Z 0 , W ± . As for Z 0 , the difference leads to an asymmetry in the angular distribution of positively and negatively charged leptons and/or quarks produced in Z boson decays. This asymmetry, known as the forward-backward asymmetry [1][2][3][4], depends on the Weinberg angle or the weak mixing angle and can enable independent determinations of the neutral-current couplings of these fermions. Due to the quark confinement it is relatively difficult to determine the electroweak interactions of quarks. Heavy quarks (c or b) can be determined by tagging the decays of corresponding hadrons containing them. However, light quarks require other flavor separation method. The difficulty in describing the weak interactions of quarks lies in the description of the quark fragmentation process. Thanks to the asymptotic freedom of QCD, the fragmentation process can be studied in the factorization theorem framework [5] in the parton model. Factorization theorem tells that measurable quantities, e.g. cross section, can be separated by the calculable hard parts from the non-perturbative soft parts. If only the fragmentation process is taken into consideration, the non-perturbative soft parts are usually factorized as fragmentation functions. Fragmentation functions (FFs) are most important physical quantities in describing hadron productions in high energy reactions. They quantify the hadronization process of quarks and/or gluon in high energy reactions where hadrons are produced. Quantities therefore can be expressed in terms of FFs in the annihilation and other fragmentation processes.
In this paper, we consider the vector meson production in the inclusive electron positron annihilation process and calculate the forward-backward asymmetry in the hadronic level, i.e. the asymmetry in the angular distribution of the produced vector meson. The calculations are carried out in the parton model at leading order twist-4 by applying the collinear expansion formalism [6][7][8][9]. Collinear expansion is a powerful tool to calculate higher twist effects by taking into account multiple gluon exchange contributions. On the one hand, gauge links will be obtained automatically which make the calculation explicitly gauge invariant. On the other hand, collinear expansion gives a very simple factorization form which consists of calculable hard parts and FFs. This will greatly simplify the systematic calculation of higher twist contributions. After obtaining the differential cross section, we introduce the definition of the forward-backward asymmetry for the production hadron. We finally present these asymmetry results in terms of FFs.
The rest of the paper is organized as follows. In Sec. II, we first introduce the general definition of the forwardbackward asymmetry of the muon pair in the electron positron annihilation process and show some conventions used in this paper. In Sec. III, we present the formalism of vector meson production annihilation process where the differential cross section is given in terms of structure functions. The study in the parton model formulism is given in IV, where we present a detailed calculation of how to obtain the hadronic tensor and the cross section at leading order twist-4. In Sec. V, we present the results for the structure functions and forward-backward asymmetries in terms of the gauge invariant FFs. A brief summary is given in Sec. VI.

II. INTRODUCTION TO THE FORWARD-BACKWARD ASYMMETRY
A simple exercise for the fermion pair production in the electron positron annihilation process is to calculate the muon pair production process e + (l )+e − (l) → µ + (k )+ µ − (k). It gives fruitful information about the annihilation reactions. By considering the EW theory, the differential cross section of the this process can be written as where θ is the scattering angle in the lepton center-ofmass frame or the gauge boson rest frame, α em = e 2 /4π is the fine structure constant and Q 2 = q 2 = (l + l ) 2 .
where M Z and Γ Z are respectively the mass and decay width of Z-boson, θ W is the Weinberg angle or the weak mixing angle. c e 1 = (c e V ) 2 + (c e A ) 2 and c e 3 = 2c e V c e A , c e V and c e A are defined in the weak interaction current . Similar notations are also used for muon and quarks where we use a superscript µ and q to replace e.
The forward-backward asymmetry in the angular distribution of positively and negatively charged muons is defined as where dσ θ = dσ/d cos θ given in Eq. (1). Using the definition in Eq. (4) and the differential cross section in Eq. (1), we have At the low-energy limit (Q 2 << M 2 Z ), this asymmetry is given approximately by where G F is the Fermi constant. Similar results can also be obtained for quarks as long as we replace the corresponding couplings for muon by that for quarks. For example, as the low-energy limit, we have where e q is the electric charge of the quark with flavor q. From Eqs. (5)- (7), we can see that forwardbackward asymmetries depend on weak couplings for certain fermions, they would give independent determinations of these couplings. In the following context, we extend the results to the hadron production process. We note that the definition of the forward-backward asymmetry in Eq. (4) will be slightly modified for calculating that for hadrons. It will be shown in Sec. V.

III. THE GENERAL FORM OF THE CROSS SECTION IN TERMS OF STRUCTURE FUNCTIONS
At first sight, the vector boson production inclusive electron positron annihilation process can not be calculated because of the lack of perturbative description of the fragmentation process. Therefore, we consider the general decomposition of the hadronic tensor and give the general form of the cross section in terms of structure functions. We show this hadron production annihilation process in Fig. 1. To be explicit, we write down the differential cross section as Here we use the notations as illustrated in Fig. 1. The standard variables z = 2p · q/Q 2 and y = p · l /p · q. The subscript r denotes γγ, γZ and ZZ corresponding respectively to the electromagnetic, interference and weak contributions to this process. The summation of r denotes the summation of these cross sections, i.e.
The leptonic tensors for the electromagnetic, interference and weak contributions are respectively given by L γZ µν (l, l ) = c e V (l µ l ν + l ν l µ − g µν l · l ) + ic e A ε µνll , (11) L ZZ µν (l, l ) = c e 1 (l µ l ν + l ν l µ − g µν l · l ) + ic e 3 ε µνll , where ε µνAB ≡ ε µναβ A α B β . The hadronic tensors are respectively given by where S denotes the polarization of the hadron and J ZZ µ (x) is the quark weak current.
To deal with the hadronic tensor, it is convenient to construct it with known quantities, e.g. momenta. First of all we show the general decomposition of the hadronic tensor by dividing it into a symmetric and an anti-symmetric part, W µν = W S µν + iW A µν , where we have omitted the subscript r = γγ, γZ, ZZ for simplicity. Each of them is given by a linear combination of a set of basic Lorentz tensors (BLTs), i.e., where h µν andh µν represent the space reflection even and space reflection odd BLTs, respectively. The subscript σ specifies the polarization. In the general decomposition of the hadronic tensor, we require that hadron tensors corresponding to the electromagnetic, interference and weak contributions have the same form. Therefore, they can by summed together. For inclusive reactions, the unpolarized or the spinindependent BLTs can only be constructed by momentum vectors, q, p. There are in total 3 unpolarized BLTs given by The subscript U denotes the unpolarized part. Here p µ q ≡ p µ − q µ (p · q)/q 2 which satisfies p q · q = 0. This notation ensures that the hadron tensor satisfies the current conservation.
The vector polarization dependent BLTs are given by where There are 7 such vector polarized BLTs in total.
The tensor polarized part is composed of S LL -, S LTand S T T -dependent parts. More discussions of polarizations of spin one particles can be found in ref. [10]. For simplicity, we do not show them in this paper. The tensor polarized part can be taken as a product of the unpolarized BLTs and polarization dependent Lorentz scalar(s) or pseudo-scalar(s). They are given by Substituting Eqs. (18)-(31) into Eqs. (16)-(17) and contracting with the leptonic tensor yield the differential cross section. The forward-backward asymmetry is defined in the lepton center-of-mass frame. To be consistent with the definition, we show this cross section in the same frame in which After making Lorentz contraction with the leptonic tensor, we obtain the general form for the cross section, where we use F andF to denote the parity conserved and parity violated parts, respectively. These explicit expressions are given by where F andF with subscripts U, L, LL, T, LT and T T are known as structure functions. We have in total 19 inclusive structure functions. Here we have defined with y = (1 + cos θ)/2. We can see that all the θ dependent terms are given explicitly. From Eq. (1) we see that the differential cross section at the quark level depends on A(y) and B(y) only. This implies that F U1,3 ,F L1,3 and F LL1,3 are leading twist structure functions (may have higher twist corrections) while the other terms are higher twist ones.

IV. THE CROSS SECTION IN THE PARTON MODEL
As mentioned in the introduction, the difficulty in describing the weak interactions of quarks lies in the description of the fragmentation process. The parton model which is applicable to any hadronic cross section involving a large momentum transfer can be used to describe this. Measurable quantity is then factorized as a convolution of the hard part and the non-perturbative soft part. If only the fragmentation process is taken into consideration, the non-perturbative soft parts are usually factorized as fragmentation functions. This is the case in this paper. In this section, in the parton model framework, we present a detailed calculation of how to obtain the hadronic tensor and the cross section at leading order twist-4.
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.
A. The general forms of hadronic tensors in the parton model The parton model gives a reliable approximate description of the hadronic interactions and an opportunity to calculate the forward-backward asymmetry at hadronic level. By applying this model, we limit our considerations at the tree level or leading order of the QCD and consider the series of diagrams illustrated in Fig.  2, where diagrams with exchange of j gluon(s) ( j = 0, 1, 2, · · · ) are included.
To obtain the explicit expression of the hadronic tensor, we use the collinear expansion formalism. It provides not only the correct formalism where the differential cross section or the hadronic tensor is given in terms of gauge invariant FFs, but also very simplified expressions so that even twist-4 contributions can be calculated. After the collinear expansion, the hadronic tensor is obtained as [11][12][13][14] where c = L, R, M denotes different cuts for left, right and middle, respectively. TheW ( j,c) µν is a trace of the collinearexpanded hard part and gauge invariant quarkj-gluonquark correlator. In other words, the hadronic tensor is written as an explicit factorized form. To be explicit, we haveW where we have omitted the arguments for simplicity. The hard parts or simplified scattering amplitudes are given byĥ (0) The corresponding quarkj-gluon-quark correlators are given bŷ where D ρ = −i∂ ρ + gA ρ are the transverse covariant derivative, L(0, y) is the gauge link. The argument ξ in the quark filed operator ψ and gauge link represents (0, ξ − ). We note that the leading power contribution of W ( j) µν is twist-( j+2). Therefore the second term in Eq. (55) has no contribution up to twist-4 because of of the factor p σ in the definition ofΞ (2 ) ρ given by Eq. (63). The leading power contribution of this term is twist-5.

B. Decompositions of correlators
In the previous part, hadronic tensors are given in the explicit factorization forms where hard parts and nonperturbative soft parts are naturally separated. These soft parts are correlators shown in Eqs. (60)-(64). Correlators can not be calculated with perturbative theory because they contain the hadronization information. However, they are 4 × 4 matrices in Dirac space and can be decomposed in terms of Γ matrices, i.e., Γ = {I, iγ 5 , γ α , γ 5 γ α , iσ αβ γ 5 }. The decomposition can be writ-ten explicitly aŝ In the inclusive electron positron annihilation process, only the chiral even quantities are involved due to no spin flip. Thus we only need to consider the γ α -and the γ 5 γ αterm in the decomposition of the correlators in terms of the Γ-matrices and corresponding coefficient functions, such asΞ (0) = Ξ (0) α γ α +Ξ (0) α γ 5 γ α + · · · . We first consider the quark-quark correlatorΞ (0) . At twist-4, the coefficient functions are given by HereS T α = ε ⊥βα S β T . D's and G's represent the γ α -and γ 5 γ α -type FFs, respectively. The digit j in the subscript denotes twist-( j + 1); the capital letter such as L, T , LL and LT denote hadron polarizations.
For the quark-gluon-quark correlatorΞ (1) ρ , the chiral even parts are where we use subscript d to denote FFs which are defined throughΞ (1) ρ . For the quark-gluon-gluon-quark correlatorsΞ (2) ρσ and Ξ (2,M) ρσ , we require that the decomposition ofΞ (2,M) ρσ takes exactly the same form as that ofΞ (2) ρσ . We just add an additional superscript M to distinguish them in the following context from each other. For the chiral even part, the corresponding coefficient function are given by where we use dd in the subscript to denote FFs defined viaΞ (2) ρσ . From Eqs. (66)-(71), we see that for the twist-4 parts, the decomposition of Ξ and that ofΞ have exact one to one correspondence. For each D 3 , there is a G 3 corresponding to it. They always appear in pairs. Because of the Hermiticity ofΞ (0) andΞ (2,M) ρσ , FFs defined via them are real. For those defined viaΞ (1) ρ andΞ (2) ρσ , there is no such constraint so that they can be complex.
Not all the FFs defined in Eqs. (66)-(71) are actually independent. We can eliminate the correlated terms by using the QCD equation of motion γ · Dψ = 0. With this equation, the quarkj-gluon-quark correlators is related to the quark-quark correlator. For the two transverse components Ξ (0)ρ ⊥ andΞ (0)ρ ⊥ , we have Equations (72) and (73) lead to a set of relationships between twist-3 FFs which can be given in the unified form where S = T and LT whenever applicable. Similarly, for the minus components of Ξ (0) α andΞ (0) α , we have From Eqs. (75) and (76), we obtain a set of relationships between twist-4 FFs defined viaΞ (0) ,Ξ (1) andΞ (2,M) , where D ± ≡ D ± G such as D −3d ≡ D 3d − G 3d and so on. We note here that only the one-dimensional or collinear FFs are shown. In fact the relationships between threedimensional or transverse momentum dependent FFs can be obtained in the same way [15], we do not repeat them in this paper.

C. The hadronic tensor at twist-4
It is straightforward to calculate the hadronic tensor with FFs and the corresponding hard parts at hand. The important step is calculating these traces. To be explicit, we calculate the leading twist, twist-3 and twist-4 contributions in turn. The leading twist contributions only come from the quark-quark correlator Ξ (0) . The corresponding traces are simple and given by Tr Tr ĥ (0) µν γ 5 / n = 4 p + c q 3 g ⊥µν + ic q 1 ε ⊥µν .
whereq = q − 2p/z. It it can be shown thatW t3µν satisfies the current conservation q µW t3µν = q νW t3µν = 0. Here we can see that consideration of the quark(-j)-gluon-quark correlator is also a requirement of the current conservation.
To obtain the contributions from the left-and right-cut parts, one need to very carefully to calculate the following traces, The hadronic tensor is given by We defineW (2) t4µν =W (2)L t4µν + W (2)L t4νµ * . Using Eqs. (77)-(79) to eliminate the non-independent FFs yields where The first four of the four-quark diagrams where no multiple gluon scattering is involved. In (a), we have k 1 = k 1 − k and k 2 = k 2 − k; in (b) we have the interchange of k 1 with k 1 ; in (c) we have the interchange of k 2 with k 2 ; in (d) we have both interchanges of k 1 with k 1 and k 2 with k 2 .
In this part we obtain the complete hadronic tensor up to twist-4 level. In Eq. (82) we show the leading twist hadronic tensor while show the twist-3 hadronic tensor in Eq. (89). The twist-4 hadronic tensors are given in Eqs. (99) and (103). All these hadronic tensors satisfy the current conservation law.

D. Contributions from the four-quark correlator
At twist-4, there are also contributions from diagrams involving the four-quark correlator [6,7,16] except for those from quark-j-gluon-quark correlators. Following the previous discussion, we consider the four-quark correlator in this part. The general operator definition of the four-quark correlator is given bŷ Some example of the four-quark diagrams are shown in Fig. 3. We note that if the cut is given at the middle we have contributions from the gluon jet. If the cut at the left and/or right, we have contributions from the quark jet. Both of them contribute to the vector meson production annihilation process, in this case we consider them together.
There are eight structure functions which have twist-3 contributions, and are given by

B. The forward-backward asymmetries
We have emphasized in the introduction, the main focus of this paper is calculating the forward-backward asymmetries for the produced hadron in the inclusive annihilation process. The forward-backward asymmetry is introduced to describe the angle distribution of the fermions from Z 0 decays as introduced in Sec. II. Here we redefine the asymmetry at the hadonic level to illustrate the angle distribution of the produced hadron in the electron positron annihilation process. Comparing to Eq. (4), we define the forward-backward asymmetry for a hadron as where [dσ] = dσ/dzd cos θ while [dσ] U denotes the differential cross section for unpolarized case at leading twist. In Eq. (117), the differential cross section is given in terms of y instead of cos θ, it is then convenient to rewrite the forward-backward asymmetry A FB in the following form, where (dσ) = dσ/dzdy and (dσ) U denotes the unpolarized differential cross section at leading twist only.
Using the definition in Eq. (148) and the corresponding differential cross section, we obtain,  rem and/or the parton model. They can be used to test the electroweak and strong interactions simultaneously. These forward-backward asymmetries can also be expressed with structure functions. We do not show them for simplicity.
To have an intuitive impression of the hadron forwardbackward asymmetry shown above, we present the numerical values of A FB,U and A FB,L in Fig. 4. The produced hadron is chosen as Λ hyperon. Only leading twist contributions are considered. We do not show other asymmetries due to lack of proper parametrizations. The parametrization of the unpolarized FF D 1 is taken from AKK08 [17]. Only the light valence quarks (u, d, s) and gluon are considered here while sea quarks and heavy quarks are ignored (We found that they have limited influences on the numerical results.). The QCD evolution of the FF starts from Q = 2GeV and is limited at leading order.
We use the same parametrization of the longitudinal spin transfer FF G 1L given in ref. [18]. We use for the s−quark FF and G q→Λ 1L (z) = Nz a D q→Λ for the u−, d−quark FF, where superscript q = u, d. We fix the parameters as a = 0.5 and N = −0.1. The evolution function and polarized splitting functions can be found in ref. [18][19][20]. We do not show them here for simplicity. For comparison, we draw asymmetries for quark (u, d, s) as well as that for the produced hadron in the same panel. We find that they have the same behaviors but different numerical values. This is because the forward-backward asymmetry which arises from the difference of Z 0 couplings for left-and right-quarks is dominated by the energy and couplings. At the same time, the longitudinal spin transfer FF G 1L (z) satisfies |G 1L (z)| ≤ D 1 (z). The same goes for momentum fractions, z = 0.20, 0.30, 0.40. We here only consider the collinear FFs. Parametrizations of the transverse momentum dependent polarizing FF for Λ can be found, e.g. in refs. [22,23].

C. Parity-violating asymmetries
With the advent of highly polarized electron beams, parity violation measurements have become a standard tool for probing a variety of phenomena. In this part, we calculate the parity-violating asymmetries in the inclusive annihilation process. Parity-violating asymmetry usually describes the difference of the cross section for respectively the right-and left-handed electrons in the deeply inelastic scattering process [24,25]. In this paper, we consider the unpolarized lepton beam and calculate the parity-violating asymmetries with the polarized produced hadron. The definition of the parity-violating asymmetry is given by where S denote the hadon spin, dσ denotes the unpolarized differential cross section, i.e. dσ = dσ/dzdy. This definition is different from that in ref. [26] where asymmetry was given with respect to the unpolarized electromagnetic cross section. Different definitions in principle do not change the physical meanings. However, numerical result shows the definition in Eq. (159) is more reasonable.
First of all, we present the two asymmetries given by the longitudinal polarized FFs, they are A PV,L = χT q 1,ZZ + χ int T q 1,γZ G 1L e 2 q T q 0,γγ + χT q 0,ZZ + χ int T q 0,γZ D 1 , We can see that they are leading twist asymmetries with twist-4 corrections. We use the same parametrization of the longitudinal spin transfer FF G 1L shown before and present the numerical values of A PV,L in Fig. 5. Correspondingly, there are two twist-4 asymmetries which are given by 2 χc e 1 c q 1 + χ int c e V c q V D 2 (y)D 3LL z 2 e 2 q T q 0,γγ + χT q 0,ZZ + χ int T q 0,γZ D 1 .
We can also calculate the parity-violating asymmetries for the transversely polarized hadron case, they are all twist-3 asymmetries.

VI. SUMMARY
In this paper, we consider the vector meson production in the inclusive electron positron annihilation process and calculate the forward-backward asymmetry in the hadronic level, i.e. the asymmetry in the angular distribution of the produced vector meson. The asymmetry arises from the difference of Z 0 couplings for left-and right-handed fermions. Measurements of this asymmetry can enable independent determinations of the neutralcurrent couplings of these fermions. To deal with the non-perturbative fragmentation process, we present a factorized form of the hadronic tensor by using the collinear expansion method in the parton model. Results are finally expressed in the factorized forms, see Eqs. (150)-(156). The explicit factorized forms provide a direct demonstration of the factorization theorem and/or the parton model. We can see that Eq. (150) is similar to Eq. (5) except for the FF D 1 (D 1 ). This process provides not only a tool for analyzing the hadronic weak interactions but also an opportunity for understanding the parton model of the strong interaction. In other words, the results can be used to test the electroweak and strong interactions simultaneously. In addition to the forward-backward asymmetries, we also calculate parity-violating asymmetries and structure functions at leading order twist-4.