Azimuthal and spin asymmetries in e^{+}e^{-}\to V\pi X at high energies and 3D fragmentation functions

# Azimuthal and spin asymmetries in e+e−→VπX at high energies and 3D fragmentation functions

## Abstract

We present the systematical decomposition results of three-dimensional (3D) fragmentation functions (FFs) from parton correlators for spin-1 hadron. We choose one of the best process to study the 3D and tensor polarization dependent FFs. By making a general kinematic analysis we show that the cross section is expressed by 81 independent structure functions, and get the results of the azimuthal and spin asymmetries. We also present the parton model results for this process and express the results in terms of the 3D FFs. Based on this formalism, we also give the numerical results of hadron longitudinal polarizations and predict the energy dependence.

###### pacs:
13.87.Fh,13.88.+e,13.66.Bc, 13.60.Le,13.60.Rj,12.38.-t,12.38.Bx, 12.39.St,13.85.Ni

## I Introduction

Parton distribution functions (PDFs) and FFs are both important inputs for describing high energy reactions. PDFs are used to describe hadron structure and the FFs describe the hadronization mechanism. They should be studied in parallel to promote our understanding of non-perturbative quantum chromodynamics (QCD). In inclusive high energy reactions, only longitudinal momentum dependent PDFs and/or FFs are involved. For describing semi-inclusive reactions, 3D or transverse momentum dependent (TMD) PDFs and/or FFs are needed to explain the phenomenon such as azimuthal asymmetries of the hadron. This will generalize our knowledge of hadron structure and hadronization mechanism to three dimensional case. See e.g. Ref. Chen:2015tca () for a recent brief review.

Based on the parallel talk given at the 22nd International Spin Symposium and Ref. Chen:2016moq (), in this proceeding we first summarize the results for 3D FFs defined via quark-quark correlator for spin-1 hadrons systematically. Then we choose the process which is one of the best places to study these 3D and tensor polarization dependent FFs. We give a general kinematical analysis of this process and get the differential cross section expression in terms of structure functions. We also present the parton model calculation results for this process in leading order perturbative QCD and up to twist-3. We get the structure functions results as well as the hadron azimuthal asymmetries and polarizations in terms of the convolution of the TMD FFs. We also present the numerical results of energy dependence of the hadron polarizations based on this formalism.

## Ii TMD FFs from quark-quark correlator

A complete parameterization for the TMD PDFs for spin- hadron can be found e.g., in Goeke:2005hb (). Here, we summarize the results for TMD FFs of spin- hadron production case.

The leading contribution of the TMD quark fragmentation is defined by the TMD quark-quark correlator, i.e.,

 ^Ξ (0)(z,k⊥;p,S)=∑X∫p+dξ−2πd2ξ⊥e−i(p+ξ−/z−→k⊥⋅→ξ⊥) ×⟨0|L†(0;∞)ψ(0)|p,S;X⟩⟨p,S;X|¯ψ(ξ)L(ξ;∞)|0⟩, (1)

where and denote the 4-momenta of the quark and the hadron respectively, denotes the spin of the hadron; is the light-cone momentum fraction of the hadron; is the famous gauge link making this definition gauge invariant. The correlator satisfies hermiticity and parity conservation, which are important constraints when we decompose it. TMD FFs are obtained from Eq. (1) by decomposing it in two steps. First, we expand in terms of the -matrices, i.e., (we omit the arguments for simplicity)

 ^Ξ(0)=Ξ(0)+iγ5~Ξ(0)+γαΞ(0)α+γ5γα~Ξ(0)α+iσαβγ5Ξ(0)αβ. (2)

The corresponding coefficient functions are Lorentz scalar, pseudo-scalar, vector, axial-vector and tensor respectively. Second, we make a Lorentz structure decomposition of these coefficient functions using the available variables at hand. The coefficients are expressed as the sum of the basic Lorentz covariants multiplied by scalar functions of and , which are defined as TMD FFs.

Take the unpolarized part as an example. The only Lorentz scalar we can build is or the hadron mass ; no pseudoscalar can be built; we have three vectors , , and one axial-vector ; three tensors , and are available. So we can immediately write down the decomposition of the TMD FFs as

 zΞU(0)=ME,z~ΞU(0)=0,z~ΞU(0)α=−~k⊥αG⊥, (3) zΞU(0)α=p+¯nαD1+k⊥αD⊥+M2p+nαD3, (4) zΞU(0)ρα=−p+M¯n[ρ~k⊥α]H⊥1+Mε⊥ραH−Mp+n[ρ~k⊥α]H⊥3. (5)

Here and in the following, we will omit the arguments for the FFs for simplicity. We see that there are two leading twist TMD FFs, i.e., the number density and Collins function , and two corresponding twist-4 addenda and . Other four are twist-3 FFs. To save space, for the vector polarization and tensor polarization dependent parts, we refer interested readers to Ref. Chen:2016moq () for the complete decomposition results which include 72 TMD FFs, as well as the discussions of the naming system and the properties of the TMD FFs.

Higher twist FFs can also be defined via quark-gluon(s)-quark correlators. However, they are often not independent because of QCD equation of motion . We also refer to Ref. Chen:2016moq () for the complete decomposition results of the twist-3 TMD FFs from quark-gluon-quark correlator. Here, we just emphasize that a unified relationships between the twist-3 FFs defined from quark-quark and quark-gluon-quark correlator can be obtained as, for the chiral-even part and for the chiral-odd part. The subscript and the superscript denote different spin and transverse momentum dependence respectively. The contributions from the quark-gluon-quark correlator can then be replaced by using these relations.

## Iii Kinematic analysis of e+e−→VπX

The best process for studying FFs is electron positron annihilation, because there is no initial PDFs involved. Inclusive hadron production process is the simplest case to study one dimensional FFs. In order to study transverse momentum dependence, we must go to semi-inclusive process, see e.g., Wei:2013csa (); Wei:2014pma () for these discussions. To access both chiral-even and chiral-odd as well as the tensor polarization dependent FFs, the two-particle semi-inclusive -annihilation with vector meson production process as illustrated in Fig. 1 is the best choice.

We consider the general weak interaction process . The differential cross section is given by

 2E1E2dσd3p1d3p2=α2χsQ4Lμν(l1,l2)Wμν(q,p1,S,p2), (6)

where is the well-known leptonic tensor, and the hadronic tensors is defined as

 Wμν (q,p1,S,p2)=1(2π)4∑X(2π)4δ4(q−p1−p2−pX) ×⟨0|Jν(0)|p1,S,p2,X⟩⟨p1,S,p2,X|Jμ(0)|0⟩. (7)

The first step we should do is to give a general kinematic analysis for this reaction, which is to construct the general form of the hadronic tensor, and then calculate the cross section expression in terms of the structure functions.

### iii.1 General form of the hadronic tensor and cross section

The hadronic tensor in Eq. (7) satisfies constraints imposed by Hermiticity, current conservation. We emphasize that if we consider annihilation via virtual photon, then parity conservation is also a constraint, but it is not valid here for weak interaction. We need to construct the basic Lorentz tensors satisfy these constraints using the kinematic variables, and then decompose under these basic tensors.

A systematic analysis of this decomposition for double spin-1/2 hadrons production has been presented in Pitonyak:2013dsu (). Here, we extend the analysis to include spin-1 hadron production, and show the interesting similarities between the unpolarized and polarization dependent parts.

For the unpolarized part, the basic Lorentz tensors can only be constructed from , and , after some careful analysis, we found that the independent basic tensors can be written as

 hSμνUi= (8) ~hSμνUi= {ε{μqp1p2(p1q, p2q)ν}}, (9) hAμνU= p[μ1qpν]2q, (10) ~hAμνUi= {εμνqp1, εμνqp2}, (11)

where ’s are parity conserved (-even) tensors, while ’s are parity violated (-odd) tensors. The superscript or denotes symmetric or anti-symmetric under exchange of Lorentz index, the subscript denotes the unpolarized part. A 4-momentum with a subscript denotes satisfying , so the current conservation is manifest. Some shorthand notations are used to make the expressions more concise, e.g., means and . We see that there are 9 basic tensors in the unpolarized case.

For the vector polarization dependent part, we have

 ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩hSμνVi~hSμνVihAμνVi~hAμνVi⎫⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎬⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎭=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩[(q⋅S),(p2⋅S)]⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣~hSμνUihSμνUi~hAμνUihAμνU⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦,εSqp1p2⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣hSμνUj~hSμνUjhAμνU~hAμνUj⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦⎫⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎬⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎭ (12)

There are in total 27 -dependent basic tensors, corresponding to 3 independent vector polarization modes. We see that they share exactly the same structures as the unpolarized part. We only need to multiply in front the spin dependent scalar or pseudoscalar.

For the tensor polarized part, they also have the same structures as the unpolarized part. For the -dependent basic tensors, because is a scalar, they are just given by the unpolarized tensors multiplied by . Therefore, we have 9 such tensors in this case. The -dependent part is given by

 (13)

There are 18 such tensors in total, corresponding to the two independent -components. The -dependent part takes exactly the same form as the -dependent part. They are given by changing to in Eq. (13). So we have in total 81 basic Lorentz tensors for , 41 of them are -even and 40 are -odd.

The hadronic tensor is then expressed as a sum of all these basic Lorentz tensors multiplied by corresponding coefficients, i.e., , with . The subscript denotes , , , and for different polarizations, and all the coefficients are real scalar functions of the Lorentz scalars , , and .

Substitute into Eq. (6), we will get the cross section in Lorentz invariant form. Here, we write down the expressions in a special reference frame. We choose the helicity frame of ( along positive -direction) which is suitable for studying the vector meson polarization. We also choose center of mass frame of the leptons, with the lepton-hadron plane as plane, as Fig. 2 shows. This is a particular Gottfried-Jackson frame Gottfried:1964nx () which we will refer to as “Helicity-GJ frame”.

In this frame, the cross section can be expressed as

 E1E2dσd3p1d3p2= α2χ2s2[(FU+~FU)+λ(FL+~FL) + |ST|(FT+~FT)+SLL(FLL+~FLL) + |SLT|(FLT+~FLT)+|STT|(FTT+~FTT)], (14)

For the unpolarized part, we have

 FU= (1+cos2θ)F1U+sin2θF2U+cosθF3U +cosφ[sinθFcosφ1U+sin2θFcosφ2U] +cos2φsin2θFcos2φU, (15) ~FU= sinφ[sinθ~Fsinφ1U+sin2θ~Fsinφ2U] +sin2φsin2θ~Fsin2φU, (16)

’s and ’s are called structure functions and are all scalar functions of , and . Among them, the 6 ’s correspond to parity conserving terms of the cross section, while the other 3 ’s are parity odd part.

For the polarized part, since and are just pseudoscalar and scalar, the structure functions related to them have one to one correspondence to the unpolarized part, i.e.,

 (17)

The transverse polarized parts also share the similar structure as the unpolarized part, the only difference is the transverse polarization angle dependence. We just show here the parity conserved dependent part for an example, i.e.,

 FT =sinφS[sinθFsinφS1T+sin2θFsinφS2T] +sin(φS+φ)sin2θFsin(φS+φ)T +sin(φS−φ)[(1+cos2θ)Fsin(φS−φ)1T +sin2θFsin(φS−φ)2T+cosθFsin(φS−φ)3T] +sin(φS−2φ)[sinθFsin(φS−2φ)1T+sin2θFsin(φS−2φ)2T] +sin(φS−3φ)sin2θFsin(φS−3φ)T. (18)

We see the additional polarization angle dependence due to terms of and .

### iii.2 The azimuthal asymmetries and hadron polarizations

Once we get the cross section results, we can derive physical observables such as hadron azimuthal asymmetries and polarizations straightforwardly.

For measuring azimuthal asymmetries, people often sum up the hadron polarizations, which correspond to the unpolarized hadron production case. In this case we have four azimuthal asymmetries due to the dependent terms in the cross section, i.e.,

 ⟨cosφ⟩U=(sinθFcosφ1U+sin2θFcosφ2U)/2FUt, (19) ⟨cos2φ⟩U=sin2θFcos2φU/2FUt, (20) ⟨sinφ⟩U=(sinθ~Fsinφ1U+sin2θ~Fsinφ2U)/2FUt, (21) ⟨sin2φ⟩U=sin2θ~Fsin2φU/2FUt, (22)

where denotes the sum of the structure functions averaged over , i.e, . We point out that the cosine-asymmetries correspond to parity conserving part, while the sine-asymmetries correspond to parity violating part which only appear in weak interaction process.

For measuring the polarization of the vector meson, we often average over the hadron azimuthal angle. In this case, we have, for the longitudinal polarization

 ⟨λ⟩=23FUt[(1+cos2θ)~F1L+sin2θ~F2L+cosθ~F3L], (23) ⟨SLL⟩=12FUt[(1+cos2θ)F1LL+sin2θF2LL+cosθF3LL]. (24)

We see that they correspond to parity violated and conserved structures respectively. For the transverse polarizations, we can measure the components with respect to the lepton-hadron plane, but a more convenient choice is to measure the components normal or tangent to the hadron-hadron plane, i.e., the two transverse directions defined as and , and then average the value over azimuthal angle . In this case, the results are give by

 ⟨SnT⟩= 23FUt[(1+cos2θ)Fsin(φS−φ)1T +sin2θFsin(φS−φ)2T+cosθFsin(φS−φ)3T], (25) ⟨StT⟩= 23FUt[(1+cos2θ)~Fcos(φS−φ)1T +sin2θ~Fcos(φS−φ)2T+cosθ~Fcos(φS−φ)3T], (26) ⟨SnLT⟩= 23FUt[(1+cos2θ)~Fsin(φLT−φ)1LT +sin2θ~Fsin(φLT−φ)2LT+cosθ~Fsin(φLT−φ)3LT], (27) ⟨StLT⟩= 23FUt[(1+cos2θ)Fcos(φLT−φ)1LT +sin2θFcos(φLT−φ)2LT+cosθFcos(φLT−φ)3LT], (28) ⟨SnnTT⟩= −23FUt[(1+cos2θ)Fcos(2φTT−2φ)1TT +sin2θFcos(2φTT−2φ)2TT+cosθFcos(2φTT−2φ)3TT], (29) ⟨SntTT⟩= 23FUt[(1+cos2θ)~Fsin(2φTT−2φ)1TT +sin2θ~Fsin(2φTT−2φ)2TT+cosθ~Fsin(2φTT−2φ)3TT]. (30)

It is interesting to see that the transverse polarizations just correspond to the structure functions of - or -terms. We also see that in this case , and are parity conserved, while the other three are parity violated.

From the above kinematic analysis results, we see that we can study the azimuthal asymmetries in the unpolarized case, or study the longitudinal hadron polarization in the helicity frame and transverse polarizations w.r.t. the hadron-hadron plane averaged over the azimuthal angle . By measuring these quantities experimentally we can obtain the information of the corresponding structure functions.

## Iv Parton model results

### iv.1 Structure functions results in terms of TMD FFs

The above kinematic analysis results are model independent. We can also calculate the hadronic tensor using parton model so that the differential cross section is expressed using FFs. We make the calculations at leading order in pQCD but up to twist-3 level. Similar calculations for double polarized spin-1/2 hadron production has been investigated e.g. in Boer:1997mf (). To this end, we need to consider the Feynman diagrams shown in Fig. 3. The leading twist contribution comes from Fig. 3. Twist-3 contribution comes from both Fig. 3 and Fig. 3 There are also three similar diagrams as Fig. 3 with the gluon attached to the quark line and the complex conjugate contribution, we do not show them here.

Corresponding to these Feynman diagrams, the hadronic tensor is written as . For the leading twist contribution from Fig. 3, we have

 ~W(0)μν= 1p+1p−2∫d2k⊥(2π)2d2k′⊥(2π)2δ2(k⊥+k′⊥−q⊥) ×Tr[^Ξ(0)(z1,k⊥,p1,S)Γμ^¯Ξ(0)(z2,k′⊥,p2)Γν]. (31)

Corresponding to Fig. 3, we have the twist-3 contribution

 ~W(1b)μν= −1√2Qp+1p−2∫d2k⊥(2π)2d2k′⊥(2π)2δ2(k⊥+k′⊥−q⊥) × Tr[Γμ^¯Ξ(0)(z2,k′⊥,p2)γρ⧸¯nΓν^Ξ(1)ρ(z1,k⊥,p1,S)], (32) Δ~W(0b)μν =1√2Qp+1p−2∫d2k⊥(2π)2d2k′⊥(2π)2δ2(k⊥+k′⊥−q⊥) ×kρ⊥ Tr[Γμ^¯Ξ(0)(z2,k′⊥,p2)γρ⧸¯nΓν^Ξ(0)(z1,k⊥,p1,S)]. (33)

We see that the hadronic tensor is expressed by the quark-quark and quark-gluon-quark correlator. Substitute the correlator’s decomposition results and carry out the trace, we get the hadronic tensor in terms of TMD FFs. Then making Lorentz contraction with the leptonic tensor, we obtain the cross section and by comparing with the kinematic analysis results, we will get the structure functions results in terms of the TMD FFs. Here, we directly show the structure functions results which have leading twist correspondence.

For the unpolarized part, we have

 F1U1=2ce1cq1C[D1¯D1], (34) F3U1=4ce3cq3C[D1¯D1], (35) Fcos2φU1=−8ce1cq2C[whhH⊥1¯H⊥1], (36)

where we have defined the convolution of the TMD FFs, e.g.,

 C[wD¯D]=1z1z2∫ d2k⊥(2π)2d2k′⊥(2π)2δ2(k⊥+k′⊥−q⊥) ×w(k⊥,k′⊥)D(z1,k⊥)¯D(z2,k′⊥), (37)

and is a dimensionless scalar weight function of and . Here, we introduce a second digital “1” in the structure function’s subscript to denote it has leading twist correspondence. We note in particular that among the azimuthal angle dependent terms, only the term has twist-2 contribution due to Collins function Collins:1992kk ().

For the longitudinal vector polarization dependent part, we have also three nonzero structure functions at twist-2 and they are very much similar to the unpolarized part, i.e.,

 ~F1L1=−2ce1cq3C[G1L¯D1], (38) ~F3L1=−4ce3cq1C[G1L¯D1], (39) Fsin2φL1=−8ce1cq2C[whhH⊥1L¯H⊥1]. (40)

For the transverse vector polarization dependent part, we have

 Fsin(φS−φ)1T1=2ce1cq1C[w1D⊥1T¯D1], (41) Fsin(φS−φ)3T1=4ce3cq3C[w1D⊥1T¯D1], (42) ~Fcos(φS−φ)1T1=2ce1cq3C[w1G⊥1T¯D1], (43) ~Fcos(φS−φ)3T1=4ce3cq1C[w1G⊥1T¯D1], (44) Fsin(φS+φ)T1=−8ce1cq2C[¯w1H⊥1T¯H⊥1], (45) Fsin(φS−3φ)T1=−8ce1cq2C[wthhH⊥1T¯H⊥1]. (46)

We see that there are 6 non-zero transverse polarization dependent structure functions at twist-2, 4 of them are parity conserved and the other 2 are parity violated.

For the tensor polarization dependent part, first of all, the -dependent part is exactly the same as the unpolarized part. We only need to change the corresponding subscript to to get the results. The -dependent part is very much similar to the -part. The main difference comes from parity, since is a vector. The 6 non-zeros are given by

 Fcos(φLT−φ)1LT1=−2ce1cq1C[w1D⊥1LT¯D1], (47) Fcos(φLT−φ)3LT1=−4ce3cq3C[w1D⊥1LT¯D1], (48) ~Fsin(φLT−φ)1LT1=−2ce1cq3C[w1G⊥1LT¯D1], (49) ~Fsin(φLT−φ)3LT1=−4ce3cq1C[w1G⊥1LT¯D1], (50) Fcos(φLT+φ)LT1=−8ce1cq2C[¯w1H⊥1LT¯H⊥1], (51) Fcos(φLT−3φ)LT1=8ce1cq2C[wthhH⊥1LT¯H⊥1]. (52)

The -dependent part is similar to the -part but the weights are different, the results are

 Fcos(2φTT−2φ)1TT1=2ce1cq1C[wttddD⊥1TT¯D1], (53) Fcos(2φTT−2φ)3TT1=4ce3cq3C[wttddD⊥1TT¯D1], (54) ~Fsin(2φTT−2φ)1TT1=2ce1cq3C[wttddG⊥1TT¯D1], (55) ~Fsin(2φTT−2φ)3TT1=4ce3cq1C[wttddG⊥1TT¯D1], (56) Fcos(2φTT−4φ)TT1=−4ce1cq2C[wtthhH⊥1TT¯H⊥1], (57) Fcos2φTTTT1=8ce1cq2C[w2H⊥′1TT¯H⊥1], (58)

To summarize, we have in total 27 nonzero structure functions at leading twist level for process. These leading twist structure functions are all related to the , and angular distribution, which is a direct result corresponding to the parton level reaction . The twist-3 level results can be found in the appendix of Ref. Chen:2016moq ().

### iv.2 Azimuthal asymmetries and hadron polarizations

From the parton model results of the structure functions we can easily derive the hadron azimuthal asymmetries and polarizations in terms of TMD FFs.

At leading twist and for unpolarized case, there is only one azimuthal asymmetry as given by Eq. (20), i.e., (here and in the following, the summation over quark flavor is implicit for the numerator and denominator)

 ⟨cos2φ⟩(0)U=−C(y)ce1cq2C[whhH⊥1¯H⊥1]Tq0(y)C[D1¯D1], (59)

this corresponds to the well-known Collins effect Collins:1992kk () and is the result of transverse spin correlation. At twist-3, we have another two azimuthal asymmetries, i.e.,

 ⟨cosφ⟩(1)U=−F(1){Tq2(y)(M1C[w1D⊥z2¯D1]+M2C[¯w1z1D1¯D⊥′]) +Tq4(y)(M1C[¯w1Hz2¯H⊥1]+M2C[w1z1H⊥1¯H⊥′])}, (60) ⟨sinφ⟩(1)U=F(1){Tq3(y)(M1C[w1G⊥z2¯D1]−M2C[¯w1z1D1¯G⊥]) +2ce3cq2(M1C[¯w1Ez2¯H⊥1]−M2C[w1z1H⊥1¯E])}, (61)

where . The asymmetry correspond to the well-known Cahn effect in DIS process Cahn:1978se (), and the asymmetry is the parity violating counterpart only appears in weak interaction process.

For hadron polarizations, we have the longitudinal polarization at twist-2 as

 ⟨λ⟩(0)=23Pq(y)Tq0(y)C[G1L¯D1]Tq0(y)C[D1¯D1], (62) (63)