Semi-inclusive production of two back-to-back hadron pairs in $e^+e^-$ annihilation revisited

The cross section for back-to-back hadron pair production in $e^+e^-$ annihilation provides access to the dihadron fragmentation functions (DiFF) needed to extract nucleon parton distribution functions from the semi-inclusive deep inelastic scattering (SIDIS) experiments with two detected final state hadrons. Particular attention is given to the so-called interference DiFF (IFF), which makes it possible to extract the transversity parton distribution of the nucleon in the collinear framework. However, previously unnoticed discrepancies were recently highlighted between the definitions of the IFFs appearing in the collinear kinematics when reconstructed from DiFFs entering the unintegrated fully differential cross sections of SIDIS and $e^+e^-$ annihilation processes. In this work, to clarify this problem we rederive the fully differential cross section for $e^+e^-$ annihilation at the leading-twist approximation. We find a mistake in the definition of the kinematics in the original expression that systematically affects a subset of terms and that leads to two significant consequences. First, the discrepancy between the IFF definitions in the cross sections for SIDIS and $e^+e^-$ annihilation is resolved. Second, the previously derived azimuthal asymmetry for accessing the helicity dependent DiFF $G_1^\perp$ in $e^+e^-$ annihilation vanishes, which explains the nonobservation of this asymmetry in the recent experimental searches by the ${\tt BELLE}$ Collaboration. We discuss the recently proposed alternative option to extract $G_1^\perp$.


I. INTRODUCTION
The understanding of the complete spin-dependent structure of the nucleon has been at the forefront of studies in nuclear physics in recent decades. Particular attention has been given to studying the so-called transversity parton distribution function (PDF), which describes the correlation of the transverse polarization of the nucleon with the transverse polarization of its constituent partons (see e.g. [1]). The chiral-odd nature of the transversity PDF makes it much harder to measure compared to the unpolarized and helicity dependent PDFs. Two approaches have been recently employed in phenomenological extractions of the transversity [2][3][4][5]. The first method uses the Collins effect [6], that describes the correlation between the transverse momentum of a produced hadron with the transverse polarization of an initial quark in the hadronization process. The convolution upon the transverse momenta of initial and final partons of the transversity and the Collins fragmentation function (FF) can be measured in a SIDIS process with a single measured final state hadron [7], while the convolution of two Collins FFs are accessible from the semi-inclusive production of two back-to-back hadrons in e þ e − annihilation [8]. The second method, based on DiFFs, leverages the correlation between the relative transverse momenta of two produced hadrons with the transverse polarization of a quark in its hadronization, which is quantified by the IFF H ∢ 1 . Similarly to the previous method, here again the SIDIS process with two final state hadrons being measured is used to access a structure function containing the transversity PDF and an IFF [9][10][11][12], while the semi-inclusive production of two back-to-back hadron pairs in e þ e − annihilation provides access to IFFs [13][14][15]. The advantage of the dihadron method compared to using the Collins effect is that it is possible to work in the collinear framework where the corresponding SIDIS structure function factorizes in a simple product of the transversity PDF and the IFF, while for the single hadron case the transversity is convoluted with the Collins function via an integral involving their transverse momentum dependences. The same is true for the structure functions containing the IFF and the Collins FF, respectively, in the e þ e − annihilation cross section. Moreover, in the collinear framework the same combination of transversity PDF and IFF can be explored also in proton-proton collisions leading to the semi-inclusive production of dihadron pairs [16,17], while this possibility is in principle precluded for the Collins effect due to factorization breaking contributions. Finally, the evolution equations connecting the IFF at different scales of the various processes have a simple standard form [18], while the evolution of a transverse-momentum dependent PDF is more complicated and depends on non perturbative parameters [19].
A major experimental effort to measure the various azimuthal asymmetries involved in extracting the transversity PDF using the dihadron way has been made by several collaborations, such as HERMES [20], COMPASS [21,22], and BELLE [23,24]. The IFFs from e þ e − measurements at BELLE were fitted in Refs. [15,25]. In turn these were used in Refs. [4,25,26] to successfully extract the transversity PDF using HERMES and COMPASS data. Recently, the STAR collaboration released also dihadron data for azimuthal asymmetries in proton-proton collisions with a transversely polarized proton [27,28] which can be included in an attempt of extracting the transversity PDF from a global fit [29].
Recently, systematic model calculations of both FFs and DiFFs for unpolarized hadrons have been performed within the extended quark-jet model, which for the first time provides a self-consistent description for the hadronization of a quark with an arbitrary polarization [30][31][32][33]. The two DiFFs, H ∢ 1 and H ⊥ 1 , describing the correlations between the relative and the total transverse moment of the hadron pair with the transverse polarization of the quark, respectively, were studied in Ref. [33]. There, it was observed that the integrated IFF built from the DiFFs entering the unintegrated SIDIS cross section is different from the one that is built from the corresponding unintegrated cross section for e þ e − annihilation derived in [13]. In particular, in SIDIS the integrated IFF contains both the zeroth Fourier cosine moment of the fully unintegrated H ∢ 1 , along with the first Fourier cosine moment of H ⊥ 1 . This admixture of H ⊥ 1 did not appear in the original derivation in Ref. [11] but was later included in Ref. [34]. On the other hand, the integrated H ∢ 1 in e þ e − annihilation in Ref. [13] contains only the zeroth Fourier cosine moment of the unintegrated H ∢ 1 . The model estimates of these two definitions of IFFs in Ref. [33] produced almost a factor of two discrepancy between them.
Another prediction of Ref. [13] concerned a particular azimuthal modulation that provides access to the first Fourier cosine moment of the quark helicity dependent DiFF G ⊥ 1 . However, the recent preliminary results from the BELLE collaboration showed no signal for this modulation within the experimental uncertainties [35,36]. The recent COMPASS studies [37] also yielded no significant signal for SIDIS. Even though the model calculations of Ref. [32] suggest that the integrated G ⊥ 1 appearing in Ref. [13] is naturally smaller in magnitude than the H ∢ 1 , this was still a surprise given the precision achieved in the BELLE analysis.
In this work, we rederive the unintegrated cross section for the semi-inclusive production of two back-to-back hadron pairs in e þ e − annihilation, first performed in Ref. [13]. We then recalculate the azimuthal asymmetries used for extracting the IFFs and the helicity dependent DiFF in order to resolve the above discrepancies.
This paper is organized in the following way. In the next section we briefly review the formalism for DiFFs. In Sec. III, we describe the kinematics of two hadron pair production in e þ e − annihilation and rederive the corresponding cross section. In Sec. IV, we rederive both azimuthal asymmetries involving H ∢ 1 and G ⊥ 1 . We present our conclusions in Sec. V.

II. FIELD-THEORETICAL DEFINITIONS OF THE DIFFS
The fragmentation of a quark q of an arbitrary polarization s into two unpolarized hadrons h 1 , h 2 is fully described at the leading twist approximation by four DiFFs, see Refs. [9][10][11]13,34]. The relevant kinematics is described by the momentum k and mass m of the quark q, and the corresponding momenta P 1 , P 2 and masses M 1 , M 2 of the h 1 , h 2 pair. In the definitions of the DiFFs, the momenta P 1 and P 2 of the individual hadrons are replaced by their total, P ≡ P h , and relative, R, momenta with P 2 h ¼ M 2 h the squared invariant mass of the pair. Theẑ axis is defined along the spatial component of the total momentum P h and the components of three-vectors perpendicular to theẑ direction are denoted by subscript T , as schematically shown in Fig. 1.
The light-cone momentum fractions of the hadrons are defined as the ratios of the plus components 1 of their four vectors to the quark momentum, z i ¼ P þ i =k þ . The following light-cone momentum fractions are used in the definitions of the DiFFs The two-hadron fragmentation of a quark is described by a quark-quark correlator [9,11,13,38] Δ ij ðk; P h ; RÞ which, for the case of unpolarized hadron pair and at the leading twist approximation, is parametrized via four DiFFs where D 1 is the unpolarized DiFF, G ⊥ 1 is the helicity dependent DiFF, H ∢ 1 is the IFF, and H ⊥ 1 is the analogue of the Collins function for the dihadron case. The lightlike vectors n − and n þ are defined as for any 4-vector a, namely a AE ¼ a · n ∓ , and n þ n − ¼ 1, n 2 þ ¼ n 2 − ¼ 0. All four DiFFs are functions of z; ξ; jk T j; jR T j, and k T · R T ¼ jk T jjR T j cosðφ k − φ R Þ, where φ R and φ k denote the azimuthal angles of the vectors R T and k T . Thus, the DiFFs only depend on the cosine of the difference of the azimuthal angles φ k − φ R , that we denote as φ KR . The DiFFs can be further expanded in an infinite series of Fourier moments with respect to angle φ KR , as done in Ref. [33] (see also Ref. [39] for an alternative expansion). It is clear, that all the sine terms vanish, as the DiFFs are even functions of φ KR .
For D 1 we have and similarly for the other DiFFs. The invariant mass of the hadron pair M h is used to replace the magnitude of R T These Fourier decompositions will prove valuable when examining the azimuthal dependence of various structure functions of the e þ e − cross section which we rederive in the next section.

III. THE e + e − CROSS SECTION
In this section we rederive the e þ e − → h 1 h 2 þh 1h2 þ X cross section at the leading twist approximation, following the framework set out in the original work of Boer et al. [8,13,40]. First, we briefly lay out the kinematics in the next subsection, followed by the evaluation of the cross section itself in the subsequent subsection.

A. Kinematics
A schematic depiction of the kinematic setup is shown in Fig. 2. Here, the electron with momentum l annihilates with FIG. 1. The dihadron fragmentation coordinate system, where theẑ axis is taken along the total 3-momentum of the two hadrons, P. The components of 3-momenta perpendicular toẑ axis are denoted with a subscript T .
FIG. 2. The kinematics of e þ e − annihilation. 1 The light-cone components of a 4-vector a are defined as a positron of momentum l 0 , creating a quark-antiquark pair. The time like momentum of the intermediate boson in this hard process is denoted as q ¼ l þ l 0 and we define q 2 ¼ Q 2 . In this work we use Q as the hard scale and will ignore all the contributions of order 1=Q. The quark and antiquark hadronize, producing two back-to-back jets. We choose a hadron pair h 1 , h 2 with momenta P 1 , P 2 and masses M 1 , M 2 from one of the jets. From the other jet, we choose the second hadron pairh 1 ,h 2 with momentaP 1 ,P 2 and massesM 1 ,M 2 . Here again we define the total and relative transverse momenta for each pair, as done in Eqs. (1), (2), and denote the corresponding momenta for theh 1 ,h 2 pair asP h andR. In the "leading hadron approximation," where we assume that a significant fraction of the energy in each jet is carried by the two pairs, we can write P h ·P h ∼ Q 2 . Then we decompose the momenta P h ,P h and q in light-cone coordinates in a frame where P hT ¼ 0 andP hT ¼ 0, to identify the corresponding dominant terms where and −q 2 We can project the components of 4-vectors transverse to n AE using the tensors where g μν is the metric tensor.
The coordinate system in Fig. 2 is defined by taking theẑ axis opposite to the 3-momentumP h , while the components of the vectors perpendicular toẑ are denoted with a subscript ⊥ in a frame where q ⊥ ¼ 0. It can be easily shown, that P h⊥ ¼ −zq T , up to negligible correction of order Q 2 T =Q 2 ≪ 1. We can then define the two orthogonal unit vectors in ⊥ direction where the following convention is used ϵ 0123 ¼ þ1.
To keep consistency, we will define all the azimuthal angles with respect to the lepton frame. Then, we can parametrize these two vectors using the azimuthal angle ϕ 1 ofĥĥ so that the azimuthal angle ofĝ is simply The lepton plane in Fig. 2 is spanned by theẑ axis and the transverse component l ⊥ of the electron momentum l. The unit vectorl ⊥ can be parametrized using the lepton plane angle φ L in the laboratory frame. However, all the following results are independent of the orientation of the scattering plane with respect to the laboratory frame, hence the φ L dependence will be ignored. Here we can also define the associated normalized 4-vector Similar to the light-cone frame, we can now define a set of orthogonal normalized 4-vectorŝ where the spacelike vectorv is denoted asẑ in Refs. [8,13,40]. Here we changed the notation to avoid any possible confusion with the notation of theẑ axis. The orthogonal projections of the 4-vectors can be again achieved using the tensors The two perpendicular projection tensors can be related In this work we neglect all terms of order Q T =Q, M h =Q, M h =Q. Thus we also neglect the differences between the T and ⊥ components of vectors.

B. Cross section
The cross section for this process is given by the convolution of leptonic and hadronic tensors where and BðyÞ ¼ yð1 − yÞ; ð30Þ The last equality holds in the center-of-mass frame, where θ 2 is the angle between the 3-momentum of the electron l and theẑ axis.
The hadronic tensor is defined as 4 δðq − P X − P h −P h Þ × h0jJ μ ð0ÞjP X ; P h ; R;P h ;RihP X ; P h ; R;P h ;RjJ ν ð0Þj0i: ð33Þ Using the parton picture, we can decompose the hadronic tensor in terms of the quark-quark correlators Δ andΔ for the production of the two hadron pairs in the fragmentation of the quark and the antiquark where a denotes the flavor of the fragmenting quark and the prefactor is the number of active colors N C ¼ 3.
Following the transformation of the phase space factor detailed in Ref. [13], the cross section expression can be written as Up until this point we have followed the same formalism and definitions as in Ref. [13]. The next step is to evaluate the trace in Eq. (34) and contract the resulting expression for the hadronic tensor with the leptonic tensor in Eq. (28). The resulting expression follows where the convolution F is defined as There are several important differences between the expression in Eq. (36) and the original expression in Eq. (19) of Ref. [13], apart from the different mass normalization. First, the terms multiplyingĝ are multiplied by a factor of −1 in our expression. Second, the factor AðyÞ in front of the G ⊥a 1Ḡ ⊥ā 1 terms is also multiplied by a factor of −1. Lastly, the dependence on angle φ L vanishes altogether, as in this work all the azimuthal angles are defined with respect to the lepton plane.
These differences allow us to rewrite the cross section in a much more compact form We obtain the cross section in collinear kinematics by integrating upon d 2 q T . This integration trivially breaks up the convolution between k T andk T in Eq. (37). In the last line, we have the product of two terms of the following form Z dφ k sinðφ KR ÞG ⊥a 1 ðz; ξ; jk T j; jR T j; cosðφ KR ÞÞ ¼ 0; where and similarly for the barred functions. Following Ref. [12], we can expand the above DiFFs in the relative partial waves of the hadron pair system. In the centerof-mass (c.m.) frame of the pair, we can change the ξ dependence to ζ ¼ 2ξ − 1 ¼ a þ b cos θ, where a, b are functions only of M 2 h and θ is the angle between the direction of the back-to-back emission in the c.m. frame and the direction of P h in the target rest frame. The Jacobian of the transformation is dξ ¼ jRj=M h d cos θ. Then, we have If we insert these expansions in Eq. (40) where R T ¼ R sin θ (and similarly forR T ), dσ 0 is the unpolarized cross section, and A is the so-called Artru-Collins asymmetry. The above expression is identical (up to a numerical factor) to the one used in Ref. [15] to extract the IFF from the BELLE experimental data for the Artru-Collins asymmetry [23]. The same IFF occurs also in the SIDIS cross section for the semi-inclusive production of hadron pairs off transversely polarized targets [12], and it is used to extract the transversity distribution from a suitable singlespin asymmetry [4,25,26]. Without expanding the DiFFs in relative partial waves and by directly computing the cosðφ R þ φRÞ moment of the cross section in Eq. (40), the resulting Artru-Collins asymmetry is also formally identical to that in Eq. (23) of Ref. [13] (see next section). The crucial difference is in the definition of Eq. (42), namely in how the integrated IFF entering the asymmetry is built in terms of unintegrated DiFFs. Starting from the correct cross section of Eq. (36), the expression in Eq. (42) (multiplied by jR T j) is now consistent with the definition of IFF entering the azimuthal asymmetry in the SIDIS cross section [33] (see also Ref. [34]). The same consistency could not be achieved from the cross section in Eq. (19) of Ref. [13]. Thus, the discrepancy is indeed resolved.

IV. THE AZIMUTHAL ASYMMETRIES
In this section, we will review and discuss the azimuthal asymmetries that allow to extract the IFF and the helicity dependent DiFF from the cross section listed in Eq. (38). For this purpose, we define the average of an arbitrary function I as We first calculate the integral of the unweighted cross section, that appears as denominator in all of the azimuthal asymmetries. Following the same steps leading to Eq. (40), we have

A. Artru-Collins asymmetry
In Ref. [13], the Artru-Collins asymmetry is defined as Following the same steps leading to Eq. (40), we have where which is identical to Eq. (23) of Ref. [13], but now H ∢a 1 ðz; M 2 h Þ is given by Eq. (51) consistently with the definition entering the azimuthal asymmetry in the SIDIS cross section [33] (and similarly forH ∢ā 1 ðz;M 2 h Þ).

B. The asymmetry for the helicity dependent DiFF
Another important consequence of the new expression for the cross section in Eq. (38) is that the so-called longitudinal jet handedness azimuthal asymmetry, suggested in Ref. [13] to address the helicity dependent DiFF, identically vanishes. This asymmetry is defined as The contributions to hcosð2ðφ R − φRÞi from terms in Eq. (38) involving BðyÞ vanish, which is easy to check using similar steps to those used in the derivations of Eq. (40), where we quickly end up with an expression multiplied by The only remaining contribution is by the last term in Eq. (38), which we can again transform to a much simpler form by redefining φ k → φ KR , φ¯k → φKR after integrating upon dq T : Thus, the asymmetry of Eq. (53) identically vanishes. In fact, any moment of the cross section that depends only on angles φ R and φR would get no contribution from the terms involving G ⊥ 1 , as can readily be seen from the derivation in Eq. (55) since the integration upon dq T already yields a zero.
It is interesting to investigate if there is a specific moment that allows to single out the helicity dependent DiFF G ⊥ 1 . If we include in the weight information on jq T j, following the same steps as before for example we get ðz; ξ; jk T j; jR T jÞ; ð58Þ are k 2 T -moments of order p of the Fourier cosine moments of order n of the involved DiFFs (and similarly for the barred functions). Note, that this definition of G ⊥ 1 ðz; M 2 h Þ is different than that in Ref. [13]. Therefore, weighing the cross section with a function of φ R , φR and q 2 T is not enough to isolate its contribution coming from the helicity dependent DiFF.
Such new weight has been recently proposed in Ref. [41], that also involves the azimuthal angle φ q ¼ φ 1 þ π of q T to exactly cancel out the contributions from the unpolarized term in the cross section: where G ⊥ 1 ðz; M 2 h Þ is defined in Eq. (59) (and similarly forḠ ⊥ 1 ðz;M 2 h Þ). Finally, it is worth noticing that since hq 2 T cosðφ R − φRÞi ≠ 0 and hcosðφ R − φRÞi ¼ 0, the latter moment can contain terms that survive the integration upon φ q but vanish because of the integration upon the modulus jq T j. If we perform all the integrations indicated in Eq. (46) except for the one upon djq T j, the only surviving contribution is (see Appendix) If hcosðφ R − φRÞi ¼ 0 vanishes because of the integration upon the modulus jq T j, it means that this moment, when considered as a function of q 2 T , must have a node. Indeed, some preliminary measurements from the BELLE collaboration indicate a non vanishing hcosðφ R − φRÞi which could be due to the limited coverage in q 2 T [42]. However, it is not evident which combination of moments of DiFFs in Eq. (62) is responsible for a node in Eq. (61). In principle, both terms could contribute in changing the sign of hcosðφ R − φRÞi because the Fourier cosine moment D is not necessarily a positive definite function.

V. CONCLUSIONS
The DiFFs provide a very rich source of information concerning the hadronization process. Moreover, in recent years they have been used to explore the structure of the nucleon using two-hadron semi-inclusive electroproduction. The information about the DiFFs extracted from the two back-to-back hadron pair semi-inclusive production in e þ e − annihilation plays an absolutely vital role in these studies. The fully unintegrated cross section for this process and the relevant azimuthal asymmetries for accessing the different DiFFs were first derived in Ref. [13].
We recently observed in Ref. [33] that the integrated IFF built from the DiFFs entering the unintegrated SIDIS cross section is apparently different from the one that is built from the corresponding unintegrated cross section for e þ e − annihilation obtained in Ref. [13]. In this work we rederived these quantities following the same kinematic setup of Ref. [13]. In Sec. III B, we found a mistake in the definition of the kinematics that impacts a subset of terms in the cross section having significant implications for the relevant asymmetries. The most important result derived in Sec. IVA is that with the corrected cross section the apparent discrepancy between the definitions of the integrated IFF in terms of unintegrated DiFFs occurring in the SIDIS and e þ e − cross sections is resolved. Although the procedure used in the extraction of the transversity PDF using the dihadron method in Refs. [4,25,26] is formally correct, it is nevertheless important to have a consistent underlying formalism, which has been established here.
The second important result, derived in Sec. IV B, is that azimuthal asymmetry previously proposed for accessing the helicity dependent DiFF G ⊥ 1 actually vanishes. The reason is the complete decoupling of the quark and antiquark transverse momenta in these asymmetries, as a consequence of which the modulations of their respective hadron productions are lost. This naturally explains the absence of the corresponding signal in the recent analysis at BELLE [35,36]. Further, we discussed the azimuthal asymmetry recently proposed in Ref. [41] that allows to access G ⊥ 1 . We have also analyzed another azimuthal asymmetry based on the relative azimuthal orientation of the planes containing the two back-to-back hadron pair momenta. Interestingly, this asymmetry vanishes independently of the various angular integrations, because it displays a node as a function of the size of the imbalance between the transverse momenta of the two back-to-back jets. As a consequence, incomplete integration on the imbalance size would generate a nonvanishing result, as well as including also the imbalance size as an additional weight.
An important next step is to extend these calculations to beyond the leading-twist contributions, both in the kinematic factors and the DiFFs themselves. The need for this is motivated by the upcoming and planned next generation experiments.