Two hadron production in e+e- annihilation to next-to-leading order accuracy

We discuss the production of two hadrons in e+e- annihilation within the framework of perturbative QCD. The cross section for this process is calculated to next-to-leading order accuracy with a selection of variables that allows the consideration of events where the two hadrons are detected in the same jet. In this configuration we contemplate the possibility that the hadrons come from a double fragmentation of a single parton. The double-fragmentation functions required to describe the transition of a parton to two hadrons are also necessary to completely factorize all collinear singularities. We explicitly show that factorization applies to next-to-leading order in the case of two-hadron production.


Introduction
The production of one hadron in e + e − annihilation has been studied in much detail in perturbative QCD [1]. The corresponding cross section for the process e + e − → γ * (Q) → H(P ) + X is usually expressed as a function of the variable representing the energy fraction carried by the hadron. In this case the cross section can be written as a convolution of the (perturbative computable) partonic cross section σ i and the (nonperturbative) fragmentation functions D H i (x) giving the probability of finding a hadron in the parton with momentum fraction x, as The cross section has been computed to next-to-leading order (NLO) accuracy in [1] and to next-to-next-to-leading order (NNLO) accuracy in [2]. Furthermore, several analyses of the available data have been performed in the last years and, as a result, fragmentation functions for several hadrons have been extracted with very good precision.
Higher order QCD corrections (NLO) to the cross section for the production of two hadrons in e + e − annihilation have been computed in [1] in the particular case when the two hadrons H 1 and H 2 are selected form different parton jets. While a symmetric extension of the one-hadron case to two hadrons would correspond to expressing the differential cross section in terms of the momentum fractions of each hadron defined by the authors in [1] introduced a different set of variables While z in Eq.(4) coincides with z 1 , the momentum fraction of hadron H 1 , the second variable u depends on both the momentum fraction of hadron H 2 and the angle θ 12 between the hadrons observed from the center of mass system as such that u is approximately zero when the angle between the hadrons is small. Therefore, configurations where both hadrons are in the same parton jet corresponds to u ≈ 0. Consequently, by considering events were z and u are not too small one can ensure that the two hadrons are produced from the hadronization of different partons and the cross section can be reduced to the product of the fragmentation functions D H i associated to each hadron [1]. In this way, the possibility of a double fragmentation from a single parton is excluded and the expression for the cross section gets simplified.
In this work we are interested in extending the calculation for the two-hadron cross section in the full phase space, including the configurations were both hadrons are produced collinearly. In order to be able to consider those events we will express the cross section in terms of the momentum fractions in Eq.(3).
With the use of these variables it is possible to contemplate simultaneously two extreme configurations: the first one corresponds to the case when the two hadrons are produced in opposite directions (or at least with a clear angular separation) and therefore belonging to different jets ( Figure 1a). Hadrons in this configuration can only be originated from the fragmentation of different partons. The second one corresponds to the case of both hadrons produced in the same direction, such that they are detected in the same jet ( Figure 1b). In the last case, hadrons could be originated from the fragmentation of two collinear partons or by the double fragmentation of the single parton. The price one has to pay to fully account for the second configuration is the introduction of a new set of non-perturbative phenomenological functions [3,4,5] to describe the possibility of the transition from a single parton to two hadrons. One needs to introduce, then, the double-fragmentation functions DD H 1 H 2 p (x 1 , x 2 ) 2 , as the probability that a parton p fragments into the hadrons H 1 and H 2 with energy fractions x 1 and x 2 . The cross section for the production of two hadrons in e + e − annihilation can therefore be written in the following schematic way where σ ij is the partonic cross section for the production of partons i and j and σ i the cross section for parton i. The cross section is separated in two terms corresponding to the contribution of the mechanisms responsible for two-hadron production: single fragmentation of two partons, and double fragmentation of a single parton.
At leading order, the first term only contributes to the first configuration, since the two partons that undergo hadronization are produced back-to-back. At next to leading order there is one extra parton which could be emitted collinearly to one of the others, giving also origin to hadrons in the second configuration. Therefore, at order α s and beyond, hadrons in the second configuration could be originated from any of the two fragmentation mechanisms, being not possible to separate the contribution of each term in Eq.(6), unless an additional (unphysical) scale is introduced. Only the sum of both contributions has physical sense.
The presence of collinear partons at order α s gives origin to collinear singularities in the cross section, which are manifested in the form of poles in ǫ = (4−N)/2 when dimensional regularization is used. By means of the usual redefinition of the the D H i functions, singularities due to collinear partons that give origin to hadrons in the first configuration can be absorbed. However, there appear singularities corresponding to hadrons belonging to the second configuration, originated from collinear partons emitted in the same direction. Since at lowest order the D H i functions only participate in processes associated with the first configuration, such singularities can not be absorbed in the single fragmentation term. We will show that with the redefinition of the DD H 1 H 2 i functions in the double fragmentation term all singularities are factorized. In this sense, the role played by the DD H 1 H 2 i functions in e + e − annihilation is similar to the one of fracture functions in DIS processes [7,8,9,10,11]. For a formal point of view, it is possible to interpret the double-fragmentation functions as the time-like version of fracture functions.
Double-fragmentation functions DD H 1 H 2 i fulfill sum rules in analogy to the sum rules for the usual fragmentation functions [1]. Momentum conservation requires being Q the initial total momentum, where the right hand side is proportional to the total free momentum available for the production of hadron H 2 . In particular, energy conservation implies [4,5] relating the second moment of the double-fragmentation function to the single one.
2 Two-hadron production in e + e − In order to formalize the convolution products in Eq.(6) we define the partonic energy fractions associated to each fragmentation mechanism. For the single fragmentation term two partons fragment independently with a fraction of the parent parton energy given by with p i the momentum of the parton i = 1, 2. At leading order both variables are fixed to one since no extra gluon radiation is allowed.
The convolution product in the single fragmentation term of Eq. (6) is expressed in terms of a double integral in x 1 and x 2 with integration intervals determined by the kinematical region allowed for the partonic process. This implies The integration zone for the single fragmentation term has to be divided into the two regions A and B indicated in Fig.2. In the case of the double fragmentation term only one parton fragments. We define the partonic variable as usual by with p being the momenta of the fragmenting parton. With this, it is possible to write the second term of Eq.(6) as a single convolution product with integration limits coming from the request z 1 /x + z 2 /x ≤ 1.
Using those conditions we can express Eq. (6) as where to NLO accuracy and K = A, B indicating the integration zone in the single fragmentation term. In Eq. (12) we have considered the case when 1 − z 2 ≥ z 1 . This implies z 1 + z 2 ≤ 1, which corresponds to the kinematical region where the double fragmentation mechanism can also contribute. If z 1 + z 2 > 1 the cross section is reduced only to the second term of Eq. (12).
Some of the partonic cross sections obey symmetry relations that allow to reduce the number of independent quantities to be computed. Due to invariance under charge conjugation To NLO accuracy it is necessary to obtain only three different partonic cross section dσ qq /dx 1 dx 2 , dσ qg /dx 1 dx 2 and dσ gq /dx 1 dx 2 .
At leading order the only non-vanishing terms are 3 dσ q dx where σ 0 = 4πα 2 s 3Q 2 . The partonic cross section at order α s is obtained by evaluating the real and virtual diagrams indicated in Fig.3 and integrating over the phase space of the final partons expressed in terms of x 1 , x 2 , such that dσ ij = dσ R + dσ V . We compute the metric and longitudinal contributions to the partonic cross section obtained, as usual, by replacing the sum over the polarization states of the virtual photon by the corresponding projectors The longitudinal contribution has been calculated projecting in the direction of hadron H 2 . In the following, we will present in detail the results for the metric contribution, since at NLO  Figure 3: Virtual and real diagrams contributing to order α s .
singularities of interest occur only on that projection of the cross section. Using dimensional regularization [12,13] we obtain for the real part with and µ being the dimensional regularization scale. The result for the virtual contribution is In the previous equations we have labeled the quark as parton 1 and the anti-quark as parton 2. The virtual part does not exhibit singularities beyond those already regularized in the form of poles in ǫ. In the real part the divergences appear when the denominators of the function F (x 1 , x 2 ) vanish. These infrared divergences can be regularized by means of the usual + prescription, which can be easily implement by multiplying and dividing by (1−x i ) 1+ǫ and considering the distribution where The range of integration is indicated as a subscript; furthermore, the subtraction point is underlined.
For dσ qq (1) /dx 1 dx 2 the singularities of the function F (x 1 , x 2 ) occur at x 2 = 1 in zone A, and at x 1 = 1 and x 2 = 1 in zone B. Applying the + prescription as indicated, the following expression is reached are the usual Altarelli-Parisi splitting kernels [14] and the functions f M ij K are presented in the Appendix.
The qg partonic cross section dσ qg(1) /dx 1 dx 2 can be obtained from the qq one relabeling the parton indexes by a x 2 → 2 −x 1 −x 2 substitution in the matrix elements. In this case F (x 1 , x 2 ) → F (x 1 , 2 − x 1 − x 2 ) which develops singularities at x 1 = 1 in zone A, and at x 1 + x 2 = 1 in zone B. Proceeding like in the previous case and ignoring terms containing distributions without support in the analyzed zones, we obtain where the functionsP k ji are the real LO Altarelli-Parisi kernels, with the index k labeling the third particle in the vertex i → jk.
In the gq partonic cross section dσ gq(1) /dx 1 dx 2 , x 1 is assigned to the gluon and x 2 to the quark. Performing the substitution x 2 ↔ x 1 in the matrix element used in dσ qg (1) which develops singularities at x 2 = 1 and x 1 + x 2 = 1 in zone A, and at x 2 = 1 in zone B. The singularities in zone A will give origin to two different distributions, one associated to the singularity at x 2 = 1 and another to the singularity at x 1 = 1 − x 2 . The result can be expressed as This result completes the presentation of the partonic cross sections that participate in the single fragmentation term.
For the double fragmentation term, the cross sections for the production of a single parton dσ i /dx are required. These are exactly the same as the ones appearing in one-hadron production [1]. As a cross-check of our calculation we have re-obtained those coefficients by applying the momentum conservation relation in Eq.(7) resulting in

Factorized fragmentation functions
The factorization of the bare fragmentation functions D H i at NLO is done in the MS factorization scheme in the standard way [1]. The expression for the bare functions in terms of the factorized ones at the scale M 2 is where the factorized distributions are labeled by the upper index (NLO).
It is easy to notice that not all the singularities in the partonic cross-section are canceled after the factorization of the fragmentation functions is performed. Singularities belonging to dσ qg(1) /dx 1 dx 2 and dσ gq(1) /dx 1 dx 2 in zone A still remain. They correspond to terms with 1/ǫ poles proportional to δ(x 1 + x 2 − 1), arising from the hadronization of a gluon being emitted collinear to a quark (or anti-quark) that also undergoes hadronization, giving as a final product two hadrons in the same jet. Those singularities clearly cannot be absorbed by the factorization of the D H i functions, since a configuration with two collinear hadrons is not allowed in the single fragmentation term at the lowest order. However, this is exactly the configuration corresponding to the double fragmentation term, indicating that these singularities could be absorbed by the appropriate factorization of the DD by requiring that the bare functions DD H 1 H 2

Conclusion.
In this work, the cross section for the production of two hadrons in e + e − annihilation is calculated to order α s considering events that include the possibility that both hadrons appear in the same jet. For this purpose it is necessary to extend the fragmentation model including a new type of functions, the double-fragmentation functions DD H 1 H 2 i , that describe the transition of a parton into two hadrons. These functions, along with the single-fragmentation function D H i , allow an unified treatment for the description of two-hadron production in e + e − annihilation.
While at leading order the DD H 1 H 2 i functions are necessary to contemplate the possibility of the double fragmentation, at next-to-leading order and beyond, they are required to perform the factorization of divergences that cannot be absorbed in the single-fragmentation functions. As a result, they obey the inhomogeneous evolution equations in Eq.(30), where the two mentioned mechanisms of fragmentation are involved. We showed, for the first time, that introducing the double-fragmentation functions the usual factorization procedure can be enlarged consistently for the production of two hadrons to order α s , reobtaining the evolution equations originally proposed in [4,6].