Evolution equations for extended dihadron fragmentation functions

We consider dihadron fragmentation functions, describing the fragmentation of a parton in two unpolarized hadrons, and in particular extended dihadron fragmentation functions, explicitly de-pendent on the invariant mass, M h , of the hadron pair. We ﬁrst rederive the known results on M h -integrated functions using Jet Calculus techniques, and then we present the evolution equations for extended dihadron fragmentation functions. Our results are relevant for the analysis of experimental measurements of two-particle-inclusive processes at diﬀerent energies.


I. INTRODUCTION
The fragmentation of partons into hadrons has been studied in detail in semi-inclusive processes with one hadron detected in the final state, such as e + e − annihilation, Semi-Inclusive Deep-Inelastic Scattering (SIDIS) or hadronhadron collisions. Factorization theorems (see, e.g., [1,2,3]) allow to separate perturbatively calculable short-range coefficient functions from long-distance nonperturbative fragmentation functions D i→h 1 (z), describing the "decay" of the hard parton i into an observed hadron h with fractional energy z, provided a hard scale is available. This is the case of the e + e − → hX reaction, where the coefficient function is known at least to O(α s ) [4,5], with α s the running strong coupling constant. The same fragmentation functions, D i→h 1 , occur in the factorized formula for SIDIS at O(α s ) [4,5], combined with the specific process-dependent coefficient functions at the same accuracy. For hadronic collisions, factorization is usually assumed, but has not been proven yet.
When considering semi-inclusive processes with two detected hadrons in the final state, e.g., e + e − → h 1 h 2 X, a new class of fragmentation functions, the so-called Dihadron Fragmentation Functions (DiFF), needs to be introduced to guarantee factorization of all collinear singularities [6]. From this perspective, DiFF are analogous to fracture functions in the space-like regime [7,8]. The DiFF evolution equations have been recently reanalyzed in Ref. [9] and the cross section for e + e − → h 1 h 2 X has been calculated to O(α s ) in Ref. [10]. At O(α 0 s ), the production of two hadrons h 1 , h 2 with fractional energies z 1 , z 2 and belonging to the same jet, is described by a DiFF, D i→h1h2 1 (z 1 , z 2 ), i.e., the fragmentation of a single parton i into the two hadrons. At O(α s ), hadrons produced in the same jet could either come from the fragmentation of a single parton into two hadrons or by the fragmentation of two collinear partons, i and j, into single hadrons. This implies that evolution equations for DiFF contain an inhomogeneous term of the form D i→h1 [10]. All these studies focused on DiFF as functions of the energy fractions z 1 and z 2 , integrated over all the other kinematical variables of the produced hadron pair, including their invariant mass M h . However, the largest amount of experimental information related to DiFF consists of invariant mass spectra of hadron pairs produced in e + e − annihilation [11,12,13], Semi-Inclusive Deep-Inelastic Scattering (SIDIS) [14,15,16] and proton-proton collisions [17,18,19]. In this paper, using the techniques of Jet Calculus [6,20] we deduce the evolution equations for DiFF with an explicit dependence on M h . In analogy with Ref. [21], we address them as extended Dihadron Fragmentation Functions (extDiFF).
DiFF turn out to have important applications in polarization studies, since they can act as spin-analyzers of the fragmenting quark [22]. In particular, the transverse polarization s T of the fragmenting quark can be related to the azimuthal orientation of the plane containing the two hadron momenta P 1 and P 2 , through the mixed product P 1 ×P 2 ·s T . The strength of this relation is described by the DiFF H < ) i→h1h2 1 [23]. In SIDIS with transversely polarized targets, this DiFF appears in combination with the transversity distribution function [24,25,26], thus providing a way to constrain this elusive partonic distribution (for a review on transversity, see Ref. [27]). The HERMES [28] and COMPASS [29] collaborations have reported preliminary measurements of the induced spin asymmetry (at Q 2 ≈ 2.5 GeV 2 ). In the meanwhile, the BELLE collaboration is planning to perform the extraction of the fragmentation function H < ) 1 in e + e − annihilation [30], but at the higher scale √ s ≈ 10 GeV [31]. The invariant-mass dependence of this fragmentation function is unknown and nontrivial, as shown, e.g., by model calculations [25,26,32]. Therefore, the study of the evolution properties of extDiFF is also timely.
The paper is organized as follows. In Sec. II, using Jet Calculus we recover the inhomogeneous evolution equations for DiFF derived in Ref. [10]. In Sec. III, following the same lines we deduce the evolution equations for the corresponding extDiFF. Finally, in Sec. IV we draw our conclusions.

II. INTEGRATED DIHADRON FRAGMENTATION FUNCTIONS
The cross section at order O(α s ) for the e + e − → hX process, where a hadron has momentum P h and energy fraction z = 2P h · q/Q 2 with respect to the center-of-mass (cm) energy Q 2 ≡ q 2 , can be written formally as where D i→h 1 describes the "decay" at leading twist of the hard parton i into the observed hadron h, the sum on i running over all possible partons species i = q,q, g. The process-dependent coefficient function σ i can be calculated and regularized in perturbation theory and it is known at least at O(α s ) [4,5]. The D i→h 1 at O(α s ), i.e., which absorbs all the collinear singularities, can be accurately parametrized [33,34], except for the z → 0 portion of phase space.
The generalization of Eq. (1) to the process e + e − → h 1 h 2 X is not straightforward, if one wants to cover the whole phase space accessible to (h 1 , h 2 ). The differential cross section, again at O(α s ), has been recently calculated in Ref. [10] and reads, with obvious notations, where the DiFF D i→h1h2 1 contains information on the fragmentation, at leading twist and O(α s ), of the hard parton i directly into the observed hadron pair h 1 , h 2 . At order O(α 0 s ), the first term of Eq. (2) would correspond to the back-to-back emission of a parton and an antiparton, eventually fragmenting in the hadrons h 1 and h 2 belonging to two well separated jets. The second term would apply instead to the case where the hadron pair is produced very close in phase space and it is detected inside the same jet while the other jet is inclusively summed over. However, at order O(α s ) a new kind of collinear singularities arises in the partonic cross section; it corresponds to the configuration where each hadron is obtained from the fragmentation of a single parton, the two partons being almost collinear, i.e., with a very small relative transverse momentum r T . These 1/r 2 T singularities cannot be reabsorbed in each D i→h 1 , because they do not correspond to the back-to-back configuration. Hence, they must be reabsorbed in D i→h1h2 1 , thus making the two terms in Eq. (2) indistinguishable [10].
As a consequence, after integrating over r T the DiFF must satisfy the following evolution equation [10] where here, and in the following, a sum over repeated parton indices is understood. The first term in the righthand side represents the usual homogeneous evolution for the DiFF D i→h1h2 1 in complete analogy with the case of single-hadron fragmentation: the probability for the parton i to fragment into the hadrons h 1 , h 2 , is affected by the probability of emitting a parton j with momentum fraction u through the Altarelli-Parisi splitting vertex P ji (u) [35], listed in Eqs. (A.1)-(A.4) of the appendix for convenience. The second term is a new inhomogeneous contribution that corresponds to the probability for the parton i to split in the two partons j and k with momentum fractions u and (1 − u), respectively, each one fragmenting in one of the two observed hadrons. TheP i jk (u) are the Altarelli-Parisi splitting functions without virtual contributions [20], again listed in Eqs. (A.6)-(A.9) of the appendix for convenience. From this point of view, the situation is similar to the DIS case in the target fragmentation region, since the DiFF can be conceived as the time-like version of the fracture functions in the space-like domain [7,8].
In the following, we will make use of Jet Calculus [6,20] and recover the evolution equation (3) within this formalism. We will consider the semi-inclusive production of two hadrons, h 1 and h 2 , belonging to the same jet and neglecting the emission of wide-angle hard partons (and related jets). Therefore, we will not perform a fixed-order calculation of the e + e − → h 1 h 2 X cross section. Rather, we will consider a parton i with a large virtuality ǫQ 2 (0 < ǫ < 1), which fragments in two hadrons h 1 and h 2 inside the same jet. The virtuality can be reconstructed from the invariant mass of the jet by a suitable jet-finding algorithm [36]. The phase-space structure of collinear singularities singled out in fixed-order calculations can be translated in Jet Calculus as a degeneracy in all possible competing mechanisms, which could realize the desidered final state configuration [6,20]. Thus, the cross section is the sum of all production mechanisms, as in Eq. (2).
We use Q 2 as evolution scale, instead of ǫQ 2 . In Leading Logarithmic Approximation (LLA), this substitution induces only subleading corrections and thus is fully justified within this approximation. Moreover, it is convenient to replace this variable with the evolution variable also named the evolution imaginary time [20]. In Eq. (4), µ 2 R is the renormalization scale and β 0 = (11N c −2N f )/(12π) is the one-loop β function with N c , N f , the number of colors and flavors, respectively. In LLA, the running of α s is taken into account at one loop by where Λ 2 QCD is the infrared scale. Hence, the differential evolution length is just Let us define the variable y as with Q 2 0 and Q 2 two arbitrary scales, and introduce the perturbative parton-to-parton time-like evolution function E i j (x, y), which expresses the probability of finding a parton j at the scale Q 2 0 with a momentum fraction x of the parent parton i at the scale Q 2 . The function E i j (x, y) can be shown to satisfy standard evolution equations [6,20] d dy that can be iteratively solved by using the initial condition with δ ji the Kronecker symbol. The E i j (x, y) resums leading logarithms of the type α n s ln n (Q 2 /Q 2 0 ), which show up in the collinear limit of perturbative calculations at the partonic level. As a crosscheck, we can expand Eq. (8) at order O(α s ) with the initial condition (9); neglecting for simplicity the running of α s in Eq. (7), we get Leading logarithmic contributions are therefore automatically accounted for at all orders through the function E. Consider now the fragmentation process i → h 1 h 2 X where h 1 and h 2 are detected within the same jet and with relative transverse momentum R 2 T Q 2 . The corresponding cross-section, normalized to the jet-cross section, can be written as The first term "A" is the convolution of the DiFF at some arbitrary (but still perturbative) factorization scale y 0 with the parton-to-parton evolution function E i j . The second term "B", instead, represents the two separate single-hadron fragmentations, integrated over all possible generic intermediate scales y at which the branching j → kl at partonic level might occur. Integration limits in Eq.(11) are fixed by momentum conservation. Both terms are depicted in Fig. 1. In order to recover Eq. , we get for the "A" term The derivative d/dY of the "B" contribution in Eq. (11) produces two terms, since there is an explicit Y -dependence in the upper integration limit. Using Eq. (9) and the same procedure as before, we get Summing up Eqs. (12) and (13), we get which is exactly Eq. (3), after changing the scaling variable Y back to the more familiar Q 2 . This derivation is useful to adjust the formalism of Jet Calculus from Refs. [6,20] to the calculation of the e + e − → h 1 h 2 X cross section at O(α s ) from Ref. [10]. However, we want to stress again that even if the two expressions are formally identical, they have been derived from rather different approaches. Eq. (3) from Ref. [10] applies to the full phase space for the production of two hadrons. Instead, Eq. (14) gives the evolution of the parton i in a jet within which we identify the two detected hadrons h 1 and h 2 ; thus, it is valid only in the portion of phase space defined by the jet. We recall that if Q 2 is the cm energy of the e + e − annihilation, the event is characterized by a certain number of jets resulting also from a large-angle hard parton emission; this is not included in the LLA used here. Therefore, the evolution scale in Eq. (14) must be intended as the invariant mass of the considered jet, i.e., ǫQ 2 with 0 < ǫ < 1.

III. EXTENDED DIHADRON FRAGMENTATION FUNCTIONS
In the previous section, we considered the inclusive production of two hadrons h 1 and h 2 inside the same jet, summing up all possible values of their invariant mass M h . If the process starts from the hard scale Q 2 of the fragmenting parton (or, equivalently, ǫQ 2 in the case of the jet), there is no intermediate scale that allows to distinguish the two contributions in Eq. (11): the y 0 in the "A" term is arbitrary, and the scale y for the partonic branching in the "B" term is summed over.
However, most of the experimental information on unpolarized DiFF consists of invariant mass spectra of hadron pairs [11,12,13,14,15,16,17,18,19]. In addition, effects related to the partial-wave expansion of DiFF [37] are best explored when the latter explicitly depend on M 2 h . Hence, in this section we will address the evolution equations for DiFF at the fixed scale M 2 h . We will indicate these objects as extended Dihadron Fragmentation Functions (extDiFF), as the time-like analogue of the extended fracture functions that were introduced in Ref. [21] for the space-like SIDIS in the target fragmentation region.
The dependence of the extDiFF upon M 2 h can be easily mapped into R 2 T , the square of the relative transverse momentum of the hadron pair. In fact, it results [37] with M 1 and M 2 the masses of the hadrons h 1 and h 2 , respectively. From the first line of Eq. (11), we get the obvious definition The inhomogeneous "B" term describes the time-like branching of parton i in two partons k and l with transverse relative momentum r T , eventually fragmenting in the two hadrons h 1 and h 2 with transverse relative momentum R T . If the R 2 T scale is fixed and in the perturbative regime, the scale at the partonic branching is no longer arbitrary as in Eq. (11) [6,20]. In fact, if u and (1 − u) are the fractional momenta of partons k and l, the parton virtualities are related by Hence, fixing r T at the partonic level determines in turn the branching scale k 2 i . At the hadronic level, this is not guaranteed and some assumptions must be made. We will suppose that in the fragmentations k → h 1 and l → h 2 the parton virtualities are negligible, i.e., k 2 k ≃ k 2 l ≃ 0, meaning that, once the branching i → kl has occurred, both the perturbative and nonperturbative transverse momenta generated in the fragmentation of the partons k and l are negligible (incidentally, perturbatively generated transverse momenta can be taken into account by using time-like evolution equations depending on transverse momentum [38]). Consequently, the transverse relative momentum r T between k and l should not be substantially altered in the fragmentation, implying Corrections to the above relation affect our final result only at subleading level. Instead, the above assumption is also consistent with the approximation we are working with. Leading logarithms are known to manifest themselves when the transversa momenta of the emitted partons are strongly ordered along the ladder, e.g., Q 2 ≫ r 2 T,1 ≫ . . . ≫ r 2 T,n ≫ Q 2 0 . Hadron pairs with large relative transverse momentum R T are thus produced earlier in the imaginary time Y than hadron pairs with small R T , as appropriate for time-like kinematics.
If R 2 T is thus in the perturbative domain, in analogy with Eq. (4), we can define the variable y T as or, in differential form, where σ i are the same coefficient functions found in the single-hadron inclusive cross section of Eq. (1). In our above derivation, we used the techniques of Jet Calculus [6,20], where the factorization of collinear singularities can be automatically accommodated through the use of the parton-to-parton evolution function E. Exchanges of soft particles are, however, not accounted for.
Eq. (24) can be conveniently diagonalized using a double Mellin transformation. We define and the anomalous dimension With simple algebra manipulations, it is easy to verify that The above results can be extended also to polarized extDiFF, in particular to the only one surviving when the hadron pair is collinear, i.e., H < ) i→h1h2 1 [23,37]. The evolution equations for this function have the same form of the unpolarized case, namely where the splitting functions δP ji for a transversely polarized fragmenting parton are used [39]. The δP ji are listed in Eqs. (A.10)-(A.13) of the appendix. These evolution equations can be conveniently used for phenomenological analyses, since they can connect experimental data taken at different energies.

IV. CONCLUSIONS
We have shown that in leading logarithm approximation the so-called extended Dihadron Fragmentation Functions (extDiFF), describing the inclusive production of two hadrons inside the same jet at fixed invariant mass M h , satisfy evolution equations of the same homogeneous type as in the single-hadron fragmentation case. We stress that the explicit dependence on the scale M 2 h is required to break the degeneracy at O(α s ) between the fragmentation from a single parton or after the branching in two collinear partons. While the first contribution pertains to the nonperturbative regime, in the latter the transverse relative momentum R T of the two hadrons can be traced back to the transverse relative momentum of the two collinear partons after the branching, and, ultimately, to the hard scale of the originating parton. The analysis of the corresponding contribution to extDiFF shows that the dependence on this perturbative scale can be predicted.
In our derivation, we used the techniques of Jet Calculus [6,20]. Factorization of collinear singularities can be automatically accommodated through the use of the parton-to-parton evolution function E. On the basis of the simple result for the evolution equations of extDiFF, we argue that the cross section at order O(α s ) for the inclusive production of the two hadrons h 1 , h 2 , inside the same jet and with invariant mass M h , can be expressed in a factorized form involving the same coefficient functions as in the single-hadron inclusive cross section. A complete proof of this statement would require however the inclusion of soft particle exchanges, which are not accounted for in Jet Calculus approach.
Evolution equations can be extended also to polarized extDiFF, in particular to H < ) i→h1h2 1 [23]. Such fragmentation function can be extracted from e + e − annihilation [30], evolved to the scale of semi-inclusive deep inelastic scattering measurements and allow the extraction of the transversity distribution function [24,25,26].

APPENDIX: SPLITTING FUNCTIONS
The unpolarized leading-order splitting functions P (u) read [35] P qq (u) = C F 2 where N f is the number of flavors, C A = 3, C F = 4/3, T R = 1/2; the " + " prescription is defined as usual by The unpolarized leading-order splitting functionsP are readily obtained from the previous ones by dropping virtual contributions at the endpoint. In our notations, they read: The transversely polarized leading-order splitting functions δP read [39,40] δP qq (u) = C F 2 1 1 − u + Due to angular momentum conservation, there is no mixing between quarks and gluons. The last splitting function can be used to evolve the DiFF for linearly polarized gluons, δĜ < ) [41].