arXiv:1012.0054v1 [hep-ph] 30 Nov 2010
Dihadron fragmentation functions and their
relevance for transverse spin studies
A. Courtoy1, A. Bacchetta1,2and M. Radici1
1INFN-Sezione di Pavia, 27100 Pavia, Italy.
2Dipartimento di Fisica Nucleare e Teorica, Universit` a di Pavia, 27100 Pavia, Italy.
into two hadrons plus other undetected hadrons.
fragmentation functions describe the azimuthal asymmetry of the dihadron distribution when
the quark is transversely polarized. They can be used as tools to probe the quark transversity
distribution in the nucleon. Recent studies on unpolarized and polarized dihadron fragmentation
functions are presented, and we discuss their role in giving insights into transverse spin
Dihadron fragmentation functions describe the probability that a quark fragments
In particular, the so-called interference
Our knowledge on the hadron structure is incomplete. We know that the Parton Distribution
Functions (PDFs) describe the one-dimensional structure of hadrons. At leading order, the PDFs
are three: number density, helicity and transversity. However the experimental knowledge on the
latter is rather poor as it is a chiral-odd quantity not accessible through fully inclusive processes.
Semi-inclusive production of two hadrons [1, 2] offers an alternative way to access transversity,
where the chiral-odd partner of transversity is represented by the Dihadron Fragmentation
Functions (DiFF) H∢
1, which relates the transverse spin of the quark to the azimuthal
orientation of the two-hadron plane. Since the transverse momentum of the hard parton is
integrated out, the cross section can be studied in the context of collinear factorization. This
peculiarity is an advantage over the pT-factorization framework, where the cross sections involve
convolutions of the relevant functions instead of simple products.
The transversely polarized DiFF has been computed only in a spectator model . Recently,
the HERMES collaboration has reported measurements of the asymmetry containing the product
1. The COMPASS collaboration has presented analogous preliminary results . The
BELLE collaboration has also presented preliminary measurements of the azimuthal asymmetry
in e+e−annihilation related to the DiFF .
Our present goal is to extract transversity through this channel. To this end, we need an
expression for the chiral-odd DiFF H∢
knowledge of the unpolarized DiFF D1. Hence, as a first step, we present here a parameterization
of the unpolarized DiFF D1 as given from the Monte Carlo generator (MC) of the BELLE
1obtained from e+e−data. This in its turn requires a
2. Two-hadron Inclusive DIS: towards Transversity
We consider the SIDIS process e(l)+N↑(P) → e(l′)+π+(P1)+π−(P2)+X, where the momentum
transfer q = l − l′is space-like, with l,l′, the lepton momenta before and after the scattering.
The two pions coming from the fragmenting quark have momenta P1and P2, respectively, and
invariant mass Mh, which is considered to be much smaller than the hard scale of the process.
We introduce the vectors Ph= P1+ P2 and R = (P1− P2)/2. We describe a 4-vector a as
[a−,a+,ax,ay], i.e. in terms of its light-cone components a±= (a0± a3)/√2 and its transverse
spatial components. We introduce the light-cone fraction z = P−
the nucleon target with mass M. We refer to Refs. [4, 3] for details and kinematics.
The spin asymmetry Asin(φR+φS) sinθ
pion pairs in the angles φSand φR, which represent the azimuthal orientation with respect to the
scattering plane of the target transverse polarization and of the plane containing the pion pair
momenta, respectively. The polar angle θ describes the orientation of P1, in the center-of-mass
frame of the two pions, with respect to the direction of Phin the lab frame. The asymmetry is
h/k−. P is the momentum of
h) is related to an asymmetric modulation of
h) ∝ −|R|
where the x-dependence is given by the PDFs only. The z and Mhdependence are governed by
the DiFFs whose functional form we need to determine. The procedure allowing us to give the
required parameterizations for the DiFFs is detailed in the following sections.
3. The Artru-Collins Asymmetry
We further consider the process e+(l)e−(l′) → (π+π−)jet1(π+π−)jet2X, with (time-like)
momentum transfer q = l + l′. Here, we have two pairs of pions, one originating from a
fragmenting parton and the other one from the related antiparton.1
The differential cross sections also depend on the invariant y = Ph· l/Ph· q which is related,
in the lepton center-of-mass frame, to the angle θ2= arccos(le+ · Ph/(|le+||Ph|)), with le+ the
momentum of the positron, by y = (1 + cosθ2)/2.
The dihadron Fragmentation Functions are involved in the description of the fragmentation
process q → π+π−X, where the quark has momentum k. They are extracted from the correlation
Since we are going to perform the integration over the transverse momentum?kT, the Wilson
lines U can be reduced to unity using a light-cone gauge. The only fragmentation functions
surviving after integration over the azimuthal angle defining the position of the lepton plane
w.r.t. the laboratory plane .
h) = 4π Tr[∆q(z,cosθ,M2
h)= 4π Tr[∆q(z,cosθ,M2
1Variables with an extra “bar” refer to the pair coming from the antiquark.
We perform an expansion in terms of Legendre functions of cosθ (and cosθ) and keep only the
s- and p-wave components of the relative partial waves of the pion pair. By further integrating
upon dcosθ and dcosθ, we isolate only the specific contributions of s and p partial waves to the
The azimuthal Artru-Collins asymmetry A(cosθ2,z, ¯ z,M2
φ¯ R) modulation in the cross section for the process under consideration. It can be written in
terms of DiFF in the following way,
h)  corresponds to a cos(φR+
h, ¯ z,¯
1 + cos2θ2
with |R| =Mh
know the function D1.
1 − 4m2
h. To extract a parameterization of the function H∢
1, we need to
4. Electron-Positron Annihilation: The Unpolarized Cross-Section from BELLE
A model independent parameterization of a function means a huge freedom on the functional
form one will choose. First, one can guess the causes of the shape of the data from physical
arguments. One can get inspired in comparing the model results with the data: here, we take
into account the results of Ref.  —including a critical eye on its shortcomings— in defining
the shape of the MC histograms for the unpolarized cross section.
In the process q → π+π−X, the prominent channels for an invariant mass of the pion pair
ranging 2mπ< Mh? 1.5 GeV are, basically :
• the fragmentation into a ρ resonance decaying into π+π−, responsible for a peak at Mh∼
770 MeV ;
• the fragmentation into a ω resonance decaying into π+π−, responsible for a small peak
at Mh∼ 782 MeV plus the fragmentation into a ω resonance decaying into π+π−π0(π0
unobserved), responsible for a broad peak around Mh∼ 500 MeV ;
• the continuum, i.e. the fragmentation into an “incoherent” π+π−pair, is probably the most
important channel. It is also the most difficult channel to describe with purely model-based
In addition to the channel decomposition, one has to take into account the flavor
decomposition of the cross section. This further decomposition is particularly important if
one wants to be able to use the resulting parametrization in another context, e.g., SIDIS. For
the time being, the MC data provided by the BELLE collaboration are additionally separated
into flavors, i.e., uds contributions and c contributions. The experimental analyses conclude that
the charm contribution to the unpolarized cross section is non-negligible at BELLE’s energy.2
The main considerations one can do, before fitting the data, are the following. First, the
most important contribution from the charm is in the continuum and cannot be neglected.
The determination of a functional form for D1consists then in four parallel steps, i.e. the 2-
dimensional parameterization of the ρ and ω channels and of the continuum for uds and only
of the continuum for the charm. Second, it can be deduced that both the ρ and ω channels
play a role at high z values, while it seems that the ρ is less important at lower z values, as it
can be seen in Fig. 1. On the other hand, the continuum decreases with z, and this behavior is
different for the uds and the c flavors. Also, it can be observed, e.g. in Fig. 1, that the behavior
in Mhchanges from z-bin to z-bin. Those are signs that the dependence on z and Mhcannot
2R. Seidl’s talk at TMD workshop, ECT∗, June 2010.
0.2 0.4 0.6 0.8 1.0
1.2 1.4 1.6 1.8
0.2 0.4 0.6 0.8 1.0
1.2 1.4 1.6 1.8
0.2 0.4 0.6 0.8 1.0
1.2 1.4 1.6 1.8
0.2 0.4 0.6 0.8 1.0
1.2 1.4 1.6 1.8
Figure 1. Number of events N for the unpolarized e+e−annihilation into 2 pions in a jet (plus
anything else) at BELLE, normalized by the integrated luminosity 647.26pb−1. We show only
the resonant channel for the ρ production.The data are represented by the dots. The error on the
data (not plotted here) is assumed to be√N. The dashed lines represent the parameterization,
and the band its errorband. Mhin GeV.
Following Eq. (6), the unpolarized cross section that we are considering here is differential in
h). The θ2-dependence is provided by the BELLE collaboration, and the set
of variables (¯ z,¯
Mh) is integrated out within the experimental bounds. The methodology is as
follows. The unpolarized cross section, differential in Mhand z, is
Q2?1 + cos2θ2? z2Da
with the integration limits to be modified according to the experiment, and where D1= Dss+pp
For both the uds and c flavors, the fitted function takes the form
Q2?1 + cos2θ2?
form appearing in fa
chosen to be in agreement with the condition Mh<< Q. The DiFFs for quarks and antiquarks
are related through the charge conjugation rules described in Ref. .
The determination of a functional form fD1is done by fitting, by means of a χ2goodness-
of-fit test, the MC histograms (4 z-bins and about 300 Mh-bins) for each channel. The best-fit
functional forms lead to interesting results. The most important point is that there is no way
the z and the Mhdependence can be factorized. Moreover, we have realized that no acceptable
zbindz means that we average over the z-dependence each z-bin, and with our functional
D1(z,Mh) = 2Mhz2Da
h). The upper integration limit in Eq. (7) is
fit would be reached with a trivial functional form for the continua. In Fig. 1 we show, as an
example, the MC of the ρ production together with its parametrization. In the depicted case,
the joint χ2/d.o.f. is ∼ 1.25 . We quote the χ2values for the other channels: ω-production
(χ2/d.o.f. ∼ 1.3) ; uds-background (χ2/d.o.f. ∼ 1.4) ; c-background (χ2/d.o.f. ∼ 1.55) .
The propagation of errors gives rise to the 1-σ error band shown in light blue.
5. Towards an extraction of H∢
The DiFF H∢
1(z,Mh) can be extracted from the Artru-Collins asymmetry. The preliminary
data from the BELLE collaboration  will be our starting point. Those data are binned in
(z, ¯ z) and (Mh,¯
Mh). While we have stated in the previous section that no factorization of the
(z,Mh) variables is possible for D1, the data do not allow us to make a similar statement for
The next step consists in the determination of a functional form, e.g.,
1(z,Mh, ¯ z,¯
Even if we expect the H∢
on the interplay of the (z,Mh) variables in the asymmetry. We opt for the simpler functional
form (8) instead. Given the large uncertainties —we sum statistical and systematic errors in
quadrature— on the asymmetry as well as the shape of the (z, ¯ z) dependence, it is easily realized
that more than one functional form could fit the data. We are currently working in improving
our fitting procedure in order to get as much information as we can from the data.
Once we will have determined the z as well as the Mh-dependence of the H∢
have to face the flavor decomposition problem. This step will crucially influence the extraction
of transversity, see Eq. (1).
We conclude by highlighting the importance of DiFFs in the extraction of transversity. We are
eagerly looking forward to analyzing the published data on e+e−from the BELLE collaboration
and to going through the described methodology.
1to arise from an sp-wave interference, we presently have no guidance
1DiFF, we will
We are thankful to the BELLE collaboration for useful information on the data.
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