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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.

E-mail: aurore.courtoy@pv.infn.it

Abstract.

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

distributions.

Dihadron fragmentation functions describe the probability that a quark fragments

In particular, the so-called interference

1. Introduction

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[3], 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 [4]. Recently,

the HERMES collaboration has reported measurements of the asymmetry containing the product

h1H∢

1[5]. The COMPASS collaboration has presented analogous preliminary results [6]. The

BELLE collaboration has also presented preliminary measurements of the azimuthal asymmetry

in e+e−annihilation related to the DiFF [7].

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

collaboration.

1obtained from e+e−data. This in its turn requires a

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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

UT

(x,y,z,M2

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

expressed as

h/k−. P is the momentum of

h) is related to an asymmetric modulation of

Asin(φR+φS) sinθ

UT

(x,y,z,M2

h) ∝ −|R|

Mh

?

?

qe2

qe2

qhq

qfq

1(x) H∢sp

1(x) Dss+pp

1,q(z,M2

(z,M2

h)

1,q

h)

,(1)

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

function [8]

∆q(z,cosθ,M2

h,φR) =

z|?R|

16Mh

?

d2?kTdk+∆q(k;Ph,R)

???k−=P−

h/z, (2)

where

∆q(k,Ph,R)ij

=

?

X

?

4ξ

(2π)4e+ık·ξ?0|Un+

(−∞,ξ)ψq

i(ξ)|Ph,R;X??Ph,R;,X|¯ψq

j(0)Un+

(0,−∞)|0?. (3)

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 [10].

Dq

1(z,cosθ,M2

h) = 4π Tr[∆q(z,cosθ,M2

h,φR)γ−], (4)

ǫij

TRTj

Mh

H∢q

1(z,cosθ,M2

h)= 4π Tr[∆q(z,cosθ,M2

h,φR)iσi−γ5]. (5)

1Variables with an extra “bar” refer to the pair coming from the antiquark.

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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

respective DiFFs.

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,¯

M2

h) [9] corresponds to a cos(φR+

A(cosθ2,z,M2

h, ¯ z,¯

M2

h)=

sin2θ2

1 + cos2θ2

π2

32

|R||R|

MhMh

?

qe2qDss+pp

qe2

qH∢sp

1,q(z,M2

(z,M2

h)H∢sp

h)Dss+pp

1,q

1,q(z,M2

h)

?

1,q

(z,M2

h)

, (6)

with |R| =Mh

know the function D1.

2

?

1 − 4m2

π/M2

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. [4] —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

physical arguments.

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

be factorized.

2R. Seidl’s talk at TMD workshop, ECT∗, June 2010.

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0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.2 0.4 0.6 0.8 1.0

Mh

1.2 1.4 1.6 1.8

dσ [pb]

0.2<z<0.3

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

0.2 0.4 0.6 0.8 1.0

Mh

1.2 1.4 1.6 1.8

dσ [pb]

0.3<z<0.4

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.2 0.4 0.6 0.8 1.0

Mh

1.2 1.4 1.6 1.8

dσ [pb]

0.4<z<0.55

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.2 0.4 0.6 0.8 1.0

Mh

1.2 1.4 1.6 1.8

dσ [pb]

0.55<z<0.75

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

(cosθ2,z,M2

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

h,z,M2

dσU

2MhdMhdz=

?

a,a

e2

a

1

3

6α2

Q2?1 + cos2θ2? z2Da

1(z,M2

h)

?1

0

dz

?Mmax

2mπ

h

2MhdMhz2Da

1(z,M2

h),

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

1

.

FFU

=

1

3

6α2

Q2?1 + cos2θ2?

?

a

e2

a

?

zbin

dzfa

D1(z,Mh)

?1

0.2

dz

?1.5GeV

2mπ

dMhf¯ a

D1(¯ z,¯

Mh),(7)

where?

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. [4].

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

1(z,M2

h). The upper integration limit in Eq. (7) is

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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 [11]. 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) [11].

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 [7] 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

H∢

1.

The next step consists in the determination of a functional form, e.g.,

1

fH∢

1(z,Mh, ¯ z,¯

Mh)∝f(z)f(¯ z)g(Mh)g(¯Mh).(8)

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

Acknowledgments

We are thankful to the BELLE collaboration for useful information on the data.

References

[1] J. C. Collins, S. F. Heppelmann and G. A. Ladinsky, Nucl. Phys. B 420 (1994) 565 [arXiv:hep-ph/9305309].

[2] R. L. Jaffe, X. m. Jin and J. Tang, Phys. Rev. Lett. 80 (1998) 1166 [arXiv:hep-ph/9709322].

[3] M. Radici, R. Jakob and A. Bianconi, Phys. Rev. D 65 (2002) 074031 [arXiv:hep-ph/0110252].

[4] A. Bacchetta and M. Radici, Phys. Rev. D 74 (2006) 114007 [arXiv:hep-ph/0608037].

[5] A. Airapetian et al. [HERMES Collaboration], JHEP 0806 (2008) 017 [arXiv:0803.2367 [hep-ex]].

[6] A. Martin [COMPASS Collaboration], Czech. J. Phys. 56 (2006) F33 [arXiv:hep-ex/0702002].

[7] A. Vossen, R. Seidl, M. Grosse-Perdekamp, M. Leitgab, A. Ogawa and K. Boyle, arXiv:0912.0353 [hep-ex].

[8] A. Bacchetta and M. Radici, Phys. Rev. D 67 (2003) 094002 [arXiv:hep-ph/0212300].

[9] X. Artru and J. C. Collins, Z. Phys. C 69 (1996) 277 [arXiv:hep-ph/9504220].

[10] D. Boer, R. Jakob and M. Radici, Phys. Rev. D 67 (2003) 094003 [arXiv:hep-ph/0302232].

[11] A. Bacchetta, A. Courtoy and M. Radici, in preparation.