Measurement of the cos(φh) and cos(2φh) AzimuthalMoments of the Deep Inelastic Scattering Cross-section

Rebecca Lamb a and Francesca Giordano b

(on behalf of the HERMES Collaboration)aUniversity of Illinois at Urbana-ChampaignbINFN & Universita degli studi di Ferrara

Abstract

The cross section for hadron production in deep-inelastic lepton scattering contains azimuthalmodulations which can be related to transverse momentum dependent (TMD) distribution andfragmentation functions. The former provide a picture of how the quarks are moving withinnucleons. Specifically, the cos φh and cos 2φh modulations of the unpolarized cross section relatequark spin and quark transverse momentum. These moments have been carefully measured atthe HERMES experiment in a fully differential way, as a function of x, y, z, and Ph⊥ for positiveand negative hadrons produced from hydrogen and deuterium targets. These measurements givenew access to the flavor dependent TMDs via their charge and target dependence.

Key words: semi-inclusive DIS, azimuthal modulation, intrinsic transverse momentum, spin

PACS: 13.88.+e, 13.60.-r

1. Introduction

In semi-inclusive deep-inelastic scattering (SIDIS) the incident lepton with momentumk emits a virtual photon that interacts with the target nucleon to produce a hadron withmomentum Ph and transverse momentum Ph⊥. This process is depicted in Figure 1 whereadditionally the φh angle is defined as the angle between the lepton scattering plane andthe hadron production plane. As the unpolarized semi-inclusive cross section can dependon five independent variables, the measurements in this work were made as a functionof four kinematic variables in addition to the φh angle. The kinematic variables used are

Email addresses: rlamb2@illinois.edu (Rebecca Lamb), giordano@fe.infn.it (Francesca

Giordano).

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doi:10.1016/j.nuclphysa.2009.05.043

the Bjorken scaling variable x and the fractional energy of the virtual photon y, alongwith the semi-inclusive variables describing the fractional energy of the observed hadronz and its transverse momentum Ph⊥.

At leading order and next-to-leading twist, the unpolarized (UU) cross-section reads [1]:

dσ

dx dy dz dP 2h⊥ dφh

={C [R f1D1] + cos 2φhC

[S h⊥

1 H⊥1

]

+ cos φh2M

QC

[T h⊥

1 H⊥1 + U f1D1 + ...

]} (1)

where R, S, T and U are kinematic factors and C denotes a convolution integral. Wedefine the moments 2〈cos nφh〉UU =

∫cos nφhd5σ/

∫d5σ.

Fig. 1. Schematic of SIDIS indicating

the lepton (k, k′) scattering plane and

the hadron (Ph) production plane, with

hadron transverse momentum Ph⊥ and

azimuthal angle φh

The cos φh moment contains the “Cahn”term, proportional to the unpolarized distribu-tion and fragmentation functions, f1D1, a kine-matic effect arising from non-zero transversequark momentum and first discussed by Cahn[2]. Observations of the Cahn term can provideinformation on the quarks’ intrinsic transversemomentum distributions. A second term con-tains the unmeasured Boer-Mulders distribu-tion function h⊥

1 and the measured Collins frag-mentation function H⊥

1 [3]. The sign and mag-nitude of h⊥

1,q reflects the correlation betweenthe spin Sq and orbital angular momentum Lq

of a quark of flavor q within the nucleon.The cos 2φh modulation arises solely from a Boer-Mulders ⊗ Collins term at twist-2,

providing direct access to h⊥1 . However, at twist-4 the Cahn term appears again.

2. Results and Interpretation

Determining the cos φh and cos 2φh moments of the unpolarized SIDIS cross sectionrequires a careful treatment of experimental effects. The finite acceptance of the spec-trometer, detector smearing, and QED radiative effects can cause false moments anddistort the measured distributions via bin migration. To correct for these kinematic-dependent effects, the data were analyzed in a 5-dimensional grid of 4800 bins in thevariables x,y,z,Ph⊥, and φh. A 10-dimensional smearing matrix was populated by MonteCarlo simulation and incorporated into the fitting procedure (see Ref. [4] for further de-tail). The analysis presented here is based on data collected during 2000, 2005, and the2006 electron beam period.

The HERMES results are presented in Figure 2. The cosφh moment on hydrogen isnear zero for negative hadrons and significantly negative for positive hadrons. The cos 2φh

moment is significantly positive for negative hadrons and near zero for positive hadrons.The deuterium target gives strikingly similar results in all cases.

The difference between the 〈cos φh〉UU moments for positive and negative hadrons, if itis the result of the Cahn term, may indicate that the average quark transverse momentumchanges with quark flavor. For the 〈cos 2φh〉UU moments, the similarity between the

R. Lamb, F. Giordano / Nuclear Physics A 827 (2009) 225c–227c226c

x-110 1

UU

⟩)h

φco

s(

⟨2

-0.3

-0.2

-0.1

0

0.1

0.2-h+h

Hydrogen

x-110 1

UU

⟩)hφ

co

s(2

⟨2

-0.2

-0.1

0

0.1

0.2-h+h

Hydrogen

y0.4 0.6 0.8

y0.4 0.6 0.8

z.2 0.4 0.6 0.8 1

z.2 0.4 0.6 0.8 1

[GeV]hP0.2 0.4 0.6

HERMES Preliminary

[GeV]hP0.2 0.4 0.6

HERMES Preliminary

x-110 1

-h+h

Deuterium

x-110 1

-h+h

Deuterium

y0.4 0.6 0.8

y0.4 0.6 0.8

z.2 0.4 0.6 0.8 1

z.2 0.4 0.6 0.8 1

[GeV]hP0.2 0.4 0.6

HERMES Preliminary

[GeV]hP0.2 0.4 0.6

HERMES Preliminary

Fig. 2. HERMES Preliminary 〈cos φh〉UU (top) and 〈cos 2φh〉UU (bottom) for negative (circles) and

positive (squares) hadrons. The error bars represent the statistical error while the bands represent the

systematic error. The left (right) panel show the results from a hydrogen (deuterium) target. The open

points at high z are not included in the projections over the other variables.

targets and the difference between positive and negative hadrons, in combination withexisting data on the Collins fragmentation function [3], seem to indicate that the Boer-Mulders distribution function − and so the correlation between spin and orbital angularmomentum − has the same sign for u and d quarks.

Comparison with theoretical predictions shows that the cos 2φh moments calculatedin the diquark spectator model by Gamberg et al. [5] is in qualitative agreement withthe data, while the calculations of Barone et al. [6] seem to indicate that the twist-4Cahn term is suppressed. The Cahn model predictions of Anselmino et al. [7] for cosφh

give kinematic dependences consistent with the data but larger predicted magnitudes,possibly indicating a significant contribution from h⊥

1 H⊥1 and additional terms.

3. Conclusions

HERMES measurements of the 〈cos φh〉UU and 〈cos 2φh〉UU moments for positive andnegative hadron production from hydrogen and deuterium targets have been presented.The 〈cos 2φh〉UU results indicate that u and d quarks have spin-orbit correlations ofthe same sign, while the 〈cos φh〉UU results may point to different average transversemomenta for u and d.

References

[1] A. Bacchetta et al., JHEP 0702, 093 (2007).[2] R.N. Cahn, Phys. Lett. B 78, 269 (1978); Phys. Rev. D 40, 3107 (1989).[3] A. Airapetian et al., HERMES Collaboration, Phys. Rev. Lett. 94, 012002 (2005);

V. Alexakhin et al., COMPASS Collaboration, Phys. Rev. Lett. 94, 202002 (2005);

R.Seidl et al., Belle Collaboration, Phys. Rev. Lett. 96, 232002 (2006);

M. Anselmino et al., Phys. Rev. D 75, 054032 (2007)[4] F. Giordano, Transversity 2008 proceedings, to be published by World Scientific.[5] L.P. Gamberg et al., Phys. Rev. D 69, 071504 (2003); arXiv:0708.0324 (2007).[6] V. Barone et al., Phys. Rev. D 78, 045022 (2008).[7] M. Anselmino et al., Phys. Rev. D 71, 074006 (2005); Eur. Phys. J. A 31, 373 (2007).

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