Table Of ContentMeasurement of azimuthal asymmetries of the
unpolarized cross section at HERMES
Francesca Giordano and Rebecca Lamb†
∗
(on behalf of the HERMES Collaboration)
9
0
INFN&UniversitàdeglistudidiFerrara,[email protected]
0 ∗
†UniversityofIllinois,[email protected]
2
n
a Abstract. Amulti-dimensional(x,y,z,Ph )extractionofcosf handcos2f hazimuthalasymmetries
J ofunpolarizedSemi-InclusiveDeepInela⊥sticScatteringatHERMESisdiscussed.Theuseofdata
6 taken with hydrogen and deuterium targets and the separation of positive and negative hadrons
1 allow to access flavor-dependentinformation about quark intrinsic transverse momenta and spin-
orbitcorrelations.Thisflavorsensitivityallowsforadiscriminationbetweentheoreticalmodelsin
] theHERMESkinematicregime.
x
e Keywords: Semi-InclusiveDIS;Azimuthalasymmetries;Intrinsictransversemomentumandspin.
-
PACS: 13.88.+e,13.60.-r
p
e
h
[ INTRODUCTION
1
v InDeepInelasticScattering (DIS), thestructureofthenucleonisprobedbytheinterac-
8
tionofahighenergy leptonandatarget nucleon,viatheexchangeofavirtualboson.If
3
4 atleastoneoftheproducedhadronsisdetectedincoincidencewiththescatteredlepton,
2 thereaction iscalled Semi-InclusiveDeep InelasticScattering (SIDIS):
.
1
0
l(k)+ N(P) l (k)+h(P ) +X(P ), (1)
9 → ′ ′ h X
0
: wherel (l )istheincident(scattered)beamlepton,N isthetargetnucleon,hthedetected
v ′
hadron and X the target remnant. The quantities in parentheses are the corresponding
i
X
four-momenta.
r
a If thecross sectionis unintegratedoverthehadron momentumcomponenttransverse
to thevirtual photondirection P (Fig. 1), an azimuthal dependence around theoutgo-
h
inghadrondirectionexists[1]: ⊥
ds a 2 g 2
= (1+ ) A(y)F +B(y)F +
dxdydzdPh2 df h xyQ2 2x { UU,T UU,L (2)
⊥ C(y)cosf Fcosf h+B(y)cos2f Fcos2f h ,
h UU h UU }
where f is theazimuthal angleof thehadron plane around the virtual-photondirection
h
(Fig. 1). Here Q2 and y are respectively the negative squared four-momentum and
the fraction of the lepton’s energy transferred to the virtual photon, x is the Bjorken
scalingvariableandzisthevirtualphoton’sfractionalenergytransferredtotheproduced
hadron. For the structure functions F, the subscriptUU denotes Unpolarized beam and
FIGURE 1. Definition of the azimuthal angle f between the scattering plane (grey) and the hadron
h
productionplane(yellow).
Unpolarized target, T (L) indicates the Transverse (Longitudinal) polarization of the
virtualphoton,a istheelectromagneticcouplingconstant,g =2Mx/QwithMthetarget
mass,A(y) (1 y+1/2y2),B(y) (1 y) andC(y) (2 y)√1 y.
≈ − ≈ − ≈ − −
Two mechanisms are expected to give important contributions to the azimuthal de-
pendence of the unpolarized cross section in the hadron transverse momentum range
accessed at HERMES:
- the Cahn effect, a pure kinematic effect, generated by the non-zero intrinsic
transversemotionofquarks,already pointedout byR. Cahn in1978[2];
- the Boer-Mulders effect, introduced more recently (1997) by D. Boer and
P. J. Mulders [3]; this mechanism originates from a coupling between quark transverse
momentumand quark transversespin.
The HERMES experiment
ThefixedtargetHERMESexperimentwasoperatedformorethan10yearsuntil2007
attheelectron-positronstorageringofHERAatDESY.TheHERMESspectrometer[4]
wasaforwardangleinstrumentconsistingoftwosymmetrichalves(top,bottom)above
andbelowthehorizontalplane.Itwas characterized byveryhighefficiency(98 99%)
÷
in electron-hadron separation, provided by a transition radiation detector, a preshower
scintillation counter and an electromagnetic calorimeter. In addition, a dual-radiator
Ring-ImagingCHerenkovdetector(RICH)providedgoodhadron-typeidentificationfor
momentaabove2 GeV/c.
MULTI-DIMENSIONAL UNFOLDING AND RESULTS
Thecross sectionazimuthalmodulationscan bemeasuredviathe cosnf -moments:
h
h i
cosnf = R cosnf hd5s , (3)
h hi R d5s
wheren=1,2 andR d5s standsforR dxdydzdPh2 df hdxdyddz5dsP2 df .
⊥ h h
Theextractionofthesecosinemomentsfromdataischallengin⊥gbecausetheycouple
toanumberofazimuthalmodulationsthatareduetoexperimentalsources,e.g.detector
UU0.2 h+ 0.2 0.2 0.2 HERMES Preliminary
æf)cos(h0.01 HDyedurteorgiuenm 0.01 0.01 0.01
Æ2
-0.1 -0.1 -0.1 -0.1
-0.2 -0.2 -0.2 -0.2
-0.3 -0.3 -0.3 -0.3
10-1 1 0.4 0.6 0.80.2 0.4 0.6 0.8 1 0.2 0.4 0.6
x y z Ph [GeV]
UU0.2 h- 0.2 0.2 0.2 HERMES Preliminary
æf)cos(h0.01 HDyedurteorgiuenm 0.01 0.01 0.01
Æ2
-0.1 -0.1 -0.1 -0.1
-0.2 -0.2 -0.2 -0.2
-0.3 -0.3 -0.3 -0.3
10-1 1 0.4 0.6 0.80.2 0.4 0.6 0.8 1 0.2 0.4 0.6
x y z Ph [GeV]
FIGURE 2. The cosf moments for positive (upper panel) and negative (lower panel) hadrons, ex-
h
tractedfromhydrogen(circles) anddeuterium(squares)data, shownas projectionversusthe kinematic
variablesx,y,zandP .
h
⊥
geometrical acceptance and higher-order QED effects (radiative effects). Moreover,
typically the event sample is binned only in one variable (1-dimensional analysis), and
integrated over the full range of all the other ones, while the structure functions F used
inequation2andtheinstrumentalcontributionsdependonallthekinematicvariablesx,
y, zand P simultaneously.
h
Therefo⊥re, in order to determine the cosine moments corrected for radiative and
detector smearing, an unfolding procedure [6] was used, in which the event sample is
binnedsimultaneouslyinall therelevantvariables(multi-dimensionalanalysis1).
Theunfoldingalgorithmisbasedontherelationbetweentheunknowndistributionof
BornyieldsB(j)and thedistributionofmeasuredyieldsX(i):
n
X(i)= (cid:229) b S(i, j)B(j)+b (i). (4)
j=1
wheren isthetotalnumberofbinsandb (i)isavectorthatcontainstheeventssmeared
b
into the measured sample from outside the acceptance. The Smearing matrix S(i, j)
describestheprobabilitythataneventoriginatingfromtheBornbin j,correspondingto
the original kinematics (free from experimental distortions), is actually observed in the
measuredbini.Boththebackgroundb (i)andthesmearingmatrixS(i, j)aredetermined
byadetailed MonteCarlo simulationoftheexperimentalapparatus.
Assuminganon-singularS(i, j)matrixoneobtains:
n
B(j)= (cid:229) b S−1(j,i)(cid:2)X(i) b (i)(cid:3). (5)
−
i=1
The extraction of cosine moments from the Born yields B(j) can be performed by
linear regression that takes into account the correlations between bins introduced by
the smearing. In this way one pair of moments can be obtained in each kinematic bin
( cosf , cos2f ), whichrepresents resultsthatare fullydifferentialinall variables.
h h
h i h i
1 Foramoredetaileddiscussionabout1-andmulti-dimensionalanalysissee[5].
UU0.2 h+ 0.2 0.2 0.2 HERMES Preliminary
æf)cos(2h0.1 HDyedurteorgiuenm 0.1 0.1 0.1
Æ2 0 0 0 0
-0.1 -0.1 -0.1 -0.1
-0.2 10-1 -0.21 0.4 0.6 0-.08.20.2 0.4 0.6 0.8 -0.21 0.2 0.4 0.6
x y z Ph [GeV]
UU0.2 h- 0.2 0.2 0.2 HERMES Preliminary
æf)cos(2h0.1 HDyedurteorgiuenm 0.1 0.1 0.1
Æ2 0 0 0 0
-0.1 -0.1 -0.1 -0.1
-0.2 10-1 -0.21 0.4 0.6 0-.08.20.2 0.4 0.6 0.8 -0.21 0.2 0.4 0.6
x y z Ph [GeV]
FIGURE 3. The cos2f moments for positive (upper panel) and negative (lower panel) hadrons,
h
extractedfromhydrogen(circles)anddeuterium(squares)data,shownasprojectionversusthekinematic
variablesx,y,zandP .
h
⊥
The dependence of a moment on a single variable can be obtained by projecting the
fully differential result onto the variable under study by weighting the moment in each
bin k with the corresponding unpolarized 4p cross section s 4p (defined by a Monte
k
Carlo), for instancein case of the ℓ-th x-bin: cosnf (x )=(cid:229) s 4p cosnf /(cid:229) s 4p ,
h hi ℓ k k h hik k k
wherek runsoverall then binscorrespondingto x .
b ℓ
The cosf momentsfrom hydrogenand deuteriumdata are shownin figure 2 as pro-
h
jectionsversustherelevantkinematicvariablesforbothhadron charges.Both hydrogen
anddeuteriumdatashowsimilarbehavior:the cosf momentsarefoundtobesizable
h
h i
and negative for positive hadrons. The signal increases with P and with the hadron
h
energy fraction z, except in the very high z range, where the pa⊥rtonic interpretation of
thecross section is no longer valid2. Thesignal for the negativehadrons is significantly
lower,butthedependenceversuszand P exhibitssimilarfeatures.
h
Figure3showsthecos2f momentstha⊥tarefoundtobeslightlynegativeforpositive
h
hadrons, and slightly positive for negative hadrons. Different results for positive and
negativehadronsarenotunexpectedbecausebothexperimentalevidenceandtheoretical
modelspredict oppositeBoer-Mulders contributionsfordifferentlycharged hadrons.
REFERENCES
1. A.Bacchettaetal.,JHEP0702,093(2007).
2. R.N.Cahn,Phys.Lett.B78,269(1978),R.N.Cahn, Phys.Rev.D40,3107(1989).
3. D.BoerandP.J.Mulders, Phys.Rev.D57,5780(1998).
4. K.Ackerstaffetal.(HERMEScollaboration), Nucl.Instr.Meth.A417,230(1998).
5. F. Giordano, Proceedings of Transversity 2008 workshop, May 28-31, 2008, Ferrara, Italy, to be
publishedbyWorldscientific.
6. G.Cowan,StatisticalDataAnalysis,ClarendonPress,Oxford,1998.
2 Thehighestz-binisplottedforcompletenessbutitisnotusedforprojectingmomentsontothesingle
variables.
