Transverse spin effects of sea quarks in unpolarized nucleons

0.27 MB·English

by  Zhun Lu

USM-TH-187
Transverse spin effects of sea quarks in unpolarized nucleons
Zhun Lu,1 Bo-Qiang Ma,2,∗ and Ivan Schmidt1,†
1Departamento de F´ısica, Universidad T´ecnica Federico Santa Mar´ıa, Casilla 110-V, Valpara´ıso, Chile
2School of Physics, Peking University, Beijing 100871, China
We calculate the non-zero Boer-Mulders functions of sea quarks inside the proton in a meson-
baryon fluctuation model. The results show that the transverse spin effects of sea quarks in an
unpolarizednucleonaresizable. UsingtheobtainedantiquarkBoer-Muldersfunctions,weestimate
the cos2φ asymmetries in the unpolarized pp and pD Drell-Yan processes at FNAL E866/NuSea
experiments. The prediction for the cos2φ asymmetries in the unpolarized ppDrell-Yan process at
theBNL Relativistic Heavy Ion Collider (RHIC)is also given.
PACSnumbers: 12.38.Bx,13.85.-t,13.85.Qk,12.39.Ki
7
0
I. INTRODUCTION process. Its advantage, however, is that the spin struc-
0
tureofhadronscanbestudiedwithoutinvokingbeamor
2
The transverse spin phenomena appearing in high en- target polarization. In fact, the cos2φ asymmetries due
n
totheBoer-Muldersfunctionofvalencequarkshavebeen
a ergy scattering processes [1] are among the most inter-
studied theoretically for the unpolarized pp¯ [15, 25, 26]
J esting issues of spin physics. Several kinds of asymme-
andπN [19,27]Drell-Yanprocesses,whichcanbe mea-
0 triesassociatedwithtransversespinhavebeenobserved.
suredinthe future PAX [28]orCOMPASS experiments,
3 One is the single spin asymmetry in the semi-inclusive
deeply inelastic scattering (SIDIS) processes [2, 3, 4, 5], respectively. Measurements ofthe cos2φasymmetries in

1 the unpolarized pN Drell-Yan process can be performed
which requires the target nucleon to be transversely po-
v bytheFNALE866/NuSeacollaborationandattheBNL
larized. Another is the cos2φ asymmetry in the unpo-
5 Relativistic Heavy Ion Collider (RHIC) [29]. This pro-
larized Drell-Yan process [6, 7], where φ is the angle be-
5
cess is dominated by the annihilation of valence and sea
2 tween the lepton plane and the hadron plane. It has
quarks from the two incident nucleons. In this work we
1 been demonstrated that these asymmetries can be ac-
0 counted for by specific leading-twist k -dependent dis- will investigatethe role ofthe Boer-Muldersfunctions of
T
7 tribution functions, i.e., the Sivers function [8, 9] can the nucleon sea, and reveal their impact on the cos2φ
0 asymmetries in the unpolarized pN Drell-Yan process.
explain [10] the single spin asymmetry in SIDIS pro-
/ We calculate the Boer-Mulders functions of the intrinsic
h cesses,while the Boer-Muldersfunction [11] canaccount
p for [12] the cos2φ asymmetry in unpolarized Drell-Yan u¯ and d¯ inside the proton using a meson-baryon fluc-
- processes. Despite their naive time reversal-odd prop- tuation model. Based on the resulting sea quark Boer-
p
Mulders functions, we estimate the cos2φ asymmetries
e erty [13], calculations [10, 14, 15, 16, 17, 18, 19] have
in unpolarized pp and pD Drell-Yan processes at E866,
h shown that these functions can be non-zero, due to the
: gauge-links [20, 21, 22] appearing in their operator defi- and the cos2φ asymmetries in unpolarized pp Drell-Yan
v
nitions. The factorizationtheorem[23, 24]involvingk - process at RHIC, respectively. The magnitude of the
i T
X dependent distribution functions has been worked out asymmetry is of several percent, and is sensitive to the
choice of the Boer-Mulders functions of valence quarks.
r recently.
a Therefore the investigation of the asymmetry in the un-
Sivers or Boer-Mulders functions correspond to a cor-
polarizedpN processescannotonlyprovideinformation
relationbetweenthetransversespinofthenucleonorthe
on the Boer-Mulders functions of intrinsic sea quarks,
quark and the transversemomentum of the quark inside
but also on those of valence quarks.
the nucleon, respectively. Therefore, the investigation
on them can provide information on the transverse spin
property of the nucleon at the quark level. The experi-
mental study of the Sivers function requires an incident
II. BOER-MULDERS FUNCTIONS OF SEA
nucleon transversely polarized, with the advantage that QUARKS INSIDE THE PROTON
this functioncouples withusualunpolarizeddistribution
or fragmentation functions, and therefore it can be ex-
The intrinsic sea quark content of the nucleon [30] is
tracted more easily from experimental data. In the case
important for understanding its parton structure. Both
of the Boer-Mulders function, it always couples with it-
experimentalmeasurementsandmodelcalculationshave
self oranother chiral-oddfunction in the hardscattering
shown that there is a sizable proportion of intrinsic sea
quarks inside the nucleon, including uu¯, dd¯and ss¯ sea
quarks, which can not be explained by the naive quark
model. A qualitatively successful model that accounts
∗Corresponding author. Electronic address:
[email protected] for a number of significant features of the intrinsic sea
†Correspondingauthor. Electronicaddress: [email protected] quarksisthemeson-baryonfluctuationmodel[31],which
2
is based on the light-cone Fock state expansion of the proton (denoted as h⊥,u¯ and h⊥,d¯). In fact, accord-
1 1
nucleon. The main assumption of this model is that the ing to the meson-baryon fluctuation model, there are
intrinsic sea quarks are multi-connected to the valence pion components in the intermediate state of the pro-
quarksandcanexistoverarelativelylonglifetimewithin ton. On the other hand, as shown in Ref. [19], non-
the nucleon bound state, therefore the intrinsic qq¯pairs zeroBoer-Muldersfunctions of valence quarksinside the
canarrangethemselves together with the valence quarks pion (denoted by h⊥,q) can been calculated using the
1π
of the target nucleon into the most energetically-favored quark-spectator-antiquark model. Therefore, a convolu-
meson-baryonfluctuations. Itis easyto understandthat tion form similar to Eq. (1) can be applied to calculate
the most important fluctuations are most likely to be h⊥,u¯ and h⊥,d¯, as follows:
those closest to the energy shell and thus have mini- 1 1
malinvariantmass. Suchfluctuations are necessarypart 1 dy x
of any quantum-mechanical description of the hadronic h⊥,u¯(x,k2) = f (y)h⊥,u¯ ,k2 (,6)
1 ⊥ Z y π− ∆++π− 1π−(cid:18)y ⊥(cid:19)
boundstateinQCDandhavealsobeenincorporatedinto x
the cloudy bag model andSkyrme solutionto chiralthe- h⊥,d¯(x,k2) = 1 dyf (cid:14) (y)h⊥,d¯ x,k2 . (7)
ories. Therefore,inthe meson-baryonfluctuationmodel, 1 ⊥ Z y π+ nπ+ 1π+(cid:18)y ⊥(cid:19)
x
the nucleon can fluctuate into an intermediate two-body (cid:14)
system of a baryon and a meson which are in turn com- The transverse momentum of the antiquark in the nu-
posite systems of quarks and gluons. For example, the cleonshouldbeasuperpositionofthetransversemomen-
protoncanfluctuateintoanπ+ or∆++π− configuration, tum of the antiquark in the pion and of the transverse
and the d¯or u¯ inside π+ or π− can be viewed as d¯or u¯ momentum of the pion in the nucleon. For simplicity we
distributed in the proton. neglect here the transverse momentum of the pion. As-
Basedonthe meson-baryonfluctuation modelone can sumingthattheprobabilitiesoftheprotonfluctuatingto
alsocalculatethemomentumdistributionoftheintrinsic nπ+ and ∆++π− (denoted as Pp→nπ+ and Pp→∆++π−,
seaquarksinside theproton,fromthe followingconvolu- respectively)areboth 12%, andusing the functions h⊥,q
1π
tion form given in Ref. [19], we obtain the numerical results for
qin(x) = 1 dyf (y)qM x , (1) hof⊥1,fu¯u(nxc,tkioT2n)saanrdehg⊥1iv,d¯e(nx,,kcT2or)r,esshpoowndnining Ftoig.th1e. cThwooiceseotsf
Z y M BM (cid:18)y(cid:19)
x Gaussian(upperpanel)andpower-lawtype(lowerpanel)
(cid:14)
where f (y) is the probability of finding the meson wavefunctions of the meson-baryonsystem, respectively.
M BM The parameters used in the calculation are: α = 0.33
M intheB(cid:14)M statewithlight-conemomentumfractiony, D
GeV, P = 3.5 (as pointed out in Ref. [32], perturba-
andq (x/y)istheprobabilityoffindingaquarkqinthe
M tive QCD predicts a nominal power-law fall off at large
meson M with light-cone momentum fraction x/y. The
k corresponding to P = 3.5), following the choice in
T
function f (y) can be determined from the form
M BM Ref. [31]. The quark mass used in the calculation is
(cid:14) m= 0.3 GeV. For the consistency here we use the same
f (y)= d2k Ψ (y,k )2, (2) valueforthemassofthequarkinsidetheprotonandthe
T BM T
M BM Z | | pion, which is different from that in Ref. [19] where we
(cid:14) use 0.1 GeV for the quark mass of the pion and 0.3 GeV
whereΨ (x,k )isthe light-conetwo-bodywavefunc-
BM T for the quark mass of the proton. This difference on the
tionofthebaryon-mesonsystem,forwhichwewillchoose
value of quark mass in the pion will bring a quantita-
a Gaussian or a power-law behavior,
tive difference on the numerical result, but the result of
theasymmetryandthemainconclusionarequalitatively
Ψ (x,k ) = A exp( M2/8α2 ), (3)
BM T D − D notchanged. Itisinterestingto pointoutthattheBoer-
ΨBM(x,kT) = A′D(1+M2/α2D)−P, (4) Muldersfunctionofu¯ord¯,calculatedfromtheGaussian
type wavefunction, qualitatively agrees with that from
respectively. HereM2 =(M2 +k2)/x+(M2+k2)/(1
M T B T − thepower-lawtypewavefunction. Weshouldnoticethat
x) is the invariant mass squared of the baryon-meson
this result is based on the assumption that the numbers
system, αD sets the characteristic internal momentum of the sea quark pairs uu¯ and dd¯inside the nucleon are
scale, and A or A′ is the normalization constant and
can be fixedDby D Athsewsaemken,oswintcheawtethaesrseumsheoPulpd→bnπe+an=ePxpc→es∆s+o+fπd−d¯hpeareir.
overuu¯pairintheproton,whichisaconsequenceofGot-
dxd2k Ψ (x,k )2 =1. (5)
T BM T tfriedsumrule[33],innextsectionwewillapplyanother
Z | |
set of p→nπ+ and p→∆++π−, which is constrained by
P P
We point out that Eqs. (3) and (4) are boost-invariant the existing parametrizationthat can produce the flavor
light-cone wavefunctions which emphasize multi-parton asymmetry of the sea quarks, and investigate the conse-
configurations of minimal invariant mass. quent asymmetries in the unpolarized pp and pD Drell-
Eq.(1)canbeextendedtocalculatethe Boer-Mulders Yan processes. Boer-Mulders functions for sea quarks
functions of the intrinsic u¯ and d¯antiquarks inside the may be obtained from other models that also allow sea
3
0.2 0.2
0.15 0.15
0.1 0.4 0.1 0.4
0.05 0.05
0 0
k HGeVL k HGeVL
0.2 T 0.2 T
00..22 00..22
00..44 00..44
xx xx
0 0
0.2 0.2
0.15 0.15
0.1 0.4 0.1 0.4
0.05 0.05
0 0
0.2kTHGeVL 0.2kTHGeVL
00..22 00..22
00..44 00..44
xx xx
0 0
FIG. 1: The Boer-Mulders functions of u¯ quark h⊥,u¯(x,k2) (left column) and d¯quark h⊥,d¯(x,k2) (right column) inside the
1 ⊥ 1 ⊥
protonastwo-dimensionaldensities. Theupperpanelandlowerpanelcorrespondtothechoiceofgaussiantypeandpower-low
typelight-cone wave functions for themeson-baryon system, respectively.
quarks in the nucleon, such as the chiral quark and me- polarized pp Drell-Yan process is
soncloudmodels. Inthesemodelstherearealsoπmeson
dσ(pp l¯lX) α2
components in the Fock state expansion of the nucleon. dΩdx d→x d2q = 12eQm2(1+cos2θ) e2qF[f1q(x1,p2⊥)
1 2 ⊥ qX=u,d
fq¯(x ,k2)]+(q q¯), (9)
× 1 2 ⊥ ↔
III. THE cos2φ ASYMMETRIES IN THE where we have used the notation [34]:
UNPOLARIZED pp AND pD DRELL-YAN
PROCESSES [ ]= d2p d2k δ2(p +k q ) , (10)
⊥ ⊥ ⊥ ⊥ ⊥
F ··· Z − ×{···}
Inthissection,wewillcalculatethecos2φasymmetries and the transverse momentum dependence of the cross-
inthe unpolarizedppandpD Drell-Yanprocesses,based section is implicit.
on the Boer-Mulders functions of sea quarks obtained ThecorrespondingcrosssectionfortheunpolarizedpD
above. The generalform of the angular differential cross Drell-Yan process is then
section for the unpolarized Drell-Yan process is
dσ(pD l¯lX) α2
dΩdx d→x d2q = 12eQm2(1+cos2θ)F[(e2uf1u(x1,p2⊥)
1 dσ 3 1 1 2 ⊥
= 1+λcos2θ+µsin2θcosφ +e2fd(x ,p2)) (fu¯(x ,k2)
σdΩ +4πν2λsi+n23θc(cid:0)os2φ . (8) +fd1d¯(1x2,k12⊥)⊥)]+×(q ↔1 q¯)2, ⊥ (11)
(cid:17)
Where we have used the isospin relation fu/D fu/p+
1 ≈ 1
Theφindependentdifferentialcross-sectionfortheun- fu/n =fu+fd.
1 1 1
4
Eqs. (9) and (11) are lowest order parton model ex- Boer-Mulders functions of valence quarks inside the nu-
pressions, where it is extended in order to include trans- cleon. Since there are no experimental measurements on
verse parton momenta. those functions yet, we use some model results for them,
Thecos2φdependentcross-sectionsforunpolarizedpp suchasthosegiveninRef.[17],withinaquarkandspec-
and pD Drell-Yan processes, in terms of Boer-Mulders tator diquark model of the nucleon. We will take two
functions, are different extreme options as follows.
dσ(pp l¯lX) OptionI:Considerthelimitwithonlythespectator
= dαΩ2emdx1sdin→x22θd2cqos⊥2(cid:12)(cid:12)(cid:12)(cid:12)φcos2φ e2 [χ(p ,k )h⊥,q(x ,p2) • thsoc⊥1a,rlSeaprardneisdqeunhat⊥1rtk,hdein=ccol0un.tdreiHdb,uertweiohnwicsehtuomsteehaehn⊥1sB,toSheara-tnMdhu⊥1hl,du⊥1e,=rVs
12Q2 qX=u,d qF ⊥ ⊥ 1 1 ⊥ functions with only spectator scalar diquark and
vector diquark included, respectively. This option
h⊥,q¯(x ,k2)]+(q q¯); (12)
× 1 2 ⊥ ↔ was also applied in Ref. [19]
dσ(pD l¯lX)
→ Option II: Consider the case with both the specta-
dΩdx1dx2d2q⊥(cid:12)(cid:12)cos2φ • tor scalar diquark and vector diquark included. In
(cid:12)
= 1α22eQm2 sin2θcos2(cid:12)φF[χ(p⊥,k⊥)(e2uh⊥1,u(x1,p2⊥) athnidsothpetiyonhahv⊥1e,uop=po23shit⊥1e,Ss+ign21.h⊥1,V andh1⊥,d =h⊥1,V,
+e2dh⊥1,d(x1,p2⊥))(h⊥1,u¯(x2,k2⊥)+h⊥1,d¯(x2,k2⊥)] The scalar and vector components of the proton are [17]
+(q q¯), (13)
↔ 4 M(xM +m)
h⊥,S(x,k2) = α N (1 x)3 ,(17)
respectively, where 1 T 3 s s − [L2(L2+k2)3]
s s T
4 M(2xM +m)
χ(p⊥,k⊥)=(2hˆ·p⊥hˆ·k⊥−p⊥·k⊥)/Mp2, (14) h⊥1,V(x,k2T) = −3αsNv(1−x)3 [L2(L2+k2)3(]1,8)
v v T
here hˆ =qT/QT and Mp is the mass of the proton. respectively,whereNs/v isanormalizationconstant,and
Therefore we can express the cos2φ asymmetry coeffi-
cient defined in Eq. (8) as (λ=1, µ=0) L2 =(1 x)Λ2+xM2 x(1 x)M2. (19)
s/v − s/v− −
2 e2 [χh⊥,qh⊥,q¯]+(q q¯) HereΛisacutoffappearinginthenucleon-quark-diquark
ν = q=u,d qF 1 1 ↔ , (15) vertex and M is the mass of the scalar diquark.
p P e2 [fqfq¯]+(q q¯) d
q=u,d qF 1 1 ↔ With the following kinematical cuts
2 [Pχ(e2h⊥,u+e2h⊥,d)(h⊥,u¯+h⊥,d¯)]+(q q¯)
ν = F u 1 d 1 1 1 ↔ , 4.5GeV<Q<9GeV and Q>11GeV,
D F[(e2uf1⊥,u+e2df1⊥,d)(f1⊥,u¯+f1⊥,d¯)]+(q ↔q¯) 0.1<x1 <1, 0.015<x2 <0.4,
(16)
wecalculatetheQ -dependentcos2φasymmetriesinthe
T
wherewehaveomittedtheargumentsofthedistribution unpolarized pp and pD Drell-Yan processes at E866, as
functions. shown in Fig. 2. In the calculation we adopt the anti-
Eqs. (15) and (16) give the explanation for the cos2φ quark Boer-Mulders functions h⊥,q¯ calculated from the
1
asymmetry observed in the unpolarized Drell-Yan pro- Gaussian type wavefunction of the baryon-meson sys-
cess from the view of Boer-Mulders function. We note tem. One observation from Fig. 2 is that the asymme-
that there are other theoretical approaches that have try calculatedaccordingto optionI is of severalpercent,
beenproposedtointerpretthisasymmetry,suchashigh- while that according to option II is around one percent,
twist [35, 36] and QCD vacuum effects [37]. In Ref. [38] indicating that the size of the asymmetry is sensitive
detailed comparison between the QCD Vacuum effect to the choice of the Boer-Mulders functions of valence
and Boer-Mulers effect has been done. quarks. Nevertheless, our calculation predicts a quite
The E866 Collaboration at FNAL employs a 800 smaller cos2φ asymmetry in the unpolarized pN Drell-
GeV/c proton beam colliding on protons or deuterons Yan process compared to that in the unpolarized πN
fixed targets to measure the unpolarized Drell-Yan pro- Drell-Yan case [6, 19]. Again, comparing the asymme-
cesses pp µ+µ−X and pD µ+µ−X [39]. The main tries calculated from the two different options, we find
→ →
goal of E866 is to study the asymmetry of the nucleon thatν andν havesimilarsizesinoptionI,while those
p D
sea distribution [40, 41]. These experiments can also be of ν and ν in option II can be very different, and even
p D
appliedtostudythecos2φasymmetryoftheleptonpair, have opposite signs. Therefore, the investigation of the
especially the role of the sea quarks contribution to the cos2φ asymmetry in the unpolarized pN Drell-Yan pro-
asymmetry. In the following we will estimate the asym- cess, can not only give information on the Boer-Mulders
metry ν and ν for the E866 experiment. In order to functions of sea quarks, but also about those of valence
p D
calculatetheasymmetrywealsoneedthespecificformof quarks.
5

0.08 0.08
0.07
0.06
0.06
0.04
0.05
0.02
0.04
0.00
0.03
0.02 -0.02
0.01 -0.04
0.00 -0.06
-0.01
-0.08
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
QT GeV QT GeV
FIG.2: Thecos2φasymmetriesνp (thicklines)andνD (thin FIG. 4: Same as Fig. 2, but using Set II for the sea quark
lines) for the E866 experiment, as functions of QT, with distributionswhichproducetheflavorasymmetryofseaquark
kinematical cuts 4.5GeV < Q < 9GeV and Q > 11GeV,
indicated bythe Gottfried sum rule.
0.1 < x1 < 1, 0.015 < x2 < 0.4. The dashed and solid lines
show the asymmetries calculated from options I and II, re-
spectively.
and ∆++π− are 12% (here we denote it as Set I for the
sea quark distributions). From this set we calculate the
flavor asymmetry x(d¯ u¯), as shown by the dashed line
−
in Fig. 3, and compare it with that from the CTEQ6L
0.08
parametrization at Q=0.6 GeV (shown by the solid line
in the same figure). None-zero asymmetry of the sea
(a)
quarkdensitiesisfoundinthiscase,butisquantitatively
)0.06
)
x much lower than the known parametrization. In a fur-
(
r
a ther consideration, we will take into account the flavor
b
ar(x)-u0.04 aansydmPmp→et∆ry++oπf−th=e 1se%a(qdueanrokteadndasaSdeotptIIP),pm→noπti+va=ted15b%y
b theexpectationthatthereshouldbeanexcessofuu¯pair
(d0.02 over dd¯ pair, which is a consequence of Gottfried sum
*
x
rule [33]. This set has also been adopted in Ref. [42].
0.00 We show the asymmetry x(d¯ u¯) from Set II by the
−
dotted line in Fig. 3, and find that it agrees with the
flavor asymmetry from CTEQ6L parametrization fairly
-0.02 well. From the sea quark distributions of Set II, we give
0.2 0.4 0.6 0.8 1.0
x the cos2φasymmetriesinthe unpolarizedpp(thick line)
and pD (thin line) Drell-Yan process at E866, as shown
FIG. 3: The flavor asymmetry of the sea quark distribu-
tions x(d¯(x) u¯(x)). The solid curve shows the result from in Fig. 4. Two options of the valence quark distribution
− are applied in this calculation. Again, the asymmetries
CTEQ6L parametrization at Q = 0.6 GeV. The dashed and
dotted line correspond to the results from Set I and II, re- νp and νD have similar sizes in option I, and those in
spectively. option II are very different. In Fig. 4, a larger negative
ν from option II is predicted compared to that given in
p
Fig 2.
Aswedon’tknowtheexactformofBoer-Muldersfunc-
The main success of the baryon-meson fluctuation tions of valence quarks,we admit that our prediction, in
model is that it can qualitatively describe a variety of a certain aspect, relies on functions given in [17], where
asymmetries of the nucleon sea distributions, such as the Boer-Muldersfunctionsfortheuanddquarksareof
the flavorasymmetry d¯(x) u¯(x), the strange sea asym- the opposite sign. Actually different models or theoreti-
−
metry s(x) s¯(x), and the polarized sea asymmetry cal considerations predict different flavor dependence of
−
∆s(x) ∆s¯(x). Intheprevioussectionwehaveassumed valence Boer-Mulders functions. A calculation based on
that th−e probabilities of the proton fluctuating to nπ+ theMITbagmodelgave[16]h⊥,d = 1h⊥,u,whilethear-
1 2 1
6
to calculate the asymmetries at E866, we estimate the
cos2φ asymmetry ν at RHIC with the kinematical con-
p

straints√s=200GeV, 1<y <2,hereyistherapidity
defined as y = 1lnx1. −The Q -dependent asymmetries
0.06 2 x2 T
for Q = 4 (thin line) and Q = 20 GeV (thick line) are
0.04 shown in Fig. 5. Here we do not averagethe asymmetry
over a range of Q, as we have done for the asymmetry
0.02
atE866,sincewewouldliketochoosetwodifferentfixed
valuesofQtostudytheQ-dependenceoftheasymmetry
0.00
at RHIC. Since the evolution equations of unintegrated
-0.02 partondistributions,especiallythatofthe Boer-Mulders
functions, are unknown, we assume that they scale with
-0.04 Q. Therefore the different asymmetries at different Q
values, as shownin Fig. 5, is not a consequence ofevolu-
-0.06
tion, but just a kinematical effect.
0.0 0.5 1.0 1.5 2.0 2.5 3.0
QT
IV. CONCLUSION
FIG. 5: The QT-dependent cos2φ asymmetries νp at RHIC,
with the kinematical conditions √s=200 GeV, 1<y <2. In summary, we have applied a meson-baryon fluctu-
−
The thin and thick lines show the asymmetry at Q = 4 and ation model to calculate the Boer-Mulders functions of
20 GeV, respectively. The dashed lines show the asymmetry u¯ and d¯inside the proton, and provided a first estimate
from option I and Set I or II (in option one the asymme-
of the transverse spin effect of sea quarks in the unpo-
tries in Set I and II are the same), and solid lines show the
larized nucleon. In the calculation we adopted both the
asymmetriescalculatedfromoptionsIIandSetI,dottedlines
Gaussian and the power-law type wavefunctions for the
show the asymmetries calculated from options II and Set II,
respectively. meson-baryonsystem. Wefoundthesizesofh⊥,u¯ orh⊥,d¯
1 1
fromthesetwotypesofwavefunctionstobequalitatively
similar. FromtheobtainedantiquarkBoer-Muldersfunc-
tions, we analyzed the cos2φ asymmetries in the unpo-
gument based on large-Nc showed that h⊥1,d =h⊥1,u [43] larizedppandpD Drell-YanprocessesatE866. Twosets
modulo 1/Nc corrections. Recent lattice work [44] and of the sea quark distributions are applied in the calcula-
works [45] on impact parameter representation of gener- tion,inwhichSetIisflavorsymmetricandSetIIreflects
alized parton distributions (GPDs) state that h⊥,u and the sea quark flavor asymmetry indicated by Gottfried
1
h⊥,d areofthesamesign. Inthosecasestheasymmetries sum rule. The size of the asymmetries is at most sev-
1
may be different from the prediction given in this work. eralpercent,inthecasethattheBoer-Muldersfunctions
In our recent paper [27], it has been suggested to study of valence quarks are chosen with only spectator scalar-
theflavordependenceofvalenceBoer-Muldersfunctions, diquark contributed. The size of the asymmetries, as
throughπpandπD Drell-Yanprocessesthatcanbe per- well as the relative size between the asymmetries in the
formedinthecomingyearsbyCOMPASSCollaboration. unpolarizedppandpDDrell-Yanprocesses,reliessignifi-
In this paper, We havetakentwo options for the valence cantlyonthechoiceofBoer-Muldersfunctionsofvalence
Boer-Muldersfunctionstostudy howtheasymmetryde- quarks. We also estimated the cos2φ asymmetry in the
pends on them. Nevertheless, the measurement of the unpolarized pp Drell-Yan process at RHIC. This investi-
cos2φ asymmetries in the unpolarized pp and pD pro- gationsuggeststhatunpolarizedpN Drell-Yanprocesses
cessesmayprovideusefulconstraintontheBoer-Mulders on hadron colliders are very helpful for a better under-
function of valence quarks. standingabouttheroleoftransversespinofbothvalence
Another promising test ground of cos2φ asymme- and sea quarks in the unpolarized nucleon.
tries in the unpolarized pN Drell-Yan process is BNL Acknowledgements. We acknowledge helpful dis-
RHIC[29],wheretheDrell-Yanprocesspp l+l−X can cussions with J.C. Peng and L. Zhu. This work is
→
bemeasured. Therearealreadynumerousstudies[12,46, partially supported by National Natural Science Foun-
47, 48] on the single spin and double spin asymmetries dation of China (Nos. 10421003, 10575003, 10505011,
in Drell-Yan processes at RHIC, which require at least 10528510), by the Key Grant Project of Chinese Min-
one incident proton to be polarized. The unpolarized istry of Education (No. 305001), by the Research Fund
experiment is also feasible at RHIC. Taking the same for the Doctoral Program of Higher Education (China),
sets of functions (Set I and II for sea quark distributions by Fondecyt (Chile) under Project No. 3050047 and
andOptionIandIIforvalencequarkdistributions)used No. 1030355.
7
[1] For a review on tranverse polarization phenomena, see [25] G. R.Goldstein, L. Gamberg, hep-ph/0506127.
V. Barone, A. Drago, P.G. Ratcliffe, Phys. Rep. 359, 1 [26] V. Barone, Z. Lu and B.-Q. Ma, to appear in European
(2002). physical journal.
[2] A.Bravaretal.,SMCCollaboration, Nucl.Phys.A666, [27] Z. Lu, B.-Q. Ma and I. Schmidt, Phys. Lett. B 639 494
314 (2000) . (2006).
[3] A.Airapetianetal.,HERMESCollaboration,Phys.Rev. [28] PAX Collaboration, V.Barone, et al., hep-ex/0505054.
Lett.94, 012002 (2005) . [29] G. Bunce, N. Saito, J. Soffer and W. Vogelsang, Ann.
[4] V.Yu.Alexakhinet al., COMPASSCollaboration, Phys. Rev. Nucl.Part. Sci. 50, 525 (2000).
Rev.Lett. 94, 202002 (2005). [30] Forreviews, see,e.g.,S.Kumano,Phys.Rep.303(1998)
[5] M. Diefenthaler, HERMES Collaboration, in Pro- 183;
ceedings of DIS 2005, Madison, Wisconsin (USA), G.T. Garvey, J.C. Peng, Prog. Part. Nucl. Phys. 47
hep-ex/0507013. (2001) 203.
[6] S. Falciano et al. NA10 Collaboration, Z. Phys. C 31, [31] S.J. Brodsky and B.-Q. Ma, Phys. Lett. B 381, 317
513 (1986); M. Guanziroli et al. NA10 Collaboration, Z. (1996).
Phys.C 37, 545 (1988). [32] G.P. Lepage and S.J. Brodsky, Phys. Rev. D 22, 2157
[7] J.S. Conway et al. Phys. Rev.D 39, 92 (1989). (1980).
[8] D. Sivers, Phys. Rev. D 41, 83 (1990); D. Sivers, Phys. [33] K. Gottfried, Phys. Rev.Lett. 18, 1174 (1967).
Rev.D 43, 261 (1991). [34] J.C. Collins and D.E. Soper, Phys. Rev. D 16, 2219
[9] M. Anselmino, M. Boglione, F. Murgia, Phys. Lett. B (1977).
362, 164 (1995). [35] A. Brandenburg, S.J. Brodsky, V.V. Khoze, and D.
[10] S.J.Brodsky,D.S.Hwang,I.Schmidt,Phys.Lett.B530, Mu¨ller, Phys. Rev.Lett. 73, 939 (1994).
99 (2002). [36] K.J.Eskola,P.Hoyer,M.V¨anttinen,andR.Vogt,Phys.
[11] D.Boer, P.J. Mulders, Phys. Rev.D 57, 5780 (1998). Lett. B333, 526 (1994).
[12] D.Boer, Phys. Rev.D 60, 014012 (1999). [37] A.Brandenburg,O.Nachtmann,andE.Mirkes,Z.Phys.
[13] J.C. Collins, Nucl.Phys. B 396, 161 (1993). C 60, 697 (1993).
[14] G.R. Goldstein, L. Gamberg, Talk given at 31st Inter- [38] D. Boer, A. Brandenburg, O. Nachtmann and A. Uter-
national Conference on High Energy Physics (ICHEP mann, Eur. Phys. J. C 40 55 (2005) .
2002), Amsterdam, The Netherlands, 24-31 July 2002, [39] J.C.Webbetal.,(E866Collaboration), hep-ex/0302019.
hep-ph/0209085. [40] E.A.Hawkeret al.,(FNALE866/NuSeaCollaboration),
[15] D. Boer, S.J. Brodsky, D.S. Hwang, Phys. Rev. D 67, Phys. Rev.Lett. 80, 3715 (1998).
054003 (2003). [41] R.S. Towell et al., (FNAL E866/NuSea Collaboration),
[16] F. Yuan,Phys.Lett. B 575, 45 (2003). Phys. Rev.D 64, 052002 (2001).
[17] A.Bacchetta,A.Sch¨afer,J.-J.Yang,Phys.Lett.B578, [42] B.-Q.Ma,I.SchmidtandJ.-J.Yang,Eur.Phys.J.A12,
109 (2004). 353 (2001).
[18] Z. Lu and B.-Q. Ma, Nucl. Phys. A 741, 200 (2004); [43] P.V. Pobylitsa, hep-ph/0301236.
Z. Lu,B.-Q. Ma, Phys.Rev.D 70, 094044 (2004). [44] M. Gockeler et al., Nucl. Phys. Proc. Suppl. 153, 146
[19] Z. Lu,B.-Q. Ma, Phys.Lett. B 615, 200 (2005). (2006).
[20] J.C. Collins, Phys.Lett. B 536, 43 (2002). [45] M. Burkardt, Phys. Rev. D 72, 094020 (2005); B.
[21] X.JiandF.Yuan,Phys.Lett.B543,66(2002);A.V.Be- Pasquini, M. Pincetti and S. Boffi, Phys. Rev. D 72
litsky,X. Ji, F. Yuan,Nucl. Phys.B 656, 165 (2003). 094029 (2005).
[22] D. Boer, P.J. Mulders, and F. Pijlman, Nucl. Phys. B [46] J. Soffer, M. Stratmann and W. Vogelsang, Phys. Rev.
667, 201 (2003). D 65, 114024 (2002).
[23] X. Ji, J.P. Ma and F. Yuan, Phys. Lett. B 597, 299 [47] J.C. Collins et al., Phys.Rev.D 73, 094023 (2006).
(2004). [48] M.Anselmino,U.D’AlesioandF.Murgia,Phys.Rev.D
[24] J.C. Collins, A. Metz, Phys. Rev. Lett. 93, 252001 67, 074010 (2003).
(2004).