Polarized Hyperons from pA Scattering in the Gluon Saturation Regime

7 Pages·0.15 MB·English

by  Daniel Boer

Polarized Hyperons from pA Scattering in the Gluon Saturation Regime
Dani¨el Boera and Adrian Dumitrub,c
aDepartment of Physics and Astronomy, Vrije Universiteit, De Boelelaan 1081, NL-1081 HV Amsterdam, The Netherlands
bInstitut fu¨r Theoretische Physik, Universit¨at Frankfurt, Postfach 111932, D-60054 Frankfurt am Main, Germany
cPhysics Department, Brookhaven National Lab, Upton, NY 11973, USA
WestudytheproductionoftransverselypolarizedΛhyperonsinhigh-energycollisionsofprotons
withlargenuclei. Thelargegluondensityofthetargetatsaturationprovidesanintrinsicsemi-hard
scale which should naturally allow for a weak-coupling QCD description of the process in terms of
a convolution of the quark distribution of the proton with the elementary quark-nucleusscattering
cross section (resummed to all twists) and a fragmentation function. In this case of transversely
polarized Λ production we employ a so-called polarizing fragmentation function, which is an odd
functionofthetransversemomentumoftheΛrelativetothefragmentingquark. Duetothiskt-odd
3 nature,theresultingΛpolarizationisessentiallyproportionaltothederivative ofthequark-nucleus
0 cross section with respect to transverse momentum, which peaks near the saturation momentum
0 scale. Suchprocesses might therefore providegeneric signatures for high parton density effectsand
2
for theapproach to the“black-body”(unitarity) limit of hadronic scattering.

n
a
J
It has been known for over 25 years that Λ’s produced in collisions of unpolarized hadrons exhibit polarization
7
perpendiculartotheproductionplane. As ofyet,suchdataarenotavailableforveryhighenergieswhereoneexpects
1
that hadronic cross sections are close to their geometrical values (the “black body limit”). However, the BNL-RHIC

2 colliderwill sooncollide protonsanddeuterons ongoldnuclei atenergiesof 200GeV inthe nucleon-nucleoncenter
∼
v of mass frame; later on, much higher energies will be accessible at the CERN-LHC. In this letter, we demonstrate
0 that the polarization of Λ hyperons produced in the forward region in high-energy collisions of protons and heavy
6
nuclei may generically be a sensitive probe of high-density effects and gluon saturation in the target.
2
The wave function of a hadron (or nucleus) boosted to large rapidity exhibits a large number of gluons at small x,
2
1 which is the fraction of the light-cone momentum carriedby the gluon. The density of gluons is expected to saturate
2 when it becomes, parametrically, of the order of the inverse QCD coupling constant αs [1]. The parton density at
0 saturation is denoted by Q2, the so-called saturation momentum. This provides an intrinsic momentum scale [2]
s
/ which grows with atomic number and with rapidity because more gluons can be radiated in the initial state when
h
p phase space is big. For sufficiently high energies and/orlarge nuclei, the saturationmomentum Qs canbecome much
- larger than ΛQCD, such that weak coupling methods are applicable.
p
Forward Λ production in pA collisions is dominated by high-x quarks from the proton traversing the high gluon
e
density region of the heavy nucleus. The quarks typically experience interactions with momentum transfers of the
h
: orderofthe saturationmomentum. Thus, for largegluondensities inthe target,suchthatthe saturationmomentum
v isintheperturbativeregime,Q > 1GeV,thecoherenceoftheprojectileislost,andthe scatteredquarks(havingan
i s
X average transverse momentum pr∼oportional to Q ) fragment independently [3]. While nonperturbative constituent-
s
quarkanddiquarkscatteringandhadronizationmodels[4]havebeenemployedtounderstandhyperonpolarizationin
r
a collisionsofprotonswithdilutetargets,weexpectthatinthehigh-energylimitthepresenceoftheintrinsicsemi-hard
scale Q should naturally allow for a weak-coupling QCD description of the process. One can thus calculate the
s
cross section for qA scattering in this kinematical domain within pQCD [5], and the deflected, outgoing quark will
subsequently fragment into hadrons, which is described by a fragmentation function.
In order to explain the transverse Λ polarization in unpolarized hadron collisions within such a factorized pQCD
description, it has been suggested that unpolarized quarks can fragment into transversely polarized hadrons, for
instance Λ hyperons. The associated probability [6,7] is described by a so-called polarizing fragmentation function,
sometimes also called Sivers (effect) fragmentation function. Its main properties are that it is an odd function of the
transverse momentum relative to the quark, ~k , and that the Λ polarization is orthogonal to ~k , because of parity
t t
invariance. The polarizing fragmentation function is defined as [7]1:
∆ND (z,~k ) Dˆ (z,~k ) Dˆ (z,~k )=Dˆ (z,~k ) Dˆ (z, ~k ), (1)
h↑/q t ≡ h↑/q t − h↓/q t h↑/q t − h↑/q − t
anddenotesthe differencebetweenthedensitiesDˆ (z,~k )andDˆ (z,~k )ofspin-1/2hadronsh,withlongitudinal
h↑/q t h↓/q t
1AnothercommonlyusednotationforthepolarizingfragmentationfunctionisD1⊥T,butwithaslightlydifferentdefinition[6].
1
momentum fraction z, transverse momentum ~k and transverse polarization or , in a jet originating from the
t
↑ ↓
fragmentation of an unpolarized parton q. Clearly, this k -odd function vanishes when integrated over transverse
t
momentum and also when the transverse momentum and the transverse spin are parallel. In order to set the sign
convention for the Λ polarization we define
P~ (~q ~k )
∆ND (z,~k ) ∆ND (z, ~k ) h· × t , (2)
h↑/q t ≡ h↑/q | t| ~q ~k
t
| × |
where ~q is the momentum of the unpolarized quark that fragments and P~ is the direction of the polarization vector
h
of the hadron h (the direction). Fig. 1 shows the kinematics of the process under consideration and indicates the
↑
direction of positive Λ polarization for each quadrant in the Λ production plane.
P >0
^ Λ
y
^
x
Λ
^
z
p A
FIG.1. Kinematics of the pA→ΛX process. Thedirection of positiveΛpolarization isindicated for each quadrantin the
Λ production plane.
It should be emphasized that such a nonzero probability difference ∆ND (z,~k ) is allowed by both parity and
h↑/q t
time reversalinvariance. Generallyitisexpectedto occurduetofinalstateinteractionsinthefragmentationprocess,
where the direction of the transverse momentum yields an oriented orbital angular momentum compensated by the
transverse spin of the final observed hadron. This polarizing fragmentation function is the analogue of the so-called
Sivers effect for parton distribution functions [8], which yields different probabilities of finding an unpolarized quark
inatransverselypolarizedhadron,dependingonthedirectionsofthetransversespinofthehadronandthetransverse
momentum of the quark. The Sivers effect can lead to single spin asymmetries, for instance in p↑p πX, a process
→
for which such (large) asymmetries have been observed in several experiments.
Recently, such a single spin asymmetry in ep↑ e′πX has been calculated in a one-gluon exchange model [9].
→
Shortly afterwards it was understood [10] as providing a model for the Sivers effect distribution function. A similar
calculationhas recently been performedby Metz [11]for the productionof polarizedspin-1/2hadrons in unpolarized
scattering, which can be viewed as providing a model for the polarizing fragmentation function. Here we will not
employsuchamodelcalculation,butratheruseaparametrizationforthepolarizingfragmentationfunctionsobtained
fromafittodata[7]. However,thesemodelcalculationsdodemonstratethatnonzeroSiverseffectfunctions canarise
in principle.
Duetothek -oddnatureofthepolarizingfragmentationfunctionitisaccompaniedbyadifferentpartofthepartonic
t
crosssection(essentiallythefirstderivativew.r.t.k )comparedtotheordinary,unpolarizedΛfragmentationfunction,
t
whichisk -even. Thecharacteristicsofthe resultingΛ polarizationwillturnouttobe ratherdifferentfrompresently
t
available data for hadronic collisions at moderately high energies and with “dilute” targets. These data show a Λ
polarizationthat increases approximately linearly as a function of the transverse momentum l of the Λ, up to l 1
t t
∼
GeV/c, after which it becomes flat, up to the highest measured l values: l 4 GeV/c. No indication of a decrease
t t
∼
at these high l values has been observed. Furthermore, the polarization increases with the longitudinal momentum
t
fractionξ andis to alargeextent√s independent. These featuresdo notchangewith increasingA [12–14]. The only
A dependence observed is a slight overall suppression of the Λ polarization for large A and higher energies. For Cu
andPbfixedtargets,probedwitha 400GeV/c protonbeam[13,14],the magnitude ofthe polarizationis about30%
lowerthanforlightnuclei. ThiseffectisusuallyattributedtosecondaryΛproductionthroughπ−N interactions[14].
Theslightsuppressionshowsnoevidenceforadependenceonl intheinvestigatedrange0.9<l <2.6GeV/c,albeit
t t
with rather low statistical accuracy. It is clear that this data on heavy nuclei is not in the kinematic region where
2
saturationis expected to play a dominantrole andthe main differences to the results presentedbelow are that in the
saturationregimethe transverseΛpolarizationwill dependonthecollisionenergyandnoplateauregionis expected.
We shall now present our calculation of Λ polarization in the gluon saturation regime, following ref. [7] regarding
the treatment of the polarizing fragmentation functions.
Asmentionedabove,inthecalculationoftheqAcrosssectiononeisdealingwithsmallcouplingifthetargetnucleus
isverydense;however,thewellknownleading-twistpQCDcannotbeusedwhenthedensityofgluonsislarge. Rather,
scatteringamplitudeshavetobe resummedtoallordersinα2 times thedensity. Whenthetargetisprobedatascale
s
< Q , scattering cross sections approachthe geometrical“black body” limit, while for momentum transfer far above
s
∼Q the target appears dilute and cross sections are approximately determined by the known leading-twist pQCD
s
expressions.
At high energies, and in the eikonal approximation, the transverse momentum distribution of quarks is essentially
given by the correlation function of two Wilson lines V running along the light-cone at transverse separation r (in
t
the amplitude and its complex conjugate),
d2q dq+ 1 2
σqA = t δ(q+ p+) tr d2z eiq~t·~zt[V(z ) 1] . (3)
Z (2π)2 − *Nc (cid:12)Z t t − (cid:12) +
(cid:12) (cid:12)
Here, P+ is the large light-cone component of the momentum(cid:12) of the incident proton,(cid:12)and that of the incoming quark
(cid:12) (cid:12)
isp+ =xP+ (q+ fortheoutgoingquark). ThecorrelatorofWilsonlineshastobeevaluatedinthebackgroundfieldof
the target nucleus. A relatively simple closed expression can be obtained [5] in the “Color Glass Condensate” model
ofthe small-xgluondistributionofthe dense target[2]. Inthat model, the small-xgluonsaredescribedas a classical
non-abelian Yang-Mills field arising from a stochastic source of color charge on the light-cone which is averagedover
with a Gaussian distribution. The quark q distribution is then given by [5]
t
dσqA q+ p+ q+ 1
q+ = δ − C(q ) ,
dq+d2q d2b P+ P+ (2π)2 t
t (cid:18) (cid:19)
d2p 1 d2p 1
C(q )= d2r eiq~t·~rt exp 2Q2 t (1 exp(ip~ ~r )) 2exp Q2 t +1 . (4)
t t − s (2π)2p4 − t· t − − s (2π)2p4
Z (cid:26) (cid:20) Z t (cid:21) (cid:20) Z t(cid:21) (cid:27)
This expression is valid to leading order in α (tree level), but to all orders in Q since it resums any number of
s s
scatterings of the impinging quark in the strong field of the nucleus. The saturationmomentum Q , as introduced in
s
eq. (4), is related to χ, the total color charge density squared (per unit area) from the nucleus integrated up to the
rapidity y of the probe (i.e. the projectile quark), by
N2 1
Q2 =4π2α2 c − χ . (5)
s s N
c
In the low-density limit, χ is related to the ordinary leading-twist gluon distribution function of the nucleus, see
for example [15]. From BFKL evolution, Q2 evolves as exp(λy) xλ, with the intercept λ 0.3 [16]. Thus, if
Q2 10 GeV2 at the proton beam rapiditys(i.e. x = 1)∼and for A ∼200 targets [5], then Q ≃3 GeV at x = 0.6,
descr≃easing to Q 2 GeV at x = 0.05; furthermore, assuming Q2 ≃ A1/3 scaling, then at xs=≃0.6, Q drops from
s ≃ s ∼ s
3 GeV to 2 GeV when the atomic number A of the targetdecreasesfrom200 20. It is clear therefore that in order
→
tobesensitivetohigh-densityeffects,experimentallyoneshouldstudyhigh-energypAcollisionsintheforwardregion
(where the polarizationis largestanyway,see below) and with large target nuclei, and then compare to pp collisions.
Below, we shall focus on polarized Λ production in a relatively small rapidity interval in the forward region, and so
take Q as a constant of order 2 3 GeV.
s
−
The integrals overp in eq. (4) are cut off in the infraredby some cutoff Λ, which we assume is of orderΛ . We
t QCD
denote the momentum ofthe producedΛ by~l=z~q+~k, with~k the transversemomentum relative to the fragmenting
quark. Assuming parity conservation in the hadronization process, only the component of~k in the production plane
contributes to the polarization , thereforein orderto simplify the kinematicswe choosek =0 aswasdone in Ref.
Λ y
[7]. For forwardkinematics, q+P q , one then finds zq l k . The polarized cross section is given by
t t t t
≫ ≃ −
dσ q+ dz d2k q+ q+
(l ,ξ)ξ = d f (x,Q2) t ∆ND (z,Q2,~k ) δ x C(q ) (6)
PΛ t dξd2l d2b P+ z2 q/p (2π)2 Λ↑/q t P+ P+ − t
t Z (cid:18) (cid:19)Z Z (cid:18) (cid:19)
dz d2k
= f (x,Q2) t ∆ND (z,Q2,~k )xC(q ) (7)
z2 q/p (2π)2 Λ↑/q t t
Z Z
1
x d2k ξ
= dx f (x,Q2) t ∆ND ,Q2,~k C(q ) , (8)
ξ q/p (2π)2 Λ↑/q x t t
Z Z (cid:18) (cid:19)
ξ
3
where ξ = l /P xz is the longitudinal momentum fraction carried by the Λ. We assume that ∆ND (z,~k ) is
z z ≃ Λ↑/q t
strongly peaked around an average~k0 lying in the production plane, such that [7]
t
d2k ∆ND (z,Q2,~k )F(~k ) ∆ND (z,Q2) F(k0) F( k0) . (9)
t Λ↑/q t t ≃ Λ↑/q t − − t
Z
(cid:2) (cid:3)
Notethatk0 isafunctionofz,seebelow. Alternatively,onecouldconsiderGaussiandistributionsoverk [17],though
t t
the above simplified treatment is sufficient for our purposes.
Considering the unpolarized cross section, we can safely neglect k -smearing from hadronization, which is of order
t
Λ , while the quarks are typically scattered to much larger transverse momenta, namely of order Q . In our
QCD s
numerical results shown below we also include the contribution from anti-quarks and gluons to the unpolarized cross
section,althoughthis is asmallcorrectionforξ > 0.1. For the polarizingfragmentationfunctions, onlycontributions
from u, d, and s quarks (the valence quarks of t∼he Λ) are considered [7]. Thus, we obtain
1dxxf (x,Q2)∆ND ξ,Q2 C x(l k0) C x(l +k0)
ξ q/p Λ↑/q x ξ t− t − ξ t t
(l ,ξ)= . (10)
PΛ t R 1dxxf (x(cid:16),Q2)D(cid:17)h (cid:16)ξ,Q2 C (cid:17)xl (cid:16) (cid:17)i
ξ q/p Λ/q x ξ t
(cid:16) (cid:17) (cid:16) (cid:17)
R
The factorization scale is chosen to be the saturation momentum of the dense nucleus, Q2 =Q2. A parametrization
s
for ∆ND (z) in terms of the unpolarized fragmentationfunction D (z) was given in ref. [7]. It was obtained by
Λ↑/q Λ/q
performing a fit to available pA Λ↑X data (for light nuclei only), where the transverse momentum l was required
t
→
to be largerthan 1GeV/c, inorderto justify the applicationofa factorizedexpressionandofpQCD forthe partonic
cross section. Although doubts have arisen about the applicability of pQCD in the kinematic region covered by the
available data [17], the resulting functions do exhibit reasonable features. Here, we shall employ those functions as
an Ansatz to investigate the dependence of the Λ polarization on the saturation momentum Q , which turns out
s
not to depend on the detailed parameterization of the polarizing fragmentation functions. Rather, it is the k -odd
t
structure(and the factthatit is peakedaroundanaveragenonzerotransversemomentum) thatis responsiblefor the
dependence. Of course, future parameterizations can be easily implemented.
To be explicit, we use
D (z,Q2)
∆ND (z,Q2) N zcq(1 z)dq Λ/q , (11)
Λ↑/q ≡ q − 2
where
N =N = 28.13, N =57.53, c =11.64, d =1.23. (12)
u d s q q
−
The averagetransverse momentum k0 acquired in the fragmentation is parameterized as
t
k0 =0.66z0.37(1 z)0.50GeV/c. (13)
t −
For the unpolarized fragmentation function D in eq. (11) the parametrization of ref. [18] is to be used; strictly
Λ/q
speaking, that parameterization holds for the fragmentation into Λ+Λ¯. However, in the forward region (ξ > 0.1),
oneexpectsPΛ+Λ¯ ≈PΛ. Furthermore,theparameterizationof[18]assumesSU(3)symmetry: DΛ/u =DΛ/d =∼DΛ/s.
However, the polarizing fragmentation functions ∆ND reduce the flavor symmetry to SU(2), since N = N
Λ↑/q u,d 6 s
(N = N was imposed in [7] to reduce the number of fit parameters). But even though ∆ND > ∆ND ,
u d Λ↑/s | Λ↑/u,d|
the overall Λ polarization in the process under consideration is in fact dominated by the valence-like quarks of the
proton, not by the strange quark.
The polarizing fragmentation function describes the probability of an unpolarized quark to fragment into a trans-
versely polarized Λ. Here, no difference is made as to whether the Λ is produced directly or as a secondary particle,
for instance as a decay product of heavier hyperons like the Σ0 or Σ+∗. This second type is usually expected to have
adepolarizingeffect,whichmeansthatthedegreeofpolarizationishigherforthedirectlyproducedΛ’s. Thenumber
of directly produced Λ’s is estimated to be roughly 75 % of the total, such that the depolarizing effect could be on
the order of 30 %. The polarizing fragmentation functions of ref. [7] thus effectively account for the depolarizing
effect from decays, since they were obtained by a fit to data that does not discriminate between direct and decay
contributions either.
4
0 0
−0.02 −0.02
−0.04 −0.04
PΛ −0.06 PΛ −0.06
−0.08 −0.08
ξ = 0.1
Q = 2 GeV ξ = 0.2
−0.1 Qs = 3 GeV −0.1 ξ = 0.3
s ξ = 0.5
−0.12 −0.12
0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7
l (GeV) l (GeV)
t t
FIG.2. Transverse momentum distribution of thetransverse Λ polarization. Left: at fixed longitudinal momentum fraction
ξ=0.5 and varying target saturation scale, Qs=2,3 GeV, respectively. Right: For Qs =2 GeV and various ξ.
A numerical evaluation of eq. (10) is shown in Fig. 2, using the CTEQ5L LO parton distribution functions for the
proton [19]. Generically, one observes that is negative (due to the fact that u and d quarks dominate); it first
Λ
P
increases with transverse momentum, then peaks at l Q , and asymptotically approaches zero again. The fact
t s
≃
that peaks at l Q has its origin in the k -odd nature of the polarizing fragmention function: from eq. (10),
Λ t s t
coPrresponds to th∼e difference of the cross sections taken with “intrinsic” transverse momentum k0 parallel and
aPnΛtiparalleltothequarktransversemomentumq . Sincek0 issmall, isessentiallyproportionaltothtederivativeof
dσqA/d2q , the differentialquark-nucleuscrosssetction,whtichvariesmPoΛstrapidly atq Q (see alsoeq.(17)below).
t t s
∼
Consequently, exhibits a maximum at such transversemomentum. This conclusionis independent of the details
Λ
|P |
of the polarizing fragmentation functions; only the k -odd nature and the fact that they are strongly peaked about
t
an averagek0 matters. In contrast, k -even distribution and fragmentation functions only probe the qA cross section
t t
itself but not its derivative with respect to q .
t
The behavior of (l ) is qualitatively rather different when the quark cross section is taken at leading twist. In
Λ t
P
that case, not only is the magnitude of the polarization larger, but moreover in the forward region peaks about
Λ
small transverse momentum < 1 GeV. This can be understood by noting thatPthe derivative of the qA cross section
at leading twist peaks in the∼infrared, contrary to eqs. (4,17). For a more quantitative evaluation of polarized Λ
production in the “dilute regime” (hadronic collisions far below the unitarity limit, e.g. pp collisions at RHIC) we
refer to Ref. [7].
Tounderstandthebehaviorofeq.(10)inmoredetail,considerfirstlargetransversemomentum,q Q . Here,the
t s
≫
last two terms of eq. (4) can be dropped, since they contribute only via a δ(q ) term. At large transverse momentum
t
the phase factor exp(i~q ~r ) in eq. (4) effectively restricts the integral over d2r to the region r < 1/q 1/Q ; the
t t t t t s
· ≪
first exponential can then be expanded order by order to generate the usual power series in 1/q∼2. The leading and
t
subleading twists are (see also [20])
Q2 4Q2 q Q2
C(q )=2 s 1+ s log t + s . (14)
t q4 π q2 Λ O q2
t (cid:20) t (cid:18) t (cid:19)(cid:21)
This expression is valid to leading logarithmic accuracy. The first term corresponds to the perturbative one-gluon
t-channelexchangecontributiontoqg qg scattering[20]. Toleadingorderink0/l ,thepolarizationgivenineq.(10)
→ t t
thus becomes
1dxxf (x,Q2)∆ND ξ,Q2 x−4 1+ 6ξ2 Q2s loglt k0/l
ξ q/p Λ↑/q x πx2 l2 Λ t t
(l ,ξ)=8 t . (15)
PΛ t R 1dxxf (x,Q2)D (cid:16) ξ,Q2(cid:17) x−4h1+ 4ξ2 Q2s loglti
ξ q/p Λ/q x πx2 l2 Λ
t
R (cid:16) (cid:17) h i
It is known [7] that the polarization (for large l ) is a higher-twist effect, i.e. it is suppressed by powers of the
t
“intrinsic” transverse momentum at hadronization, k0, over the external momentum scale l . Eq. (15) shows that
t t
despite a partialcancellation the first power-suppressedcorrectionto the quark-nucleus cross section (the subleading
terms in the square brackets) enhance at large l , in agreement with the behavior at l > 5 GeV in Fig. 2.
Λ t t
P
∼
5
Regarding the scaling of at the peak, consider the quark-nucleus cross section for q Q Λ. Again, the
Λ t s
P ∼ ≫
last two terms of eq. (4) can be dropped, while in the leading logarithmic approximation the argument of the first
exponential reads
Q2r2 1
s t log + (Q2r2) . (16)
− 4π r Λ O s t
t
The phase factor effectively cuts off the integral at r 1/q 1/Q , and so 1/r Λ is large. We therefore replace
t t s t
∼ ∼
1/r Q in the argument of the above logarithm, since it is slowly varying and formally makes the expression
t s
→
well-behaved at large r . The remaining integral leads to
t
4π2 πq2
C(q ) exp t . (17)
t ≃ Q2log Q /Λ −Q2log Q /Λ
s s (cid:18) s s (cid:19)
This approximation reproduces the behavior of the full expression (4) about q Q reasonably well. Expres-
t s
∼
sions (16,17) are useful only when the cutoff Λ Q , that is, when color neutrality is enforced on distance scales of
s
≪
order 1/Λ 1/Q . If, however, color neutrality in the target nucleus were to occur on distances of order 1/Q [21]
s s
≫
then Λ Q and one would have to go beyond the leading-logarithmic approximation.
s
From∼eq. (17), is given by (to leading order in k0/l )
PΛ t t
(l ,ξ)=4π lt2 ξ1dxxfq/p(x,Q2)∆NDΛ↑/q xξ,Q2 xξ22 exp −Q2sloπgl2tQs/Λxξ22 kt0/lt . (18)
PΛ t Q2slog Qs/ΛR ξ1dxxfq/p(x,Q2)DΛ/(cid:16)q xξ,Q(cid:17)2 exp −(cid:16)Q2sloπgl2tQs/Λxξ22 (cid:17)
R (cid:16) (cid:17) (cid:16) (cid:17)
Thus, at the peak scales approximately with 1/(Q √logQ /Λ), as indeed seen in Fig. 2. The strong dependence
Λ s s
P
on the target gluon density, as parametrized by Q , is rather different from leading-twist perturbation theory.
s
As mentioned above, there is a related k -odd effect in processes with one transversely polarized hadron in the
t
initial state (the Sivers effect), which can lead to asymmetries in p↑p πX [22], for example. Since at RHIC
polarized proton beams are also available (for recent preliminary p↑p →πX data from STAR, see Ref. [23]), one
could investigate the process p↑A πX in the saturation regime. Simi→lar signatures should arise in that process as
for pA Λ↑X pointed out here. →
→
Insummary,wehavestudiedtransverseΛpolarizationinpAcollisionsathighenergiesandwithdensetargets. The
resultingΛpolarizationisquitedifferentfromthatobservedinpAandppcollisionstodate,whichpresumablydidnot
probe the saturation regime yet. To study the high-density limit, we have performed a weak coupling analysis of the
hardqAscattering,determinedbythesaturationmomentumQ ,anddescribedtheunpolarizedquarkfragmentation
s
into a transversely polarized Λ hyperon by the so-called polarizing fragmentation functions. We observe that the Λ
polarizationpeaksattransversemomentum Q ,whereitalsoscalesapproximatelyas1/(Q √logQ /Λ)andhence
s s s
∼
is collision energy dependent. Moreover, no plateau region for larger transverse momenta is present. These features
are independent of the details of the polarizing fragmentationfunctions, but rather occur due to their k -odd nature.
t
Similar effects are expected in the process of p↑A πX in the saturation region. Both processes can be studied, in
→
principle, at the BNL-RHIC collider, and perhaps in the future at the CERN-LHC.
Acknowledgement:
We thank Francois Gelis and Werner Vogelsang for helpful discussions and for contributing some of their computer
codes. A.D. gratefully acknowledges support by the German Minister for Education and Research (BMBF) and by
the U.S. Department of Energy under contract number DE-AC02-98CH10886. The research of D.B. has been made
possible by financial support from the Royal Netherlands Academy of Arts and Sciences.
[1] A.H. Mueller, Nucl.Phys. B 558, 285 (1999).
[2] for a recent review see E. Iancu, arXiv:hep-ph/0210236, and references therein.
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