Lambda polarization in polarized proton-proton collisions at BNL RHIC

PUBLISHED VERSION

1. Boros, Csaba Ladislaus Laszlo; Londergan, J. Timothy; Thomas, Anthony William

2000; 62(1):014021

© 2000 American Physical Society

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27

^th

March 2013

Lambda polarization in polarized proton-proton collisions at BNL RHIC

C. Boros

Department of Physics and Mathematical Physics, and Special Research Center for the Subatomic Structure of Matter, University of Adelaide, Adelaide 5005, Australia

J. T. Londergan

Department of Physics and Nuclear Theory Center, Indiana University, Bloomington, Indiana 47408 A. W. Thomas

Department of Physics and Mathematical Physics, and Special Research Center for the Subatomic Structure of Matter, University of Adelaide, Adelaide 5005, Australia

共Received 8 February 2000; published 5 June 2000兲

We discuss⌳ polarization in inclusive proton-proton collisions, with one of the protons longitudinally polarized. The hyperfine interaction responsible for the ⌬-N and ⌺-⌳ mass splittings gives rise to flavor asymmetric fragmentation functions and to sizable polarized non-strange fragmentation functions. We predict large positive⌳polarization in polarized inclusive proton-proton collisions at large rapidities of the produced

⌳. The effect of⌺⁰and⌺*decays is also discussed. Forthcoming experiments at RHIC will be able to test our predictions.

PACS number共s兲: 13.87.Fh, 13.85.Ni

I. INTRODUCTION

Measurements of the polarization dependent structure function, g₁, in deep inelastic scattering 关1兴 have inspired considerable experimental and theoretical effort to under- stand the spin structure of baryons. While most of these stud- ies concern the spin structure of nucleons, it has become clear that similar measurements involving other baryons would provide helpful, complementary information 关2–9兴. The Lambda baryon plays a special role in this respect. It is an ideal testing ground for spin studies since it has a rather simple spin structure in the naive quark parton model. Fur- thermore, its self-analyzing decay makes polarization mea- surements experimentally feasible.

Forthcoming experiments at the BNL Relativistic Heavy Ion Collider 共RHIC兲 could measure the polarization of ⌳ hyperons produced in proton-proton collisions with one of the protons longitudinally polarized, p^↑p→⌳^↑X. The polar- ization dependent fragmentation function of quarks and glu- ons into⌳hyperons can be extracted from such experiments.

These fragmentation functions contain information on how the spin of the polarized quarks and gluons is transferred to the final state⌳. The advantage of proton proton collisions, as opposed to e^⫹e^⫺ annihilation, where ⌳ production and polarization is dominated by strange quark fragmentation, is that ⌳’s at large positive rapidity are mainly fragmentation products of up and down valence quarks of the polarized projectile. Thus, the important question, intimately related to our understanding of the spin structure of baryons, of whether polarized up and down quarks can transfer polariza- tion to the ⌳ can be tested at RHIC关5兴.

In a previous publication, we have shown that the hyper- fine interaction, responsible for the ⌬-N and ⌺⁰-⌳ mass splittings, leads to non-zero polarized non-strange quark fragmentation functions 关12兴. These non-zero polarized up and down quark fragmentation functions give rise to sizable

positive ⌳ polarization in experiments where the strange quark fragmentation is suppressed. On the other hand, pre- dictions based either on the naive quark model or on SU(3) flavor symmetry predict zero or negative⌳ polarization关2兴. In Sec. II, we briefly discuss fragmentation functions and show how the hyperfine interaction leads to polarized non- strange fragmentation functions. We fix the parameters of the model by fitting the data on⌳ production in e^⫹e^⫺annihila- tion. In Sec. III, we discuss⌳ production in p p collisons at RHIC energies. We point out that the production of ⌳’s at high rapidities is dominated by the fragmentation of valence up and down quarks of the polarized projectile, and is ideally suited to test whether non-strange quarks transfer their polar- ization to the final state⌳. We predict significant positive⌳ polarization at large rapidities of the produced ⌳’s.

II. FRAGMENTATION FUNCTIONS

Fragmentation functions can be defined as light-cone Fou- rier transforms of the matrix elements of quark operators 关10,11兴:

1

zD_q^⌫_⌳共z兲⫽1

4

兺

_n

冕

^d2␰␲^{⫺e⫺i P}

⫹␰^⫺/z

⫻Tr兵^⌫具^0兩␺共0兲兩⌳共PS兲;n共p_n兲典

⫻具⌳共PS兲;n共p_n兲兩␺^¯共␰^⫺兲兩0典其^{, 共1兲}

where ⌫ is the appropriate Dirac matrix, P and p_n refer to the momentum of the produced ⌳ and of the intermediate system n, S is the spin of the ⌳ and the plus projections of the momenta are defined by P^⫹⬅(1/

冑

^{2)( P0}⫹P³). z is the plus momentum fraction of the quark carried by the pro- duced⌳.

Translating the matrix elements, using the integral repre- sentation of the delta function and projecting out the light- cone plus and helicity⫾ components we obtain

1

zD_q^⫾_⌳共z兲⫽ 1

2

冑

²

兺

_n ^{␦关共1/z⫺1兲P⫹⫺pn⫹兴}

⫻兩具⁰兩␺_⫹^⫾共0兲兩⌳共PS_储兲;n共p_n兲典兩². 共2兲 Here, ␺_⫹^⫾⫽¹2␥_⫺␥_⫹¹2(1⫾␥5)␺, and we have defined ␥_⫾

⫽(1/

冑

²⁾⁽␥0⫾␥3). The fragmentation function of an anti- quark into a⌳is given by Eq.共2兲, with␺_⫹replaced by␺_⫹^{† :}

1

zD¯_q⫾⌳共z兲⫽ 1

2

冑

²

兺

_n ^{␦关共1/z⫺1兲P⫹⫺pn⫹兴}

⫻兩具⁰兩␺_⫹^†⫾共0兲兩⌳共PS_储兲;n共p_n兲典兩². 共3兲 D_q^⫾_⌳ can be interpreted as the probability that a quark with positive-negative helicity fragments into a ⌳ with positive helicity and similarly for antiquarks.

The operator ␺⫹ (␺_⫹^†) either destroys a quark 共an anti- quark兲or it creates an antiquark共quark兲when acting on the

⌳ on the right hand side in the matrix elements. Thus, whereas in the case of quark fragmentation the intermediate state can be either an anti-diquark state, q¯ q¯, or a four-quark- antiquark state, qq¯ q¯q¯, in the case of antiquark fragmenta- tion, only four-antiquark states, q¯ q¯q¯q¯, are possible assuming that there are no antiquarks in the ⌳. 共Production of ⌳’s through coupling to higher Fock states of the ⌳ is more complicated and involves higher number of quarks in the intermediate states. As a result it would lead to contributions at lower z values.兲Thus, we have

共1a兲q→qqq⫹¯ q¯⫽q ⌳⫹¯ q¯q 共1b兲q→qqq⫹qq¯ q¯q¯⫽⌳⫹qq¯ q¯q¯, for the quark fragmentation, and

共2兲¯q→qqq⫹¯ q¯q¯q¯q ⫽⌳⫹q¯ q¯q¯q¯, for the antiquark fragmentation.

While in case 共1a兲 the initial fragmenting quark is con- tained in the produced ⌳, in case 共1b兲 and 共2兲, the ⌳ is mainly produced by quarks created in the fragmentation pro- cess. Therefore, we not only expect that ⌳’s produced through 共1a兲 usually have larger momenta than those pro- duced through共1b兲or共2兲but also that⌳’s produced through 共1a兲 are much more sensitive to the flavor spin quantum numbers of the fragmenting quark than those produced through 共1b兲 and共2兲. In the following we assume that 共1b兲 and共2兲lead to approximately the same fragmentation func- tions. In this case, the difference, D_q⌳⫺D¯_q⌳, responsible for leading particle production, is given by the fragmentation functions associated with process共1a兲.

Similar observations also follow from energy-momentum conservation built into Eqs. 共2兲 and 共3兲. The delta function implies that the function, D_q(z)/z, peaks at关12兴

z_max⬇ M

M⫹M_n. 共4兲

Here, M and M_n are the mass of the produced particle and the produced system, n, and we work in the rest frame of the produced particle. We see that the location of the maxima of the fragmentation function depends on the mass of the sys- tem n. While the high z region is dominated by the fragmen- tation of a quark into the final particle and a small mass system, large mass systems contribute to the fragmentation at lower z values. The maxima of the fragmentation functions from process共1a兲are given by the mass of the intermediate diquark state and that of the the fragmentation functions from the processes 共1b兲and共2b兲 by the masses of interme- diate four quark states. Thus, the contribution from process 共1a兲is harder than those from共1b兲and共2兲.

Energy-momentum conservation also requires that the fragmentation functions not be flavor symmetric. While the assertion of isospin symmetry, D_u⌳⫽D_d⌳, is well justified, SU(3) flavor symmetry is broken not only by the strange quark mass but also by the hyperfine interaction. Let us dis- cuss the fragmentation of a u 共or d) quark and that of an s quark into a⌳through process共1a兲. While the intermediate diquark state is always a scalar in the strange quark fragmen- tation, it can be either a vector or a scalar diquark in the fragmentation of the non-strange quarks. The masses of the scalar and vector non-strange diquarks follow from the mass difference between the nucleon and the ⌬ 关13兴, while those of the scalar and vector diquark containing a strange quark can be deduced from the mass difference between ⌺ and⌳ 关14兴. They are roughly m_s⬇650 MeV and m_v⬇850 MeV for the scalar and vector non-strange diquarks, and m_s⬘

⬇890 MeV and m_v⬘⬇1010 MeV for scalar and vector di- quarks containing strange quarks, respectively 关14,12兴. Ac- cording to Eq. 共4兲, these numbers lead to soft up and down quark fragmentation functions and to hard strange quark fragmentation functions.

Energy-momentum conservation, together with the split- ting of vector and scalar diquark masses, has the further im- portant consequence that polarized non-strange quarks can transfer polarization to the final state⌳. To see this we note that the probabilities for the intermediate state to be a scalar or vector diquark state in the fragmentation of an up or down quark with spin parallel or anti-parallel to the spin of the ⌳ can be obtained from the SU(6) wave function of the⌳:

⌳^↑⫽ 1

2

冑

^{3关2s↑共ud兲0,0⫹}

冑

^2d↓共us兲1,1⫺d^↑共us兲1,0⫹d^↑共us兲0,0

⫺

冑

^2u↓共ds兲1,1⫹u^↑共ds兲1,0⫺u^↑共ds兲0,0兴. 共5兲 While the u or d quarks with spin anti-parallel to the spin of the ⌳ are always associated with a vector diquark, u and d quarks with parallel spin have equal probabilities to be ac- companied by a vector or scalar diquark. The fragmentation functions of non-strange quarks with spin parallel to the ⌳ spin are harder than the corresponding fragmentation func- tions with anti-parallel spins. Thus, ⌬D_u⌳ is positive for

C. BOROS, J. T. LONDERGAN, AND A. W. THOMAS PHYSICAL REVIEW D 62 014021

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large z values and negative for small z. Their total contribu- tion to polarized⌳ production might be zero or very small.

Nevertheless, ⌬D_u⌳ and ⌬D_d⌳ can be sizable for large z values, since both D_u⌳and⌬D_u⌳are dominated by the spin- zero component in the large z limit. Furthermore, they will dominate polarized ⌳ production whenever the production from strange quarks is suppressed.

The matrix elements can be calculated using model wave functions at the scale relevant to the specific model and the resulting fragmentation functions can be evolved to a higher scale to compare them to experiments. In a previous paper 关12兴, we calculated the fragmentation functions in the MIT bag model and showed that the resulting fragmentation func- tions give a very reasonable description of the data in e^⫹e^⫺ annihilation. Since the mass of the intermediate states con- taining more than two quarks is not known, we only calcu- late the contributions of the diquark intermediate states in the bag model. The other contributions have been determined by performing a global fit to the e^⫹e^⫺ data. For this, we used the simple functional form

D_q¯⌳共z兲⫽N_q¯z^␣共1⫺z兲^␤ 共6兲

to parametrize D_q^¯_⌳⫽D^¯_u⌳⫽D^¯_d⌳⫽•••D^¯_b⌳ and also set D_g⌳⫽0 at the initial scale, ␮⫽0.25 GeV.

The fragmentation functions have to be evolved to the scale of the experiment,␮. The evolution of the non-singlet fragmentation functions in leading order 共LO兲 is given by 关15,16兴

d

d ln␮^{2关Dq⌳⫺D¯q⌳兴共z,}␮²兲⫽

冕

^z1dzz⬘^Pqq

冉

^zz⬘

冊

^{关Dq⌳⫺D¯q⌳兴}

⫻共z⬘^,␮²兲. 共7兲

The singlet evolution equations are

d

d ln␮²

兺

_q ^{Dq⌳共z,␮2兲⫽}

冕

z 1dz⬘

z⬘

冋

^Pqq

冉

^zz⬘

冊 兺q Dq⌳共z,␮2兲

⫹2n_fP_gq

冉

^zz⬘

冊

^{Dg⌳共z⬘,␮2兲}

册

d

d ln␮^2Dg⌳共z,␮²兲⫽

冕

z 1dz⬘

z⬘

冋

^Pqg

冉

^zz⬘

冊 兺q Dq⌳共z,␮2兲

⫹P_gg

冉

^zz⬘

冊

^{Dg⌳共z⬘,␮2兲}

册

^{, 共8兲}

where the splitting functions are the same as those for the evolution of quark distributions. n_f is the number of flavors.

We used the evolution package of Ref. 关17兴 suitably modi- fied for the evolution of fragmentation functions共interchang- ing the off-diagonal elements in the singlet case兲.

TABLE I. Fit parameters obtained by fitting the e^⫹e^⫺ data. We also parametrized Dq⌳⫺Dq¯⌳ and

⌬D_q⌳⫺⌬D¯_q⌳, calculated in the bag.

Parameter D_s⌳⫺D¯_s⌳ D_u⌳⫺D¯_u⌳ D_q¯⌳ ⌬D_s⌳⫺⌬D¯_s⌳ ⌬D_u⌳⫺⌬D¯_u⌳

N 5.81⫻10⁹ 1.60⫻10¹⁷ 99.76 3.73⫻10¹⁸ ⫺6.25⫻10¹⁰

␣ 21.55 30.49 1.25 21.21 32.48

␤ 13.60 28.34 11.60 13.38 27.72

␥ — — — — 0.52

FIG. 1. Inclusive⌳production in e^⫹e^⫺annihilation. The solid lines are the result of the global fit. They contain two parts, the fixed contributions from D_q⌳⫺D_q¯⌳ calculated in the bag共dashed line only shown for the Aleph data兲and D¯q⌳obtained from the fit 共dash-dotted line兲. xE is defined as xE⫽2E_⌳/冑^{s where E}⌳ is the energy of the produced⌳in the e^⫹e^⫺center of mass frame and冑^s

is the total center of mass energy.

The results of the LO fit to e^⫹e^⫺data关18–23兴are shown in Fig. 1. The parameters of our fits are given in Table I. The bag model calculations for D_q⌳⫺D¯_q⌳ and ⌬D_s⌳⫺⌬D¯_s⌳

were parametrized using the functional form of Eq.共6兲. The fragmentation functions, ⌬D_u⌳⫺⌬D_u¯⌳⫽⌬D_d⌳⫺⌬D¯_d⌳, change sign at some value of z; hence we parametrized them using the form

⌬D_q⌳共z兲⫺⌬D¯_q⌳共z兲⫽N_q¯z^␣共1⫺z兲^␤共␥⫺z兲. 共9兲

These parameters are also given in Table I. We also per- formed a fit using flavor symmetric fragmentation functions which we shall need for the discussion of ⌳ production in p p collisions. Since the ⌬D_q^¯_⌳ were assumed to be flavor symmetric in the first fit, we use the same parameters for them in the second fit. The fit parameters are given in Table II. In Fig. 2, we show the calculated fragmentation functions at Q²⫽M_Z². We note that our fragmentation functions de- scribe the asymmetry in leading and non-leading particle production, as well as the⌳ polarization measured in e^⫹e^⫺ annihilation at the Z pole, very well — as has been shown in Ref. 关12兴.

III. POLARIZED PROTON-PROTON COLLISION Spin-dependent fragmentation of quarks can be studied in proton-proton collisions with one of the protons polarized 关5兴. Here, many subprocesses may lead to the final state⌳so that one has to select certain kinematic regions to suppress the unwanted contributions. In particular, in order to test whether polarized up and down quarks do fragment into po- larized ⌳ the rapidity of the produced ⌳ has to be large, since at high rapidity,⌳’s are mainly produced through va- lence up and down quarks.共We count positive rapidity in the direction of the polarized proton beam.兲

The difference of the cross sections to produce a ⌳ with positive helicity through the scattering of a proton with positive-negative helicity on an unpolarized proton is given in leading order perturbative QCD 共LO PQCD兲by¹

E_C⌬d␴

d³p_C共AB→C⫹X兲

⫽E_C d␴

d³p_C共A^↑B→C^↑⫹X兲⫺E_C d␴

d³p_C共A^↓B→C^↑⫹X兲

⫽_abcd

兺 冕

^{dxadxbdzc⌬fAa共xa,␮2兲fBb共xb,␮2兲}

⫻⌬D_cC共z_c,␮²兲 sˆ

␲^zc 2

⌬d␴

d tˆ 共ab→cd兲␦共sˆ⫹tˆ⫹uˆ兲. 共10兲

1Since the relevant spin dependent cross sections on the parton level are only known in LO, we perform a LO calculation here.

FIG. 2. Fragmentation functions. The solid and dashed lines stand for the calculated fragmentation functions of up and strange quarks into ⌳ baryons through production of a ⌳ and an anti- diquark and correspond to Du⌳⫺D¯u⌳and Ds⌳⫺D¯s⌳, respectively.

The dash-dotted line represents the contributions from higher inter- mediate states, and is obtained by fitting the e^⫹e^⫺data and corre- sponds to D¯_q⌳. The short dashed line is the gluon fragmentation function. The light and heavy lines are the fragmentation functions at the scales Q²⫽␮² and Q²⫽M_Z², respectively. Note that D_g⌳

⫽0 at Q²⫽␮².

FIG. 3. z_cas a function of x_aand y for two different transverse momenta, p_⬜⫽10 GeV 共left兲 and p_⬜⫽30 GeV 共right兲, and for two different values of x_b, x_b⫽x_bmin⫹0.01共top兲 and x_b⫽x_bmin

⫹0.1共bottom兲. TABLE II. Fit parameters obtained by fitting the e^⫹e^⫺data and

assuming that the fragmentation functions are flavor symmetric.

Parameter D_q⌳⫺D¯_q⌳ D_q¯⌳

N 1.92⫻10⁴ 99.76

␣ 7.47 1.25

␤ 8.06 11.60

C. BOROS, J. T. LONDERGAN, AND A. W. THOMAS PHYSICAL REVIEW D 62 014021

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Here, ⌬f_Aa(x_a,␮^{2) and f}Bb(x_b,␮²) are the polarized and unpolarized distribution functions of partons a and b in pro- tons A and B, respectively, at the scale␮^{. x}a and x_b are the corresponding momentum fractions carried by partons a and b. ⌬D_cC(z_c,␮²) is the polarized fragmentation function of parton c into baryon C, in our case C⫽⌳. z_cis the momen- tum fraction of parton c carried by the produced⌳.⌬d␴^{/d tˆ} is the difference of the cross sections at the parton level between the two processes a^↑⫹b→c^↑⫹d and a^↓⫹b→c^↑

⫹d. The unpolarized cross section is given by Eq.共10兲with the ⌬’s dropped throughout.

The Mandelstam variables at the parton level are given by sˆ⫽x_ax_bs, tˆ⫽⫺x_ap_⬜

冑

^se⫺y/zc, uˆ⫽⫺x_bp_⬜

冑

^sey/zc

共11兲 where y and p_⬜are the rapidity and transverse momentum of the produced ⌳ and

冑

s is the total center of mass energy.

The summation in Eq. 共10兲 runs over all possible parton- parton combinations, qq⬘→qq⬘^{, qg}→qg, qq¯→qq, . . . . The elementary unpolarized and polarized cross sections can be found in Refs.关24,25兴. Performing the integration in Eq.

共10兲over z_cone obtains

E_C⌬d␴

d³p_C共AB→C⫹X兲

⫽_abcd

兺 冕

x_amin 1

dx_a

冕

x_bmin 1

dx_b⌬f_Aa共x_a,␮²兲f_Bb

⫻共x_b,␮²兲⌬D_cC共z_c,␮²兲 1

␲^zc

⌬d␴

d tˆ 共ab→cd兲 共12兲 with

z_c⫽x_⬜

2x_be^⫺y⫹ x_⬜

2x_ae^y, x_bmin⫽ x_ax_⬜e^⫺y 2x_a⫺x_⬜e^y,

x_amin⫽ x_⬜e^y

2⫺x_⬜e^⫺y, 共13兲 where x_⬜⫽2 p_⬜/

冑

^s.

In order to elucidate the kinematics, in Fig. 3, we have plotted z_c as a function of x_a and y for two different trans- verse momenta, p_⬜⫽10 GeV 共left兲 and p_⬜⫽30 GeV 共right兲, and for two different values of x_b, x_b⫽x_bmin⫹0.01 共top兲 and x_b⫽x_bmin⫹0.1共bottom兲. Note that z_c is maximal both for x_b⫽x_bmin and x_a⫽x_amin. With increasing rapidity, y, both the lower integration limit of x_a, x_amin, and the momentum fraction of the fragmenting quark transferred to the produced ⌳, z_c, increase. Hence, large rapidities probe the fragmentation of mostly valence quarks into fast⌳. The dependence on the transverse momenta is also shown in Fig.

3. With increasing p_⬜, the kinematic boundary is shifted to smaller rapidities and the fragmentation of the valence quarks can be studied at lower rapidities. This is important

since the available phase space is limited by the acceptance of the detectors at RHIC. However, the cross section also decreases with increasing transverse momenta, leading to lower statistics.

In Fig. 4, we show the contributions of the various chan- nels to the cross section for two different transverse mo- menta, both for inclusive⌳, Fig. 4共a兲, and inclusive jet pro- duction, Fig. 4共b兲. gq→gq stands, for example, for the contribution to the cross section coming from the subprocess involving a gluon g and a quark q in the initial and final states. In qq⬘^→qq⬘, the quarks have different flavors and qq¯⬘^→qq¯⬘ is also included. Although the kinematics are not exactly the same for these two processes,²one can study the role played by the fragmentation functions by comparing in- clusive⌳and inclusive jet production. In particular, the con- tributions from channels containing two gluons in the final state are suppressed in inclusive ⌳ production due to the smallness of D_g⌳. We note that D_g⌳has been set to zero at the initial scale and is generated through evolution. Thus, while qg→qg is the dominant channel in inclusive⌳ pro- duction, both qg→qg and gg→gg are equally important in inclusive jet production. There is some ambiguity due to our poor knowledge of the gluon fragmentation — larger prob-

2While there is only one integration variable, xa, in inclusive jet production, once p_⬜ and y are fixed, both x_a and x_b have to be integrated over the allowed kinematic region in inclusive⌳produc- tion, since the produced ⌳ carries only a fraction of the parton’s momentum.

FIG. 4. Contributions from the various channels共a兲to the inclu- sive⌳production cross section ( p p→⌳⫹X) and共b兲to the inclu- sive jet production cross section ( p p→jet⫹X) at p_⬜⫽10 GeV 共left兲and p_⬜⫽30 GeV共right兲at冑^s⫽500 GeV.

abilities for g→⌳ will enhance the contributions from glu- ons in inclusive ⌳ production. However, the contribution from the process gg→gg falls off faster than that from qg

→qg with increasing rapidity, since g(x_a) decreases faster than q(x_a) with increasing x_a and the ⌳’s are produced mainly from valence up and down quarks at high rapidities.

Our analysis of the kinematics and the various contribu- tions to inclusive ⌳ production already indicate that⌳ po- larization measurements in p p collisions at high rapidities are ideally suited to test whether polarized up and down quarks may fragment into polarized ⌳’s. We calculated the

⌳ polarization using our flavor asymmetric fragmentation functions for RHIC energies and for p_⬜⫽10 GeV. Increas- ing the transverse momentum gives similar results, with the only difference being that P_⌳ starts to increase at lower ra- pidities. We used the standard set of Glu¨ck-Reya-Stratmann- Vogelsang共GRSV兲LO quark distributions for the polarized parton distributions关26兴and the LO CTEQ4 distributions for the unpolarized quark distributions关27兴. The scale,␮^{, is set} equal to p_⬜. We also checked that there is only a very weak dependence on the scale by calculating the polarization using

␮⫽p_⬜/2 and ␮⫽2 p_⬜. The predicted ⌳ polarization is shown in Fig. 5a. It is positive at large rapidities where the contributions of polarized up and down quarks dominates the production process. At smaller rapidities, where x_a is small, strange quarks also contribute. However, since the ratios of the polarized to the unpolarized parton distributions are small at small x_a, the ⌳ polarization is suppressed. The result also depends on the parametrization of the polarized

quark distributions. In particular, the polarized gluon distri- bution is not well constrained. However, it is clear from the kinematics that the ambiguity associated with the polarized gluon distributions only effects the results at lower rapidities.

This can be seen in Fig. 5b where we plot the contribution from gluons, up plus down quarks and strange quarks to the

⌳ polarization.

Next, we contrast our prediction with the predictions of various SU(3) flavor symmetric models which use

D_u⌳⫽D_d⌳⫽D_s⌳. 共14兲 We fitted the cross sections in e^⫹e^⫺ annihilation using Eq.

共14兲and the functional form given in Eq.共6兲. For the polar- ized fragmentation functions, we discuss two different sce- narios: The model, SU(3)_A 共cf. Fig. 5a兲, corresponds to the expectations of the naive quark model that only polarized strange quarks can fragment into polarized⌳:

⌬D_u⌳⫽⌬D_d⌳⫽0, ⌬D_s⌳⫽D_s⌳. 共15兲 It gives essentially zero polarization because the strange quarks contribute at low rapidities where the polarization is suppressed. Model, SU(3)_B 共cf. Fig. 5a兲, which was pro- posed in Ref.关2兴, is based on deep inelastic scattering共DIS兲 data, and sets

⌬D_u⌳⫽⌬D_d⌳⫽⫺0.20D_u⌳, ⌬D_s⌳⫽0.60D_s⌳. 共16兲 This model predicts negative ⌳ polarization. These SU(3) symmetric models have also been discussed by de Florian et al. in connection with the RHIC experiment关5兴. They con- sider a further SU(3) symmetric scenario with the ad hoc assumption that all polarized fragmentation functions are positive and equal, independent of the flavor of the fragment- ing quark. In contrast, our fragmentation functions are calcu- lated by using a model for hadron structure that gives a rea- sonable, qualitative description of the properties of the ⌳. Flavor asymmetric fragmentation functions and positive po- larized up and down quark fragmentation functions are direct consequences of this model.

Finally, we address the problem of ⌳ produced through the decay of other hyperons, such as ⌺⁰and⌺*. In order to estimate the contribution of hyperon decays we assume, in the following, that 共1兲 the ⌳’s produced through hyperon decay inherit the momentum of the parent hyperon and 共2兲 and that the total probability to produce⌳,⌺⁰ or⌺*from a certain uds state is given by the SU(6) wave function and is independent of the mass of the produced hyperon.

Further, in order to estimate the polarization transfer in the decay process we use the constituent quark model. The polarization can be obtained by noting that the boson emitted in both the ⌺⁰→⌳␥ ^{and the} ⌺*_→⌳␲ decay changes the angular momentum of the nonstrange diquark from J⫽1 to J⫽0, while the polarization of the spectator strange quark is unchanged. Then, the polarization of the⌳is determined by the polarization of the strange quark in the parent hyperon, since the polarization of the ⌳ is exclusively carried by the strange quark in the naive quark model.

FIG. 5. Lambda polarization at RHIC.共a兲The solid line repre- sents our prediction. The predictions of SU(3) symmetric fragmen- tation models are shown for comparision. The model labeled as SU(3)_Ais based on the quark model expectation that only the po- larized strange quark may fragment into polarized⌳, while SU(3)B

is based on DIS data.共b兲Contributions of different flavors to the⌳ polarization. The light dashed, dash-dotted and heavy dashed lines stand for the contributions from up plus down, from strange and from gluon fragmentation, respectively, as calculated here. The es- timated polarization including both⌺⁰and⌺* _共lower dotted line兲 and only⌺*_共upper dotted line兲decays is also shown. See text for further details.

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First, let us discuss the case when the parent hyperon is produced by a strange quark. Since the strange quark is al- ways accompanied by a vector ud diquark, in both ⌺⁰ and

⌺* the fragmenation functions of strange quarks into these hyperons are much softer than the corresponding fragmenta- tion function into a⌳. Thus, in the high z limit, the contri- butions from the processes, s→⌺⁰→⌳and s→⌺*_→⌳, are negligible compared to the direct production, s→⌳. Further- more, both channels, s→⌺⁰→⌳ and s→⌺*_→⌳, enhance the already positive polarization from the direct channel, s

→⌳.

This is different in the case when the parent hyperon is produced by an up or down quark. Both ⌳ and ⌺⁰ can be produced by an up 共down兲 quark and a scalar ds (us) di- quark — a process which dominates in the large z limit.共The component with a vector diquark can be neglected in this limit.兲 Furthermore, the up and down fragmentation func- tions of the⌺*are as important as those of the⌳and⌺⁰ in the large z limit. This is because the u fragmentation function of ⌺* peaks at about 1385/(1010⫹1385)⬇0.58 which is almost the same as the peak of the scalar components of the

⌳ and ⌺, which are 1115/(890⫹1115)⬇0.57 and 1190/(890⫹1190)⬇0.57, respectively. Thus, for the up and down quark fragmentation, it is important to include the⌳’s from these decay processes.

The relevant probabilities to produce a ⌳ with positive and negative polarization from a fragmenting up quark with positive polarization and an ds diquark are shown in Table III. We assumed that all spin states of the ds diquark are produced with equal probabilities. The final weights which are relevant in the large z limit are set in boldface. We find that if we include all channels which survive in the large z limit, the polarization of the ⌳ is reduced by a factor of 10/27 compared to the case where only the directly produced

⌳’s are included. Since the⌺*decay is a strong decay, it is sometimes included in the fragmentation function of the ⌳. Including only⌺*, the suppression factor we obtain is 49/81.

共Only the decay of neutral ⌺*’s is included in this discus- sion. Inclusion of the charged ⌺* decays would result in a prediction between the two dotted lines in Fig. 5b. We also neglected the small deviation of the branching ratio for ⌺*

→⌳␲ from unity.兲 Note that our model predicts that

u^↑(ds)_0,0→⌳, u^↑(ds)_0,0→⌺⁰ and u^↑(ds)_1,0→⌺* have ap- proximately the same z dependence and are approximately equal 共up to the Clebsch-Gordon factors兲 since the ratios, M /( M⫹M_n), have roughly the same numerical values.

Thus, the effect of the ⌺⁰ and⌺* decays can be taken into account by a multiplicative factor.

In order to illustrate the effect of these decays on the final

⌳ polarization, we multiplied our results by these factors.

The results are shown in Fig. 5b as dotted lines. We note that our implementation of this correction relies on the assump- tions that the produced ⌳ carries all the momentum of the parent hyperon and that all states are produced with equal probabilities. Since neither of these assumptions is strictly valid, we tend to overestimate the importance of hyperon decays. Note also that the inclusion of ⌺⁰ decay in the SU(3) symmetric models makes the resulting polarization more negative. As a result, even if effects of ⌺ decays are included, large discrepancies still persist between our predic- tions and those of SU(3) symmetric models.

IV. CONCLUSIONS

Measurements of the⌳ polarization at RHIC would pro- vide a clear answer to the question of whether polarized up and down quarks can transfer polarization to the final state

⌳. We predict positive⌳polarization at high rapidities. Pre- dictions based either on the non-relativistic quark model or on SU(3) and DIS data give zero or negative⌳polarization.

Our prediction is based on the same physics which led to harder up than down quark distributions in the proton and to the ⌬-N and ⌺-⌳ mass splittings. We also estimated the importance of ⌺⁰ and⌺* decays which tend to reduce the predicted⌳ polarization.

ACKNOWLEDGMENTS

This work was partly supported by the Australian Re- search Council. One of the authors共J.T.L.兲was supported in part by National Science Foundation research contract PHY- 9722706. One author共C.B.兲wishes to thank the Indiana Uni- versity Nuclear Theory Center for its hospitality during the time part of this work was carried out.

TABLE III. Different channels for the production of⌳hyperons from a positively polarized up quark and a ds diquark. It is assumed that all spin states of the ds diquark are produced with the same probabilities.⌺*^↑and⌺*^⇑stand for the 1/2 and 3/2 spin components of the⌺*. See text for further details.

u(ds) states u^↑(ds)0,0 u^↑(ds)1,1 u^↑(ds)1,0 u^↑(ds)1,⫺1

Relative weights ¹₄ ¹₄ ¹₄ ¹₄

Products ⌳^↑ ⌺^0↑ ⌺*^0↑ _⌺*^0⇑ _⌳^↑ _⌺^0↑ _⌺*^0↑ _⌳^↓ _⌺^0↓ _⌺*^0↓

Relative weights ¹₄ ³₄ 0 1 ¹_{4 12}^{1 2}₃ ¹₂ ¹₆ ¹₃

Decay products ⌳^↑ ⌳^↑ ⌳^↓ ⫺ ⌳^↑ ⌳^↓ ⌳^↑ ⌳^↓ ⌳^↑ ⌳^↓ ⌳^↑ ⌳^↓ ⌳^↑ ⌳^↓ ⌳^↑ ⌳^↓ ⌳^↑ ⌳^↓

Relative weights 1 ¹₃ ²₃ 0 1 0 1 0 ¹₃ ²₃ ²₃ ¹₃ 0 1 ²₃ ¹₃ ¹₃ ²₃

Final weights ₁₆¹ ₁₆^{1 1}₈ 0 ¹₄ 0 ₁₆¹ 0 ₁₄₄¹ ₇₂^{1 1}_{9 18}¹ 0 ¹_{8 36}¹ ₇₂¹ ₃₆¹ ₁₈¹

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