Lambda polarization from unpolarized quark fragmentation

⌳ polarization from unpolarized quark fragmentation

M. Anselmino,1D. Boer,2 U. D’Alesio,3and F. Murgia3

1_{Dipartimento di Fisica Teorica, Universita` di Torino and INFN, Sezione di Torino, Via P. Giuria 1, I-10125 Torino, Italy} 2_{RIKEN-BNL Research Center, Brookhaven National Laboratory, Upton, New York 11973}

3_{Istituto Nazionale di Fisica Nucleare, Sezione di Cagliari and Dipartimento di Fisica, Universita` di Cagliari,} Casella Postale 170, I-09042 Monserrato (CA), Italy

共Received 17 August 2000; published 8 February 2001兲

The long-standing problem of explaining the observed polarization of⌳ hyperons, inclusively produced in the high-energy collisions of unpolarized hadrons, is tackled by considering spin- and k_⬜-dependent quark-fragmentation functions. The data on⌳’s and ⌳¯ ’s produced in p-N processes are used to determine simple phenomenological expressions for these new ‘‘polarizing fragmentation functions,’’ which describe the experi-ments remarkably well.

DOI: 10.1103/PhysRevD.63.054029 PACS number共s兲: 13.85.Ni, 12.38.Bx, 13.88.⫹e, 14.20.Jn

I. INTRODUCTION

It has been well known for a long time that⌳ hyperons produced with x_Fⲏ0.2 and p_Tⲏ1 GeV/c in the collision of two unpolarized hadrons, AB→⌳↑X, are polarized perpen-dicularly to the production plane as allowed by parity invari-ance; a huge amount of experimental information, for a wide energy range of the unpolarized beams, is available on such single spin asymmetries 关1兴,

P_⌳⫽

d␴AB→⌳↑X⫺d␴AB→⌳↓X

d␴AB→⌳↑X_⫹d␴AB→⌳↓X. 共1兲

Similar effects have been observed for several other hyper-ons but we shall chyper-onsider here only⌳’s and ⌳¯ ’s.

Despite the wealth of data and the many years they have been known, no convincing theoretical explanation or under-standing of the phenomenon exists 关2,3兴. The perturbative QCD dynamics forbids any sizable single spin asymmetry at the partonic level关4兴; the polarization of hyperons resulting from the strong interaction of unpolarized hadrons must then originate from nonperturbative features, presumably in the hadronization process. A number of models attempting some understanding of the data in this perspective 关2–9兴 only achieve partial explanations.

In the last years other single spin asymmetries observed in p↑p→␲X reactions 关10兴 have attracted a lot of theoretical activity 关11–20兴; a phenomenological description of such asymmetries appears possible now with the introduction of new distribution 关21,11,14,22兴 and/or fragmentation 关12,19,23兴 functions that are spin- and k⬜-dependent; k⬜

de-notes either the transverse momentum of a quark inside a nucleon or of a hadron with respect to the fragmenting quark. In particular, the effect first discussed by Collins 关12兴— that is, the azimuthal angle dependence of the number of hadrons produced in the fragmentation of a transversely po-larized quark—has been recently observed 关24,25兴; were such results confirmed the role of these new fragmentation functions would be of great phenomenological importance.

We consider here an effect similar to that suggested by Collins, namely, a spin and k_⬜ dependence in the

fragmen-tation of an unpolarized quark into a polarized hadron: a function describing this mechanism was first introduced in Ref.关23兴 and denoted by D⬜_1T. This function is introduced in a frame defined by two lightlike four-vectors n_⫹ and n_⫺, satisfying n_⫹•n_⫺⫽1, and by the plane transverse to them. The four-momentum P of the outgoing hadron—a ⌳ hy-peron in the present investigation—is in the n_⫺direction共up to a mass term correction兲. The function D⬜_1T is then defined as 共displayed in the n_⫹•A⫽0 gauge兲

⑀T i j kTiST j Mh D⬜_1T共z,k_⬜兲 ⬅

兺

X

冕

dy⫹d2_y T 4z共2␲兲3e ik•y ⫻Tr

具

0兩␺共y兲兩P,S_T;X

典具

P,S_T;X兩␺¯共0兲␥⫺兩0

典

兩_y⫺_⫽0, 共2兲 where the final state depends on the transverse part (ST) of

the spin vector S of the produced ⌳ only, i.e., one should interpret it as兩P,ST;X

典

⬅(兩P,S⫽ST;X

典

⫺兩P,S⫽⫺ST;X

典

)/

2, such that the contribution from unpolarized fragmentation cancels out. Furthermore, k_⬜⫽兩k_⬜兩 is the modulus of the transverse momentum of the hadron in a frame where the fragmenting quark has no transverse momentum. More de-tails on this type of definition of fragmentation 共or decay兲 functions can be found in Refs. 关26,12,23兴.

In the notations of Ref. 关19兴 a similar function is defined as

⌬N_D

h↑/a共z,k⬜兲⬅Dˆh↑/a共z,k⬜兲⫺Dˆh↓/a共z,k⬜兲

⫽Dˆh↑/a共z,k⬜兲⫺Dˆh↑/a共z,⫺k⬜兲, 共3兲 and denotes the difference between the density numbers Dˆ_h↑/a(z,k⬜) and Dˆh↓/a(z,k⬜) of spin 1/2 hadrons h, with longitudinal momentum fraction z, transverse momentum

by the fragmentation of an unpolarized parton a. From the above definition it is clear that the k_⬜integral of the function vanishes.

The exact relation between D⬜_1Tand⌬NDh↑/a is given by 共notice that also D⬜1T should have labels h and a, which are often omitted兲: ⌬N_D h↑/a共z,k_⬜兲⫽2 k_⬜ z Mh sin␾D⬜_1T共z,k_⬜兲, 共4兲 where ␾ is the angle between k_⬜ and the transverse polar-ization vector of the hadron, which shows that the function ⌬N_D

h↑/a(z,k⬜) vanishes in case the transverse momentum and transverse spin are parallel.

In the following we shall refer to ⌬N_D

h↑/a and D⬜1T as polarizing fragmentation functions.

In analogy to Collins’s suggestion for the fragmentation of a transversely polarized quark we write 关12,27兴

Dˆh↑/q共z,k⬜兲⫽ 1 2Dˆh/q共z,k⬜兲 ⫹1₂⌬N_D h↑/q共z,k⬜兲 Pˆh•共pq⫻k_⬜兲 兩pq⫻k⬜兩 共5兲

for an unpolarized quark with momentum pq that fragments

into a spin 1/2 hadron h with momentum ph⫽zpq⫹k⬜ and

polarization vector along the ↑⫽Pˆh direction; Dˆh/q(z,k⬜)

⫽Dˆh↑/q(z,k⬜)⫹Dˆh↓/q(z,k⬜) is the k⬜-dependent unpolar-ized fragmentation function. Notice that Pˆ_h•(p_q⫻k_⬜) ⫽pq•(k_⬜⫻Pˆh)⬃sin␾.

A QCD factorization theorem gives for the high-energy and large-pT process AB→⌳↑X at leading twist with

collin-ear parton configurations

d␴AB→⌳↑X⫽

兺

a,b,c,d fa/A共xa兲丢fb/B共xb兲 丢d␴ˆab→cd丢D_⌳↑/c共z兲 共6兲 and d␴AB→⌳↓X⫽

兺

a,b,c,d fa/A共xa兲丢fb/B共xb兲 丢d␴ˆab→cd丢D_⌳↓/c共z兲. 共7兲 Here and in the sequel we shall fix the scattering plane as the x-z plane, with hadron A moving along⫹zˆ and the detected ⌳ produced in the first x-z quadrant; the ↑, ↓ directions are then respectively⫹yˆ and ⫺yˆ.

In the absence of intrinsic k_⬜ 共or rather when integrated over兲 the fragmentation functions D_⌳↑/c(z) ⫽兰d2_k

⬜Dˆ⌳↑/c(z,k⬜) 关or D⌳↓/c(z)] cannot depend on the hadron polarization, so that one has d␴↑⫽d␴↓, which im-plies P_⌳⫽0.

However, by taking into account intrinsic k_⬜ in the had-ronization process and assuming that the factorization

theo-rem holds also when k_⬜’s are included关12兴, one has, using Eq. 共5兲 instead of D_⌳/c(z) in Eqs.共1兲, 共6兲, and 共7兲

E_⌳d3␴AB→⌳X d3_p ⌳ P_⌳ ⫽

兺

a,b,c,d

冕

dxadxbdz ␲z2 d 2_k ⬜fa/A共xa兲fb/B共xb兲 ⫻sˆ␦共sˆ⫹tˆ⫹uˆ兲d␴ˆ ab_→cd d tˆ 共xa,xb,k⬜兲⌬ N_D ⌳↑/c共z,k⬜兲 共8兲 where sˆ, tˆ, and uˆ are the Mandelstam variables for the el-ementary process: for collinear configurations, sˆ⫽xaxbs,

tˆ⫽x_at/z, and uˆ⫽x_bu/z and the modifications due to intrin-sic k_⬜ will be taken into account in the numerical evalua-tions. E_⌳d3␴AB→⌳X/d3p_⌳ is the unpolarized cross section

E_⌳d3␴AB→⌳X d3p_⌳ ⫽

兺

a,b,c,d

冕

dxadxbdz ␲z2 d 2_k ⬜fa/A共xa兲fb/B共xb兲 ⫻sˆ␦共sˆ⫹tˆ⫹uˆ兲d␴ˆ ab→cd d tˆ 共xa,xb,k⬜兲Dˆ⌳/c共z,k⬜兲. 共9兲 In Eq.共8兲 k_⬜is considered only where its absence would lead to zero polarization, that is, leading collinear configura-tions are assumed for partons a and b inside unpolarized hadrons A and B, while transverse motion is considered in the fragmentation process. The final hadron, detected with momentum p_⌳, is generated by the hadronization of a parton c whose momentum pc⫽(p⌳⫺k⬜)/z varies with k⬜; also the

corresponding elementary process ab→cd depends on k_⬜. P_⌳ is a function of the hyperon momentum p_⌳⫽pL⫹pT

and is normally measured in the AB c.m. frame as a function of xF⬅2pL/

冑

s and pT.

Notice that, in principle, there might be another contribu-tion to the polarizacontribu-tion of a final hadron produced at large pT

in the high-energy collision of two unpolarized hadrons; in analogy to Sivers’s effect关11,14兴 one might introduce a new spin- and k_⬜-dependent distribution function

⌬N_f

a↑/A共xa,k⬜a兲⬅ fˆa↑/A共xa,k⬜a兲⫺ fˆa↓/A共xa,k⬜a兲

⫽ fˆa↑/A共xa,k⬜a兲⫺ fˆa↑/A共xa,⫺k⬜a兲,

共10兲 i.e., the difference between the density numbers fˆa↑/A(xa,k⬜a) and fˆa↓/A(xa,k⬜a) of partons a, with

longitu-dinal momentum fraction xa, transverse momentum k⬜a,

and transverse polarization ↑ or ↓, inside an unpolarized hadron A.

This idea was first applied to unpolarized lepto-production关22兴 and to single spin asymmetries in pp↑ scat-tering 关28兴; the corresponding function, related to ⌬N_f

a↑/A(xa,k_⬜a), was denoted by h⬜1 . In the present case of transversely polarized⌳ production this function would en-ter the cross-section accompanied by the transversity frag-mentation function H1(z) 共or ⌬Dh↑/a↑). We shall not con-sider this contribution here, not only because of the problems concerning ⌬Nfa↑/A(xa,k⬜a) discussed below, but also

be-cause the experimental evidence that the hyperon polariza-tion is somewhat independent of the nature of the hadronic target suggests that the mechanism responsible for the polar-ization is in the hadronpolar-ization process.1 A clean test of this should come from a measurement of P_⌳in unpolarized deep inelastic scattering共DIS兲 processes, lp→⌳↑X 关30兴.

The k_⬜-dependent functions considered in Refs. 关11,14,19,22,12,23兴 (⌬N_f

a/A↑, ⌬Nfa↑/A, ⌬NDh/a↑, and

⌬N_D

h↑/a or, respectively, f⬜1T, h⬜1, H⬜1, and D⬜1T) have the common feature that the transverse momentum direction is correlated with the direction of the transverse spin of either a quark or a hadron, via a sin␾ dependence, as can be seen from Eq. 共2兲, for example. The reason for considering these functions is that this ‘‘handedness’’ of the transverse mo-mentum compared to the transverse spin is displayed by the single spin asymmetry data in, for instance, p p↑→␲X and p p→⌳↑X. However, the problem of using such functions is that naively they appear to be absent due to time reversal invariance. This conclusion would be valid if the hadronic state appearing in the definition of such functions is treated as a plane-wave state. One can then show that the functions are odd under the application of time reversal invariance, whereas hermiticity requires them to be even. If, however, initial- or final-state interactions are present, then time rever-sal symmetry will not prevent the appearance of nonzero 共naively兲 T-odd functions. In the case of a state like 兩Ph,Sh;X

典

, final-state interactions are certainly present and

nonzero fragmentation functions ⌬N_D

h/a↑ and ⌬NDh↑/a are expected. However, for distribution functions this issue poses severe problems and since we will only consider frag-mentation functions here, we refer to Refs. 关12,14,22兴 for more detailed discussions on this topic.

The main difference between the function ⌬NDh/a↑ as

originally proposed by Collins and the function under present investigation ⌬ND_h↑/a is that the former is a so-called chiral-odd function, which means that it couples quarks with left- and right-handed chiralities, whereas the latter function is chiral even. Since the perturbative 共pQCD兲 interactions conserve chirality, chirodd functions must al-ways be accompanied by a mass term or appear in pairs.

Both options restrict the accessibility of such functions and make them harder to be determined separately. On the other hand, the chiral-even fragmentation function can simply oc-cur accompanied by the unpolarized 共chiral-even兲 distribu-tion funcdistribu-tions, which are the best determined quantities, al-lowing for a much cleaner extraction of the fragmentation function itself. Moreover, since chiral-even functions can ap-pear in charged current mediated processes 共as opposed to chiral-odd functions兲, more methods of extraction are avail-able关31兴.

As it was studied in Ref.关32兴 the Collins function H⬜₁ 共or ⌬N_D

h/a↑) satisfies a sum rule arising from momentum

con-servation in the transverse directions. The same holds for the other k_⬜-odd, T-odd fragmentation function D⬜_1T 关33兴,

兺

h

冕

dz

冕

dk_⬜2 k_⬜2 z Mh

D⬜1T共z,k⬜兲⫽0, 共11兲

or in terms of the function⌬NDh↑/a,

兺

h

ChMh⬅

兺

h

冕

dz

冕

d2k_⬜k_⬜sin␾⌬NDh↑/a共z,k⬜兲⫽0, 共12兲 which is equivalent to Eq.共11兲 via Eq. 共4兲.

Notice that the above sum rule can be written as

兺

h

冕

dz

冕

d2k_⬜k_⬜Dˆh↑/a共z,k⬜兲⫽0, 共13兲

and the same holds independently for Dˆh↓/a(z,k_⬜). Equation 共13兲 has a clear nontrivial physical meaning: for each polar-ization direction (↑ or ↓) the total transverse momentum carried by spin 1/2 hadrons2 is zero.

The sum over hadrons prevents a straightforward applica-tion of the sum rule to the case of ⌳ production alone. It cannot be used as a constraint on the parametrization of the function to be fitted to the data. However, we note that the integral ChMhfor each hadron type h is the same function of

the energy scale 共implicit in all expressions兲 apart from as yet unknown normalization. In this sense it closely resembles the tensor charge. In other words, the running of the func-tions are the same for any type of hadron and there is no mixing with other functions. The ratios for different types of hadrons are constants, which allow for checks of consistency between sets of data obtained at different energies, without the need for evolution. These constants are universal, if in-deed the T-odd fragmentation functions are universal. This universality is the main point of interest here: one wants to see if ⌳ polarization from unpolarized quark fragmentation is independent of the initial state, as is implicit when writing down the factorized cross sections Eqs. 共8兲 and 共9兲. At the 1_{This is not in contradiction with the observed spin transfer (D}

NN)

in p p↑→⌳↑X 关29兴, which in the factorized approach can be

de-scribed in terms of the ordinary transversity distribution and frag-mentation functions, rather than in terms of⌬NDh↑/a.

2_{Strictly speaking, the sum over h is over all hadrons that carry}

transverse polarization, which might be true also for higher spin hadrons.

present time, this cannot be verified due to lack of data but some predictions can be given 关30兴 that will allow tests of this universality.

At present it is unclear up to what extent is factorization in Eqs.共8兲 and 共9兲 valid and how large nonfactorizable con-tributions could be in the kinematical range of the available data. However, extension of factorization theorems with in-clusion of k_⬜ is a natural ansatz关12兴 and further corrections might be small for ratios of cross sections. We are attempting a description of ⌳ polarization in unpolarized hadronic pro-cesses, based on perturbative QCD factorized elementary dy-namics and nonperturbative spin- and k_⬜-dependent frag-mentation functions: even qualitative agreement could be considered as substantial progress and as a starting point for further refinements and developments. Use of the same methods and the same results obtained here for proton-nucleon processes to predict⌳ polarization in other reactions will provide a clear and crucial test of our approach: it will reveal whether or not we have taken into account the main source of the observed polarization.

We only consider the quark fragmenting into a⌳ and not into other hyperons like the⌺0. The latter actually produces a significant amount 共20–30 %兲 of the ⌳’s via the decay ⌺0_{→⌳␥. The reason we do not introduce a separate ⌺}0 fragmentation function at this stage is that the factorized ap-proach by itself does not address such a separation 共it is about a generic spin-1/2 hadron of type h), unless one intro-duces some additional input like a model based on SU(3) or unless one applies it to separate sets of data for each hyperon 共which are not available yet兲 关34兴. Apart from that, for each quark flavor such a⌺0 fragmentation function would evolve in the same way as the ⌳ fragmentation function, which implies that their relative fractions stay constant. In this way we can view the ⌳ fragmentation function as an effective fragmentation function that includes the contamination due to⌺0’s. Indeed, the fragmentation functions we shall use in the next section have been obtained from fits to inclusive ⌳ productions in e⫹e⫺processes independently of their origin. In this respect the sum over h that appears in Eqs.共11兲–共13兲 should exclude hadrons already taken into account in the effective ⌳ polarizing fragmentation function.

At a later stage, one might make the distinction that the ⌺0 _{fragmentation is a different fraction of the effective ⌳} fragmentation function for different quark flavors. One can insert all this information with hindsight and correct for it, but the present approach cannot be used to acquire this in-formation unless the data could clearly distinguish the ⌳’s coming from ⌺0’s. Our approach of using an effective ⌳ fragmentation function would be more problematic if the⌺0 would decay into final states other than ⌳␥ 共where the branching ratio happens to be 100%兲: then only a part of the total ⌺0 fragmentation function would be included into the effective⌳ fragmentation function and this would be energy dependent.

We shall now consider both ⌳ and ⌳¯ production and attempt a determination of the polarizing fragmentation func-tions ⌬N_D

⌳↑/q by comparing results for P⌳ and P⌳¯ from Eqs. 共8兲 and 共9兲 with data from 关35–39兴.

II. NUMERICAL FITS OF DATA ON P_⌳FROM pN\⌳X

PROCESSES

Equation 共8兲 for proton-nucleon processes can be rewrit-ten as E_⌳d3␴pN→⌳X d3p_⌳ P⌳ ⫽

兺

a,b,c,d

冕

(⫹k_⬜) d2k_⬜

冋

冕

z_min 1 dz

冕

x_amin 1 dxa 1 ␲z x ¯ b 2 s 共⫺t⌽t兲 ⫻ fa/ p共xa兲fb/N共x¯b兲 d␴ˆab→cd d tˆ 共xa,x ¯_b,k⬜兲 ⫺兵k_⬜→⫺k_⬜其

册

⌬ND_⌳↑/c共z,k⬜兲, 共14兲 which deserves several comments and some explanation of notations.

In deriving Eq. 共14兲 from Eq. 共8兲 we have used the fact that ⌬ND_⌳↑/c(z,k⬜), Eq. 共3兲, is an odd function of k⬜; the (⫹k_⬜) integration region of k_⬜ runs over one half plane of its components.

The xb integration has been performed by exploiting the

␦(sˆ⫹tˆ⫹uˆ) function; the resulting value of xb is given by

x

¯_b⫽⫺ xat⌽t xazs⫹u⌽u

, 共15兲

where⌽t and⌽u are defined below.

k_⬜ could have any direction in the plane perpendicular to

p_c; however, due to parity conservation in the hadronization process, Eq. 共5兲, the only k_⬜ component contributing to the polarizing fragmentation function is that perpendicular to

P_⌳, i.e., the component lying in the production plane, the x-z plane in our conventions. To simplify the kinematics we shall then consider only the leading contributions of k_⬜ vec-tors in the x-z plane.

s, t, and u are the Mandelstam variables for the pN

→⌳X process; in the simple planar configuration discussed

above (p_c and k_⬜ both lying in the x-z production plane兲 they are related to the corresponding variables for the el-ementary processes by sˆ⫽2pa•pb⫽xaxbs, tˆ⫽⫺2pa•pc⫽共xa/z兲t⌽t共⫾k⬜兲, 共16兲 uˆ⫽⫺2p_b•p_c⫽共x_b/z兲u⌽_u共⫾k_⬜兲, with t⌽_t共⫾k_⬜兲⫽g共k_⬜兲

再

t⫿2k_⬜

冑

stu t⫹u ⫺关1⫺g共k⬜兲兴 t⫺u 2

冎

共17兲

u⌽u共⫾k⬜兲⫽g共k⬜兲

再

u⫾2k⬜

冑

stu t⫹u ⫹关1⫺g共k⬜兲兴 t⫺u 2

冎

共18兲 where g(k_⬜)⫽

冑

1⫺(k_⬜/ p_⌳)2 and where⫾k_⬜ refer respec-tively to the configuration in which k_⬜points to the left or to the right of pc. Up to the leading order in k⬜/ p⌳ one has

⌽t共k_⬜兲⫽1⫺ k_⬜ p_⌳

冑

u t ⌽u共k⬜兲⫽1⫹ k_⬜ p_⌳

冑

t u. 共19兲 The lower integration limits in Eq. 共14兲 are given by

xa min⫽⫺ u⌽_u共⫾k_⬜兲 zs⫹t⌽t共⫾k_⬜兲 , z⭓⫺t⌽t共⫾k⬜兲⫹u⌽u共⫾k⬜兲 s . 共20兲

Notice that the integration limits are slightly different for ⫹k⬜ and ⫺k⬜; when replacing k⬜ with ⫺k⬜ inside the

square bracket of Eq. 共14兲, one should not forget to change accordingly also the z and xaintegration limits and the value

of x¯_b Eq. 共15兲, although the results are only marginally af-fected by this.

Equation共14兲 can be schematically written as d␴pN→⌳XP_⌳ ⫽d␴pN→⌳↑X_⫺d␴pN→⌳↓X ⫽

兺

a,b,c,d fa/ p共xa兲丢fb/N共xb兲丢关d␴ˆab→cd共xa,xb,k⬜兲 ⫺d␴ˆab_→cd共x a,xb,⫺k_⬜兲兴丢⌬ND_⌳↑/c共z,k_⬜兲, 共21兲 which shows clearly that P_⌳ is a higher twist effect, despite the fact that the polarizing fragmentation function⌬N_D

h↑/ais a leading twist function: this is due to the difference in the square brackets 关d␴ˆ (⫹k_⬜)⫺d␴ˆ (⫺k_⬜)兴⬃k_⬜/ p_T similar to what happens for the single spin asymmetries in p↑p→␲X 关14,19兴.

In the computation of the unpolarized cross section E_⌳d3␴pN→⌳X/d3p_⌳intrinsic-transverse motion is significant only in limited phase space regions: we have checked that the values obtained in most of the kinematical regions of available data do not vary whether or not we take into ac-count k_⬜. Notice that when taking into account k_⬜, the ex-pression for E_⌳d3␴pN→⌳X/d3p_⌳is the same as Eq.共14兲 with the ⫺ inside the square brackets replaced by a ⫹ and ⌬N_D

⌳↑/c(z,k⬜) replaced by Dˆ⌳/c(z,k⬜).

When computing the cross sections for scattering off nu-clei pA→⌳↑X for which plenty of data are available, we have adopted the most simple incoherent sum neglecting nuclear effects. That is, for the scattering off a nucleus with A nucleons and Z protons we use

d␴pA→⌳X⫽Zd␴p p→⌳X⫹共A⫺Z兲d␴pn→⌳X. 共22兲

We have checked that different ways of taking into account nuclear effects leave results for P_⌳—a ratio of cross sections—almost unchanged. The partonic distribution func-tions in a neutron are obtained from the usual proton distri-bution functions by applying isospin invariance.

Equation共14兲 holds for any spin 1/2 baryon; we shall use it also for⌳¯ ’s using charge conjugation invariance to obtain q

¯_→⌳¯ _{fragmentation properties from q→⌳ properties,} which implies D_{⌳¯ /q¯}⫽D_⌳/q and⌬ND¯_⌳↑/q¯⫽⌬ND⌳↑/q.

We now use Eq.共14兲 in order to see whether or not it can reproduce the data and in order to obtain information on the new polarizing fragmentation functions. To do so we intro-duce a simple parametrization for these functions and fix the parameters by fitting the existing data on P_⌳ and P_⌳¯ 关35–

39兴.

We assume that⌬ND_⌳↑/c(z,k⬜) is strongly peaked around an average value k_⬜0 lying in the production plane, so that we can expect

冕

(⫹k_⬜) d2k_⬜⌬ND_⌳↑/c共z,k⬜兲F共k⬜兲⯝⌬0 N D_⌳↑/c共z,k_⬜ 0_兲F共k ⬜0兲. 共23兲 Consistently, since in this case F(k_⬜) depends weakly on k_⬜ when k_⬜ⰆpT, in the computation of the unpolarized cross

section we use

冕

(⫹k_⬜)

d2k_⬜Dˆ_⌳/c共z,k_⬜兲F共k_⬜兲⯝1

2D⌳/c共z兲F共k⬜ 0_{兲. 共24兲}

The average k_⬜0 value depends on z and we parametrize this dependence in a most natural way,

k_⬜0共z兲 M ⫽Kz

a_共1⫺z兲b_{, 共25兲}

where M is a momentum scale ( M⫽1 GeV/c); in perform-ing the fit we demand that K, a and b are constrained so that they satisfy the kinematical bound pq

2_⫽(p

⌳

2_⫺k

⬜2)/z2⭓p⌳2 ,

from which k_⬜2⭐(1⫺z2) p_⌳2 and

k_⬜0共z兲ⱗ共p_⌳兲min

冑

1⫺z⯝共1 GeV/c兲

冑

1⫺z, 共26兲 which implies a⭓0 and b⭓0.5. The values of K, a, and b resulting from our best fit will have a clear physical meaning.

We parametrize⌬₀ND_⌳↑/c(z,k_⬜ 0

) in a similar simple form: we know that it has to be zero when k_⬜⫽0 and z⫽1; in addition, the positivity of the fragmentation functions, Eq. 共5兲, requires, at any k_⬜ value, 兩⌬NDh↑/q(z,k_⬜)兩 ⭐Dˆh/q(z,k⬜). However, according to Eqs.共23兲 and 共24兲 and

to take into account the mentioned 关see comment after Eq. 共20兲兴 difference of the integration regions (⫹k⬜) and

(⫺k_⬜), which is significant at the boundaries of the kine-matical ranges共when pT⯝k⬜and when pT⯝pT max), we

pre-fer to impose the more stringent bound 兩⌬₀ND_⌳↑/c(z,k_⬜ 0 )兩 ⭐D⌳/c(z)/2. Following Ref. 关40兴, this is automatically

⌬0 N D_⌳↑/q共z,k_⬜ 0_兲 ⫽Nq

⬘

k_⬜0共z兲 M

冋

z␣q共1⫺z兲␤q c_qcq_d q d_q /共cq⫹dq兲cq⫹dq

册

D_⌳/q共z兲 2 ⫽Nq

⬘

K zcq共1⫺z兲dq c_qcq_d q d_q /共c_q⫹d_q兲cq⫹dq D_⌳/q共z兲 2 ⬅Nqzcq共1⫺z兲dq D_⌳/q共z兲 2 , 共27兲

where we have used Eq. 共25兲, and we require cq⫽a⫹␣q

⬎0, dq⫽b⫹␤q⬎0, and 兩Nq

⬘

兩K⭐1.

We are now almost fully equipped to compute P_⌳ and P_⌳¯; let us briefly discuss the remaining quantities appearing

in Eq.共14兲.

We sum over all possible elementary interactions com-puted at lowest order in pQCD. The polarizing fragmentation functions—parametrized as in Eq. 共27兲—are supposed to be nonvanishing only for ⌳ valence quarks, u, d, and s; simi-larly for⌳¯ valence antiquarks u¯, d¯, and s¯. All contributions to the unpolarized fragmentation functions—from quarks, antiquarks, and gluons—are taken into account.

We adopt the unpolarized distribution functions of Martin-Roberts-Stirling-Thorne 关41兴. We have explicitly checked that a different choice makes no difference in our conclusions. We fix the QCD hard scale of distribution共and fragmentation兲 functions at 2 (GeV/c)2, corresponding to an average value pT⯝1.5 GeV/c. Since the range of pT

val-ues of the data is rather limited, no evolution effect would be visible anyway.

We use the set of unpolarized fragmentation functions of Ref.关42兴, which allows a separate determination of D_⌳/qand D_⌳¯ /q, and which includes ⌳’s both directly and indirectly produced; it also differentiates between the s quark contribu-tion and the u and d contribucontribu-tions: the nonstrange fragmen-tation functions D_⌳/u⫽D_⌳/d are suppressed by an SU(3) symmetry breaking factor␭⫽0.07 as compared to D_⌳/s. In our parametrization of⌬₀ND_⌳↑/q(z,k_⬜0), Eq.共27兲, we use the same D_⌳/qas given in Ref.关42兴 keeping the same parameters ␣qand␤q (cqand dq) for all quark flavors but allowing for

different values of Nu⫽Nd and Ns.

We can now use pQCD dynamics together with the cho-sen distribution and fragmentation functions and the param-etrized expressions of the polarizing fragmentation functions in Eq.共14兲 to compute P_⌳and P_⌳¯; by comparing with data

we obtain the best fit values of the parameters K, a, b, Nu

⫽Nd, Ns, cq, and dq introduced in Eqs. 共25兲 and 共27兲.

Notice that we remain with seven free parameters, c_q and d_q being the same for all flavors.

Our best fit results 关␹2/degree of freedom共DOF兲⫽1.57兴, taking into account all data关35–39兴, are shown in Figs. 1 –5. They correspond to the best fit parameter values

K⫽0.69, a⫽0.36, b⫽0.53, 共28兲

Nu⫽⫺4.30, Ns⫽1.13, cq⫽6.58, dq⫽0.67.

共29兲

Let us comment in some details on our results.

In Figs. 1 and 2 we present our best fits to P_⌳as a func-tion of pTfor different xF values as indicated in the figures:

the famous approximately flat pTdependence, for pTgreater

than 1 GeV/c, is well reproduced. Such a behavior, as ex-pected, does not continue indefinitely with pT and we have

explicitly checked that at larger values of pT the values of

P_⌳ drop to zero: the shape of such a decrease, contrary to what happens in the pT region of the data shown here,

strongly depends on the assumptions about the nuclear cor-rections. It may be interesting to note that this falloff has not yet been observed experimentally, but is expected to be seen in the near-future BNL Relativistic Heavy Ion Collider data. FIG. 1. Our best fit to P_⌳data from p-Be reactions as a function of pTand for different xFbins as indicated in the figure. Only some

of the bins are shown, see Fig. 2 for complementary bins. The experimental results关37–39兴 are collected at two different c.m. en-ergies,

冑

s⯝82 GeV and

冑

s⯝116 GeV. For each xF-bin, the cor-responding theoretical curve is evaluated at the mean xF value in

the bin, and at

冑

s⫽80 GeV; the results change very little with the

energy. See the text for further details.

FIG. 2. Our best fit to P_⌳data from p-Be reactions as a function of pTand for different xFbins as indicated in the figure. Only some

of the bins are shown, see Fig. 1 for complementary bins. The experimental results关37–39兴 are collected at two different c.m. en-ergies,

冑

s⯝82 GeV and

冑

s⯝116 GeV. For each xF-bin, the cor-responding theoretical curve is evaluated at the mean xF value in

the bin and at

冑

s⫽80 GeV; the results change very little with the

Also the increase of兩P_⌳兩 with x_F at fixed p_Tvalues can be well described as shown in Fig. 3; the two curves corre-spond to pT⫽1.5 GeV/c 共solid兲 and pT⫽3 GeV/c

共dashed-dotted兲.

The best fits of Figs. 1 and 2 are compared to p-Be data 关36–39兴, these are collected at two different energies in the p-Be c.m. frame,

冑

s⯝82 GeV 关36–38兴 and

冑

s ⯝116 GeV 关39兴. Our calculations are performed at

冑

s ⫽80 GeV; we have explicitly checked that by varying the energy between 80 and 120 GeV, our results for P_⌳vary in the kinematical range considered here at most by 10%, in agreement with the observed energy independence of the data.

Some data from p-p collisions are also available; in Ref. 关35兴 a linear fit to P⌳(xF) is performed collecting all data

with pT⭓0.96 GeV/c for an average value

具

pT

典

⫽1.1 GeV/c. In Fig. 4 we show these data and the linear fit 共central line兲, the upper and lower lines show the fit error band. The solid line is our computation at pT⫽1.1 GeV/c

with all parameters fixed as in Eqs. 共28兲 and 共29兲; our fit reproduces the data with good accuracy.

In Fig. 5 we show our best fit results for P_⌳¯ as a function

of p_T for different x_F values, as indicated in the figure: in this case all data关36,38兴 are compatible with zero, with large errors, and the measured xFrange is not as wide as for P⌳as

expected from the lack of overlapping between ⌳¯ and nucleon valence quarks.

The resulting values of the parameters, Eqs.共28兲 and 共29兲, are very realistic; notice, in particular, that b essentially reaches its kinematical limit 0.5 and the whole function共25兲 giving the average k_⬜value of a⌳ inside a jet turns out to be very reasonable.

Mostly u and d quarks contribute to P_⌳, resulting in a

FIG. 4. Experimental results for P_⌳in p-p reactions as a func-tion of xFfrom Ref.关36兴. All data with pT⭓0.96 GeV/c are

col-lected and具pT典⫽1.1 GeV/c. Also shown is a linear fit to the data

taken from Ref.关36兴 共central line兲; the upper and lower dot-dashed lines show the corresponding fit error band. The solid curve shows the theoretical computation at pT⫽1.1 GeV/c with all parameters fixed as in Eqs.共28兲 and 共29兲.

FIG. 6. Plot of兩⌬₀ND_⌳↑/u兩共⫽ 兩⌬0

N

D_⌳↑/d兩) and⌬0

N

D_⌳↑/sas given

by Eq.共27兲 with the best fit parameters 共28兲 and 共29兲. For compari-son we also show the unpolarized fragmentation functions

D_⌳/u共⫽D_⌳/d) and D_⌳/s关42兴. FIG. 3. P_⌳data for p-Be reactions as a function of xFand for

different pTbins as indicated in the figure. The data are collected at

two different c.m. energies,

冑

s⯝82 GeV and

冑

s⯝116 GeV 关37–

39兴. The two theoretical curves, evaluated at

冑

s⫽80 GeV,

corre-spond to pT⫽1.5 GeV/c 共solid兲 and pT⫽3 GeV/c 共dot-dashed兲.

FIG. 5. Our best fit to P_⌳¯data from p-Be reactions as a function

of pT and for different xF bins as indicated in the figure. The

ex-perimental results 关36,38兴 are collected at the c.m. energies

冑

s

⯝82 GeV. For each xF-bin, the corresponding theoretical curve is evaluated at the mean xFvalue in the bin and at

冑

s⫽80 GeV; the

negative value of Nu; instead, u, d, and s quarks all

contrib-ute significantly to P_⌳¯ and opposite signs for N_u and N_sare

found, inducing cancellations.

In Fig. 6 we plot 兩⌬₀ND_⌳↑/u,d兩 and⌬0

N

D_⌳↑/s as given by Eq.共27兲 with the best fit parameters 共28兲 and 共29兲. We show, for comparison, also D_⌳/u,d and D_⌳/s: notice how a tiny value of the polarizing fragmentation functions in a limited large z region is enough to allow a good description of the data. This also shows that taking into account only valence quark contributions to ⌬ND_⌳↑/q, as we have done, is a jus-tified assumption.

A different set of unpolarized fragmentation functions can be found in the literature 关43兴: it holds for the quark frag-mentation into⌳⫹⌳¯ and gives D_{(⌳⫹⌳¯ )/q}rather than a sepa-rate expression of D_⌳/qand D_{⌳¯ /q}; it would be appropriate to compute the⌳⫹⌳¯ single spin asymmetry,

P_⌳⫹⌳¯⫽ d␴⌳↑⫹d␴⌳¯↑⫺d␴⌳↓⫺d␴⌳¯↓ d␴⌳↑⫹d␴⌳¯↑⫹d␴⌳↓⫹d␴⌳¯↓ ⫽

冉

P_⌳⫹d␴ ⌳¯ d␴⌳P⌳ ¯

冊冉

1⫹ d␴⌳¯ d␴⌳

冊

⫺1 . 共30兲

However, since one knows from experiments on p-N reac-tions that in the kinematical range of interest d␴⌳¯Ⰶd␴⌳ 共and this is confirmed in our scheme, simply due to the dominance of q over q¯ in a nucleon兲, one can safely assume P_⌳⫹⌳¯⯝P_⌳. 共31兲

This set—differently from the one of Ref. 关42兴—assumes SU(3) symmetry and takes D_⌳/u⫽D_⌳/d⫽D_⌳/s.

We have also computed P_⌳with this second set of frag-mentation functions; as in the previous case we have param-etrized⌬ND_⌳↑/q according to Eq.共27兲 with the same values of cq and dq for each flavor but different values of Nu,dand

Ns. We can equally well (␹2/DOF⫽1.85) fit the data on P⌳

obtaining fits almost indistinguishable from those of Figs. 1–4, with the best fit values

K⫽0.66, a⫽0.37, b⫽0.50, 共32兲

Nu⫽⫺28.13, Ns⫽57.53, cq⫽11.64, dq⫽1.23.

共33兲 Notice that also in this case of SU(3) symmetric fragmenta-tion funcfragmenta-tions D_⌳/q, and using only data on P_⌳, one reaches similar conclusions about the polarizing fragmentation func-tions ⌬ND_⌳↑/q: Nu,d⫽” Ns and not only is there a difference

in magnitude, but once more one finds negative values for ⌬0

N_D

⌳↑/u,dand positive ones for⌬0

N_D

⌳↑/s. This seems to be a well-established general trend. Plots analogous to those of Fig. 6, would show also in this case, Eqs. 共27兲 and 共33兲, ⌬0

N_D

⌳↑/s⬎兩⌬0

N_D

⌳↑/u,d兩.

Very recently, new sets of quark and antiquark fragmen-tation functions into a ⌳ based on a bag model calculation and a fit to e⫹e⫺ data have been published 关44兴. Both a SU(3) flavor symmetric and a SU(3) symmetry breaking

set are available, although at a rather too low energy scale (␮2⫽0.25 GeV2). Nevertheless, we have also tried using these sets in our scheme to fit the data on P_⌳and P_⌳¯: once

more we can fit the data better with the symmetric than with the asymmetric set, and again negative⌬₀ND_⌳↑/u,dand posi-tive ⌬₀ND_⌳↑/sare found with⌬0

N_D

⌳↑/s⬎兩⌬0

N_D

⌳↑/u,d兩.

III. CONCLUSIONS

A phenomenological approach has been developed to-wards a consistent explanation and predictions of transverse single spin effects in processes with inclusively produced hadrons; we work in a kinematical region where pQCD and the factorization scheme can be used, but pT is not much

larger than intrinsic k_⬜so that higher twist contributions are still important. This applies to several processes for which data are available, like p↑p→␲X 关16,19兴 and pN→⌳↑X. The single spin effect originates in the fragmentation pro-cess, either of a transversely polarized quark into an unpo-larized hadron—Collins’s effect 关12兴—or of an unpolarized quark into a transversely polarized hadron—the polarizing fragmentation functions 关23兴. Single spin effects in quark distribution functions 关11兴 have also been discussed 关14,17兴. We have considered here the well-known and long-standing problem of the polarization of ⌳ hyperons, pro-duced at large p_Tin the collision of two unpolarized hadrons. We have assumed a generalized factorization scheme, with the inclusion of intrinsic transverse motion, with pQCD dy-namics; the new spin- and k_⬜-dependent polarizing fragmen-tation functions⌬ND_⌳↑/qhave been parametrized in a simple way and data on p Be→⌳↑X, p Be→⌳¯↑X and p p→⌳↑X have been used to determine the values of the parameters that give a best fit to the experimental measurements.

The data can be described with remarkable accuracy in all their features: the large negative values of the⌳ polarization, the increase of its magnitude with xF, the puzzling flat pT

ⲏ1 GeV/c dependence, and the

冑

s independence; data from p-p processes are in agreement with data from p-Be interactions and also the tiny or zero values of⌳¯ polarization are well reproduced. The resulting functions ⌬ND_⌳↑/q are very reasonable and realistic.

Different sets of unpolarized fragmentation functions— either SU(3) symmetric or not—lead to very similar conclu-sions about these new polarizing fragmentation functions de-scribing the hadronization process of an unpolarized quark into a polarized ⌳: they have opposite signs for u and d quarks as compared with s quarks and their magnitudes are larger for s quarks. They are sizeable with respect to the unpolarized fragmentation functions only in limited z re-gions, yet they can describe the experiments remarkably well.

These polarizing fragmentation functions have a partonic interpretation and a formal definition, Eq.共2兲; they are free from the ambiguities related to initial-state interactions that might affect analogous distribution functions and we expect them to be universal, process-independent functions. Our pa-rametrization of ⌬N_D

⌳↑/q should allow us to give a predic-tion for ⌳ polarization in other processes; a study of lp

ACKNOWLEDGMENTS

M.A. thanks RIKEN BNL Research Center for support during a Workshop last year when this work was initiated.

D.B. thanks RIKEN, Brookhaven National Laboratory and the U.S. Department of Energy for support under Contract No. DE-AC02-98CH10886. U.D. and F.M. thank COFINANZIAMENTO MURST-PRIN for partial support.

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