The structure of the strange sea of the proton
TThe structure of the strange sea of the proton
C. ´Avila, ∗ L. Salazar-Garcia, and J.C. Sanabria † Departamento de F´ısica, Universidad de los Andes, A.A. 4976, Bogota, D.C., Colombia
J. Magnin ‡ Centro Brasileiro de Pesquisas F´ısicas, Rua Dr. Xavier Sigaud 150, Urca 22290-180, Rio de Janeiro, Brazil
In this work we study the strange sea of the proton using a version of the Meson Cloud Modelcontaining both, efective and perturbative degrees of freedom. We construct the s and ¯ s partondistributions functions at the initial energy scale, Q , where QCD evolution starts. The initial s and¯ s pdfs depend on a number of parameters which we fix by comparison to parameterizations of thestrange sea of the nucleon obtained in a recent global fit to experimental data, allowing for a s − ¯ s asymmetry. We show that the model describes well the strange sea of the proton and argue thatit can be a phenomenologically motivated alternative to the usual input parameterizations used infits to experimental DIS data. PACS numbers: 14.20.Dh, 14.65.Bt
I. INTRODUCTION
The composite nature of hadrons in term of quark andgluon degrees of freedom is today firmly stablished. It isalso firmly believed that the internal dynamics of hadronsis determined by the strong interactions between quarksand gluons, as governed by Quantum Chromodynamics(QCD). However, a detailed theoretical description ofthe hadron structure is still missing because QCD canonly be solved in the perturbative regime, correspond-ing to the short distance domain probed in hard colli-sions, whereas the long distance part of the interactionrequires a non-perturbative treatment usualy supplied byeffective models, latice simulations, etc. It is worth not-ing that is just the long distance part of the interactionwhich is responsible for the hadron as a bound state ofquarks and gluons. Indeed, hadrons are made up of afixed number of valence quarks plus a varying numberof sea quark-antiquark pairs and gluons, being these lastthe “glue” which keeps valence quarks together forminga hadron. As a matter of fact, the sea q − ¯ q pairs andgluons which bring together the valence quarks into abound state are the so-called intrinsic sea [1], which hasto be distinguished from the extrinsic sea generated byQCD evolution and consequently, dependent on the en-ergy scale Q .From an experimental point of view, the struc-ture of nucleons is by far the best known hadronstructure, existing today a variety of parton distribu-tion function (pdf) parameterizations extracted fromdata. Although the analysis of the existing data hasconfirmed the impresive success of the Quark Par-ton Model and the Dokshitzer-Grivov-Lipatov-Altarelli-Parisi (DGLAP) evolution equations [2] when describing ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] the nucleon structure, in the Bjorken regime, in terms ofpdf’s given as a function of x , the fraction of the nucleonmomentum carried by partons, and Q , the energy scalewhich in Deep Inelastic Scattering (DIS) experiments isidentified with the 4-momentum transfer between the lep-ton and the nucleon, several questions remain still un-solved. Among them, what is the functional form - andwhy - of the valence and sea quark and gluon pdf’s at theenergy scale Q where QCD evolution starts; what is thevalue of the initial Q energy scale; what is the structureof the so-called intrinsic sea of q − ¯ q pairs and gluons; etc.Of course, finding answers to these questions requires todeal with the long distance - confining - realm of QCD,thus, as long as there is no solution of QCD in the low Q regime, we have to rely on effective models.Furthermore, the structure of the nucleon’s intrinsicsea of quarks and gluons reflects the dynamics of non-perturbative QCD. In fact, the form of the initial pdfsat the Q initial scale, as well as the relative abundanceof the different quark flavors are the footprint of sub-tle non-perturbative QCD dynamical effects. Notice, forinstance, that the experimentally observed ¯ d/ ¯ u asym-metry [3] and the violation of the Gottfried sum rule(GSR) [4] cannot be described in terms only of perturba-tive QCD and gluon splitting [5], as the ratio ¯ d/ ¯ u wouldbecame equal to one, due to the equal probability ofgluon splitting into d ¯ d or u ¯ u pairs. The distribution ofstrange and anti-strange quarks in the nucleon sea is an-other nontrivial aspect of the nucleon structure. Fromthe experimental side, some evidence has been found ona possible s − ¯ s asymmetry coming from global fits todata [6]. Notice that there is no fundamental symme-try preventing s ( x ) (cid:54) = ¯ s ( x ) in the nucleon, provided that (cid:82) [ s ( x ) − ¯ s ( x )] = 0. On the theoretical side, specula-tions about a possible | KH (cid:105) component in the nucleonwave function, where K, H are virtual Kaon and Hy-peron states, leading to a s − ¯ s asymmetry, date since1987, with the pionering work by Signal and Thomas [7].Since then on, several models have been proposed in the a r X i v : . [ h e p - ph ] A p r literature [8, 9], with different predictions. It is worth tonote also that a s − ¯ s asymmetry in the nucleon is gener-ated perturbatively starting at Next-to-Next to LeadingOrder (NNLO) [10] because at this order the splittingfunctions P qq and P q ¯ q are different. That asymmetry, al-though very small, should compete with the s − ¯ s asym-metry generated by the non-perturbative dynamics of thebound state.In this paper we shall investigate to what extent a non-perturbatively generated s − ¯ s asymmetry can describethe strange sea asymmetry found in recent global fits toexperimental data. In order to do that, in Section IIwe will revise a model for the generation of the intrin-sic s/ ¯ s sea of the nucleon, following in Section III witha comparison of the model with experimental data andrecent parameterizations of the strange sea of the pro-ton. Section IV will be devoted to further discussion andconclusions. II. THE NON-PERTURBATIVE STRANGE SEAOF THE PROTON
We start by considering a simple picture of the nucleonin the infinite momentum frame as being formed by threedressed valence quarks - valons , v ( x ) - which carry all ofits momentum [11], v u ( x ) = 2 β ( a u + 1 , b u + 1) x a u (1 − x ) b u ,v d ( x ) = 1 β ( a d + 1 , b d + 1) x a d (1 − x ) b d , (1)with x the fraction of momentum carried by the valonwith respect to the proton.In the framework of the Meson Cloud Model (MCM),the nucleon can fluctuate to a meson-baryon bound statecarrying zero net strangeness. As a first step in sucha process, we may consider that each valon can emita gluon which, before interacting, decays perturbativelyinto a s ¯ s pair. The probability of having such a perturba-tive q ¯ q pair can be computed in terms of Altarelli-Parisisplitting functions [2] P gq ( z ) = 43 1 + (1 − z ) z ,P qg ( z ) = 12 (cid:0) z + (1 − z ) (cid:1) . (2)These functions have the physical interpretation as theprobability of gluon emision and q ¯ q creation with momen-tum fraction z from a parent quark or gluon respectively.Hence, q ( x, Q ) = ¯ q ( x, Q ) = N α st ( Q )(2 π ) × (cid:90) x dyy P qg (cid:18) xy (cid:19) (cid:90) y dzz P gq (cid:16) yz (cid:17) v ( z ) (3) FIG. 1: Sea q/ ¯ q parton distribution in the proton as given byEq. 2. is the joint probability density of obtaining a quark oranti-quark coming from subsequent decays v → v + g and g → q + ¯ q at some fixed low Q . In eq. (3), N is a normalization constant which should scale with themasses of the flavors being created so that to a heavierflavor, corresponds a smaller N . Since the valon distribu-tion does not depend on Q [11], the scale dependence ineq. (3) only exhibits through the strong coupling constant α st . The range of values of Q at which the process ofvirtual pair creation occurs in this approach is typicallybelow 1 GeV . A tipical sea quark distribution obtainedfrom eq. (3), is shown in Fig. (1), where, in the spirit ofkeeping as simple as possible the description of the pro-cess , whe have assumed v u = v d and used a u = a d = 0 . b u = b d = 2 in eqs (1).Once a s ¯ s pair is produced, it can rearrange itselfwith the remaining valons so as to form a most ener-getically favored Kaon-Hyperon bound state. In orderto obtain the Kaon and Hyperon probability densities inthe | KH (cid:105) component of the proton wave function, thewell known approach of the recombination model [12]has been used [8]. Notice however that the Kaon andHyperon probability densities obtained in this way canbe represented in terms of the simple forms [8] P K ( x ) = 1 β ( a K + 1 , b K + 1) x a K (1 − x ) b K ,P H ( x ) = 1 β ( a H + 1 , b H + 1) x a H (1 − x ) b H , (4)which are both properly normalized to unity, as it shouldbe for a bound state of one Kaon and one Hyperon andin order to cope with the zero net strangeness of theproton. These forms are also consistent with the valonmodel for a hadron made of two partons bound state.The coefficients a K , b K , a H , b H in eqs. (4) are notindependent. In fact, as the Kaon and Hyperon have toexhaust the momentum of the proton, then (cid:90) dx [ xP H ( x ) + xP K ( x )] = 1 , (5)giving the constraintΓ( a K + b K + 2)Γ( a K + 2)Γ( a K + 1)Γ( a K + b K + 3) +Γ( a H + b H + 2)Γ( a H + 2)Γ( a H + 1)Γ( a H + b H + 3) = 1 . (6)Finally, the non-perturbative strange and anti-strangesea distributions in the nucleon can be computed bymeans of the two-level convolution formulas s NP ( x ) = N (cid:90) x dyy P H ( y ) s H ( x/y ) , (7)¯ s NP ( x ) = N (cid:90) x dyy P K ( y ) ¯ s K ( x/y ) , (8)where the sources s H ( x ) and ¯ s K ( x ) are the probabil-ity densities of the strange valence quark and anti-quarkin the Hyperon and Kaon respectively, evaluated at thehadronic scale Q [7]. In principle, to obtain the non-perturbative distributions given by eqs. (8), one shouldsum over all the strange Kaon-Hyperon fluctuations ofthe nucleon but, since such hadronic Fock states are nec-essarilly off-shell, the most likely configurations are thoseclosest to the nucleon energy-shell, namely Λ K + , Σ + K and Σ K + , for a proton state. For the sake of simplicity,we will only consider a generic Kaon and Hyperon insidethe proton.For the s H ( x ) and ¯ s K ( x ) probability densities in eqs.(7) and (8) we also used the simple forms s H ( x ) = 1 β ( a sH + 1 , b sH + 1) x a sH (1 − x ) b sH , ¯ s K ( x ) = 1 β ( a sK + 1 , b sK + 1) x a sK (1 − x ) b sK , (9)according to the valon model. The coefficients a sK , b sK , a sH , b sH in eqs. (9) have to be determinedby comparison to experimental data. III. COMPARISON WITH DATA
In order to fix the coefficients of the model, we comparewith the s and ¯ s parton distribution functions[15] givenin Ref. [6], which have been recently obtained in a globalfit to DIS data. The comparison has been done by meansof a simultaneous least square fit of the model to xs ( x ) + x ¯ s ( x ) and xs ( x ) − x ¯ s ( x ) at Q = 20 GeV . The resultsof the fit are shown in Figs. (2) and (3) and in Table I.The procedure of the fit was as follows: i) a set ofinitial values for the parameters was chosen and the non-perturbative s NP and ¯ s NP strange quark distribution TABLE I: Coefficients for s ( x ) and ¯ s ( x ) obtained from simul-taneous fits to the strange parton distributions funcions inthe proton as given in Ref. [6]. a H = 2 .
889 is fixed by therequirement of momentum conservation given by eq. (6). b H . ± . a K . ± . b K . ± . a sH . ± . b sH . × − ± . a sK . ± . b sK . ± . N .
019 0 . xs ( x ) − x ¯ s ( x ) at Q = 20 GeV . Dashed line: theasymmetry obtained in Ref. [6]. Full line: the result of ourfit. The shadow area is the uncertainty of the fit. functions were calculated according to eqs. (7) and (8).Then ii) the s NP and ¯ s NP distributions were evolvedfrom Q up to Q = 20 GeV and the squared distance S = n (cid:88) i =1 (cid:34) (cid:0) y d ( x i ) − y thd ( x i ) (cid:1) σ d ( x i )+ (cid:0) y s ( x i ) − y ths ( x i ) (cid:1) σ s ( x i ) (cid:35) (10)was calculated. In eq. (10), y k ( x i ) = xs NP ( Q , x i ) ± x ¯ s NP ( Q , x i ) and y thk = xs ( Q , x i ) ± x ¯ s ( Q , x i ), withindex k = d, s refering to the difference and the sumrespectively. An arbitrary error σ d/s ( x i ) correspondingrespectively to 10% of the value of the difference and thesum at x i has been considered to perform the fit. Finallya new set of parameters was chosen and the procedurehas been repeated until a minimum in S was reached. FIG. 3: xs ( x ) + x ¯ s ( x ) at Q = 20 GeV . Dashed line: thedistribution obtained in Ref. [6]. Full line: the result of ourfit. In the insert is shown the result of our fit for 0 . < x < | KH (cid:105) Fock state of the proton. Dashed line: Kaon probability den-sity in the | KH (cid:105) . The package MINUIT has been used to perform the fit.Notice also that the QCD evolution of the combination s + ¯ s depends on the whole set of parton distributionfunctions in the proton, for which we used the pdf’s ofRef. [6].The best fit has been obtained using Q = 0 . FIG. 5: Strange (left) and anti-strange (right) quark distribu-tions in the proton at Q = 20 GeV . Our model compared tothe V. Barone et al. [6], MRST [13] and CTEQ5 [14] strangeparton distribution functions. strange parton distributions found in Ref. [6] for x > ∼ . x our results are below theresults of the global fit of Ref. [6]. This can be due to adeficit in the content of gluons, as seems to be indicatedby the fact that a good fit is only obtained for extremelylow values of Q .Concerning the model itself, the Kaon and Hyperondistributions functions in the | KH (cid:105) Fock state of the pro-ton are displayed in Fig. (4). The momentum carried bythe Hyperon in the | KH (cid:105) Fock state is xP H ( x ) = 0 . xP K ( x ) = 0 .
4, agree-ing with the common intuition that the Hyperon carriesmore momentum than the Kaon in the | KH (cid:105) componentof the proton wave-function.In Fig. (5) the strange and anti-strange quark distribu-tions at Q = 20 GeV are shown and compared to the xs and x ¯ s distributions found in Ref. [6]. We also com-pare to the MRST [13] and CTEQ5 [14] strange quarkpdfs, which have been determined imposing s = ¯ s . Asshown in the figure, while our strange quark and anti-quark pdfs and those of Ref. [6] are consistent in thefull range 0 . < x <
1, they deviate from the behaviorshown by the MRST and CTEQ5 sets at x > ∼ . IV. CONCLUSIONS
In this paper we have presented a model for the non-perturbative structure of the strange sea of the proton. Inthe model, the non-perturbative s NP and ¯ s NP intrinsicsea quark distributions of the proton are given in terms ofa convolution of Hyperon and Kaon probability densitiesand valence quark distributions inside a | KH (cid:105) compo-nent of the proton wave function. This naturally gen-erates an asymmetry in the momentum distributions ofthe strange and anti-strange quarks in the proton, sincethe Hyperon, being heaviest than the Kaon, carries moremomentum in the | KH (cid:105) wave funtion component. Themodel depends on eight parameters which have to befixed by fits to experimental data.Parameters of the model have been fixed by fits to thestrange quark distributions found in a recent global fit toDIS data [6]. As shown in section III, the model describesqualitatively well the behavior of the s − ¯ s as well as s + ¯ s distributions, being this agreement better in the region x > ∼ .
1. For lower x , the s + ¯ s distribution is below thecorresponding curve given by the distributions of Ref. [6].The probability of the | KH (cid:105) fluctuation of the proton isabout 0 . N .The fact that the model does not describe well the s + ¯ s distribution at x < . s and ¯ s distributions have been determined by fits to the s − ¯ s and s + ¯ s distributions found in [6]. This restrictthe space of parameters allowed to the fit. Second, thegluon distribution found in Ref. [6] seems to be insuffi-cient to generate enough s/ ¯ s quarks at low momentum,as evidenced by the fact that while the s − ¯ s distributionis well described in the whole range 0 < x <
1, the s + ¯ s is not. This could also explain why a good fit is obtainedonly for extremely low values of Q , the scale at whichperturbative QCD evolution starts. And third, a mean-ingful comparison of the model has to be done through a global fit to experimental data.Notice also that no NNLO effects have been includedin the QCD evolution of the strange parton distribu-tions. The inclusion of those effects should produce aslightly bigger non-perturbative contribution to the pro-ton wave funtion to compensate the opposite asymmetrywhich arises at NNLO [10].Finally we would like to emphasize that the model pre-sented here can be taken as a phenomenologically moti-vated alternative for the description of the input strangeand anti-strange quark distributions used in fits to ex-perimental data. Notice that the model allows for a fullrepresentation of the non-perturbative processes insidethe proton in terms of well known mechanisms such asthe splitting of quarks and gluons and recombination,which after all must account for the dynamics of the pro-ton as a bound state. Another interesting aspect of themodel is that it can shed light on the parton structureof real Kaons and Hyperons, since the structure of thestrange sea of the proton is related to the strange va-lence quark distributions at a low Q scale of the formerstrange mesons and baryons. Acknowledgements
Support for this work has been received from the “Fun-daci´on para la promoci´on de la Investigaci´on y la tec-nologia”, Banco de la Rep´ublica de Colombia under con-tract No. 2407 and FAPERJ, under project No. E-26/110.266/2009. J.M. acknowledges the warm hospital-ity in the Physics Department, Universidad de los Andes,where part of this work has been done. [1] S.J. Brodsky, P. Hoyer, C. Peterson and N. Sakai, Phys.Lett.
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