We analyse recent data on the production of forward neutrons in deep
inelastic scattering at HERA in the framework of a perturbative QCD description
for semi-inclusive processes, which includes fracture functions.Comment: To be published in Proceedings of the Madrid low-x Workshop,
  Miraflores de la Sierra, June 18-21, 199

de Florian, D.

Sassot, R.

English

arXiv

We analyse recent data on the production of forward neutrons in deep inelastic scattering at HERA in the framework of a perturbative QCD description for semi-inclusive processes, which includes fracture functions.We analyse recent data on the production of forward neutrons in deep inelastic scattering at HERA in the framework of a perturbative QCD description for semi-inclusive processes, which includes fracture functions

De Florian, D.

CERN Document Server

TARGET FRAGMENTATION AND FRACTURE FUNCTIONS.D. DE FLORIANTheoretical Physics Division, CERN, CH 1211 Geneva 23, SwitzerlandR. SASSOTDepartamento de Fsica, Universidad de Buenos AiresCiudad Universitaria, Pab.1 (1428) Bs.As., ArgentinaWe analyse recent data on the production of forward neutrons in deep inelasticscattering at HERA in the framework of a perturbative QCD description for semi-inclusive processes, which includes fracture functions.In the most naive quark-parton model picture, the semi-inclusive crosssection for the production of a hadron h from the deep inelastic scattering ofcharged leptons is usually taken to be given by1:dhpdx dy dzh=(1 + (1  y)2)2y2Xi=q;qcifi=p(x)Dh=i(zh) ; (1)where, fi=pis the parton distribution of avour i, Dh=iis the fragmentationfunction of a hadron h from a parton i, and ci= 4e2qi2=x(P + l)2. Thekinematical variables used to characterize these processes are:x =Q22P  q; y =P  qP  l; zh=P  hp  q=EhEp(1  x)1  cos h2; (2)where q is the transferred momentum ( q2= Q2), l and P are the incominglepton and proton momenta respectively. Eh, Epand hare the producedhadron and target nucleon energies, and the angle between the hadron and thetarget in the centre of mass of the virtual photon-proton system, respectively.Although next to leading order corrections to this cross section are also wellknown, and have been shown to give a very good description of the so-calledcurrent fragmentation region (h> =2), the target fragmentation region,which corresponds to h= 0 (zh= 0), cannot be described with this simpliedpicture. First of all, it is easy to see that, at the lowest order, hadrons canonly be produced antiparallel to the target nucleon (h= ), excluding theforward congurations. On the other hand, going to next to leading order,1the corrections to the cross section develop divergences proportional to 1=zh,related to soft emission (Eh= 0), and also to collinear congurations wherehadrons are produced in the direction of the remnant target (h= 0). Since atlowest order hadrons cannot be produced in that direction, it is not possible tofactorize the divergence, as usual, into parton distributions and fragmentationfunctions. Then, in order to describe hadrons produced in the target frag-mentation region even at the lowest order, and also to be able to perform athigher orders a consistent factorization of divergences originated in the currentfragmentation region, a new distribution has to be introduced, the so-calledfracture functions, Mi;h=N(x; (1   x)z)2;3. These distributions represent theprobability of nding a parton of avour i and a hadron h in the target N(here z = Eh=Ep(1  x)).Therefore the complete leading order expression for the cross section be-comesdhpdx dy dz=(1 + (1  y)2)2y2Xi=q;qcifi=p(x)Dh=i(z)+ (1  x)Mi;h=p(x; (1  x)z): (3)Higher order corrections to this kind of cross section can be found in refs.3;4;5.The scale dependence of fracture functions at O(s) is driven by two kindsof processes, which contribute to the production of hadrons in the remnanttarget direction: the emission of collinear partons from those found in thetarget (the usual source of scale dependence of parton distributions, oftencalled homogeneous evolution), and those where partons radiated from theone to be struck by the virtual probe, fragment into the measured hadron (theso-called inhomogeneous term). These two contributions lead at leading orderto the following equation:@@ logQ2Mi;h=p ; ;Q2=s(Q2)2Z1=(1 )duuPij(u)Mj;h=Nu; ;Q2+s(Q2)2Z=(+)du(1  u)^Pi;lj(u) fj=pu;Q2Dh=lu(1  u); Q2; (4)where Pij(u) and^Pi;lj(u) are the regularized and real Altarelli-Parisi splittingfunctions, respectively.Recently, the ZEUS Collaboration has measured DIS events identifyinghigh-energy neutrons in the nal state6, at very small angles with respect tothe proton direction (lab 0:75 mrad), in the kinematical range given by310 4< x < 610 3, 10 < Q2< 100 GeV2and high xL' z(1 x) > 0:30.2The ZEUS Collaboration have reported that events with xL 0:50 representa substantial fraction (of the order of 10%) of DIS events. In the framework ofa picture for semi-inclusive processes including fracture functions, as the oneoutlined above, the ZEUS ndings can be represented (at LO) byR1 x0:50dhpdx dy dxLdxLdpdx dyR1 x0:50Mn=p2 x; xL; Q2dxLFp2(x;Q2): (5)In eq.(5) we have also dened the equivalent to F2for fracture functions:Mn=p2(x; xL; Q2)  xPie2iMi;n=p(x; xL; Q2) and we have made explicit theintegration over a nite (measured) range of xL.In g. 1 we show the experimental outcome for this fracture function (asdened in eq. (5) and at Q2= 10 GeV2, taking advantage of the negligibleQ2dependence of the data), and we compare it to Fp2and FpL. We also showthe contribution to the same observable coming from current fragmentationprocesses, which is pure NLO and it is about 8 orders of magnitude smallerthan the experimental data.Fracture functions, as parton distributions in general, are essentially of anon-perturbative nature and have to be extracted from experiment. However,their close relation with fragmentation and structure functions allows in certainextreme cases a model estimate for them, which can then be compared withactual measurements and evolved with the corresponding evolution equations.As an example, recently, a very sensible model estimate for the production offorward hadrons in DIS7, exploiting the idea of non-perturbative Fock com-ponents of the nucleon has been proposed. In this approach the semi-inclusiveDIS cross sections, and through them the corresponding fracture functions at acertain input scale Q20, can be interpreted as the product of a ux of neutronsin the proton (integrated over p2T) times the structure function of the pionexchanged between them, i.e.Mn=p2 x; xL; Q20' n=p(xL)F+2x1  xL; Q20: (6)Using a non-perturbative computation of the ux, which is in very good agree-ment with experimental data on high energy neutron and ++production inhadron-hadron collisions and a parametrization for the pion structure function8, in g. 1 we make a comparison between the model estimate and the datafor Q20= 10 GeV2, nding a remarkable agreement between them.The success of the model estimate encourages us to go further and usethe functional dependence of fracture functions, induced by the model andcorroborated by the data, to analyse also the Q2dependence.3Figure 1: Fracture function of neutrons in protons as measured by ZEUS compared to themodel prediction and current fragmentation contributionsWe rst analyse the more familar process of neutron production. Sincethe probability of current parton fragmentation into a neutron (given by frag-mentation functions) is comparatively small with respect to that of processesoriginated in the target (fracture functions), no signicant eects are expectedin the scale evolution arising from the inhomogeneous term in this process.The evolution is mainly driven by the usual homogeneous term of the evo-lution equations leading to an almost constant ratio between the number ofneutron tagged events and that of all DIS events, as observed by ZEUS.However, the scale dependence induced in the cross section for the pro-duction of pions, at least in the kinematical region of very small x and smallxL, can be considerably aected by the inhomogeneity, given that soft pionsare produced more copiously from quarks than from neutrons. In order toanalyse these features of the evolution, we estimate the proton to pion frac-ture function at some input scale Q20using the same ideas formerly applied toneutron production, and noticing that the ux can be straightforwardly ob-tained from the one used in the last section by means of the crossing relation+=p(xL) = +n=p(1  xL). Then, the proton to pion fracture function can beapproximated byM+=p2 x; xL; Q20' +=p(xL)Fn2x1  xL; Q20: (7)In g. 2a we show the model estimates for proton to pion fracture functions4(taking Q20= 4 GeV2), integrated over two dierent bins of xL, comparedwith the contribution coming from the current fragmentation processes. Weassume here the same restrictions as in the data from the ZEUS Collaborationfor neutron production.Figure 2: a) Prediction for the fracture function of +in protons for two dierent bins ofxLand the current fragmentation contribution. b) Evolution of the fracture function of +in protons for xL= 0:50 and Q20= 1 GeV2, c) xL= 0:10, and d) Q20= 4 GeV2Of course, the model is not expected to work over the whole kinematicalregion and, in fact, any deviation from the scale dependence implied in eq.(8) (note that the ux is assumed to be Q2independent) would show thebreakdown of the approximated factorization hypotesis. However, the ansatzin eq. (8) can be taken as an eective relation, valid at some initial value ofQ20, for which the estimated ux is adequate, and therefore provides a sensibleinput distribution. As usual, the correct scale dependence is that given by theevolution equations for fracture functions, and that is the aim of our next step.In order to study the eect of the inhomogeneity in the evolution we takedierent values of xL, and keep them xed while we analyse the x and Q2dependence of fracture functions induced by both the homogeneous and thecomplete evolution equations. In g. 2b we show the result of an evolution5from Q20= 1 GeV2at xL= 0:50. Both solutions, the homogeneous (dottedline) and the complete (solid line) are superimposed, the dierence being lessthan 0:1%. These behaviours are perfectly compatible with the results ob-tained by the ZEUS Collaboration in the case of neutron production (wherethe inhomogeneity contributes about 10 times less) in the same kinematicalregion, where no dierence has been found in the evolution between F2andM2.However, for smaller xLthe situation is completely dierent. As the frag-mentation function increases with lower values of the argument, the inhomoge-neous contribution becomes much more relevant and its eect in the evolutionis sizeable. In fact, g. 2c shows the evolution result for xL= 0:10 and Q20= 1GeV2, where the full evolution results outsize the homogeneous one by a factorof 4 at small x. These corrections are smaller if the ansatz of eq. (8) is assumedto be valid at values of Q20= 4 GeV2(g. 2d) but still remain considerable.Concluding, in this paper we have analysed recent experimental data on theproduction of forward neutrons in DIS in terms of fracture functions, ndingthat the main features of the data can be fairly reproduced by this perturbativeQCD approach, once a non-perturbative model estimate for the input fracturefunctions is given. Studying the evolution properties of these fracture functionsin the specic case of forward pions in the nal state, we have found that theeects of the inhomogeneous term in the evolution equations are large andmeasurable, particularly in the kinematical region of very small x and smallxL. These eects are negligible for large values of xL, justifying the use ofthe usual homogeneous Altarelli-Parisi equations for, as an example, the t-integrated diractive structure function FD(3)2(xIP; ;Q2), which is just thefracture function of protons in protons Mp=p2 xIP; (1  xIP); Q29.References1. G.Altarelli, R.K.Ellis, G.Martinelli and S.Y.Pi, Nucl. Phys. B 160, 301(1979).2. L. Trentadue and G. Veneziano, Phys. Lett. B 323, 201 (1994).3. D. Graudenz, Nucl. Phys. B 432, 351 (1994).4. D. de Florian, C.A. Garca Canal and R. Sassot, Nucl. Phys. B 470, 195(1996).5. D. de Florian and R. Sassot, Nuc. Phys. B 488, 367 (1997).6. ZEUS Collaboration, M. Derrick et al., Phys. Lett. B 384, 388 (1996).7. H. Holtmann et al., Phys. Lett. B 338, 363 (1996).8. M. Gluck, E. Reya and A. Vogt, Z. Phys. C 53, 651 (1992).9. D. Graudenz and G. Veneziano, Phys. Lett. B 365, 302 (1996).6

Target fragmentation and fracture functions

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Target Fragmentation and Fracture Functions

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