Exclusive processes with a leading neutron in ep collisions
aa r X i v : . [ h e p - ph ] A p r LU TP 15-55December 2015
Exclusive processes with a leading neutron in ep collisions V.P. Gon¸calves , , F.S. Navarra and D. Spiering Department of Astronomy and Theoretical Physics, Lund University, SE-223 62 Lund, Sweden High and Medium Energy Group, Instituto de F´ısica e Matem´atica, Universidade Federal de PelotasCaixa Postal 354, 96010-900, Pelotas, RS, Brazil. Instituto de F´ısica, Universidade de S˜ao Paulo,C.P. 66318, 05315-970 S˜ao Paulo, SP, Brazil.
In this paper we extend the color dipole formalism to the study of exclusive processes associatedwith a leading neutron in ep collisions at high energies. The exclusive ρ , φ and J/ Ψ production,as well as the Deeply Virtual Compton Scattering, are analysed assuming a diffractive interactionbetween the color dipole and the pion emitted by the incident proton. We compare our predictionswith the HERA data on ρ production and estimate the magnitude of the absorption corrections. Weshow that the color dipole formalism is able to describe the current data. Finally, we present ourestimate for the exclusive cross sections which can be studied at HERA and in future electron-protoncolliders. PACS numbers: 12.38.-t, 24.85.+p, 25.30.-cKeywords: Quantum Chromodynamics, Leading Particle Production, Saturation effects.
I. INTRODUCTION
The study of electron - proton ( ep ) collisions at HERA has improved our understanding of the structure of theproton as well as about the dynamics of the strong interactions at high energies (For a review see e.g. Ref. [1]). Inparticular, the study of diffractive processes has been one of the most successful areas at HERA, with vector mesonproduction and Deeply Virtual Compton Scattering (DVCS) in exclusive processes ( γ ∗ p → Ep , with E = ρ, φ, J/ Ψ , γ )being important probes of the transition between the soft and hard regimes of QCD. These processes have been thesubject of intensive theoretical and experimental investigations, with one of the main motivations for these studiesbeing the possibility to probe the QCD dynamics at high energies, driven by the gluon content of the proton which isstrongly subject to non-linear effects (parton saturation) [2]. An important lesson from the analysis of the HERA dataat small values of the Bjorken - x variable is that the inclusive and diffractive processes can be satisfactorily describedusing a unified framework – the color dipole formalism. This approach was proposed many years ago in Ref. [3] andconsiders that the high energy photon can be described by a color quark - antiquark dipole and that the interactionof the dipole with the target can be described by the color dipole cross section σ dt ( x, r ), with the transverse sizeof the dipole r frozen during the interaction process. In this approach all information about the target and stronginteraction physics is encoded in σ dt ( x, r ), which is determined by the imaginary part of the forward amplitude of thescattering between a small dipole (a colorless quark-antiquark pair) and a dense hadron target, denoted by N ( x, r , b ),where the dipole has transverse size given by the vector r = x − y , with x and y being the transverse vectors of thequark and antiquark, respectively, and b = ( x + y ) / N contains all the information about non-linear and quantum effects in the hadron wave function.It can be obtained by solving an appropriate evolution equation in the rapidity Y ≡ ln(1 /x ), which in its simplestform is the Balitsky-Kovchegov (BK) equation [5, 6]. Alternatively, the scattering amplitude can be obtained usingphenomenological models based on saturation physics constructed taking into account the analytical solutions of theBK equation which are known in the low and high density regimes. As demonstrated in [7], the combination betweenthe color dipole formalism and saturation physics are quite successful to describe the recent and very precise HERAdata on the reduced inclusive cross section as well as the data on the exclusive processes in a large range of photon -proton center - of - mass energies W , photon virtualities Q and x values.HERA has also provided high precision experimental data on semi - inclusive e + p → e + n + X processes, wherethe incident proton is converted into a neutron via a charge exchange [8]. Very recently the first measurements of √ s W Wpe e nπ Xq ( x ) L (1 − x ) L t √ s Wpe e nπ Eq ( x ) L (1 − x ) L tW π FIG. 1: Semi - inclusive (left panel) and exclusive (right panel) ep processes associated with a leading neutron n production inthe color dipole formalism. exclusive ρ photoproduction associated with leading neutrons ( γp → ρ π + n ) were presented [9]. The description ofthese leading neutron processes is still a theoretical challenge. In particular, the x L (Feynman momentum) distributionof leading neutrons remains without a conclusive theoretical description [10–19]. In Ref. [20] we extended the colordipole formalism to leading neutron processes and demonstrated that the experimental data on the semi - inclusivereactions can be well described by this approach. Our goal in this paper is to further extend our previous analysis toexclusive processes and try to show that the color dipole formalism may also provide a unified description of leadingneutron processes. Using the same assumptions made in Ref. [20], we compare our predictions with the HERA data on ρ exclusive photoproduction and estimate the contribution of the absorption corrections to exclusive leading neutronprocesses. Taking into account these corrections we present predictions for the exclusive φ , J/ Ψ and γ productionassociated with a leading neutron in ep collisions at the energies of HERA and future electron - proton colliders.This paper is organized as follows. In the next Section we present a brief review of leading neutron production in ep collisions and we discuss the treatment of exclusive processes with the color dipole formalism. In Section III we analysethe dependence of our predictions on the models of the vector meson wave function, on the pion flux and on the dipolescattering amplitude. A comparison with the recent HERA data on exclusive ρ photoproduction is performed andpredictions for the exclusive φ , J/ Ψ and γ production associated to a leading neutron in ep collisions for the energiesof HERA and of the future electron - proton colliders are presented. Finally, in Section IV we summarize our mainconclusions. II. EXCLUSIVE PROCESSES ASSOCIATED WITH LEADING NEUTRON PRODUCTION IN THECOLOR DIPOLE FORMALISM
At high energies, the differential cross section for a given process (semi - inclusive or exclusive) associated with aleading neutron production can be expressed as follows: d σ ( W, Q , x L , t ) dx L dt = f π/p ( x L , t ) σ γ ∗ π ( ˆ W , Q ) (1)where Q is the virtuality of the exchanged photon, W is the center-of-mass energy of the virtual photon-protonsystem, x L is the proton momentum fraction carried by the neutron and t is the square of the four-momentum ofthe exchanged pion. Moreover, f π/p is the flux of virtual pions emitted by the proton and σ γ ∗ π ( ˆ W , Q ) is the crosssection of the interaction between the virtual-photon and the virtual-pion at center-of-mass energy ˆ W , which is givenby ˆ W = (1 − x L ) W . The pion flux f π/p ( x L , t ) (also called sometimes pion splitting function) is the virtual pionmomentum distribution in a physical nucleon (the bare nucleon plus the “pion cloud”). In general, it is parametrizedas follows [10–19] f π/p ( x L , t ) = 14 π g pπp π − t ( t − m π ) (1 − x L ) − α ( t ) [ F ( x L , t )] (2)where g pπp / (4 π ) = 14 . π pp coupling constant, m π is the pion mass and α ( t ) is the Regge trajectory of thepion. The form factor F ( x L , t ) accounts for the finite size of the nucleon and of the pion and is model dependent. Asin Ref. [20], we will consider the following parametrizations for the form factor: F ( x L , t ) = exp (cid:20) R ( t − m π )(1 − x L ) (cid:21) , α ( t ) = 0 (3)from Ref. [11], where R = 0 . − . F ( x L , t ) = 1 , α ( t ) = α ( t ) π (4)from Ref. [10], where α π ( t ) ≃ t (with t in GeV ) is the Regge trajectory of the pion. F ( x L , t ) = exp (cid:2) b ( t − m π ) (cid:3) , α ( t ) = α ( t ) π (5)from Ref. [12], where α π ( t ) ≃ t (with t in GeV ) and b = 0 . − . F ( x L , t ) = Λ m − m π Λ m − t , α ( t ) = 0 (6)from Ref. [13], where Λ m = 0 .
74 GeV. F ( x L , t ) = (cid:20) Λ d − m π Λ d − t (cid:21) , α ( t ) = 0 (7)also from Ref. [13], where Λ d = 1 . f , f , ... f , respectively. Moreover, it is important to emphasize that in the caseof the more familiar exponential (3), monopole (6) and dipole (7) forms factors, the cut-off parameters have beendetermined by fitting low energy data on nucleon and nuclear reactions and also data on deep inelastic scattering andstructure functions [21].In Ref. [20], we described the semi - inclusive leading neutron processes in the color dipole formalism. The basicidea is that at high energies, this process can be seen as a sequence of three factorizable subprocesses [See Fig. 1 (leftpanel)]: i) the photon fluctuates into a quark-antiquark pair (the color dipole), ii) the color dipole interacts with thepion, present in the wave function of the incident proton, and iii) the leading neutron is formed. Consequently, thephoton - pion cross section can be factorized in terms of the photon wave functions Ψ, which describes the photonsplitting in a q ¯ q pair, and the dipole-pion cross section σ dπ . In the eikonal approximation the dipole-proton crosssection σ dπ is given by: σ dπ (ˆ x, r ) = 2 Z d b N π (ˆ x, r , b ) , (8)where ˆ x = Q + m f ˆ W + Q = Q + m f (1 − x L ) W + Q (9)is the scaled Bjorken variable and N π ( x, r , b ) is the imaginary part of the forward amplitude of the scattering betweena small dipole (a colorless quark-antiquark pair) and a pion, at a given rapidity interval Y = ln(1 / ˆ x ). In Ref. [20] weproposed to relate N π with the dipole-proton scattering amplitude N p , usually probed in the typical inclusive andexclusive processes at HERA, assuming that N π (ˆ x, r , b ) = R q · N p (ˆ x, r , b ) (10)with R q being a constant. In the additive quark model it is expected that R q = 2 /
3, which is the ratio betweenthe number of valence quarks in the target hadrons. This model was first applied to soft hadronic reactions [22]and, in particular, it predited the following relation between the pion-proton and proton-proton total cross sections: σ πp = 2 / σ pp . This relation is observed experimentally. It refers to total cross sections and the only kinematicalvariable is the c.m.s energy √ s . In the low energy domain, where it was verified, the dependence on √ s was veryweak. As it was discussed in Ref. [23], in hard hadronic reactions, where a high energy scale is present, Eq. (10) maystill be valid, although deviations from 2 / Q ≥
10 GeV canresolve the quarks in the target and interact with each of them independently. The cross section is then proportionalto the number of quarks in the target. At increasing Q and/or collision energies quantum fluctuations become moreimportant, increasing the effective number of quarks. According to Ref. [23], this growth is stronger in the protonthan in the pion and hence 2 / → / / R q , going from 1 / /
3. As itwill be seen, our results imply that if R q = 2 / K tends to be too small. In view of the existingcalculations of K , we would conclude that R q = 2 / R q could be in the range 1 / ≤ R q ≤ /
3. Withthis basic assumption we have estimated the dependence of the predictions on the description of the QCD dynamicsat high energies as well as the contribution of gluon saturation effects to leading neutron production. Moreover, withthe parameters constrained by other phenomenological information, we were able to reproduce the basic features ofthe H1 data on leading neutron spectra [8].As mentioned in Ref. [20], one source of uncertainty in the study of inclusive leading neutron production (in Fig. 1on the left) is the fact that there are several processes which lead to the same final state. Apart from one pion emissionwe may have, for example, ρ emission. Even with pion emission we may have ∆ production with the subsequent decay∆ → n + π . The strength of these contributions is highly model dependent and their existence prevents us fromextracting more precise information on the photon-pion cross section or on the pion flux. In contrast, in ρ exclusiveproduction with a leading neutron none of these processes contributes to the exclusive reaction shown in the rightpanel of Fig. 1. This feature makes the leading neutron spectrum measured in exclusive processes a better testingground for both the determination of the photon-pion cross section and of the pion flux.In what follows we will assume that the factorization given by Eq. (1) also is valid and that the photon - pion crosssection for the production of an exclusive final state E , such as a vector meson ( E = V ) or a real photon in DVCS( E = γ ), in the γ ∗ π → Eπ process is given in the color dipole formalism by: σ ( γ ∗ π → Eπ ) = X i = L,T Z −∞ dσ i d ˆ t d ˆ t = 116 π X i = L,T Z −∞ |A γ ∗ π → Eπi ( x, ∆) | d ˆ t , (11)with the scattering amplitude being given by A γ ∗ π → EπT,L (ˆ x, ∆) = i Z dz d r d b e − i [ b − (1 − z ) r ] . ∆ (Ψ E ∗ Ψ) T,L N π (ˆ x, r , b ) (12)where (Ψ E ∗ Ψ) T,L denotes the overlap of the photon and exclusive final state wave functions. The variable z (1 − z )is the longitudinal momentum fractions of the quark (antiquark) and ∆ denotes the transverse momentum lost bythe outgoing pion (ˆ t = − ∆ ). The variable b is the transverse distance from the center of the target to the center ofmass of the q ¯ q dipole and the factor in the exponential arises when one takes into account non-forward correctionsto the wave functions [25]. In what follows we will assume that the vector meson is predominantly a quark-antiquarkstate and that the spin and polarization structure is the same as in the photon [26–29] (for other approaches see, forexample, Ref. [30]). As a consequence, the overlap between the photon and the vector meson wave function, for thetransversely and longitudinally polarized cases, is given by (For details see Ref. [31])(Ψ ∗ V Ψ) T = ˆ e f e π N c πz (1 − z ) (cid:8) m f K ( ǫr ) φ T ( r, z ) − (cid:2) z + (1 − z ) (cid:3) ǫK ( ǫr ) ∂ r φ T ( r, z ) (cid:9) , (13)(Ψ ∗ V Ψ) L = ˆ e f e π N c π Qz (1 − z ) K ( ǫr ) " M V φ L ( r, z ) + δ m f − ∇ r M V z (1 − z ) φ L ( r, z ) , (14)where ˆ e f is the effective charge of the vector meson, m f is the quark mass, N c = 3, ǫ = z (1 − z ) Q + m f and φ i ( r, z )define the scalar parts of the vector meson wave functions. We will consider the Boosted Gaussian and Gauss-LCmodels for φ T ( r, z ) and φ L ( r, z ), which are largely used in the literature. In the Boosted Gaussian model the functions φ i ( r, z ) are given by φ T,L ( r, z ) = C T,L z (1 − z ) exp " − m f R z (1 − z ) − z (1 − z ) r R + m f R . (15)In contrast, in the Gauss-LC model, they are given by φ T ( r, z ) = N T [ z (1 − z )] exp (cid:0) − r / R T (cid:1) , (16) φ L ( r, z ) = N L z (1 − z ) exp (cid:0) − r / R L (cid:1) . (17)The parameters C i , R , N i and R i are determined by the normalization condition of the wave function and by themeson decay width. In Table I we present the value of these parameters for the vector meson wave functions. It isimportant to emphasize that predictions based on these models for the wave functions have been tested with successin ep and ultra peripheral hadronic collisions (See, e. g. Refs. [7, 32, 33]). In the DVCS case, as one has a real photonat the final state, only the transversely polarized overlap function contributes to the cross section. Summed over thequark helicities, for a given quark flavour f it is given by [31], Meson M V /GeV m f /GeV ˆ e f N T C T R T /GeV − N L C L R L /GeV − R /GeV − ρ √ φ J/ψ (Ψ ∗ γ Ψ) fT = N c α em e f π (cid:8)(cid:2) z + ¯ z (cid:3) ε K ( ε r ) ε K ( ε r ) + m f K ( ε r ) K ( ε r ) (cid:9) , (18)where we have defined the quantities ε , = z ¯ z Q , + m f and ¯ z = (1 − z ). Accordingly, the photon virtualities are Q = Q (incoming virtual photon) and Q = 0 (outgoing real photon).Finally, in order to estimate the photon - pion cross section we must specify the dipole - pion scattering amplitude N π . As considered in Ref. [20] for the semi - inclusive processes, we will assume the validity of the approximationexpressed by Eq. (10), with the dipole proton scattering amplitude N p being given by the bCGC model, proposed inRef. [31] and recently updated in Ref. [7], which is based on the CGC formalism and takes into account the impactparameter dependence of the dipole - proton scattering amplitude. As demonstrated in Refs. [7, 33], this model isable to describe the vector meson production in ep and ultra peripheral hadronic collisions. In the bCGC model thedipole - proton scattering amplitude is given by [31] N p (ˆ x, r , b ) = N (cid:16) r Q s ( b )2 (cid:17) ( γ s + ln(2 /rQs ( b )) κ λ Y ) rQ s ( b ) ≤ − e − A ln ( B rQ s ( b )) rQ s ( b ) > κ = χ ′′ ( γ s ) /χ ′ ( γ s ), where χ is the LO BFKL characteristic function. The coefficients A and B are determineduniquely from the condition that N p (ˆ x, r , b ), and its derivative with respect to r Q s ( b ), are continuous at r Q s ( b ) = 2.In this model, the proton saturation scale Q s ( b ) depends on the impact parameter: Q s ( b ) ≡ Q s (ˆ x, b ) = (cid:16) x ˆ x (cid:17) λ (cid:20) exp (cid:18) − b B CGC (cid:19)(cid:21) γs . (20)The parameter B CGC was adjusted to give a good description of the t -dependence of exclusive J/ψ photoproduction.Moreover, the factors N and γ s were taken to be free. The set of parameters which will be used here is the following: γ s = 0 . κ = 9 . B CGC = 5 . − , N = 0 . x = 0 . λ = 0 . N p , we also will consider the IIMS [34, 35] and GBW[36] models, as well as the numerical solution of the BK equation obtained in Ref. [37]. Such models were discussedin detail in Ref. [20]. For these models, we assume N p (ˆ x, r , b ) = N p (ˆ x, r ) S ( b ) and σ dp (ˆ x, r ) = σ · N p (ˆ x, r ), withthe normalization of the dipole cross section ( σ ) being fitted to data, and that the ˆ t - dependence of the photon -pion cross section can be approximated by an exponential ansatz, dσ/d ˆ t = dσ/d ˆ t (ˆ t = 0) · e − B | ˆ t | , with the slope beinggiven by B = σ / π . It is important to emphasize that the conclusions obtained in [20] are not modified if the bCGCmodel is used as input in the calculations.Before discussing our results, a comment is in order. As in the semi - inclusive case, our predictions for the exclusiveprocesses associated with a leading neutron are essentially parameter free, depending only on the choices of the modelsfor the pion flux and on the dipole scattering amplitude. The main uncertainties are associated with the choice of R q (in Eq. (10)) and the magnitude of the absorption effects which can arise by soft rescatterings. These latter aredifficult to calculate [18, 19] but are expected to modify almost uniformly all the x L spectrum of the leading neutrons.As in Ref. [20], in what follows we will assume that these effects can be mimicked by a factor K , which multiplies theright side of Eq. (1) changing the normalization of the spectra and which should be estimated from the analysis ofexperimental data. In spite of the efforts made in several studies of absorptive corrections in semi - inclusive processes[16–19, 23, 38], the magnitude of these effects in exclusive processes remains an open question. III. RESULTS
Let us start our analysis considering the exclusive ρ photoproduction associated with leading neutrons as analysedby the H1 Collaboration [9]. In what follows we will assume that W = 60 GeV, Q = 0 .
04 GeV and that p T < . L d σ / dx L [ µ b ] Gaus-LCBoosted Gaussian Q = 0.04 GeV W = 60 GeV f (y)p T < 0.2 GeV 0.4 0.5 0.6 0.7 0.8 0.9x L d σ / dx L [ µ b ] Gaus-LCBoosted Gaussian Q = 0.04 GeV W = 60 GeV f (y)p T < 0.2 GeV FIG. 2: Leading neutron spectra in exclusive ρ photoproduction considering two different models for the vector meson wavefunction (Boosted Gaussian and Gauss - LC) and two different models for the pion flux ( f and f ). L d σ / dx L [ µ b ] f (y), K = 0.164f (y), K = 0.134f (y), K = 0.179f (y), K = 0.210f (y), K = 0.195H1 (p T < 0.2 GeV) bCGC FIG. 3: Leading neutron spectra in exclusive ρ photoproduction considering different models of the pion flux. Data from Ref.[9]. GeV, where p T is the transverse momentum of the leading neutron. Moreover, we will assume initially that R q = 2 / φ and J/ Ψ production. Consequently, in what follows we will consider only the Gauss-LC modelfor the vector meson wave functions. Let us now compare our predictions with the experimental data [9] consideringdifferent models for the pion flux. In order to constrain the value of the K factor associated to absorptive corrections,our strategy will be following: for a given model of the pion flux, R q and dipole cross section, we will estimate thetotal cross section. The value of K will be the value necessary to make our prediction consistent with the H1 data [9].In Fig. 3 we present our predictions for the leading neutron spectra in exclusive ρ photoproduction consideringdifferent models for the pion flux. The corresponding K values are also presented. We obtain a reasonable agreementwith the experimental data, with K values in the range 0 . < K < . K are strongly correlated with our choice for R q . For example, if instead of R q = 2 / R q = 1 / K values should be multiplied by 4, since the exclusive cross sections depend quadratically on thedipole scattering amplitude. If we assume a priori that the magnitude of the absorptive correction factor is of theorder of 0 . R q = 1 /
3. However, as the magnitude of these corrections for exclusive processes is still an open question, as well asthe value of R q , we refrain from drawing strong conclusions. Therefore, in what follows we will only present results L d σ / dx L [ µ b ] IIMS, K = 0.078bCGC, K = 0.134GBW, K = 0.077rcBK, K = 0.093H1 (p T < 0.2 GeV) f (y) 0.4 0.5 0.6 0.7 0.8 0.9x L d σ / dx L [ µ b ] IIMS, K = 0.103bCGC, K = 0.179GBW, K = 0.101rcBK, K = 0.122H1 (p T < 0.2 GeV) f (y) FIG. 4: Leading neutron spectra in exclusive ρ photoproduction considering different models for the dipole scattering amplitudeand of the pion flux. Data from Ref. [9]. assuming R q = 2 /
3, but the reader should keep in mind the quadratic correlation between K and R q , implying thatthe same fits could be obtained with much bigger values of K and smaller values of R q . With more data on differentprocesses with leading neutron production, it may be possible to disentangle K from R q . It is interesting to noticethat leading neutron production is dominated by pion emission from the proton, i.e. p → π + n . In all existingtheoretical approaches this pion is soft and takes only a small fraction of the incoming proton energy, leaving theneutron with most of it. This is the physical reason for the peak seen at x L ≃ .
75 in the x L spectrum of the leadingneutron.In Fig. 4 we analyse the dependence of our predictions on the choice of the dipole scattering amplitude for twodifferent models of the pion flux. As done before, we will constrain the value of K by adjusting the predictions of thedifferent dipole models to the experimental value of the total cross section. We find that the different predictions forthe x L spectra are very similar. However, the effective value of the absorptive correction K depends on the model ofthe dipole scattering amplitude as expected, since they predict different values for the B slope, which determines thenormalization of the photon - pion cross section. Figs. 3 and 4 are in a sense complementary, since what changes inthe former (the flux factor) is kept fixed in the latter (where the dipole model is changed) and vice-versa. In eachcurve the overall constant KR q is chosen so as to bring our calculations as close as possible to the experimental points.Comparing the curves we can conclude that the shape of the leading neutron spectrum is much more sensitive to theflux factor than to the dipole scattering amplitude N . The normalization of the spectrum is hence determined by K and R q , since N is fixed from the analysis of other data. The values used for K are significantly smaller than thosefound in theoretical estimates. Larger values of K would be more plausible implying a deviation from the valencequark scaling and the consequent change in the factor 2 /
3. A comparison between the curves in Fig. 4 favors thechoice of the pion flux f , which, unlike f , contains a t-dependent form factor. A similar preference was found inRef. [39], where a combined analysis of E866 and HERA data was performed.In what follows we will only consider the bCGC model, which successfully describes the HERA data on exclusiveprocesses. In Fig. 5 we present our predictions taking into account the experimental uncertainty present in the H1data for the total cross section [9], which in our analysis translates into a range of possible values for the K factor.These results indicate that the experimental data are better described using the pion flux f ( y ). As a cross check ofour results, we can compare our predictions with other H1 data obtained assuming p T < . · x L GeV. Assuming thesame range of values for K obtained in Fig. 5 we can see in Fig. 6 that our predictions describe these data quite well.Considering that the main inputs of our calculations have been fixed by the experimental data on exclusive ρ photoproduction we can extend our analysis to other exclusive final states. We will assume the Gauss-LC model forvector meson wave function, the bCGC dipole scattering amplitude, the f model for the pion flux and the same K values needed to describe the ρ data. Initially, let us consider the kinematical range probed by HERA. As in the ρ case, we will assume W = 60 GeV and p T < . φ and J/ Ψ production we assume Q = 0 . , while for the DVCS we consider that Q = 10 GeV . The corresponding predictions for the leading neutronspectra in exclusive φ and J/ Ψ production as well as in DVCS are presented in Fig. 7. For the HERA kinematicalrange we predict σ ( γp → φπn ) = 25 . ± .
70 nb, σ ( γp → J/ Ψ πn ) = 0 . ± .
03 nb and σ ( γ ∗ p → γπn ) = 0 . ± . K . Finally, let us presentour predictions for the kinematical range which may be probed in future ep colliders assuming p T < . L d σ / dx L [ µ b ] H1 (p T < 0.2 GeV) f (y)K max = 0.153 K min = 0.114K med = 0.134 0.4 0.5 0.6 0.7 0.8 0.9x L d σ / dx L [ µ b ] H1 (p T < 0.2 GeV) f (y)K max = 0.205K min = 0.152K med = 0.179 FIG. 5: Leading neutron spectra in exclusive ρ photoproduction obtained considering the possible range of values of the K factor and two models for the pion flux. Data from Ref. [9]. L d σ / dx L [ µ b ] H1 (p T < x L . 0.69 GeV) f (y)K max = 0.153 K min = 0.114K med = 0.134 0.4 0.5 0.6 0.7 0.8 0.9x L d σ / dx L [ µ b ] H1 (p T < x L . 0.69 GeV) f (y)K max = 0.205K min = 0.152K med = 0.179 FIG. 6: Leading neutron spectra in exclusive ρ photoproduction obtained considering the possible range of values of the K factor fixed using the other set of experimental data and two models for the pion flux. H1 data [9] obtained assuming that p T < . · x L GeV. neutron spectra increases with the energy at fixed Q and decreases with the virtuality at fixed W . In particular,for W = 1 TeV and Q = 5 GeV we predict σ ( γ ∗ p → ρπn ) = 6 . ± .
95 nb, σ ( γ ∗ p → φπn ) = 1 . ± .
25 nb, σ ( γ ∗ p → J/ Ψ πn ) = 1 . ± .
17 nb and σ ( γ ∗ p → γπn ) = 0 . ± .
02 nb. We believe that for these values of total crosssections, the experimental analysis of the exclusive processes associated with a leading neutron is feasible in future ep colliders. In particular, as the cross sections strongly increase when Q →
0, the analysis of the vector mesonphotoproduction in ep collisions can be useful to understand leading neutron spectra, which are of crucial importancein particle production in cosmic ray physics. Another possibility is the study of this process in ultraperipheral hadroniccollisions, with the leading neutron being a tag for exclusive production. In principle these processes can be studiedin the future at the LHC. Such proposition will be discussed in detail in a forthcoming publication. IV. SUMMARY
One of the important goals in particle physics is to understand the production of leading particles, i.e. the productionof baryons which have large fractional longitudinal momentum ( x L ≥ .
3) and the same valence quarks (or at leastone of them) as the incoming particles. Recent measurements of leading neutron spectra in ep collisions at HERAhave shed a new light on this subject. However, the description of the semi - inclusive and exclusive leading neutronprocesses remains without a satisfactory theoretical description. In a previous work [20], we proposed to study semi- inclusive leading neutron production using the color dipole formalism, which successfully describes both inclusiveand diffractive HERA data, taking into account the QCD dynamics and its non - linear effects, which are expected L d σ / dx L [ nb ] f (y) φ L d σ / dx L [ nb ] f (y) J/ ψ L d σ / dx L [ nb ] f (y) DVCS
FIG. 7: Predictions for the leading neutron spectra in exclusive φ , J/ Ψ and DVCS production in the HERA kinematical range: W = 60 GeV and p T < . L d σ / dx L [ nb ] W = 100 GeVW = 250 GeVW = 500 GeVW = 750 GeVW = 1000 GeV ρ Q = 5 GeV L d σ / dx L [ nb ] W = 100 GeVW = 250 GeVW = 500 GeVW = 750 GeVW = 1000 GeV φ Q = 5 GeV L d σ / dx L [ nb ] W = 100 GeVW = 250 GeVW = 500 GeVW = 750 GeVW = 1000 GeV J/ ψ Q = 5 GeV L d σ / dx L [ nb ] W = 100 GeVW = 250 GeVW = 500 GeVW = 750 GeVW = 1000 GeV DVCSQ = 5 GeV FIG. 8: Energy dependence of the leading neutron spectra in exclusive ρ , φ , J/ Ψ and DVCS production in the kinematicalrange of the future ep colliders ( Q = 5 GeV ). to be present at high energies. Making use of very simple assumptions about the relation between the dipole - pionand the dipole - proton scattering amplitudes and about the absorptive corrections, we demonstrated that the semi- inclusive data can be described by the dipole formalism and that Feynman scaling is expected at high energies. Inthis paper we have extended our analysis to exclusive processes associated with a leading neutron. Considering thesame assumptions used for the semi - inclusive case, we have analysed in detail the dependence of our predictions onthe choices of the vector meson wave function, of the dipole model and and of the pion flux. We demonstrated thatthe HERA data on the exclusive ρ photoproduction associated with a leading neutron can be quite well described bythe color dipole formalism. Assuming the validity of this approach, we have presented for the first time predictionsfor the exclusive φ , J/ Ψ and γ production in ep collisions for the energies of HERA and future colliders. Our resultsindicate that the experimental analysis of these processes is feasible and that they can be used to understand thislong standing problem in high energy physics.0 L d σ / dx L [ nb ] Q = 2 GeV Q = 4 GeV Q = 6 GeV Q = 8 GeV Q = 10 GeV ρ W = 1 TeV 0.4 0.5 0.6 0.7 0.8 0.9x L d σ / dx L [ nb ] Q = 2 GeV Q = 4 GeV Q = 6 GeV Q = 8 GeV Q = 10 GeV φ W = 1 TeV0.4 0.5 0.6 0.7 0.8 0.9x L d σ / dx L [ nb ] Q = 2 GeV Q = 4 GeV Q = 6 GeV Q = 8 GeV Q = 10 GeV J /ψ W = 1 TeV L d σ / dx L [ nb ] Q = 2 GeV Q = 4 GeV Q = 6 GeV Q = 8 GeV Q = 10 GeV DVCSW = 1 TeV
FIG. 9: Dependence on the virtuality of the leading neutron spectra in exclusive ρ , φ , J/ Ψ and DVCS production in thekinematical range of the future ep colliders ( W = 1 TeV). Finally, it is important to emphasize that the current sources of uncertainties in the computation of leading neutronspectra are: i) the strength of the absorptive corrections represented by the factor K ; ii) the validity of the additivequark model for the photon-pion cross section; iii) the strength of the contribution from direct fragmentation of theproton into neutrons; iv) the precise form of the pion flux; v) the precise form of the dipole cross section. Withsufficient experimental information, we can rule out candidates of the pion flux and of the photon-pion cross section.We believe that it is possible to constrain the unknown numbers and assumptions with the help of more experimentaldata on other processes with tagged leading neutrons, such as those on D ∗ production [40] and those with dijetproduction [41]. Work along this line is in progress. Acknowledgments
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