ЯДЕРНАЯ ФИЗИКА, 2013, том 76, № 3, с. 348-357
ЭЛЕМЕНТАРНЫЕ ЧАСТИЦЫ И ПОЛЯ
FEYNMAN SCALING VIOLATION ON BARYON SPECTRA IN pp COLLISIONS AT LHC AND COSMIC RAY ENERGIES
© 2013 G. H. Arakelyan1)*, C. Merino2)**, C. Pajares2)***, Yu. M. Shabelski3)****
Received December 20, 2011; in final form, June 6, 2012
A significant asymmetry in baryon/antibaryon yields in the central region of high energy collisions is observed when the initial state has nonzero baryon charge. This asymmetry is connected with the possibility of baryon charge diffusion in rapidity space. Such a diffusion should decrease the baryon charge in the fragmentation region and translate into the corresponding decrease of the multiplicity of leading baryons. As a result, a new mechanism for Feynman scaling violation in the fragmentation region is obtained. Another numerically more significant reason for the Feynman scaling violation comes from the fact that the average number of cut Pomerons increases with initial energy. We present the quantitative predictions of the Quark—Gluon String Model for the Feynman scaling violation at LHC energies and at even higher energies that can be important for cosmic ray physics.
DOI: 10.7868/S0044002713020025
1. INTRODUCTION
The problem of Feynman scaling violation has evident both theoretical and practical interest. In particular, this question is very important [1,2] for cosmic ray physics, where the difference from the primary radiation to the events registrated on the ground or mountain level is determined by the multiple interactions of the so-called leading particles (mainly baryons) in the atmosphere.
Despite the lack of direct measurements of Feynman scaling violation for secondary baryon spectra in nucleon—nucleon collisions at energies higher than those of ISR, some experimental information from cosmic ray experiments seems to confirm [3—5] the presence of significant Feynman scaling violation effects. Now the LHCf Collaboration has started the search [6, 7] of Feynman scaling violation effects for the spectra of photons (^0) and neutrons in the fragmentation region at LHC energies.
In principle, the violation of Feynman scaling in the fragmentation region should exist due to the
Alikhanyan National Science Laboratory, Yerevan Physics Institute, Armenia.
2)Departamento de Física de Partículas and Instituto Galego de Física de Altas Enerxías Universidade de Santiago de Compostela, Galiza, Spain.
3)Petersburg Nuclear Physics Institute, Gatchina; NRC Kur-chatov Institute, Moscow, Russia.
E-mail: argev@mail.yerphi.am E-mail: merino@fpaxp1.usc.es E-mail: pajares@fpaxp1.usc.es E-mail: shabelsk@thd.pnpi.spb.ru
energy conservation, since the spectra of charged particles increase in the central region. However, no quantitative predictions can be made without some model of the particle production.
The Additive Quark Model [8, 9] predicts the violation of Feynman scaling in the fragmentation region due to the increase of the interaction cross sections. However, to make a description of the energy dependences of the spectra as a function of xF additional assumptions and parameters are needed.
The Quark—Gluon String Model (QGSM) [10, 11] allows the calculation of the spectra of secondaries at different initial energies in the whole xF region. The QGSM is based on Dual Topological Unitarization (DTU), Regge phenomenology, and nonperturbative notions of QCD. This model is successfully used for the description of multiple production processes in hadron—nucleon [12—15], hadron— nucleus [16, 17], and nucleus—nucleus [18] collisions. The quantitative predictions of the QGSM depend on several parameters which were fixed by comparison of the calculations to the experimental data obtained at fixed target energies. The first experimental data obtained at LHC show [19, 20] that the model predictions are in reasonable agreement with the data.
We will consider the energy dependences of the spectra of secondary baryons in the projectile fragmentation region which we determine as the interval 0.05 < xF < 0.8. These values of xF are larger than the typical values for central production and smaller than the values where triple-Reggeon diagrams dominate.
In the frame of QGSM several reasons for the Feynman scaling violation in the fragmentation region exist. The first one is the increase of the average number of exchanged Pomerons with the energy, which leads to the corresponding increase of the yields of hadron secondaries in the central region and to their decrease in the fragmentation region. This effect is present even at asymptotically high energies. The preliminary estimation of this effect was provided in [21, 22].
In the case of nuclear (air) targets, the growth of the hN cross section with energy leads to the increase of the average number of fast hadron inelastic collisions inside the nucleus. Thus, the average number of Pomerons is additionally increased, resulting in a stronger Feynman scaling violation [21, 22].
In [23] these predictions were taken into account to calculate the penetration of fast hadrons into the atmosphere, leading to a better description of the cosmic ray experimental data.
The differences in the yields of baryons and antibaryons produced in the central (midrapidity) region of high energy pp interactions [15, 19, 24—27] are significant. Evidently, the appearance of the positive baryon charge in the central region of pp collisions should be compensated by the decrease of the baryon multiplicities in the fragmentation region that leads to an additional reason for Feynman scaling violation. This effect has a preasymptotical behavior and it is saturated at very high energies (see Section 4).
In the present paper we consider the effects of Feynman scaling violation, i.e. the energy dependences of the spectra of secondary protons, neutrons, and A produced in pp collisions in the fragmentation region.
In our estimations the role of the nuclear factor for air nuclei should be similar to that presented in [21, 22].
2. INCLUSIVE SPECTRA OF SECONDARY HADRONS IN THE QUARK-GLUON STRING MODEL
The QGSM [10, 11] allows us to make quantitative predictions for different features of multiparticle production, in particular, for the inclusive spectra of different secondaries, both in the central and in fragmentation regions. In QGSM high energy hadron—nucleon collisions are considered as taking place via the exchange of one or several Pomerons, all elastic and inelastic processes resulting from cutting through or between Pomerons [28].
Each Pomeron corresponds to a cylindrical diagram (see Fig. 1a), and thus, when cutting one Pomeron, two showers of secondaries are produced as it is shown in Fig. 16. The inclusive spectrum
Fig. 1. (a) Cylindrical diagram correspondingto the one-Pomeron exchange contribution to elastic pp scattering, and (b) the cut of this diagram which determines the contribution to the inelastic pp cross section. Quarks are shown by solid curves and the string junction by dashed curves.
of a secondary hadron h is then determined by the convolution of the diquark, valence quark, and sea quark distributions, u(x,n), in the incident particles, with the fragmentation functions, Gh(z), of quarks and diquarks into the secondary hadron h. These distributions, as well as the fragmentation functions, are constructed by using the Reggeon counting rules [29]. Both the diquark and the quark distribution functions depend on the number n of cut Pomerons in the considered diagram. The details of the model are presented in [10—13, 15].
For a nucleon target, the inclusive rapidity (y), or Feynman-x (xF), spectrum of a secondary hadron h has the form [10]:
dn 1 da
dy o-jnei dy
xe da
ainel dxp = ^ Wntâix) +w0^DD (x),
(1)
n=1
where xp = 2p^/ ^/s is the Feynman variable xp, and xe = 2E/ sfs, and one has then to use value of (p^) (here we have taken the value (p"t) = 0.35 (GeV/c)2) to make the transition to the values of da/dxp which are presented in the experimental paper [30].
The functions 4n(x) determine the contribution of the diagram with n cut Pomerons and wn is the relative weight of this diagram X^Li wn = 1. The last term in Eq. (1) accounts for the contribution of diffraction dissociation processes that are determined by the cuts between Pomerons (n = 0).
por pp collisions
4hp(x) = fgq(x+ ,n)fh(X-,n) +
+ fq(x+,n)fgq(X-,n) +
+ 2(n - 1)fh(x+,n)fh(x-,n),
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w,
10-
10-
10-
2.5 7.5 12.5 17.5 22.5
n
Fig. 2. The calculated probabilities for cutting different number of Pomerons at energies ^fs = 17.3 GeV (dotted curve), 200 GeV (dash-dotted curve), 8 TeV (dashed curve), and 100 TeV (solid curve).
" B
С
¿в
B
С
--B d--
С
с с с с
M M
'тв
с с
Fig. 3. QGSM diagrams describing secondary baryon B production by diquark d. (a) Central production of BB pair. Single B production in the processes of diquark fragmentation: (b) initial SJ together with two valence quarks and one sea quark, (c) initial SJ together with one valence quark and two sea quarks, and (d) initial SJ together with three sea quarks. Quarks are shown by solid curves and SJ by dashed curves.
x± = _ 2
\J4m|/s + x2 ± x
where fqq, fq, and fs correspond to the contributions of diquarks, valence quarks, and sea quarks, respectively.
These functions are determined by the convolution of the diquark and quark distributions with the fragmentation functions, e.g. for the quark one can write:
i
¡h(x+,n) = J Uq (xi,n)Gh(x+/xi)dxi.
(4)
Ж +
The fragmentation functions Gh(z) are independent of the number of cutted Pomerons n. On the contrary, the diquark and quark distributions u(x, n) (which are normalized to unity) become softer when n increases. Thus, for example in [10] it was assumed4) that the diquark distributions depend on n as:
U
(x) - (1 - x)-aR+(n-1).
(5)
(3)
If the intercept of the Pomeron trajectory is larger than unity,
ap(0) = 1 + A, A > 0, (6)
the average nu
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