”ˆ‡ˆŠĨ ŗ‹…Œ…Ł’ĨńŁ›• —Ĩ‘’ˆ– ˆ Ĩ’łŒŁłƒł Ÿ„ńĨ 2003. ’. 34. ‚›Ń. 4 “„Š 539.17.17 LIGHT FRONT VARIABLES IN HIGH-ENERGY HADRONÄHADRON AND NUCLEUSÄNUCLEUS INTERACTIONS L. Chkhaidze, T. Djobava ∗ , Yu. Tevzadze High Energy Physics Institute, Tbilisi State University, Tbilisi, Georgia V. Garsevanishvili Mathematical Institute of the Georgian Academy of Sciences, Tbilisi, Georgia INTRODUCTION. DEFINITION AND PROPERTIES OF VARI- ABLES 1033 INCLUSIVE DISTRIBUTIONS IN HADRONÄHADRON COLLI- SIONS 1037 NUCLEUSÄNUCLEUS COLLISIONS 1047 CONCLUSION 1059 REFERENCES 1060 ∗ e-mail: [email protected]
LIGHT FRONT VARIABLES IN HIGH-ENERGYHADRONÄHADRON AND NUCLEUSÄNUCLEUS
INTERACTIONSL. Chkhaidze, T. Djobava∗, Yu. Tevzadze
High Energy Physics Institute, Tbilisi State University, Tbilisi, Georgia
V. Garsevanishvili
Mathematical Institute of the Georgian Academy of Sciences, Tbilisi, Georgia
Light front variables are introduced to study inclusive spectra of secondaries in hadronÄhadronand nucleusÄnucleus interactions. It is established that the phase space of secondary pions is dividedinto two parts with signiˇcantly different characteristics. The thermal equilibrium seems to be reachedin one of these parts. Corresponding temperatures of pions in hadronÄhadron and nucleusÄnucleuscollisions are extracted. The results are compared with the results of other types of analysis. Theresults for nucleusÄnucleus collisions are compared with the predictions of the Quark-Gluon StringModel (QGSM). The QGSM satisfactorily reproduces the experimental data for light and intermediate-mass nuclei.
1. INTRODUCTION. DEFINITION AND PROPERTIES OF VARIABLES
The study of single-particle inclusive processes [1] remains one of the sim-plest and effective tools for the investigation of multiple production of secondariesat high energies. The consequences of the limiting fragmentation hypothesis [2]and those of the parton model [3] and the principle of automodelity for stronginteractions [4] have been formulated in this way.
At high energies different dynamical mechanisms contribute to spectra of sec-ondaries. Among them ®pionization¯ and fragmentation mechanisms are widely
Fig. 1. The schematic view of the surfaces of constants xF (a), x⊥ (b), and y (c) in the(pz , p⊥) plane
discussed. ®Pionization¯ means the existence of secondary pions with relativelylow momenta and �at (almost isotropic) angular distribution in the centre-of-massframe of colliding objects. The fragmentation component has sharply anisotropicangular distribution in the centre-of-mass frame. One of the principal problemsin this direction is the separation of these two components. Up to now, thereexist no unique way to separate these mechanisms. Different authors proposedifferent ways and non of them seems to be satisfactory. It will be shown thatthe presentation of inclusive spectra in terms of light front variables provides aunique possibility of separating these two components.
An important role in establishing many properties of multiple productionis played by the choice of kinematic variables in terms of which observablequantities are presented (see, e. g., [5Ä7]). The variables which are commonlyused are the following: the Feynman xF = 2pz/
s, etc. In the case ofazimuthal symmetry, the surfaces of constant xF are the planes pz = xF
√s/2,
the surfaces of constant y are the hyperboloids
p2z
[(1 + e2y
1 − e2y
)2
− 1
]− p2
T = m2,
and the surfaces of constant xT are the planes pT = xT√
s/2 in the phase space(see Fig. 1).
Uniˇed scale invariant variables for the presentation of single-particle inclu-sive distributions have been proposed [8], the properties of which are describedbelow.
Consider an arbitrary 4-momentum pµ(p0,p) and introduce light front com-binations [9]:
p± = p0 ± p3. (1)
LIGHT FRONT VARIABLES IN HIGH-ENERGY 1035
If the 4-momentum pµ is on the mass shell (p2 = m2), the combina-tions p±,pT (where pT = (p1, p2)) deˇne the so-called horospherical coordi-nate system (see, e. g., [10, 11]) on the corresponding mass-shell hyperboloidp20−p2 = m2 (u = p/m, u2
0−u2 = 1). Corresponding hyperboloid in the veloc-ity space is the realization of the curved space with constant negative curvature,i. e., the Lobachevsky space.
The following relations deˇne the horospherical coordinate system:
u0 + u3 = ea, (2)
u0 − u3 = e−a + eaγ2⊥, (3)
u⊥ = (u1, u2) = eaγ⊥, (4)
where γ⊥ = (γ1, γ2)
(u1, u2, u3) → (ea, γ⊥), (5)
dpE
= e2adadγ⊥, (6)
Edσ
dp=
1πm2
dσ
dadγ2⊥
. (7)
In particular, if a = 0, u0 + u3 = 1. Thus the horosphere in the Lobachevskyspace, realized on the upper sheet of the hyperboloid u2
0 − u2 = 1, is a two-dimensional surface with Euclidean intrinsic geometry. Coordinates (a, γ⊥) arerelated to measurable quantities (pz,p⊥) as follows:
a = lnE + pz
m= ln
√p2⊥ + p2
z + m2 + pz
m, (8)
γ⊥ =p⊥
E + pz=
p⊥√p2⊥ + p2
z + m2 + pz
, p⊥ = pT . (9)
Let us construct scale invariant variables:
ξ± = ±pc±
pa± + pb
±(10)
in terms of the 4-momenta paµ, pb
µ, pcµ of particles a, b, c, entering the inclusive
reaction a+b → c+X . The z axis is taken to be the collision axis, i. e., pz = p3.Particles a and b can be hadrons, heavy ions, leptons.
It is interesting to note the properties of ξ± variables in some limiting cases.The light front variables ξ± in the centre-of-mass frame are deˇned as follows [8]:
ξ± = ±E ± pz√s
= ±E + |pz|√s
, (11)
1036 CHKHAIDZE L. ET AL.
where s is the usual Mandelstam variable; E =√
p2z + p2
T + m2 and pz are theenergy and the z component of the momentum of produced particle. The uppersign in Eq. (11) is used for the right-hand side hemisphere; and the lower sign,for the left-hand side hemisphere.
In order to enlarge the scale in the region of small ξ±, it is convenient alsoto introduce the variables
ζ± = ∓ ln |ξ±|. (12)
The ζ+ variable is related to a via the following relation:
ζ+ = −a − lnm√s.
The invariant differential cross section in terms of these variables looks as follows:
Edσ
dp=
|ξ±|π
dσ
dξ±dp2T
=1π
dσ
dζ±dp2T
. (13)
In the limits of high pz (|pz | � pT ) and high pT (pT � |pz|), the ξ± variablesgo over to the well-known variables
ξ± → 2pz√s
= xF , ξ± → mT√s→ pT√
s=
xT
2, mT =
√p2
T + m2,
respectively, which are intensively used in high-energy physics. ξ± variables arerelated to xF , xT and rapidity y as follows:
ξ± =12
(xF ±
√x2
F + x2T
), xT =
2mT√s
, (14)
y = ±12
ln(ξ±
√s)2
m2T
. (15)
The region |ξ±| < m/√
s is kinematically forbidden for the ξ± spectraintegrated over all values of p2
T , and the region |ξ±| < mT /√
s is forbidden forthe ξ± spectra at ˇxed values of p2
T . The minimum value of ξ± = ±m/√
s wecall the threshold value.
Light front variables have been introduced by Dirac [9], and they are widelyused now in theoretical studies of relativistic composite systems (see, e. g., [12Ä26], in theoretical and experimental studies of nuclear reactions with beams ofrelativistic nuclei (see, e. g., [22, 27, 28]) and in the study of quark conˇnement inQCD (see, e. g., [29]). Combinations like Eq. (1) appear also when considering thescale transformations [30] in the theory with fundamental length (see, e. g., [31]).
LIGHT FRONT VARIABLES IN HIGH-ENERGY 1037
2. INCLUSIVE DISTRIBUTIONS IN HADRONÄHADRON COLLISIONS
Using ξ± variables, inclusive reactions in π−p interactions at 5 and 40 GeV/chave been analysed. The following reactions have been studied:
π−p → π+ + X, (16)
π−p → π− + X, (17)
π−p → γ + X, (18)
π−p → K0 + X, (19)
π−p → Λ0 + X ; (20)
and the following experimental data have been used: for the reaction (16) wehave 19500 inelastic events detected in the one-metre Hydrogen Bubble Chamberof JINR [32]; for the reactions (17) and (18) at 5 GeV/c we have 7900 eventsdetected in the one-metre Propane Bubble Chamber of JINR [33]; for the reactions(16)Ä(20) at 40 GeV/c we have 14300 inelastic events in the two-metre PropaneBubble Chamber of JINR [34].
Experimental data for π− mesons obtained at both energies are presented in
the form of invariant differential cross sections|ξ±|π
dσ
dξ±and
|ξ±|π
dσ
dξ±dp2T
in
Fig. 2. The ξ± distribution of π+ (a) and π− mesons (b) from π−p collisions at 5 (◦) and40 GeV/c (•) of incident momentum
1038 CHKHAIDZE L. ET AL.
Fig. 3. The ξ± distribution of π+ (a, c) and π− mesons (b, d) from π−p collisions atdifferent intervals of transverse momentum: a, b) 0.1 < p⊥ < 0.2 GeV/c: ◦ Å 5 GeV/c;× Å 40 GeV/c of incident momentum; c, d) 0.0 < p⊥ < 0.1 GeV/c: � Å 5 GeV/c; � Å40 GeV/c of incident momentum
Figs. 2Ä5. For comparison, in Fig. 6, the data for the reaction (16) at the sameenergies are given as function of the usual Feynman variable xF .
ξ± distributions have two features, which make them differ from correspond-ing xF distributions: 1) Existence of the forbidden region near the point ξ± = 0(cross sections vanish in the region |ξ±| ≤ mc/
√s). Corresponding threshold
values ξ±th for various types of particles in various p⊥ intervals are given in Ta-
bles 1 and 2; 2) Existence of maxima at some ξ̃± in the region of relatively small|ξ±|. Maxima in ξ± distributions become less pronounced with increasing massof the detected particle. For γ quanta and π mesons we have well pronouncedpeaks in the region of small |ξ±|, for K0 mesons, and especially for Λ0 hyperons,we have more smooth behaviour.
LIGHT FRONT VARIABLES IN HIGH-ENERGY 1039
Fig. 4. The ξ± distribution of π+ (a, c) and π− mesons (b, d) from π−p collisions atdifferent intervals of transverse momentum: a, b) 0.3 < p⊥ < 0.4 GeV/c: � Å 5 GeV/c;� Å 40 GeV/c of incident momentum; c, d) 0.2 < p⊥ < 0.3 GeV/c: ◦ Å 5 GeV/c; • Å40 GeV/c of incident momentum
Differential cross sections as functions of ξ± variables at 5 and 40 GeV/c inthe region |ξ±| > |ξ̃±| are rather close to each other. This allows one to speakabout the approximate scale invariance (automodelity) in some regions of phasespace.
In order to study the nature of these maxima, we have investigated theangular and p2
⊥ distributions of π± mesons in π−p interactions at 40 GeV/c inthe regions |ξ±| < |ξ̃±| and |ξ±| > |ξ̃±| separately [35]. Results for π+ mesonsare presented in Fig. 7. The angular distribution of particles with |ξ±| > |ξ̃±|is sharply anisotropic in contrast to the almost �at distribution of particles with|ξ±| < |ξ̃±|. The slopes of p2
⊥ distributions differ substantially.
1040 CHKHAIDZE L. ET AL.
Fig. 5. The ξ± distribution of π+ (a, c) and π− mesons (b, d) from π−p collisions atdifferent intervals of transverse momentum: a, b) 0.5 < p⊥ < 1.0 GeV/c: ◦ Å 5 GeV/c;• Å 40 GeV/c of incident momentum; c, d) 0.4 < p⊥ < 0.5 GeV/c: � Å 5 GeV/c; � Å40 GeV/c of incident momentum
Note, that the surfaces of constant ξ± are the paraboloids
pcz =
pc2
⊥ + mc2 − (ξ±√
s)2
−2ξ±√
s(21)
Table 1. Threshold values and the values corresponding to the points of maxima(ξ̃+, ξ̃−) of ξ± variables for inclusive spectra
Fig. 6. The xF distribution of π+ from π−p col-lisions: ◦ Å 5 GeV/c; • Å 40 GeV/c of incidentmomentum
in the phase space (Fig. 8). Thusthe paraboloids
pcz =
pc2
⊥ +mc2 − (ξ̃+√
s)2
−2ξ̃+√
s(22)
separate two groups of particleswith signiˇcantly different charac-teristics.
Further analysis of inclu-sive reactions in terms of lightfront variables has been performedby means of ζ± variables (seeEq. (12)). Experimental data forinvariant differential cross sections1/π dσ/dζ± in the reaction (16)at 5 and 40 GeV/c are presentedin Fig. 9. Maxima at ζ̃± are alsoobserved. However, the region|ξ±| < |ξ̃±| goes over to the re-gion |ζ±| > |ζ̃±| and vice versa(see Eqs. (11) and (12)). It is in-
Table 2. Threshold values and the values corresponding to the points of maxima of ξ±
variables for different intervals of transverse momentum pc⊥
Fig. 7. Angular and p2⊥ distributions of π+ mesons from π−p collisions at 40 GeV/c with
ξ± < ξ̃± (•) and ξ± > ξ̃± (×)
Fig. 8. The schematic view of the sur-faces of const ξ±
terersting to note that in the region |ζ±| >|ζ̃±| experimental distributions at these twovalues of s1 and s2 (square of the totalenergy in the c. m. s.) are almost parallelcurves separated from each other to the dis-tance
∆± =12
lns2
s1(23)
in the ζ scale. It seems to be interestingto check this regularity at other energiesand for other produced particles. It willbe worth to investigate this phenomenon ine+−e− annihilation and in nucleusÄnucleuscollisions.
The almost �at cos Θ distribution in the region |ξ±| < |ξ̃±| allows oneto conclude that the thermal equilibrium seems to be reached. In order tocheck this hypothesis, we have studied the inclusive reaction p̄p → π± + X
LIGHT FRONT VARIABLES IN HIGH-ENERGY 1043
Fig. 9. The ζ± distribution of π+ mesons from π−p collisions at 5 GeV/c (◦) and40 GeV/c (•) of incident momentum
at 22.4 GeV/c, which has been detected in the two-metre Hydrogen Bubble Cham-ber of JINR. Details of experiment can be found in Ref. 36. In this case it issufˇcient to study the right-hand side hemisphere only, due to the CP symmetryof the reaction.
In Fig. 10 ξ+ and ζ+ distributions of π± mesons are shown. In Fig. 11 theangular and p2
⊥ distributions in two different regions of phase space are given.
1044 CHKHAIDZE L. ET AL.
Fig. 10. (ξ+/π) (dσ/dξ+) distribution (a) and (1/π) (dσ/dζ+) distribution (b) of π±
mesons in the reaction p̄p → π±X at 22.4 GeV/c. Solid curve Å ˇt of the data in theregion ξ+ < ξ̃+(ζ+ > ζ̃+) by the BoseÄEinstein distribution; dashed-dotted curve Åˇt of the data in the region ξ+ < ξ̃+(ζ+ > ζ̃+) by the Boltzmann distribution; dashedcurve Å ˇt of the data in the region ξ+ > ξ̃+(ζ+ < ζ̃+) by the formula (1 − ξ+)n
To describe the spectra in the region ξ+ < ξ̃+(ζ+ > ζ̃+), the simpleststatistical model (see, e. g., [37]) with the Boltzmann f(E) ∼ e−E/T and theBoseÄEinstein f(E) ∼ (eE/T − 1)−1 distributions has been used.
The distributions (1/π) (dσ/dζ+), dσ/dp2T , and dσ/d cos Θ look in this
region as follows:
1π
dσ
dζ+∼
p2T,max∫0
Ef(E)dp2T , (24)
LIGHT FRONT VARIABLES IN HIGH-ENERGY 1045
Fig. 11. Angular distribution (a) and p2T distribution (b) of π± mesons in the reaction
p̄p → π±X at 22.4 GeV/c (a) and (b). Solid curve Å ˇt of the data in the regionξ+ < ξ̃+(ζ+ > ζ̃+) by the BoseÄEinstein distribution; dashed-dotted curve Å ˇt of thedata in the region ξ+ < ξ̃+(ζ+ > ζ̃+) by the Boltzmann distribution; dashed curve Å ˇtof the data in the region ξ+ > ξ̃+(ζ+ < ζ̃+) by the formula (32)
dσ
dp2T
∼pz,max∫0
f(E)dpz, (25)
dσ
d cosΘ∼
pmax∫0
f(E)p2dp, (26)
E =√
p2 + m2π, p2 = p2
z + p2T , (27)
where
p2T,max = (ξ+√s)2 − m2
π, (28)
1046 CHKHAIDZE L. ET AL.
pz,max =p2
T + m2π − (ξ̃+√s)2
−2ξ̃+√
s, (29)
pmax =−ξ̃+
√s cosΘ +
√(ξ̃+
√s)2 − m2
π sin2 Θ
sin2 Θ. (30)
Table 3. Results of the ˇts of (1/π)(dσ/dζ+),dσ/d cos Θ, and dσ/dp2
T distributions in theregion ξ+ < ξ̃+(ζ+ > ζ̃+)
T , GeV
BoseÄEinstein Boltzmann
1
π
dσ
dζ+0.134 ± 0.004 0.119 ± 0.003
dσ
d cos Θ0.091 ± 0.003 0.086 ± 0.003
dσ
dp2T
0.110 ± 0.001 0.105 ± 0.001
The experimental distributions(1/π)(dσ/dζ+), dσ/dp2
T , and dσ/
d cos Θ in the region ξ+ < ξ̃+(ζ+ >ζ̃+) have been ˇtted by Eqs. (24),(25), and (26), respectively. The re-sults of the ˇt given in Table 3 andFigs. 10, b, 11, a, 11, b show satis-factory agreement with experiment.Thus the spectra of π± mesons inthe region ξ+ < ξ̃+(ζ+ > ζ̃+) aresatisfactorily described by the for-mulae which follow from the statis-tical model. The same formulaewhen extrapolated to the region ξ+ >ξ̃+(ζ+ < ζ̃+) deviate from the data.
In the region ξ+ > ξ̃+(ζ+ < ζ̃+), ζ+ distribution has been ˇtted by theformula
1π
dσ
dζ+∼ (1 − ξ+)n = (1 − e−ζ+
)n; (31)
and the p2T distribution, by the formula
dσ
dp2T
∼ α e−β1p2T + (1 − α) e−β2p2
T . (32)
Table 4. Results of the ˇts of dσ/dζ+ and (1/π) (dσ/dp2T ) distributions in the region
ξ+ > ξ̃+(ζ+ < ζ̃+)
α β1, β2, n(GeV/c)−2 (GeV/c)−2
1
π
dσ
dζ+Å Å Å 3.7 ± 0.1
dσ
dp2T
0.8 ± 0.03 6.0 ± 0.1 2.8 ± 0.3 Å
LIGHT FRONT VARIABLES IN HIGH-ENERGY 1047
Thus, the dependence (1 − ξ+)n is in good agreement with the data in theregion ξ+ > ξ+(ζ+ < ζ̃+) and deviates from them in the region ξ+ < ξ̃+(ζ+ >ζ̃+) (see Fig. 10, b).
Fig. 12. Black-body radiation intensityas a function of x = �ω/kT : 1 ÅdEω ∼ x2dx (Wien); 2 Å dEω ∼x3e−xdx (RayleighÄJeans); 3 Å dEω ∼x3(e−x − 1)−1dx (Planck)
Note that in the region ξ+ → 1,the parameterization (31) goes over tothe well-known quarkÄparton model pa-rameterization (1 − x)n with x = xF =2pz/
√s. The results of the ˇt are given
in Table 4 and Figs. 10, b and 11, b. Sincethe dependence (1−x)n, which is derivedfor x → 1, describes the data even in theregion x → 0 (where, in general, it mustnot work), but the dependence (31) devi-ates from the data in the region of smallξ+, it seems that the analysis of data interms of ξ± and ζ± distributions is moresensitive to the phenomenological modelsof multibody production at high energiesthan the analysis in terms of xF .
It is interesting to recall the similarsituation in the study of black-body radi-ation, where the Wien formula describesthe low frequency part of the spectrumand does not describe the high frequency part, whereas the situation is reversedin the case of RayleighÄJeans formula (see, e. g., [38]). To illustrate this inFig. 12, the black-body radiation intensity according to the Wien, RayleighÄJeansand Planck formulae are plotted against the dimensionless variable x = �ω/kT .
3. NUCLEUSÄNUCLEUS COLLISIONS
The study of π− mesons produced in the relativistic nucleusÄnucleus col-lisions in terms of the light front variables [39Ä41] has been performed. Thechoice of the light front variables is due to the fact, that, as one can see from theprevoius sections, these variables seem to be more sensitive to the dynamics ofinteraction as compared to the well-known Feynman variable xF and rapidity y.The analysis has been carried out in the nucleusÄnucleus centre-of-mass systemfor π− mesons from 6147 Å He(Li, C), 902 Å CÄNe, 6261 Å MgÄMg, 1203 ÅCÄCu, and 732 Å OÄPb collisions [39, 40] obtained on the SKM-200-GIBS fa-cility of JINR and for π− mesons from 8371 Å pÄC, 13318 Å HeÄC, 20594 ÅCÄC, and 1989 Å CÄTa collisions collected with the two-metre Propane BubbleChamber (PPK-500) of JINR [41].
1048 CHKHAIDZE L. ET AL.
Table 5. Number of events, trigger and the results of the joint ˇt of the distributions(1/π)(dN/dζ+), dN/dp2
T , dN/d cosΘ of π− mesons by Eqs. (24)Ä(26) in the regionζ+ > ζ̃+ and (1/π)(dN/dζ+) distributions by Eq. (31) in the region ζ+ < ζ̃+
SKM-GIBS consists of a 2-m streamer chamber, placed in a magnetic ˇeldof 0.8 T, and a triggering system. The streamer chamber was exposed to thebeam of He, C, O, Ne, and Mg nuclei accelerated in the synchrophasotron up tothe energy of 3.7 GeV/nucleon. The thickness of the solid targets in the shape of
Fig. 13. The ξ± distribution of π− mesons from different interactions: a) ∗ Å He(Li, C);� Å CÄNe; ◦ Å OÄPb; b) from CÄCu interactions: ◦ Å the experimental data; � Å theQGSM data. The curves are the result of polynomial approximation of the experimentaldata
LIGHT FRONT VARIABLES IN HIGH-ENERGY 1049
Fig. 14. The ξ± distribution of π− mesonsfrom MgÄMg interactions: ◦ Å experimen-tal data; Å QGSM data; the curve is aresult of polynomial approximation of theexperimental data
thin discs was 0.2 ÷ 0.4 g/cm2 (for Lithe thickness was 1.59 g/cm2). Neongas ˇlling the chamber also served asa nuclear target. The triggering sys-tem allowed the selection of ®inelastic¯and ®central¯ collisions. The ®central¯trigger was selecting events with nocharged projectile spectator fragments(with P/Z > 3 GeV/c) within a coneof half-angle Θch, Θn = 2.4◦ or 2.9◦.The trigger mode for each exposure isdeˇned as T (Θch, Θn). The number ofevents for all pairs of nuclei and corre-sponding trigger modes are listed in Ta-ble 5. Due to the small statistics and av-erage multiplicities, the data of HeÄLiand HeÄC collisions have been unitedand thus He(Li, C) represents this sam-ple of the data. In Figs. 13 and 14,the ξ± distributions of π− mesons fromHe(Li, C), CÄNe, MgÄMg, CÄCu, andOÄPb interactions are presented. Thesedistributions are similar for all analysed pairs of nuclei. One can see fromFigs. 13 and 14 that the principal differences of ξ± distributions as compared tothe corresponding xF distributions (Figs. 15 and 16) are the following:
Fig. 15. The xF distribution of π−
mesons from He(Li, C) (∗), CÄCu (�)and OÄPb (◦) interactions
Fig. 16. The xF distribution of π−
mesons from MgÄMg interactions: ◦ Åexperimental data; Å QGSM data
1050 CHKHAIDZE L. ET AL.
Fig. 17. The ζ± distribution of π− mesons from different interactions: a) ∗ Å He(Li, C);� Å CÄNe; ◦ Å OÄPb; b) from CÄCu interactions: ◦ Å the experimental data; � Å theQGSM data. The curves are the result of polynomial approximation of the experimentaldata
Fig. 18. The ζ± distribution of π− mesonsfrom MgÄMg interactions: ◦ Å experimen-tal data; Å QGSM data; the curve Åresult of polinomial approximation of theexperimental data
1) existence of some forbidden re-gion around the point ξ± = 0;
2) existence of maxima at some ξ̃±
in the region of relatively small |ξ±|;3) existence of limits for |ξ±| ≤
m/√
s), similarly to hadronÄhadron col-lisions.
The experimental data for invariantdistributions (1/π)(dN/dζ±) are shownin Figs. 17 and 18. The curves are theresult of the polynomial approximationof the experimental distributions. Themaxima at ζ̃± are also observed in theinvariant distributions (1/π)(dN/dζ±).However, the region |ξ±| > |ξ̃±| goesover to the region |ζ±| < |ζ̃±| andvice versa (see Eqs. (11) and (12)).The values of maxima are observed atζ̃± = 2.0 ± 0.1 for all pairs of nuclei.The ζ̃± is the function of the energy (see
Eqs. (11), (12)) and does not depend on the mass numbers of the projectile (AP )and target (AT ).
LIGHT FRONT VARIABLES IN HIGH-ENERGY 1051
Fig. 19. The p2T distribution of π−
mesons from He(Li, C) (∗), CÄNe (�),OÄPb (◦) interactions: a) for ζ+ > ζ̃+;b) for ζ+ < ζ̃+. c) From CÄCu in-teractions for ζ+ > ζ̃+: ◦ Å experi-mental data; � Å the QGSM data; forζ+ < ζ̃+: � Å experimental data; ∗ Åthe QGSM data. The solid lines Å ˇtof the experimental data in the regionsζ+ > ζ̃+ and ζ+ < ζ̃+ by Eqs. (25)and (32), correspondingly
In order to study the nature of these maxima, the phase space has beendivided into two parts |ζ±| > |ζ̃±| (ζ̃+ = 2.0) and |ζ±| < |ζ̃±|, and the p2
T
and the angular distributions of π− mesons in these regions have been studiedseparately. The numbers of pions in these two regions are approximately equal.For example in CÄCu interactions in the region |ζ±| > |ζ̃±| the number of pionsis equal to −1987; and in |ζ±| < |ζ̃±|, to 2212. In Figs. 19Ä22 the p2
T and theangular distributions of π− mesons from He(Li, C), CÄNe, MgÄMg, CÄCu, andOÄPb interactions in different regions of ζ+ (ζ+ > ζ̃+ and ζ+ < ζ̃+) in theforward hemisphere are presented.
One can see from Figs. 19Ä22 that the p2T and the angular distributions of
π− mesons differ signiˇcantly in ζ+ > ζ̃+ and ζ+ < ζ̃+ regions. The angular
1052 CHKHAIDZE L. ET AL.
distribution of pions in the region ζ+ < ζ̃+ (Figs. 21, b, c and 22) is sharplyanisotropic in contrast to the almost �at distribution in the region ζ+ > ζ̃+
(Figs. 21, a, c and 22). The �at behaviour of the angular distribution allows oneto think that one observes a partial thermal equilibrium in the region |ζ±| > |ζ̃±|(|ξ±| < |ξ̃±|) of phase space. The slopes of p2
T distributions differ greatly indifferent regions of ζ± (Figs. 19 and 20). For example in MgÄMg interactions:〈p2
T 〉 = (0.027 ± 0.002) (GeV/c)2 in the region ζ+ > ζ̃+; 〈p2T 〉 = (0.103 ±
0.009) (GeV/c)2 in the region ζ+ < ζ̃+.
Fig. 20. The p2T distribution of π− mesons
from MgÄMg interactions: ◦ Å experi-mental data for ζ+ > ζ̃+ (ζ̃+ = 2.0);� Å the QGSM data for ζ+ > ζ̃+; � Åexperimental data for ζ+ < ζ̃+; � Å theQGSM data for ζ+ < ζ̃+; dashed lines Åˇt of the experimental data by the Boltz-mann distribution; solid lines Å ˇt of theQGSM data by the Boltzmann distribution
Thus the values of ζ̃± are the boundaries of the two regions with signiˇcantlydifferent characteristics of π− mesons. The validity of this statement can beseen from the momentum distributions of π− mesons in the laboratory frame.Figure 23 presents the momentum disribution of pions from CÄCu collisions inthe laboratory frame. The shaded area corresponds to the region of ζ+ > ζ̃+
and the nonshaded one to the region of ζ+ < ζ̃+. One can see from Fig. 23that these two regions almost do not overlap in the momentum space unlike tothe c. m. s. case (overlap ∼ 45%). The pions from the region ζ+ > ζ̃+ havesmall momentum, approximately up to 0.6 GeV/c as compared to the pions fromζ+ < ζ̃+ (the momentum of pions ranges from ∼ 0.6 to 3 GeV/c). Similarresults have been also obtained for the other pairs of nuclei. Figure 24 presentsthe dependence of 〈P 〉lab on Θlab for all analysed pairs of nuclei (He(Li, C),and CÄNe data are presented with the same symbol because of the similarityof their dependences) in the ζ+ > ζ̃+ and ζ+ < ζ̃+ regions. The shapes ofthese dependences are different in two regions of ζ+. The curves are the resultof polynomial approximation. 〈P 〉lab decreases and 〈Θ〉lab increases with theincreasing of AP , AT .
LIGHT FRONT VARIABLES IN HIGH-ENERGY 1053
Fig. 21. The cos Θ distribution of π−
mesons from He(Li, C) (∗), CÄNe (�),OÄPb (◦) interactions: a) for ζ+ > ζ̃+;b) for ζ+ < ζ̃+. c) From CÄCu inter-actions for ζ+ > ζ̃+: ◦ Å experimentaldata; � Å the QGSM data; for ζ+ < ζ̃+:� Å experimental data; ∗ Å the QGSMdata. The solid lines Å ˇt of the exper-imental data in the region ζ+ > ζ̃+ bythe Eq. (26) and in the ζ+ < ζ̃+ by thepolynom
To describe the spectra in the region ζ+ > ζ̃+, the Boltzmann
f(E) ∼ e−E/T
distribution has been used.The distributions (1/π)(dN/dζ+), dN/dp2
T , dN/d cos Θ are given by for-mulae (24)Ä(27). The experimental distributions in the region ζ+ > ζ̃+ have beenˇtted by the expressions (24)Ä(26), respectively. The results of the joint ˇt ofthe distributions (1/π)(dN/dζ+), dN/dp2
T , dN/d cos Θ are given in Table 5 andFigs. 19Ä22. They show a rather good agreement with experiment. In Table 5 thevalues of the parameter T obtained by ˇtting the data with Boltzmann distribution
1054 CHKHAIDZE L. ET AL.
are presented. In order to determine how the characteristics vary, the analysis hasbeen carried out also for ζ̃+ = 1.9 and 2.1. The results are similar, but the jointˇt of the distributions is better for ζ̃+ = 2.0 (presented in ˇgures).
The spectra of π− mesons in the region ζ+ > ζ̃+ are satisfactorily describedby the formulae which follow from the thermal equilibration. The same formulae
Fig. 22. The cos Θ distribution of π−
mesons from MgÄMg interactions: ◦ Åexperimental data for ζ+ > ζ̃+ (ζ̃+ =2.0); � Å the QGSM data for ζ+ > ζ̃+;� Å experimental data for ζ+ < ζ̃+;� Å the QGSM data for ζ+ < ζ̃+;dashed lines Å ˇt of the experimentaldata; solid lines Å ˇt of the QGSM data
Fig. 23. The momentum distribution of π− mesons from CÄCu interactions in the laboratorysystem. The shaded area corresponds to the region of ζ+ > ζ̃+
Fig. 24. The dependence of 〈P 〉lab on Θlab in the regions ζ+ > ζ̃+ (bottom data) andζ+ < ζ̃+ (top data) for different interactions: � Å He(Li, C) and CÄNe; � Å CÄCu; ∗ ÅMgÄMg; ◦ Å OÄPb; the curves Å result of polynomial approximation
LIGHT FRONT VARIABLES IN HIGH-ENERGY 1055
when extrapolated to the region ζ+ < ζ̃+ (Figs. 25 and 26) deviate signiˇcantlyfrom the data. Therefore in the region ζ+ < ζ̃+, the p2
T distributions have beenˇtted by formula (32); and the ζ+ distributions, by formula (31). The dependence(31) (1 − e−|ζ+|)n is in good agreement with experiment in the region ζ+ < ζ̃+
and deviates from it in the region ζ+ > ζ̃+ (Figs. 25 and 26). The results of theˇt are given in Table 5 and Figs. 19, 20, 25, and 26.
Thus in the ζ± (ξ±) distributions we have singled out points ζ̃± (ξ̃±) whichseparate in the phase space two groups of particles with signiˇcantly differentcharacteristics. There are no such points in the xF and y distributions.
The similar analysis of π− meson spectra produced in pÄC, HeÄC, CÄC, andCÄTa interactions at a momentum of 4.2 GeV/c/nucleon has been carried out inlight front variables [41]. The data have been obtained using the 2-m PropaneBubble Chamber of JINR (Dubna), placed in a magnetic ˇeld of 1.5 T. Thechamber, which housed tantalum targets of thickness about 1mm, was irradiatedwith protons, deuterons and by relativistic He and C nuclei of incident momentathat varied between 2 and 10 GeV/c per nucleon. From the whole ensemble of
Fig. 25. The (1/π)(dN/dζ+) distribution of π− mesons from CÄCu interactions: ◦ Åexperimental data; the solid line Å ˇt of the experimental data in the region ζ+ > ζ̃+ bythe Eq. (24); the dashed line Å ˇt of the experimental data in the region ζ+ < ζ̃+ by theEq. (31)
Fig. 26. The (1/π)(dN/dζ+) distribution of π− mesons from MgÄMg interactions: ◦ Åexperimental data; � Å QGSM data; the solid line Å ˇt of the data in the region ζ+ > ζ̃+
by the Boltzmann distribution; the dashed line Å ˇt of the data in the region ζ+ < ζ̃+ by
the formula (1 − e−|ζ+|)n
1056 CHKHAIDZE L. ET AL.
Table 6. Results of the ˇt of (1/π)(dN/dζ+), dN/dp2T , dN/d cos Θ distributions for
π− mesons by Eqs. (24)Ä(26) in the region ζ+ > ζ̃+
collisions of C nuclei in Propane Chamber, inelastic CÄC collisions have beenselected. The points have been singled out ζ̃± : ζ̃± = 2.0 for pÄC, ζ̃± = 1.8 ÅHeÄC, ζ̃± = 1.9 Å CÄC, and ζ̃± = 2.0 Å CÄTa. The spectra of pions havebeen described with the same formulae, as for SKM-200-GIBS data, and theparameters T (ζ+ > ζ̃+) and n (ζ+ < ζ̃+) have been extracted. The results ofˇts are presented in Tables 6 and 7. One can see (Tables 5Ä7) that the results ofSKM-200-GIBS and of 2-m Propane Bubble Chamber Collaboration (PPK-500)agree quite well. The angular and transverse momentum distributions of π−
mesons in various regions of variables ξ± and ζ± in pÄC, HeÄC, CÄC, and CÄTacollisions show the characteristics [41] similar to those from He(Li, C), CÄNe,MgÄMg, CÄCu, and OÄPb collisions of SKM-200-GIBS experiment.
The Quark Gluon String Model was used for the comparison with exper-imental data of SKM-200-GIBS. He(Li, C), CÄNe, MgÄMg, CÄCu, and OÄPbinteractions have been generated using Monte-Carlo generator COLLI, based onthe QGSM. The events have been traced through the detector and trigger ˇlter.The events have been generated for not ˇxed impact parameter b̃. From the impactparameter distribution the mean value of 〈b〉 has been obtained. For the obtainedvalue of 〈b〉, total samples of AP ÄAT events have been generated. The numbers
Table 7. Results of the ˇt of (1/π)(dN/dζ+), dN/dp2T distributions for π− mesons by
of generated events for all analysed pairs of nuclei are listed in Table 5. In SKM-200-GIBS experiment the following values of b are most probable: b = 1.55 fmfor He(Li, C); b = 2.20 fm for CÄNe; b = 1.34 fm for MgÄMg; b = 2.75 fm forCÄCu; b = 3.75 fm for OÄPb.
The experimental results have been compared with the predictions of theQGSM for the above-mentioned values of b, and satisfactory agreement betweenthe experimental data and the model has been found. In Figs. 13, b, 14, 17, b,and 18, the ξ± and ζ± distributions of π− mesons from the QGSM calculationsare presented together with the experimental ones for CÄCu and MgÄMg inter-actions. One can see that the QGSM well reproduces these distributions. Thesimilar results have been obtained for all analysed pairs of nuclei. The QGSMalso reproduces the p2
T and cosΘ distributions (Figs. 19, c, 20, 21, c, and 22). TheQGSM data show the similar characteristics in different regions of ζ as experi-mental ones: sharply anisotropic angular distributions in the region ζ+ < ζ̃+ andthe almost �at distribution in the region ζ+ > ζ̃+; the slopes of p2
T distributionsdiffer greatly in different regions of ζ+. The average values of 〈p2
T 〉 in these tworegions also differ: for example, for MgÄMg, 〈p2
T 〉 = (0.029 ± 0.003) (GeV/c)2
in the region ζ+ > ζ̃+; 〈p2T 〉 = (0.109± 0.009) (GeV/c)2 in the region ζ+ < ζ̃+.
The average values of 〈p2T 〉 in different regions of ζ from the experimental and
QGSM data agree respectively; the momentum distributions of pions in the lab-oratory frame in different regions of ζ+ have also different shape of spectra asexperimental ones (Fig. 23). Momentum distributions of QGSM data reproducethe corresponding experimental spectra in both regions of ζ+. The distributionsobtained by the QGSM in the region ζ+ > ζ̃+ have been ˇtted by the expressions(24)Ä(26). The results of the ˇt are given in Table 5 and Figs. 19, c, 20, 21, cand 22. In the region ζ+ < ζ̃+, the p2
T and the ζ+ distributions have been ˇttedby the formulae (32) and (31), respectively. The results of the ˇt are given inTable 5 and Figs. 19, c, 20, 25 and 26. One can see from Table 5 that the valuesof the T extracted from the experimental and QGSM data coincide within theerrors. The QGSM does not reproduce satisfactorily the OÄPb data. This maybe caused by the fact, that QGSM simpliˇes the nuclear effects, which are morepronounced for heavy nuclei. In Ref. 42, it has been indicated that the modelcan be improved by including higher mass baryon resonances and taking intoaccount a possible increase of the pion absorption cross section, σ∆∆→NN indense baryon medium, in comparison with the cross section, obtained from thedetailed balance relation.
In Fig. 27, the dependence of the parameter T from Tables 5 and 6 on(AP · AT )1/2, obtained from the experimental and QGSM data of SKM-200-GIBS and experimental results of PPK-500, is presented. One can see that Tdecreases linearly with the increasing (AP · AT )1/2, i. e., with the increasingnumber of participating nucleons. Similar behaviour is predicted by the QGSM.
1058 CHKHAIDZE L. ET AL.
Fig. 27. The dependence of the parameterT on (AP · AT )1/2 for He(Li, C), CÄNe,MgÄMg, CÄCu, and OÄPb [38, 39]: ◦ Åthe experimental data; � Å the QGSMdata; for pÄC, HeÄC, CÄC, and CÄTa [40]:∗ Å the experimental data. The dashedline is a result of linear approximation
In Ref. 43, the temperatures of pi-ons in HeÄLi, HeÄC, CÄNe, MgÄMg,CÄCu, and OÄPb interactions were ob-tained be means of inclusive kinetic en-ergy and transverse momentum spec-tra in central rapidity interval (0.5Ä2.1for light nuclei and 0.1Ä1.8 for heavyones), which corresponds to the pioniza-tion region and with the c. m. s. angles90 ± 10◦. The pion spectra for HeÄLi,HeÄC, and CÄNe have been ˇtted by oneexponent; and for MgÄMg, CÄCu, andOÄPb, by a sum of two exponents, ortwo temperatures T1 and T2 (describingthe low and high momentum part of thespectrum). The temperatures extractedby the light front analysis for light pairsof nuclei are less about (15Ä20) % ascompared to those obtained in Ref. 43.For heavy pairs of nuclei, the temper-atures are more close to the low tem-perature T1. It seems obvious that thethermal equilibrium region correspondsto lower momenta. It should be men-
tioned that the extraction procedures of T in the light-front variables and inRef. 43 are quite different and it seems, that different regions of phase space areseperated by these methods.
The temperatures of pions have been extracted in the GSI experiments (FOPI,KAON and TAPS Collaborations, see, e. g., [44Ä46]). The T in the GSI experi-ments have been obtained in the same manner as in Ref. 43. FOPI Collaboration[44, 46] has obtained that the π− spectra from NiÄNi collisions require the sumof two exponential functions with independent yields, and slope parameters Tl
and Th describe mostly the low and the high momentum parts of the spectrum,respectively: at E = 1.06 GeV/nucleon Tl = (55±3) MeV, Th = (93±5) MeV;at E = 1.45 GeV/nucleon Tl = (56 ± 3) MeV, Th = (100 ± 5) MeV; atE = 1.93 GeV/nucleon Tl = (61 ± 3) MeV, Th = (115 ± 6) MeV. TheFRS Collaboration has found that the T for π− mesons in NeÄNaF collisionsvary from (78 ± 2) MeV to (96 ± 3) MeV for projectile energies from 1.34 to1.94 GeV/nucleon.
TAPS Collaboration has found for π0 mesons T = (83 ± 3) MeV in CÄCinteractions at incident energy of E = 2 GeV/nucleon; T = (70± 1) MeV in ArÄCa interactions at incident energy of E = 1.5 GeV/nucleon; KAON Collaboration
LIGHT FRONT VARIABLES IN HIGH-ENERGY 1059
has found a value of T for π+ mesons ranging from (71±3) MeV (at energy E =1 GeV/nucleon) to (95± 3) MeV (at energy E = 1.8 GeV/nucleon). The numer-ical values of the parameter T for pions in AuÄAu collisions at 1.0 GeV/nucleon(FOPI) and our values for the heaviest colliding pair are close to each other.
CONCLUSION
A remark on the nature of maxima in ζ± distributions is in order. The ALEPHCollaboration observed the maxima in the ξ distributions (ξ = − ln p/pmax) [47]of secondary hadrons in e+e− collisions, which coincide to high precision withpredictions of the perturbative QCD (see, e. g., [48, 49]). The accuracy of coinci-dence increases when next-to-leading order corrections are taken into account. Sothe shapes of ξ distributions are related to the details of the underlying dynam-ics. Similarly, it seems that the maxima in ζ± distributions re�ect the dynamicsof the processes considered. In particular, secondary pions with |ξ±| < |ξ̃±|have almost �at angular distribution in the centre-of-mass frame, whereas pionswith |ξ±| > |ξ̃±| are produced sharply anisotropically. So the phase space ofsecondary pions is divided into two parts with signiˇcantly different characteris-tics in a unique way. Separation points are points of maxima in correspondingξ±(ζ±) spectra (or corresponding paraboloids in the phase space). Thus one cansay that the problem of separation of ®pionization¯ and fragmentation componentsseems to be solved in a unique way. An application of the method proposed to awide class of hadronic and nucleusÄnucleus reactions and e+e− annihilation intohadrons seems to be of great interest.
As is mentioned in the Introduction, light front variables deˇne the so-called horospherical coordinate system in the Lobachevsky space. It seems to beinteresting to use other coordinate systems on the mass shell hyperboloid (see,e. g., [10]) for the analysis of inclusive spectra.
In conclusion, we think that the use of light front variables can help todistinguish in between different dynamical contributions, or test basic principlesin other types of analysis, such as two-particle correlations, HBT-interferometry[50, 51] and transverse �ow studies [52].
Acknowledgements. The authors would like to thank the staffs of the 1-mPropane Bubble Chamber, 2-m Propane Bubble Chamber, 2-m Hydrogen BubbleChamber and SKM-200-GIBS Collaborations of JINR for helping in obtainingthe data, fruitful collaboration and many valuable discussions. One of the au-thors (V. G.) expresses his deep gratitude to V. Kadyshevsky and P. Bogolubovfor the warm hospitality at the Laboratory of Theoretical Physics of JINR. Manycolleagues from the High Energy Physics Institute of Tbilisi State Universitycontributed to these investigations. The investigations for hadronÄhadron colli-sions have been begun under the leadership of Prof. N. Amaglobeli. The authors
1060 CHKHAIDZE L. ET AL.
are very indepted to Prof. N. Amaglobeli and express him their deep gratitude.Special thanks are due L. Abesalashvili, B. Chiladze, N. Koutsidi, G. Kuratashvili,R. Kvatadze, Sh. Shoshiashvili, and T. Topuria for helpful collaboration.
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