KNO and Polyakov's multiplicity scaling in inelastic pp- and pp -collisions at superhigh energies
It is shown that the normalized Saleh-Teich's multiplicity charged particles distribution in inelastic pp- and pp-collisions describes the KNO scaling low in soft processes (the range of ISR energies) and can be exhibited as Polyakov's scaling low in hard processes ( (√s >> 1800 GeV)...
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| Zitieren: | KNO and Polyakov's multiplicity scaling in inelastic pp- and pp -collisions at superhigh energies / V.D. Rusov, T.N. Zelentsova, S.I. Kosenko, M.M. Ovsyanko, I.V. Sharf // Вопросы атомной науки и техники. — 2001. — № 6. — С. 161-166. — Бібліогр.: 21 назв. — англ. |
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| author | Rusov, V.D. Zelentsova, T.N. Kosenko, S.I. Ovsyanko, M.M. Sharf, I.V. |
| author_facet | Rusov, V.D. Zelentsova, T.N. Kosenko, S.I. Ovsyanko, M.M. Sharf, I.V. |
| citation_txt | KNO and Polyakov's multiplicity scaling in inelastic pp- and pp -collisions at superhigh energies / V.D. Rusov, T.N. Zelentsova, S.I. Kosenko, M.M. Ovsyanko, I.V. Sharf // Вопросы атомной науки и техники. — 2001. — № 6. — С. 161-166. — Бібліогр.: 21 назв. — англ. |
| collection | DSpace DC |
| container_title | Вопросы атомной науки и техники |
| description | It is shown that the normalized Saleh-Teich's multiplicity charged particles distribution in inelastic pp- and pp-collisions describes the KNO scaling low in soft processes (the range of ISR energies) and can be exhibited as Polyakov's scaling low in hard processes ( (√s >> 1800 GeV) but in any case the multiplicity scaling is always violated in the range of Sp − pS energies. It is supposed that the increase of transverse momentuma (or, what is the same, increase of phase volume) is the reason of the violation of any type scaling in the range of Sp − pS energies, when the part of semi-hard processes is essentially increased.
|
| first_indexed | 2025-12-02T06:10:36Z |
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| fulltext |
KNO AND POLYAKOV'S MULTIPLICITY SCALING
IN INELASTIC pp - AND pp - COLLISIONS
AT SUPERHIGH ENERGIES
V.D. Rusov, T.N. Zelentsova, S.I. Kosenko, M.M. Ovsyanko, I.V. Sharf
Odessa National Polytechnic University, Odessa, Ukraine
e-mail: siiis@te.net.ua
It is shown that the normalized Saleh-Teich's multiplicity charged particles distribution in inelastic pp- and pp-
collisions describes the KNO scaling low in soft processes (the range of ISR energies) and can be exhibited as
Polyakov's scaling low in hard processes (√s >> 1800 GeV) but in any case the multiplicity scaling is always viol-
ated in the range of pSpS − energies. It is supposed that the increase of transverse momentuma (or, what is the
same, increase of phase volume) is the reason of the violation of any type scaling in the range of pSpS − energies,
when the part of semi-hard processes is essentially increased.
PACS: 13.85.-t;12.40.Ee;25.40.Ep
1. INTRODUCTION
As its shown in [1], the multiplicity charged
particles distributions in inelastic pp- and pp-collisions
in all the range of ISR- and pSpS − energies up to en-
ergy of Fer-milab tevatron collider are well described by
Salech-Teich’s three-parametrical distribution. The ST-
distri-bution is obtained in the supposition that on the
first stage of interaction (which is described by homo-
geneous Poisson process) the random number of
primary particles <ν> (partons, pomerons, strings, etc.)
in the impact parameters bt space1 is produced, and on
the secondary stage each of primary particles generates
inhomogeneous Poisson process of a random number of
secondary particles <ε> («grey» hadrons) production on
characteristic length <∆b> equal to average length of
parton chain (or, in other words, characteristic length of
hadron jet formation). Thus the decolouration of «grey»
hadrons happens on the interval of impact parameters
[0, R], where R is average so-called radius of interaction
of primary particles [2].
We won't show here bulky expressions of Saleh-
Teich’s three-parametrical distribution [1]. Let us note
that identification and quantitative analysis of appropri-
ate parameters of experimental charged particles distri-
butions in hadron-hadron collisions is reduced to the de-
termination of parametrical triplet R/<∆b>, <ε>, <ν>}
by solution of the system of the nonlinear equations for
1 In impact parameters representation the naïve model [1] has such
physical singularities. The Eq. (4) in [1] is the convolution of the func-
tion of hadron production on the interval [0, R] (in terms of present
paper), which can be re-written as:
)()()(,)()()( RbububHdbbbhbHbh ttttttttR −−=′+⋅=′ ∫
∞
∞−
where u(bt) is single Heaviside function. In this expression the physi-
cal interpretation of the «step» H(bt) consists in the fact that in impact
parameters space it represents a model image of real function, i.e.
overlap function Gin under condition of unitary limit (Gin≤1) at s→∞.
Then the maximum value of bt is meaningful of radius of primary par-
ticles interaction R.
ST-distribution moments, which are explicitly con-
sidered in Ref. [3].
As result of the analysis of fitting data (collected in
Table) of AВCDHW [4], UA5 [5], CDF [6] Collabora-
tions experimental multiplicity distribution the follow-
ing empirical dependences of the ST-distribution para-
meters are obtained:
∆2
~ sn , (1)
( ) ( )
∆
−−×
×−+
∆
−⋅+
∆
b
b
R
sRs
2exp1
8.2ln11.12exp10066.0~
2
ε
(2)
1
42
41.023.123.1~
−
∆∆
+
+
−
∆
γγ
γ
ccb s
s
s
sR , (3)
where ∆ = 0.212, γ = (s – sc)/|s - sc|, √sc = 770 GeV.
In a way, the asymptotic expressions (1)-(3) make it
possible to consider the three-parametrical ST-distribu-
tion as "one-parametrical" multiplicity distribution
P(n,s), where the role of the parameter plays collision
energy s. Hence naturally suggests the idea to verify the
known hypothesis of the asymptotic scaling behavior of
multiplicity distributions [7,8] :
( )
=
)()(
1,
sn
n
sn
snP ψ , (4)
which is widely known as KNO scaling of the multipli-
city distributions.
For the first time such form of P(n, s) was derived
within the framework of the conformal theories in the
paper by Polyakov [7] as consequence for the strong in-
teractions of hadrons of the application of three ideas -
unitarity, analyticity and similarity. The similarity hypo-
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2001, № 6, p. 161-166. 161
mailto:siiis@te.net
thesis, i.e. hypothesis of asymptotic scale invariance
(which was induced by "strong coupling" regime in the
Pomeranchuk pole problem [9]) reduces both to scaling
form of secondary particles distribution (4) and to the
fact that the average multiplicity of hadrons in strong in-
teractions at small distances is power-law function of
collision energy s:
( ) ∆∝ 2ssn . (5)
The same form of P(n, s) was obtained by Koba,
Nielsen and Olesen [8], but it was based on Feynman's
scaling [10]. The Feyman's model notions about high-
energy collisions of hadrons (including the suppositions
of transverse momentuma limitation of produced
particles and the independence of Feynman's functions
on collision energy) led the authors [8] both to scaling
form of secondary particles distribution (4) and to the
fact that the average multiplicity of hadrons in strong in-
teractions at large distances is logarithmic function of
collision energy s:
ssn ln)( ∝ . (6)
Let us add that in strong interactions at large dis-
tances the dependence (5) was obtained earlier within
frameworks of Regge phenomenology [11] and also in
multiperipheral models [12].
The purpose of the given research are the study and
designing of the properties of normalized multiplicity
ST-distribution, which satisfy to condition of similarity
in strong interactions at small distances (Polyakov's
scaling) and at large distances (KNO scaling).
2. PHENOMENOLOGY
OF ASYMPTOTICAL BEHAVIOR 0F NAIVE
MODEL PARAMETERS
Here we will consider in what degree qualitative and
quantitative aspects of asymptotical behavior of expres-
sions (1)-(3) at energies √s ≥ 900 GeV meet to relevant
physical mechanisms. We understand relevance as a
such asymptotic properties of parameters (1)-(3), which
ensure not only optimum fits of experimental date in the
energy range √s = 20÷1800 GeV, but also make it pos-
sible physically grounded to simulate the probable beha-
vior of multiplicity secondary particle distributions,
which could be obtained in the future researches at vari-
ous ultrahigh energies [1].
First of all it concerns the physical substantiation of
evolution of the parameters <∆b>, R and their ratio (3)
with the energy √s growth (because an asymptotical be-
havior of other two parameters (1) and (2) is well
known and are explicitly considered in Sec. 3 and 4 ac-
cordingly).
So, let us consider the phenomenology of the para-
meter (3) evolution at increasing of collision energy √s.
Now it is already indisputable that the semi-hard
processes (i.e. production of hadron mini-jets with relat-
ive large characteristic transverse momenta but with
small part of an initial energy x<<1 [2, 13, 14]) become
the main source of secondary particles at increasing of
the collision energy. Such picture of hadron high ener-
getic collisions was unusual for a long time because it
was considered that dominating source of secondary
particles were the soft processes. In particular, it con-
cerns the pSpS energy range, where strong interactions
at small distances become essential and where the QCD
perturbation theory is true.
Levin and Ryskin have shown [15] that the estima-
tion of radius of interaction R0 calculated in frameworks
of LLA QCD at certain value of scale of cutting (Q0) on
transverse momentuma p⊥ has following form:
[ ] 41
0
0 2954.0~ n
Q
R , (7)
where factor 2 takes into account the fact that in our
model the average multiplicity of secondary particles
<n> is considered not for all charged particles but for a
number of meson pairs [1]. It follows that R0=21/4R.
On the other hand, the ratio R/<∆b> jumping varies
at energy 900 GeV (see Table 1), and at the subsequent
increase of energy the area of indeterminacy <∆b> of
hadron production (in one mini-jet) in the impact para-
meters space becomes essential small value in compar-
ison with radius of interaction, i.e. R >> <∆b>. It means
that just hard processes play decisive role at energy √s ≥
900 GeV. Then taking into account that, on the one
hand, p⊥⋅<∆b> ~1 and, on the other hand, that the con-
sideration of the distribution of meson pairs numbers
gives a value of effective transverse moment equal to p⊥
≥2Q0, we have following an expression (with allowance
for (7)) for hard processes:
GeVsfornR
b
900908.1 41 ≥⋅≥
∆ . (8)
Thus the expression (8) qualitatively and quantitat-
ively reflects velocity gradient of ratio R/<∆b> (3) at ul-
trahigh energies starting from √s = 900 GeV and it's
also completely determined by the value of average
multiplicity <n> of secondary particles. It means, in it's
turn, that multiplicity ST-distribution becomes practic-
ally two-parametrical at energies √s ≥ 900 GeV, i.e. it
depends only on average multiplicity <n> and average
multiplicity <ε> in one mini-jet.
3. POLYAKOV'S SCALING
In terms of moments the homogeneous form of dis-
tribution P(n,s) in (4) means the independence of nor-
malized algebraic moments on collision energy s
∑== n
qqqq
q PnnwherennC , . (9)
For clear understanding of the strategy of designing of
the properties of normalized ST-distribution, which sat-
isfies to conditions of similarity hypothesis in strong in-
teractions at small distances, let's consider the asymptot-
ic properties of second normalized algebraic moment of
ST-distribution:
162
Table 1. Parameters of multiplicity ST-distributions obtained by fitting of experimental data
and predicted by our naive model
Experiment √s, GeV <n> <ε> R/<∆b> χ2/NDF
pp – collisions
ABCDHW Collaboration [4] 30.4 5.27 4.8 0.03 6/13
ABCDHW Collaboration [4] 44.5 6.04 5.5 0.05 6/15
ABCDHW Collaboration [4] 52.6 6.38 5.8 0.06 4/17
ABCDHW Collaboration [4] 62.2 6.81 6.2 0.07 3/16
−pp collisions
UA5 Ansorge et.al. [5] 200 10.70 8.3 0.22 47/34
UA5 Alner et.al. [5] 546 14.55 10.4 0.37 63/56
UA5 Ansorge et.al. [5] 900 17.80 12.3 6.80 52/52
CDF [6] 1800 24.00 16.3 4.22 135/97
Predictions of naïve model
- 2400 27.1 18.2 4.35 -
- 6000 40.0 26.6 4.80 -
- 14000 57.3 37.9 5.25 -
- 100000 131.8 53.7 6.47 -
- 200000 176.9 116.7 6.96 -
> >
∆
+
+
< <
∆
∆
+
+
=+==
.1,
1
1
;1,
1
1)var(1 22
2
2
b
b
b
R
n
R
n
R
n
n
n
n
C
ε
ε
(10)
where as var (n) Eq. (11) from Ref. [1] is used.
As it show the analysis of fitting data (Table), the
values of ratio (R/<∆b>) are practically identical in all
the range of ISR energies and monotonically increase in
all the range of pSpS -energies up to energy √s = 1800
GeV. The asymptotic tendency of second moment (10)
to constant value at energies over √s = 1800 GeV is
stipulated by an identical asymptotic of <n> and <ε>. It
means that if average multiplicity in hadron mini-jet <ε
> and total average multiplicity of hadrons <n> behave
uniformly (i.e., according to Polyakov [7], they are
power-law functions of energy) the Polyakov's scaling
must be observed in the range of ISR energies (where R/
<∆b> << 1) and can be observed at energies much above
√s = 1800 GeV (where R/<∆b> >> 1).
The power-law dependence of average total hadron
multiplicity on energy (4) is practically beyond doubt.
This is confirmed by experimental data in the range √s =
20-1800 GeV presented in Table. At the same time the
analogous statement, which concerning the average
multiplicity in hadron mini-jet <ε>, is rather ambiguous.
Within the framework of existing experimental data, <ε
> can be fitted both by logarithmic and by power-law
dependence on energy. The mixed variant of these de-
pendences with specified "survival" one of them in the
limit of high energies is also possible. For instance, such
mixed empirical dependence (which is almost «mirror»
concerning Eq.(5)):
( )
∆
−−+⋅+
+
∆
−⋅−
∆
b
b
Rs
Rs
exp110066.0
exp)8.2ln11.1(~
2
ε
, (11)
describes the appropriate values <ε> as good as Eq. (5)
(see Table), but this dependence asymptotically tends to
power-law dependence at energies over √s = 900 Gev
(R/<∆b> >>1), i.e., <ε> ~ s∆ with s → ∞.
Explanation of power-low behavior of average mul-
tiplicity in the jet <ε> is following. In 1968 Gribov and
Migdal [9], considering the problem of the interaction
of reggeons, showed that one possibility is "strong
coupling" case in which the many-particle Green's func-
tions are power-low, not logarithmic, functions of their
arguments. On the other hand, as Polyakov has noted
[7], the power-low asymptotic forms of the Green's
functions at large momentumа mean that the theory has
invariance with a respect to change of the space-time
scale, which is manifested at very small distances. Thus
the similarity hypothesis in the strong interaction at
small distances reduces to power-low asymptotics of the
average hadron multiplicity in the jet [7], i.e., <ε> ~
const⋅ s∆.
In a way, the physical content of Eq. (8) is clear. In soft hadron-hadron collisions (i.e., in strong interactions
a) b)
c) d)
Fig. 1. Normalized multiplicity ST-distribution for the non single-diffractive pp- and pp-collisions shows the
validity of KNO scaling in soft processes (a) and predicts the approximate validity of Polyakov's scaling in semi-
hard (c), hard processes (d) and strongly violation of any type scaling in the range of pSpS -energies (b)
L ((R/<∆b>)<< 1), but in semi-hard and hard hadron-
hadron collisions (i.e., in strong interactions at small
distances) the power-law physics "survives "((R/<∆b>) >
> 1).
et us consider the behavior of Polyakov's scaling in
the energy range √s = 0.03 - 100 TeV. For this purpose
we used the values obtained by fitting of experimental
distribution of secondary particles on energy interval √
s= 0.03 - 1.8 TeV and theoretically predicted by empir-
ical expressions (1), (8) and (11) at asymptotically high
energies (see Table) as the parameters of three-paramet-
rical ST-distribution. The research of properties of such
normalized ST-distribution (which satisfy to conditions
of the similarity hypothesis in strong interactions at
small distances) has shown an anomalous behavior of
Polyakov's multiplicity scaling of charged particles pro-
duced in hadron collisions. Scaling is valid inthe range
of ISR energies (Fig.1а), it is approximately valid at su-
perhigh energies starting from the energy √s = 100 TeV
(Fig. 1d) and it is completely violated in pSpS energy
range √s = 200-1800 GeV (Fig. 1b). In the energy range
s2-100 TeV slow but stable tendency to scaling is ob-
served (Fig. 1с).
The analysis of multiplicity distributions parameters
(Table 1) and expressions (1) and (11) shows that the
parameters <n> and <ε> are monotone increasing
power-law functions of energy, whereas the parameter
R/<∆b> jumping increase just in the energy range √
s=200-1800 GeV. It is experimentally shown that an av-
erage radius of inelastic interactions R is increased in-
significantly on energy in the range 200-1800 GeV and
is on interval R~<bt
2>1/2= 0.9-1.1 Fm [16].
The theoretical estimation of average radius of in-
elastic interaction (7) is calculated in LLA of QCD for
d
large transverse momenta [15] and satisfactorily de-
scribes known experimental data for semi-hard pro-
cesses [16]. Hence, the nature of R/<∆b> change on en-
ergy interval √s = 200-1800 GeV allows to conclude
that the sharp decreasing of average length of parton
chain <∆b> is main reason of total violation of
Polyakov's scaling. Such sharp decrease of <∆b>, which
is accompanied by the simultaneous increase of average
hadron multiplicity in mini-jet <ε>, point out the sharp
increase of parton density in the chain. This, in its turn,
means that the increase of multiplicity in mini-jet hap-
pens only due to the increase of transverse momentum,
i.e. directly due to increase of phase volume [2]. Thus, it
is possible to assume that significant increase of phase
volume in semi-hard processes is the reason of total vi-
olation of Polyakov's scaling.
Let's consider the mathematical interpretation of
Polyakov's scaling violation in energy range √s=200-
1800 GeV. It is easy to show that with (<ε>R/<∆b>)→0
ST-distribution asymptotically transforms to Poisson
distribution and with (R/<∆b>) >>1 it transforms to Ney-
man type A distribution from two parameters <n> and <
ε> [3, 17]:
( )∑
∞
=
−
⋅−=
0
exp
!
1exp
!
)(
)(
k
kn nn
k
k
n
k
nP
εε
ε
ε
, (12)
which exactly coincides with compound Poisson distri-
bution [18]. Hence, the jumping increase of parameter
R/<∆b> from 0.22 to 6.8 (see Table) in indicated energy
range leads to the fact that the ST-distribution (by virtue
of its asymptotic properties on parameter R/<∆b> [3,
17]) sharply changes its type and structure, i.e., it
changes Poisson (one-humped) type at R/<∆b> <<1 to
Neyman (multi-humped) type at R/<∆b> >>1 [3, 17].
Thus, such an asymptotic transition causes the loss of
self-similarity of distribution function, and this is the
reason of Polyakov's scaling violation. Physically it usu-
ally means the change of phase states of researched sys-
tem on parameter R/<∆b>.
A test of the scaling hypothesis is provided by an ex-
amination of energy dependence of the moments of the
distribution. The moments C2-5 for the non-single-dif-
fractive events are shown in Fig. 2.
The behavior of moments confirms our expectations
concerning to anomalous behavior of Polyakov's scaling
(Fig. 1) and, what is especially important, total equiva-
lence of Polyakov's scaling and normalized Neyman
type A distribution (12) in the limit of high energies
(over 100 TeV) (see Fig. 1d).
4. KNO SCALING
As it was noted above, KNO scaling grows out of
the approach based on similarity hypothesis in strong in-
teractions at large distances, and, in a way, it reflects the
self-similar properties of logarithmic physics. The au-
thors of Ref. [8] accent attention just to such properties.
They written that theirs' result "... can be expressed by
the saying that the normalized multiplicity distribution
keeps its form independently of the beam energy and
just scales up as lns" [8].
Fig. 2. Values of the first four C-moments for the
non single-diffractive ST-distributions as a function of
the center of mass energy. The values of C-moments at
ISR-, pSpS -energies and at √s=1800 GeV are given in
Refs. [4], [5], [13], accordingly
The asymptotic analysis of algebraic moments (6) of
ST-distributions (and, in particular, an example of an
asymptotic dependence of normalized second moment
(10) on collision energy s) presented in Sec. 3 allows to
make the conclusion concerning to KNO scaling also. If
the average multiplicity in hadron mini-jet <ε> and total
average multiplicity of hadrons <n> behave uniformly
(for instance, they both are logarithmic functions of en-
ergy), then KNO scaling must be observed in the ISR
energy range (where R/<∆b> <<1) and can be observed
at energies over √s >> 1800 GeV (where R/<∆b> >> 1).
However, as it shown above (see Table), the average
hadron multiplicity <n> is power-law function of colli-
sion energy s, i.e. <n>~s∆ at s→∞. Then, if the average
hadron multiplicity in mini-jet <ε> is logarithmic depen-
dence on collision energy (for example, such as Eq. (2)),
KNO scaling must be observed in the range of ISR-en-
ergies and always must be violated at energies over ISR-
energies. Such asymptotic of <ε> and <n> and also the
reasons of KNO scaling violation are typical, for exam-
ple, in different versions of dual parton model and
quark-gluon strings model [19], which met with consid-
erable success in describing and, in some cases, predict-
ing the main feature of low-pT physics at ISR and collid-
er energies [20].
In our case, in spite of the fact that the parameters <n>
and <ε> have different functional forms on collision en-
ergy s, KNO scaling really satisfies to a good accuracy
in the range of ISR-energies (and practically does not
differ from Fig. 1а); however as energy increases KNO
scaling is strongly violated. The validity of KNO scal-
ing in the ISR energy range is explained by the fact that
the asymptotic expressions for <n> and <ε> (though
they have power-law and logarithmic forms of the de-
pendence on collision energy s) are close in this energy
range and insignificantly vary with energy growth.
5. CONCLUSIONS
The basic result of the present work consists in that
the normalized multiplicity ST-distribution describes
KNO scaling in soft processes and can be exhibited as
Polyakov's scaling in hard processes. We assume that
the increase of transverse momentuma (or what is the
same, increase of phase volume) is the reason of the vi-
olation of any type scaling low in the range of pSpS -
energies, where the part of semi-hard processes is essen-
tially increased.
If further collider pp-experiments at energies
s >>1800 will prove the Polyakov's scaling existence,
then, firstly, the importance of true reasons of any type
scaling violation in area of pSpS -energies can be
scarcely exaggerated, and, secondly, the very important
answer to very old question: «Could it be that as in the
theory of critical phenomena, while the basic physics
(Hamiltonians) between e+e- annigilation and hadronic
multiparticle production could be entirely different,
multiplicity scaling may yet still be a universal feature
shared by both ?» [21] can be obtained.
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Table 1. Parameters of multiplicity ST-distributions obtained by fitting of experimental data
|
| id | nasplib_isofts_kiev_ua-123456789-79477 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-02T06:10:36Z |
| publishDate | 2001 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Rusov, V.D. Zelentsova, T.N. Kosenko, S.I. Ovsyanko, M.M. Sharf, I.V. 2015-04-02T15:21:10Z 2015-04-02T15:21:10Z 2001 KNO and Polyakov's multiplicity scaling in inelastic pp- and pp -collisions at superhigh energies / V.D. Rusov, T.N. Zelentsova, S.I. Kosenko, M.M. Ovsyanko, I.V. Sharf // Вопросы атомной науки и техники. — 2001. — № 6. — С. 161-166. — Бібліогр.: 21 назв. — англ. 1562-6016 PACS: 13.85.-t;12.40.Ee;25.40.Ep https://nasplib.isofts.kiev.ua/handle/123456789/79477 It is shown that the normalized Saleh-Teich's multiplicity charged particles distribution in inelastic pp- and pp-collisions describes the KNO scaling low in soft processes (the range of ISR energies) and can be exhibited as Polyakov's scaling low in hard processes ( (√s >> 1800 GeV) but in any case the multiplicity scaling is always violated in the range of Sp − pS energies. It is supposed that the increase of transverse momentuma (or, what is the same, increase of phase volume) is the reason of the violation of any type scaling in the range of Sp − pS energies, when the part of semi-hard processes is essentially increased. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Electrodynamics of high energies in matter and strong fields KNO and Polyakov's multiplicity scaling in inelastic pp- and pp -collisions at superhigh energies KNO и поляковский законы скейлинга множественности в неупругих pp- и pp-столкновениях при сверхвысоких энергиях Article published earlier |
| spellingShingle | KNO and Polyakov's multiplicity scaling in inelastic pp- and pp -collisions at superhigh energies Rusov, V.D. Zelentsova, T.N. Kosenko, S.I. Ovsyanko, M.M. Sharf, I.V. Electrodynamics of high energies in matter and strong fields |
| title | KNO and Polyakov's multiplicity scaling in inelastic pp- and pp -collisions at superhigh energies |
| title_alt | KNO и поляковский законы скейлинга множественности в неупругих pp- и pp-столкновениях при сверхвысоких энергиях |
| title_full | KNO and Polyakov's multiplicity scaling in inelastic pp- and pp -collisions at superhigh energies |
| title_fullStr | KNO and Polyakov's multiplicity scaling in inelastic pp- and pp -collisions at superhigh energies |
| title_full_unstemmed | KNO and Polyakov's multiplicity scaling in inelastic pp- and pp -collisions at superhigh energies |
| title_short | KNO and Polyakov's multiplicity scaling in inelastic pp- and pp -collisions at superhigh energies |
| title_sort | kno and polyakov's multiplicity scaling in inelastic pp- and pp -collisions at superhigh energies |
| topic | Electrodynamics of high energies in matter and strong fields |
| topic_facet | Electrodynamics of high energies in matter and strong fields |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/79477 |
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