Self-similar solutions for A-dependences in relativistic nuclear collisions in the transition energy region. A.A.Baldin.

Self-similar solutions for A-dependences in relativistic nuclear collisions in the

transition energy region.

A.A.Baldin

One of the most important problems nowadays, formulated by a distinguished scientist S.Nagamia in 1994, is the determination of the conditions in which hadrons lose their identity, and sub-nucleonic degrees of freedom begin to play a leading role.

A.M.Baldin proposed a classification of applicability of the notion “elementary particle” on the basis of the variable bik introduced by him.

12122

ki

kikikikiik mm

ppEEUUUUb

1ch212 ikkiik UUb

• the region

relates to non-relativistic nuclear physics, where nucleons can be considered as point objects;

• the region

relates to excitation of internal degrees of freedom of hadrons;

• the region

should, in principle, be described by quantum chromodynamics.

2100 ikb

1~ikb

1ikb

PDG

0.1 1 10 100

101

102

tot

[mb]

b12

pp

-p

+p

K. Hagiwara et al., Phys. Rev. D 66, 010001 (2002) (http://www-pdg.lbl.gov/)

• PDG

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

101

102

QCD -> transition region<-nucleon physics

tot

[mb]

pp

p p K-p

K+p

K. Hagiwara et al., Phys. Rev. D 66, 010001 (2002) (http://www-pdg.lbl.gov/)

V.Burov at.al. Phys.Lett. B67:46,1977 , PEPAN,15:1249-1295,1984

Cumulative ProcessesCumulative Processes

Structure of the nuclei at short distances

33

3 3

2/ 2 3 3( 1)

1 2 3 4

3

0

3( ) ( )

(2 )

( ) ( ) ; ( / ) ;

10

1, 5%, 0.3%, 0.

0.75 ;

0

1

(180 )

2

,

.

2%

1

f

f

k kk

p

ff

i rk kk k

p

p

f

E GeV T GeV

pp

p

ddEd p

r e d r r r r fm

p A X

dE p pd p

m

X

f

3M

1

11 A

PX

2

22 A

PX

,...,,,,,, 3331221 ihhPSAA

( ) ( )' 'X m u X m u M u X m u X m u M ukk

k1 0 1 2 0 2 3 32

1 0 1 2 0 24

2

ij i j i j i ju u P P M M /

21

122122

21 2

2

1 XXXX

Ed

d pC A A fX X

3

3 1 1 21 2

Self-similarity is a special symmetry of solutions which consists in that the change in the scales of independent variables can be compensated by the self-similarity transformation of other dynamical variables.

This results in a reduction of the number of the variables which any physical law depend upon.

The relationship between X1 and X2 is described by the laws of conservation written in the form

Here Mn is the nucleon mass, and M3 the mass of an emitted particle. Essentially, we are using an experimentally proved correlation depletion principle in the relative four-velocity space which enables us to neglect the relative motion of not detected particles, namely the quantity

in the right-hand part of the above equation. Employing this approximation, the correlation between X1 and X2 can

conveniently be written in the form

In the case of production of antiparticle with mass M3, the mass M4 is equal to M3 as a consequence of conservation of quantum numbers. In studying the production of protons and nuclear fragments M4 = M3 as far as minimal value of corresponds to the fact that any other additional particles are not produced. The X1 and X2 obtained from the minimum are used to construct an universal description of the A-dependencies.

The analysis of the experimental data shows that the A-dependence of the inclusive production cross section can be parametrized by a universal function , were X is equal to X1 and X2 , respectively.

X M u X M u M u M X u M X u M un n k kk

1 1 1 2 2 2 3 3

2

1 1 2 24

2

2 11

kl k lk

M M

X X XM

M

M

MX

M

M

M

M

M M

Mp p p p p1 2 12 1

313

42

323

4 42

32

12

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.01E-10

1E-9

1E-8

1E-7

1E-6

1E-5

1E-4

1E-3

0.01

0.1

1

10

45 Gev 159o

Al Ti Mo W

10.14 GeV 97o

Al Cu Ta

inv A

11 A

22

[mb G

ev -

2 c3

sr -

1 ]

Cumulative processes

S.V.Boyarinov, et al. Yad. Fis. , v.57, N8, (1994) ,1452-1461.

O.P.Gavrishchuk et al. Nucl. Phys., A523 (1991) 589.

0.5 1.0 1.5 2.0 2.5 3.010-7

10-6

10-5

10-4

10-3

10-2

10-1

100

101

102

103

inv/A

1 1 A2 2

[mb GeV

-2 c3

sr-1 ]

p+C->

150

350

450

600

800

1200

p+Al->

970

p+W-> 970

1.5 2.0 2.5 3.0 3.5 4.0 4.510-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

in

v A

1

1 A

2

2 [

mb

Gev

-2 c

3 sr-1

]

_ _ A+A--->P,K+... _ GeV/n Angle P

dC 3.65 24o

CC 3.65 24o

CCu 3.65 24o

SiSi 2.0 0o

SiSi 1.65 0o

_ K

dC 2.5 24o

CC 2.5 24o

SiSi 1.0 0o

SiSi 1.4 0o

SiSi 2.0 0o

CaCa 2.0 0o

Twice cumulative

Jim Carroll Nucl. Phys. A488 (1989) 2192.A.Shor et al. Phys. Rev. Lett. 62 (1989) 2192.A.A.Baldin et al. Nucl. Phys., A519 (1990) 407.A.A.Baldin et al. Rapid Communications JINR, 3-92 (1992) 20.

1.5 2.0 2.5 3.0 3.5 4.0 4.510-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

in

v A

1

1 A

2

2 [

mb

Gev

-2 c

3 sr-1

] _ _ A+A--->P,K +...

_ GeV/n 0o

P NeCu1.9 NeSn1.89 NeSn1.69 NeBi1.87 NiNi1.85 NiNI1.66

_ K

NeCu1.9 NeSn1.89 NeSn1.69 NeSn1.49 NeBi1.87 NiNi1.85 NiNi1.66

Twice cumulative

A.Schroter et al. Z.Phys. A350, (1994), 101-113.

0.5 1.0 1.5 2.0 2.5 3.0 3.510-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

101

102p(240 GeV)+Be--->h +... (0o)

p anti p d anti d

t ( 3He)

anti t ( 3He)

iv

n A

1 A

2

[m

b G

eV -2 c

3 s

r -1 ]

Antimatter production

0.5 1.0 1.5 2.0 2.510-3

10-2

10-1

1

10

102

103

104

in

v/A1 A

2

12C+181Ta 3.65 GeV/n

100,350,450,600,1000,1200

20Ne+64Cu,119Sn,209Bi 1.5-1.9 GeV/n

58Ni+58Ni 1.7-1.9GeV/n 00

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

10-4

10-3

10-2

10-1

100

10-4in

24Mg+24Mg----> <N>=8.24

50

150

250

350

450

600

800

1000

1200

1500

inv/A

1 A

2

A.A.Baldin, E.N.Kladnitskaya, O.V. Rogachevsky, JINR Rapid Comm., (1999), N.2 [94]-99, p.20.M.Kh.Anikina, et al., Phys. Lett. B., (1997), v.397, p.30.

1.0 1.5 2.0 2.5 3.0

1

10

y

A+A ----> P +X 0o

Si+Au Si+Cu Si+Al

ivn

[mb/

GeV

2 /

c 3 ]

Rapidity (Lab Frame)

P.Stankus et al. Nucl. Phys. A544 (1992) p .603c-608c.

0.5 1.0 1.5 2.0 2.5 3.0

10-5

10-4

10-3

10-2

10-1

Au(25GeV/u)+Au->J/

inv

y

100

150

10

conclusion

• The proposed self-similar solution quantitatively describes the angular, energy and A- dependences of inclusive production cross sections of all hadrons with transverse momentum up to 2GeV. For higher transverse momenta the A-dependence becomes a function not only of X1, X2, but also of Рt (or mt).

• The analysis of inclusive spectra for the data selected in different ways shows that multiplicity in relativistic nuclear collisions has its origin basically in independent nucleon-nucleon interactions. Thus, high multiplicity at interaction of heavy nuclei is not a satisfactory criterion for search and study of collective interactions, or detection of exotic states of nuclear matter (such as quark-gluon plasma).

• It is natural to consider two types of collectivity in nuclear-nuclear collisions: the first is related to production of particles in the region kinematically forbidden for single nucleon-nucleon interactions (X1 or X2 or both greater than unity); the second is a result of collectivity of the initial state in nucleus-nucleus collisions) – high probability of a large number of independent nucleon-nucleon interactions in the collision. The analysis of multiple experimental data on the basis of the proposed self-similarity approach allows to conclude that the effect of the collectivity of the first type drops drastically with increasing energy of colliding particles and increases with increasing mass of the produced particle.

2

331

233

1

113

3

exp21

CAACpd

dE

XX

• Collective effects of the first type (cumulative) “die out” with increasing collision energy. Therefore, in order to investigate subthreshold processes, it is necessary to optimize the combination of the nucleus mass and energy to provide sufficient number of nucleons, on the one hand, and avoid

extra multiplicity, on the other hand.

21

122122

21 2

2

1 XXXX

We must remember that what we observe is not Nature itself, but Nature which uncovers in the form in which it is revealed by our way of putting questions. Scientific work in physics consists in putting questions about Nature in the language we use, and trying to get an answer in experiment performed by the means (tools) we have.

At that the words of Bohr about quantum theory come to mind: if harmony in life is sought for, one should never forget that in the game of life we are spectators and players at the same time.