_{1}

^{*}

With a view to understanding J/Ψ suppression in relativistic heavy ion collisions, we compute the suppression rate within the framework of hy-drodynamical evolution model. For this, we consider an ellipsoidal flow and use an ansatz for temperature profile function which accounts for time and the three dimensional space evolution of the quark-gluon plas-ma. We have calculated the survival probability separately as the func-tion of transverse and longitudinal momentum. We have shown that previous calculations are special cases of this model.

Relativistic heavy ion collisions vis-à-vis formation of quark-gluon plasma (QGP) and its implications have been very excited field of research [

In the present work, we consider ellipsoidal evolution and use exponential type temperature profile function. In the next section, we briefly outline various theoretical works on the understanding of J/Ψ suppression. In Section 3, we compute the suppression rate within the framework of the present model. Results are discussed and summarized in Section 4.

The formation and evolution of QGP in the relativistic heavy ion collisions is well described within the framework of Bjorken hydrodynamical evolution model [

P = 1 − S = 1 − exp ( − T ) (1)

In all the works on the J/Ψ suppression, survival probability has been calculated by parameterizing the thermodynamical quantities, namely the entropy density s [

s ( t 0 , r ) = s 0 ( 1 − r 2 / R 2 ) a ′ ; ε ( r ) = ε 0 ( 1 − r 2 / R 2 ) 2 / 3 ; T ( r ) = T ( 0 ) ( 1 − r 2 / R 2 ) b / 3 and T ( r , t ) = T 0 exp ( − b r / R ) exp ( − η z ) t − 1 / 3 (2)

where R is the projectile radius, a’ is a free parameter [_{T} has been calculated in the non transparent manner while in [

In this section, we consider ellipsoidal evolution in both time and three dimensional space within the framework of hydrodynamical model for QGP. It is pertinent to mention here that ellipsoidal evolution has been used in physical sciences and other branches of science [_{L} and p_{T} respectively.

For ellipsoidal symmetry, the four dimensional temperature profile function, in general, can be written as

T ( t , r ′ ) = T 0 f ( t , z ) T ( r ) (3)

where f ( t , z ) is a function of time and z axis along the collision direction; r ′ is a function of x, y and z; and r is a function of x and y in the transverse plane in the ellipsoidal interaction volume under consideration. Here we have considered various functions as

f ( t , z ) = t − 1 / 3 e − η z ; T ( r ) = e − α 1 x e − α 2 y and T ( r ) f ( t , z ) = T ( r ′ ) (4)

where η , α 1 and α 2 are measure of steepness of the fall of temperature along z, x and y directions. Here, α 1 = b 1 / R 1 , α 2 = b 2 / R 2 with R 1 and R 2 are radius of colliding nuclei and b_{1} and b_{2} are parameters. The time dependence of the temperature profile function given in Equation (3) is in accordance with the scaling law [

∫ T ( t , r ′ ) d τ ′ = 1 (5)

where d τ ′ is an ellipsoidal evolution volume element. Equations (3) and (4) lead to

T 0 ∫ t − 1 / 3 d t ∫ e − η z d z ∫ e − α 1 x d x ∫ e − α 2 y d y = 1 (6)

Solving integrals in Equation (6), we obtain the normalization constant as

T 0 = ( η α 1 α 2 / 12 ) [ ( t f 2 / 3 − t i 2 / 3 ) sinh ( η c ) ⋅ sinh ( α 1 a 1 − h 2 / c 2 ) ⋅ sinh ( α 2 b 1 − h 2 / c 2 ) ] − 1 (7)

where a, b and c are along x, y and z directions respectively; h within sine and cosine hyperbolic functions represents z co-ordinate such that x varies from − a 1 − h 2 / c 2 to + a 1 − h 2 / c 2 ; y varies from − b 1 − h 2 / c 2 to + b 1 − h 2 / c 2 and z varies from −c to +c. The plasma evolution ellipsoidal volume, V = ∫ F ( h ) d h = ( 4 / 3 ) π a b c where the area F ( h ) = π a b ( 1 − h 2 / c 2 ) .

Therefore, the temperature profile function becomes

T ( t , r ′ ) = ( η α 1 α 2 / 12 ) [ ( t f 2 / 3 − t i 2 / 3 ) sinh ( η c ) ⋅ sinh ( α 1 a 1 − h 2 / c 2 ) ⋅ sinh ( α 2 b 1 − h 2 / c 2 ) ] − 1 ⋅ t − 1 / 3 e − η z e − ( α 1 x + α 2 y ) (8)

As momentum p is the conjugate of position vector r ′ , we take the Fourier transforms of T ( t , r ′ ) in Equation (3) with respect to r ′ which finally leads to p L and p T dependences of survival probability S. Therefore, we write

T ( p L , p T ) = ∫ T ( t , r ′ ) e i p ′ ⋅ r ′ d t d τ ′ (9)

From Equation (9), we obtain

| T ( p L , p T ) | = ( 3 / 2 ) T 0 ( t f 2 / 3 − t i 2 / 3 ) | I z | | I x | | I y | (10)

where | I z | = I z I z * , | I x | = I x I x * , | I y | = I y I y * with I z = ∫ e i p L z e − η z d z , I x = ∫ e i p T x x e − α 1 x d x , and I y = ∫ e i p T y y e − α 2 y d y .

Therefore, we obtain

| T ( p L , p T ) | = η [ sinh 2 η c cos 2 p L c + cosh 2 η c sin 2 p L c ] 1 / 2 sinh η c ⋅ ( η 2 + p L 2 ) 1 / 2 ⋅ α 1 [ sinh 2 α 1 a ( 1 − z 2 / c 2 ) 1 2 cos 2 p T x a ( 1 − z 2 / c 2 ) 1 2 + cosh 2 α 1 a ( 1 − z 2 / c 2 ) 1 2 sin 2 p T x a ( 1 − z 2 / c 2 ) 1 2 ] 1 / 2 ( α 1 2 + p T x 2 ) 1 / 2 sinh α 1 a ( 1 − z 2 / c 2 ) 1 2 ⋅ α 2 [ sinh 2 α 2 b ( 1 − z 2 / c 2 ) 1 2 cos 2 p T y b ( 1 − z 2 / c 2 ) 1 2 + cosh 2 α 2 b ( 1 − z 2 / c 2 ) 1 2 sin 2 p T y b ( 1 − z 2 / c 2 ) 1 2 ] 1 / 2 ( α 2 2 + p T y 2 ) 1 / 2 sinh α 2 b ( 1 − z 2 / c 2 ) 1 2 (11)

From Equation (11), we obtain the value of | T ( P T ) | and | T ( P L ) | . When p L → 0 , we obtain an expression for | T ( P T ) | and the expression for | T ( P L ) | is obtained by taking p T → 0 . The obtained expressions [

| T ( p T ) | = α 1 [ sinh 2 α 1 a ( 1 − z 2 / c 2 ) 1 2 cos 2 p T x a ( 1 − z 2 / c 2 ) 1 2 + cosh 2 α 1 a ( 1 − z 2 / c 2 ) 1 2 sin 2 p T x a ( 1 − z 2 / c 2 ) 1 2 ] 1 / 2 sinh ( α 1 a ( 1 − z 2 / c 2 ) 1 2 ) ⋅ ( α 1 2 + p T x 2 ) 1 / 2 ⋅ α 2 [ sinh 2 α 2 b ( 1 − z 2 / c 2 ) 1 2 cos 2 p T y b ( 1 − z 2 / c 2 ) 1 2 + cosh 2 α 2 b ( 1 − z 2 / c 2 ) 1 2 sin 2 p T y b ( 1 − z 2 / c 2 ) 1 2 ] 1 / 2 sinh α 2 b ( 1 − z 2 / c 2 ) 1 2 ⋅ ( α 2 2 + p T y 2 ) 1 2 (12)

and

| T ( p L ) | = η sinh η c ⋅ ( η 2 + p L 2 ) 1 / 2 [ sin 2 h η c cos 2 p L c + cos 2 h η c sin 2 p L c ] 1 / 2 (13)

The expression for survival probability S of the J/Ψ suppression in the relativistic heavy ion collisions is obtained by substituting the value of | T ( P T ) | and | T ( P L ) | from Equations (12) and (13) in Equation (1). It is to be noted here that Equation (5) ensures the dimensionless character of | T ( P T ) | and | T ( P L ) | given in Equations (12) and (13) and subsequently that of S in Equation (1). It is seen that the expression for suppression rate of momentum along longitudinal direction in the ellipsoidal evolution remains the same as that in cylindrical evolution, i.e. Equation (13) is the same as Equation (6) in [

Within the framework of the Bjorken hydrodynamical model we have tried to understand the time and space evolution of QGP by considering ellipsoidal flow of the fluid and exponential fall of temperature in longitudinal and transverse directions. In the present work we have calculated the J/Ψ suppression in the relativistic heavy ion collisions not only of transverse momentum P T but also of longitudinal momentum P L . Presently available data [

The author is greatly benefited from discussion with Professor H. Satz at T I F R and he wishes to express his gratitude to him. Author also wishes to thank Prof. R. S. Kaushal and Prof. Permanand for discussions. The author also wishes to thank the management of PCCS, Dr. APJ Abdul Kalam Technical University, India where the author worked for quite a long time, i.e. from November 24, 1999 to October 03, 2018 forenoon. Thanks are also due to the Head of the Department of Physics, Dean of the School of Basic Sciences and Research, and Dean, RTDC, Sharda University for providing the facilities to complete the work. The author also wishes to thank Prof. Ashok Kumar and Suvrat Karn for helping me in the typing work.

The author declares no conflicts of interest regarding the publication of this paper.

Karn, S.K. (2020) A Theoretical Approach to Study J/Ψ Suppression in Relativistic Heavy Ion Collisions with Ellipsoidal Evolution. Open Journal of Microphysics, 10, 1-7. https://doi.org/10.4236/ojm.2020.101001