Charged cylindrical polytropes with generalized polytropic equation of state

Abstract

We study the general formalism of polytropes in the relativistic regime with generalized polytropic equations of state in the vicinity of cylindrical symmetry. We take a charged anisotropic fluid distribution of matter with a conformally flat condition for the development of a general framework of the polytropes. We discuss the stability of the model by the Whittaker formula and conclude that one of the models developed is physically viable.

Eur. Phys. J. C (2016) 76:510DOI 10.1140/epjc/s10052-016-4358-4

Regular Article - Theoretical Physics

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Web End = Charged cylindrical polytropes with generalized polytropic equation of state

M. Azam1,a, S. A. Mardan2,b, I. Noureen2,c, M. A. Rehman2,d

1 Division of Science and Technology, University of Education, Township Campus, Lahore 54590, Pakistan

2 Department of Mathematics, University of the Management and Technology, C-II, Johar Town, Lahore 54590, Pakistan Received: 26 August 2016 / Accepted: 5 September 2016 / Published online: 17 September 2016 The Author(s) 2016. This article is published with open access at Springerlink.com

Abstract We study the general formalism of polytropes in the relativistic regime with generalized polytropic equations of state in the vicinity of cylindrical symmetry. We take a charged anisotropic uid distribution of matter with a conformally at condition for the development of a general framework of the polytropes. We discuss the stability of the model by the Whittaker formula and conclude that one of the models developed is physically viable.

1 Introduction

In general relativity (GR), polytropes play a very vital role in the modeling of relativistic compact objects (COs). Over the past few decades, many researchers have been engaged in the study of polytropes due to the simple form of the poly-tropic equation of state (EoS) and the corresponding Lane Emden equation (LEE), which can be used for the description of various astrophysical phenomenons. Chandrasekhar [1] was the pioneer, who established the theory of polytropes originating with the laws of thermodynamics in the Newtonian regime. Tooper [2,3] formulated the initial framework of the polytropes for a compressible and adiabatic uid under quasi-static equilibrium condition to develop the LEE. After a few years, Kovetz [4] provided some corrections in the Chandrasekhar formalism for polytropes. The general form of the LEE in higher-dimensional space was developed by Abramowicz [5] in spherical, cylindrical, and planar geometry.

The study of anisotropy in the modeling of an astrophysical CO plays a very signicant role and many physical problems cannot be modeled without taking anisotropic stress into

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account. In 1981, a very sophisticated process of modeling an anisotropic CO was provided by Cosenza [6]. In this scenario, Herrera and Santos [7] presented a detailed study of the existence of anisotropy in a self-gravitating CO. Herrera and Barreto [8] developed a general model for Newtonian and relativistic anisotropic polytropes. Herrera et al. [9] gave a complete set of equations with anisotropic stress for a self-gravitating spherical CO. A new way to check the stability of anisotropic polytropic models by the Tolman mass was provided by Herrera and Barreto [10,11]. Herrera et al. [12] used a conformally at condition in the analysis of anisotropic polytropes to reduce the parameters involved in the LEE. He et al. [13] also discussed the implementation of cracking criteria for the stability of anisotropic polytropes.

In GR, the existence of charge considerably affects the modeling of a relativistic CO. Bekenstein [14] investigated gravitational collapse by means of a hydrostatic equilibrium equation in a charged CO. Bonnor [15,16] showed that gravitational collapse can be delayed by electric repulsions in a CO. A complete study of the contraction of a charged CO in isotropic coordinates was presented by Bondi [17]. Koppar et al. [18] developed a novel way to calculate a charge generalization of the static charged uid solution of a CO. Ray et al. [19] investigated charged CO with high density and found that they can have large amount of charge approximately 1020 coulomb. Herrera et al. [20] used structure scalars to study dissipative uids in a charged spherical CO. Takisa [21] provided models of polytropic COs in the presence of charge. Sharif [22] developed a modied LEE for a charged polytropic CO with the conformally at condition. Azam et al. [2327] discussed the cracking of different charged CO models with linear and quadratic EoS.

It is always a crucial issue to choose the proper EoS for the modeling of astronomical objects. Chavanis [28,29] proposed a modication in the conventional polytropic EoS

Pr = K1+

1n , where Pr is radial pressure, n is the poly-

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510 Page 2 of 9 Eur. Phys. J. C (2016) 76 :510

tropic index, and K is a polytropic constant. He combined a linear EoS Pr = 1o with a polytropic EoS as Pr =

1o + K1+

1n and used it to describe different cosmological situations. He developed the models of the early and the late universe for n > 0 and n < 0 through a generalized polytropic equation of state (GPEoS). Freitas [30] applied the modied polytropic EoS for the development of a model of the universe with constant energy density and discussed quantum uctuations of the universe. Azam et al. [31] provided a comprehensive study for the development of a modied form of LEE with GPEoS for spherical symmetry.

Cylindrically symmetric spacetimes have been used widely in GR to describe various physically interesting aspects. For the rst time, Kompaneets [32] provided the general form of a four-dimensional cylindrically symmetric metric. Some specic examples of a cylindrically symmetric metric which provide exact solutions to the system of Einsteins eld equations and cylindrical gravitational waves were studied in [33,34]. Thorne [34] dened the C-energy of cylindrical systems as the gravitational energy per unit specic length. The most interesting fact about these space-times is the so-called C-energy and as a result gravitational waves are thought to be carriers of energy in a gravitational eld. Whittaker [35] introduced the concept of a mass potential in GR. Herrera et al. [36] used a conformally at condition with cylindrical symmetry to give a solution of the eld equations which is completely matched to the Levi-Civita vacuum spacetime. Herrera and Santos [37] studied the matching condition for perfect uid cylindrical gravitational collapse. Debbasch et al. [38] discussed regularity and the matching condition for a stationary cylindrical anisotropic uid. Di Prisco et al. [39] studied cylindrical gravitational collapse with a shear-free condition. Sharif and Fatima [40] presented cylindrical collapse with a charged anisotropic uid. Sharif and Azam [41] studied dynamical instability of cylindrical collapse in the Newtonian and the post Newtonian regime. Ghua and Banerji [42] described dissipative cylindrical collapse with a charged anisotropic uid. Sharif and Sadiq [43] presented conformally at polytropes with an anisotropic uid for the cylindrical geometry. Mahmood et al. [44] considered a charged anisotropic uid for the discussion of cylindrical collapse and found that the presence of charge enhances the anisotropy of the collapsing system.

In this paper, we will explore charged anisotropic polytropes by using GPEoS for a cylindrical symmetric congu-ration with a conformally at condition. In Sect. 2, we present the EinsteinMaxwell eld equations and a modied hydro-static equilibrium equation. In Sect. 3, the LEE is developed for relativistic polytropes. The energy conditions, the conformally at condition, and the stability of the model is given in Sect. 4. In the last section we conclude our results.

2 Matter distribution and EinsteinMaxwell eld equations

In this section, we will describe the inner matter distribution and EinsteinMaxwells eld equations. We assume a static cylindrically symmetric spacetime,

ds2 = A2dt2 + B2dr2 + C2d2 + dz2, (1) where t (, ), r [0, ), [0, 2] and z

(, ) are the conditions on the cylindrical coordinates.

The energy-momentum tensor for a charged anisotropic uid distribution is

Ti j = (Pr + )Vi Vj (Pr Pz)Si Sj (Pr P)Ki K j

+Pr gi j +

14 (Fi Fj

14 F Fgi j), (2)

where Pr, P, Pz, and represent pressures in r, , z directions and energy density of uid inside cylindrical symmetric distribution. The four-velocity Vi and four vectors Si, Ki satisfying the following relations:

V i Vi =1, Si Si = Ki Ki = 1, Siki = V iki = V i Si = 0.

(3)

These quantities in co-moving coordinates can be written as

Vi = A0i, Ki = C2i, Si = A3i. (4) The Maxwell eld equations are

F[i j;k] = 0, Fi j;j = 4 Ji, (5) where Fi j = j,i i,j is eld tensor and i is the four-

potential and Ji is four-current. The four-potential and four-velocity are related to each other in co-moving coordinates by

i = (r)0i, Ji = V i, i = 0, 1, 2, 3, (6) with and represents scalar potential and charge density, respectively.

The EinsteinMaxwell eld equations for the line element of Eq. (1) are given by

B C

B3C

C

B2C = 8 42E2, (7) A C

AB2C = 8 Pr + 42E2, (8) A

AB2

A B

AB3 = 8 P 42E2, (9) A

AB2

A B

AB3

B C

B3C

A C

AB2C

C

B2C = 8 Pz 42E2, (10)

where denotes the differentiation with respect to r and E =

q2C with q(r) = 4

r0 BCdr represents total amount of charge per unit length of cylinder. We consider the exterior

+

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Eur. Phys. J. C (2016) 76 :510 Page 3 of 9 510

metric for the cylindrical symmetric geometry with retarded time coordinate dened as

ds2 =

Using Eq. (19), the hydrostatic equilibrium equation (17) becomes

dPrdr +

d2 2ddR

+R2(d2 + 2dz2), (11)

where M is the total mass and is an arbitrary constant.

The junction condition has a very important role to play in the theory of relativistic objects. This condition shows the feasibility of physically acceptable solutions. For the continuity and matching of two spacetimes, junction conditions on the boundary yield [4547]

m(r) M =

2M

R +

Q2

R2

8r Pr + 42r E2 1 8m + 162r E2

+ Pr

+

r + E E +

8 , l = 4C, Pr = 0. (12)

The Schwarzschild coordinate is selected as C = r [43] and

EinsteinMaxwell eld equations reduced to the form [44]

B r B3 = 8 42E2, (13)

A

AB2r = 8 Pr + 42E2, (14)

A

AB2

l8, Q2 =

ql2

8 E2r = 0. (20)

The basic theory of polytropes is established with the hypotheses of a polytropic EoS and a hydrostatic equilibrium state of the relativistic object under consideration. In the next section, we will discuss relativistic polytropes with a generalized polytropic EoS in the presence of charge for cylindrical symmetry.

3 The relativistic polytropes

In this section, we provide a comprehensive way for the development of the LEE which is the main consequence of the theory of polytropes with GPEoS in the cylindrical regime. The EoS is the union of the linear EoS Pr = 1o and the polytropic EoS Pr = K1+

1 n

A B

o . The linear EoS describes

pressureless (1 = 0) or radiation (1 =

13 ) matter. The polytropic part is related to the cosmological aspects of the early universe for n > 0 whereas it describes the late universe with n < 0 [18,19]. The cosmic behavior of the universe is demonstrated with o as the Planck density but in the relativistic regime we take it as a mass density for case 1 and the total energy density in case 2. Here, we shall develop the general formalism for relativistic polytropes with GPEoS in the presence of charge.

3.1 Case 1

Here, the GPEoS is

Pr = 1o + Ko = 1o + K1+

1 n

AB3 = 8 P 42E2, (15) A

AB2

A

AB2r = 8 Pz 42E2. (16)

Solving Eqs. (13)(15) simultaneously leads to the hydro-static equilibrium equation

dPrdr +

A B

AB3

B

B3r

A

A + Pr +

r + E E +

8 E2r = 0, (17)

where we have used = Pr P.

Thorne [34] dened the C-energy (gravitational energy per unit specic length of cylindrical geometry), in the form of a mass function,

E =

18[1 l2a ra r],

with

2 = (1)aa(1), l2 = (2)aa(2), r= l,

here r, , l represents the areal radius, the circumference

radius, and the specic length, respectively, and for the static case the expression of the C-energy can be written as

m(r) =

o . (21)

The original polytropic part remains conserved and the relationship of the mass density 0 with the total energy density is given by [7]

= o + nPr. (22) Now making the following assumptions:

=

Prc

E =

1 8

1

1 B2

gc , r =

X , o = gcn, m(r)

+ 2r2E, (18)

Differentiating Eq. (18) and using Eq. (14), we get

A

A =

2gcv()

16 Prc

, (23)

where Prc and gc represent the central pressure and mass density. Also , , and v are dened to be dimensionless

=

X2 , X2 =

8r Pr + 42r E2 1 8m + 162r E2

. (19)

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510 Page 4 of 9 Eur. Phys. J. C (2016) 76 :510

variables. Using the assumptions (23) with the EoS (21), the hydrostatic equilibrium equation (20) transforms as

1 v() + 43/2

Prc E2

1 + (n + 1)(1 + )

dd +

We get a modied LEE by using Eqs. (29) and (28) given in the appendix, see (40)), representing the relativistic charged polytropes with GPEoS.

4 Energy conditions, conformally at condition, and stability analysis

In the mathematical modeling of a CO, the energy conditions play a very peculiar role in the analysis of the developed model. The energy condition provides us with the maximum information without depending upon the EoS used in the modeling. These conditions have been developed with the understanding that the energy density is always positive; otherwise the empty space created due to positive and negative regions denitely become unstable. The energy conditions that should be satised by all the models are [48]

+

2Prc(1 + )n + E2

n1n + (n + 1)

4Prc

( + E dEd + 8 E2)(1 v() + 43/2

Prc E2)

+ (1 + (n + 1)(1 + ))(n1

n

+ (n + 1))

n Prc = 0.

(24)

Now taking the derivative of Eq. (18) with respect to r and applying the relations of Eq. (23), we obtain

dv()d = n(1 + n1 + n) +

(4 ) E

2Prc +

dE d .

(25)

The combination of Eqs. (24) and (25) results in the modied LEE (Eq. (39) in the appendix), which describe the relativistic charged polytropes with GPEoS.

3.2 Case 2

In this case the GPEoS as Pr = 1 + K1+

1n , here mass density o is replaced by total energy density in Eq. (21)

and we have the following relation [7]:

=

2

Prc

P

2 E2 > 0, Pr E2,

Pz

1. (30)

If we take case 1 of the developed model, the energy conditions (30) transform as

1 +

,

E21 + n(1 + )n

> 0, 1

E2(n 1)(1 + )

,

o

1 K1 n

P

Prc(1 + n(1 + ))n

n . (26)

We make the following assumptions:

=

1,

o

PzPrc(1 + n(1 + ))n

Prc

1, (31)

and for case 2 the energy conditions (30) emerge as

1 +

c , r =

X , = cn,

n E2

E2n + Prc Prc(1 1)

,

2cv()

16 Prc

2Prc > 0, 1

, (27)

where c means that each quantity is calculated at the center of a CO. Using GPEoS and the assumptions of Eq. (27), the hydrostatic equilibrium Eq. (20) turns out to be

1 v() + 43/2

Prc E2

1 + 1 +

m(r) =

X2 , X2 =

Pn

Pzn

Prc 1,

d d

4Prc

Prc 1. (32)

We observe that the coupled Eqs. (24) and (25) and Eqs. (28) and (29) result in a system of differential equations with three variables. Thus, we need more information to study the charged polytropic CO with GPEoS in cylindrical symmetry. So, we use a conformally at condition to reduce the systems of differential equations to two variables.

For this purpose, the Weyl scalar dened in terms of the Kretchman scalar, the Ricci tensor, and the Ricci scalar is given by [43]

C2 = R +

2Prc(1 + )n + E2

n1n + (n + 1)

+ ( + E dEd + 8 E2)(1 v() + 43/2

Prc E2)

+

(1 + 1 + )(n1n + (n + 1))

R2

n Prc = 0. (28)

Taking the derivative of Eq. (18) with respect to r and applying Eq. (27), we get

dv()d = n +

3 2R R. (33)

For our line element, the above equation becomes

C2 =

43r2 A2B6 [(A2 r AA + r2 A 2)B 2

+B2(A 2 r A A + r2 A 2) +B B (r A (A 2r A ) + A(A + r A ))]. (34)

(4 ) E

2Prc +

2

Prc

dEd . (29)

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Eur. Phys. J. C (2016) 76 :510 Page 5 of 9 510

Now applying conformal atness i.e., C2 = 0, and using the

eld equations (13)(16) and (19) in Eq. (34), we obtain the anisotropy as

= E2 +

+ 4 E Prc

( + 2( + 32 Prcn(1 + ))

3 + 32

Prc

dE d

18 (8 Pr + 42E2 + (8 Pr + 42E2)2 +(8 Pr + 42E2)(42E2 + 8)

+(42E2 + 8)2

+(1 8 Pr 82E2 + 8)

+v + 2

+8Prc (1 + v) + 2

dE d

(8 Pr + 42r E2)2

(1 8m + 162r E2)2

Prc

r(8 Pr + 42r E2)(42E2 + 8)

1 8m + 162r E2

3/21(1 + n1)1+2n

+22 Prc

3/2(1 + 2n1)2+2n

+2n3 Prc

42E2 + 8 P r + 82r E E

+ 1 8m + 162r E2

(8 Pr + 42r E2)(162E2 8m + 322r E E )

(1 8m + 162r E2)2

3/23+2n

Prc2+n dvd + nPrc1(1 + v)n

d d

(8 Pr + 42r E2)2

+ (1 8m + 162r E2)2

Prc1+n 1 dvd (1 + n)(1 + v)

d d

r(8 Pr + 42r E2)(42E2 + 8)

1 8m + 162r E2

+ E2

1283/2 P2rc1+n(1 + ) + 2

42E2 + 8 P r + 82r E E

+ 1 8m + 162r E2

Prc

(1 + 32 v

dv d )

(8 Pr + 42r E2)(162E2 8m + 322r E E )

(1 8m + 162r E2)2

)2 .(35)

Now using Eq. (23) in the above equation, we obtain anisotropy factor for case 1 given in the appendix; see Eq. (41). One can derive a modied LEE for conformally at polytropes by using Eq. (41) in (24) and coupling with Eq. (25) for case 1. Similarly, the anisotropy parameter for case 2 turns out to be Eq. (42) (see the appendix) and the modied LEE by using Eq. (42) in (28) and coupling it with Eq. (29) for case 2, which can help in the study of conformally at polytropes.

We will use a modied Whittaker [35] formula for the stability analysis of the model, which is the measure of the active gravitational mass per unit length of a cylinder dened by

mL = 8

r

2Prc(2(1 + v) + (n + 4

Prc

)2+n

+16n Prc 1n

d d

1+n +

n 4

Prc

1

d. (37)

Similarly for case 2, the Whittaker formula yields

16(1 + n)

Prc

d d

2MR + Q2R2

Prc

mL =

1

0 1 + 43/2 E2 Prc

v

0 ( + Pr + P + Pz E2)dr. (36)

For case 1, using the eld equations (13)(16), (19) and (23) in (36), we get

mL =

1

Prc

165/2 E3

Prc

+

22 E4 3 + 32

Prc

2MR + Q2R2

Prc

E2

+4E(1 + v) + 1

Prc

0 43/2 E2 Prc

Prc(1 + v)

+ 32 + 4

Prc

+ 4

Prc

v

Prc

165/22 E3

+ 22 E4

+2 Prc 4 + 64

Prc

1

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510 Page 6 of 9 Eur. Phys. J. C (2016) 76 :510

3.5 Whittaker mass per unit length for n=1.0

3

2.5

2

m L

1029

1.5

1

0.5

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Fig. 1 Case 1: mL as a function of for n = 1, curve a: = 8

1011, Q = 0.2 M , curve b: = 1010, Q = 0.4 M , curve c:

= 2 1010, Q = 0.6 M , curve d: = 4 1010, Q = 0.64 M

+2 Prc (1 + n1)

n

+2 4 Prc + n2

Prc

+64 Prc

3/2 1+n

+8 v + 2

Prc

3/22n

d. (38)

In order to test the physical viability of polytropes, we formulate the energy condition and found that these conditions are satised only for case 1 and fail in case 2 for charged anisotropic polytropes with GPEoS. We also plot the Whittaker formula (mL ) given in Eq. (37) with respect to the dimensionless radius in the radial direction for a cylinder of unit radius. The bounded behavior of (mL ) in the radial direction for various values of the charge shows that our model is stable. So, only case 1 for polytropes is physically viable.

5 Conclusion and discussion

In this article, we have formulated the general framework to study charged relativistic polytropes with GPEoS in the cylindrical regime. The GPEoS, Pr = 1 + K1+

1n , is the

(1 + )(1 + n1 + n)

combination of a linear EoS with a polytropic EoS, and it is used in cosmology for the description of eras of the universe. For the discussion of relativistic polytropes, we have

developed the generalized framework to get the expressions of the modied LEE whose solutions are called polytropes. The polytropes mainly depend on the density function and the polytropic index decides the order of the solution. These polytropes are very useful in the description of various astrophysical aspects of a CO due to its simplicity. However, this simplicity is obtained at the cost of an empirical power-law relation between density and pressure, which should hold throughout in a CO. The LEE have been obtained with the anisotropic factor in the expressions (see the appendix, Eqs. (39) and (40)) depending upon three variables. In order to reduce one variable ( ), we have used a conformally at condition and calculated the value of anisotropic factor in the form of other two variables (see the appendix, Eqs. (41) and (42)). One can easily obtain a modied LEE in two variables by substituting these values of anisotropy factor.

The stability analysis is very important in the development of mathematical models to check the physical viability. The energy conditions are very helpful in this regard, as they can entail maximum information without considering the EoS involved in the development of the model. The energy conditions have been obtained for both cases of GPEoS in the presence of an electromagnetic eld. We have used a modied form of the Whittaker formula [35] for the stability analysis of charged anisotropic relativistic polytropes. We have calculated mL , which is a measure of the active gravitational mass per unit length in the z-direction, for both the case of GPEoS for different values of parameters involved in the model (see Fig. 1) for a cylinder of unit radius. We have taken these values from our previous work on the polytropes [31]. We have plotted mL against the dimensionless radii and found that its graph remains bounded in the radial direction even for very high values of the charge Q = 0.2

M , 0.4 M , 0.6 M , and 0.64 M . Also it is clear that

more mass is concentrated near the center of cylinder and its magnitude decreases as we move away from the center in a radial direction. The highest magnitude is observed in the middle of cylindrical symmetry in the radial direction. As the gravitational mass cannot be negative, Fig. 1 shows the stable positive and bounded behavior of mL , which shows that our model is stable. The energy conditions are valid only for case 1 and fail to hold for case 2. Hence the model developed in case 2 is not physically viable due to the negation of the energy conditions.

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Eur. Phys. J. C (2016) 76 :510 Page 7 of 9 510

Appendix

LEE for case 1:

1 4

( E2 + 2Prcn(1 + )) Prc((1 + n) + n1

n)

1 +

d

d

1+n n21 + (1 + n)1+n

E2 + 2Prcn

Prc(n1 + (1 + n)

1+n)2

4(1 + n)

1 + 43/2 E2

Prc v

n 8 E2 + + dEd

d

d

2

Prc(1 + 1 + )(1 + (1 + n)(1 + ))

4 1 + 43/2 E2

Prc v)n

8 E2 + + dEd

4 1 + 43/2 E2

Prc v

n 8 E2 + + dEd

d

d

2 Prc(1 + (1 + n)(1 + ))

2

Prc(1 + 1 + )

2(1 + (1 + n)(1 + ))

4n 1 + 43/2 E2

+

2n 8 E2 + +

dE d

Prc v

1n 8 E2 + + dEd

d

d

Prc(1 + 1 + )(1 + (1 + n)(1 + ))

83/2E2

Prc

2 Prcn(1 + n1 + n) +

dE d

2(Prcn + dEd

+ E

4 + +

163/2 dE

d

Prc

d

d

+

2 8

3/2E2

Prc

+ E 4 + +

163/2 dEd

Prc

Prc(1 + 1 + )

( P2rc(1 + (1 + n)1 + (1 + n))2)

1+n(n21 + (1 + n)1+n)( E2 + 2Prcn(1 + )) ddPrc(n1 + (1 + n)

1+n)2

2

Prc v

d

d

4 1 + 43/2 E2

2

(1 + 1 + )

21+n E dEd + Prcn(n1 + (1 + n)) dd

8(1 + n)

1 + 43/2 E2

Prc v

n 8 E2 + + dEd

d

d

Prc(n1 + (1 + n)

1+n)

3

Prc(1 + (1 + n)(1 + ))

+ 4 1 + 43/2 E2

Prc v

n ( + 16 E) dEd + d d + d2Ed

4n 1 + 43/2 E2

Prc v

1n 8 E2 + + dEd

d

d

Prc(1 + 1 + )(1 + (1 + n)1 + (1 + n))

+

+ 4 1 + 43/2 E2

2

Prc(1 + (1 + n)(1 + ))

Prc v

d

2

d2

+

2 83/2E2

Prc

2(Prcn(1 + n1 + n) +

dE d

1 + 1 +

= 0. (40)

for case 1:

= E2 +

1 8

+ E 4 + +

163/2 dEd

Prc

d d

42 E2 +

8 Prcn(1 + )

(Prc(1 + (1 + n)1 + (1 + n)))

4(1 + n)

1 + 43/2 E2

Prc v

d

d

2

42 E2 +

8 Prcn(1 + )

2

2

21+n E dEd + Prcn

n1 + (1 + n)

d

d

(1 + (1 + n)(1 + ))

+ 42 E2 +

8 Prcn(1 + )

Prc(n1 + (1 + n)

1+n)

+ 4 1 + 43/2 E2

Prc v

n ( + 16 E) dEd + d d + d2Ed2

+ 42 E2 +

8 Prcn(1 + n1 + n)

2

Prc(1 + (1 + n)1 + (1 + n))

2

+ 4 1 + 43/2 E2

Prc v

d

2

d2

42 E2 +

8 Prcn(1 + n1 + n)

+ 1 + (1 + n)(1 + )

= 0. (39)

LEE for case 2:

1 4

( E2 + 2Prcn(1 + )) Prc((1 + n) + n1

n)

+

+ 1 82 E2

8 Prcn(1 + )

3/2 E2

Prc

2

+ 8 Prc

n (1+)

+

8 Prcn(1 + n1 + n)

1 + 43/2 E2 Prc

v

2

4 1 + 43/2 E2

Prc v

n 8 E2 + + dEd

3/2 E2 Prc

+ 8 Prc

n (1+)

42 E2 + 8 Prc

n (1+n1+n)

2 Prc(1 + 1 + )(1 + (1 + n)(1 + ))

+

2n 8 E2 + +

dE d

4 Prc

1 + 43/2 E2 Prc

v

2 Prcn +

dE d

+ 8 Prc

n (1+)

162 E2 + 322 E dEd dvd

83/2E2

Prc

3/2 E2

Prc

1 + 43/2 E2 Prc

v

2

+ E 4 + +

163/2 dEd

Prc

8 Prc1+n (n1+(1+n)) dd

1 + 43/2 E2 Prc

v

+

42 E2 + 82 E dEd +

( P2rc(1 + 1 + )(1 + (1 + n)1 + (1 + n)))

123

510 Page 8 of 9 Eur. Phys. J. C (2016) 76 :510

3/2 E2

Prc

+ 8 Prc

n (1+)

2

+ 8 Prc

n (1+)

162 E2 + 322 E dEd dvd

+ 1 + 43/2 E2 Prc

v

2

3/2 E2

Prc

1 + 43/2 E2 Prc

v

2

3/2 E2 Prc

+ 8 Prc

n (1+)

42 E2 + 8 Prc

n (1+n1+n)

8 Prc1+n (n1+(1+n)) dd

1 + 43/2 E2 Prc

v

2

+

42 E2 + 82 E dEd +

. (42)

4 Prc

1 + 43/2 E2 Prc

v

+ 8 Prc

n (1+)

162 E2 + 322 E dEd dvd

References

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3/2 E2

Prc

1 + 43/2 E2 Prc

v

2

2

+

42 E2 + 82 E dEd +

8 Prc1+n (n1+(1+n)) dd

1 + 43/2 E2 Prc

v

. (41)

for case 2:

= E2 +

1 8

42 E2 +

8 Prcn(1 + )

+

2

42 E2 +

8 Prcn

+

42 E2 +

8 Prcn(1 + )

42 E2 +

8 Prcn

2

+

42 E2 +

8 Prcn(1 + )

+

1 82 E2 +

8 Prcn

8 Prcn(1 + )

42 E2 + 8 Prc

n

3/2 E2 Prc

+ 8 Prc

n (1+)

4 Prc

1 + 43/2 E2 Prc

v

3/2 E2

Prc

+ 8 Prc

n (1+)

2

+ 1 + 43/2 E2 Prc

v

2

+ 8 Prc

n (1+)

162 E2 + 322 E dEd dvd

3/2 E2

Prc

1 + 43/2 E2 Prc

v

2

8 Prc1+n (n1+(1+n)) dd

1 + 43/2 E2 Prc

+

42 E2 + 82 E d Ed +

v

42 E2 + 8 Prc

n

3/2 E2 Prc

+ 8 Prc

n (1+)

+

4 Prc

1 + 43/2 E2 Prc

v

3/2 E2

Prc

+ 8 Prc

n (1+)

)2

+ 1 + 43/2 E2 Prc

v

2

123

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