Analytical Results for the Classical and Quantum Tsallis Hadron Transverse Momentum Spectra: the Zeroth Order Approximation and beyond
AAnalytical Results for the Classical and Quantum Tsallis Hadron TransverseMomentum Spectra: the Zeroth Order Approximation and beyond
Trambak Bhattacharyya ∗ and Alexandru S. Parvan
1, 2, † Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, 141980,Moscow Region, Russia Horia Hulubei National Institute of Physics and Nuclear Engineering, Bucharest-Magurele, Romania
We derive the analytical expressions for the first and second order terms in the hadronic transversemomentum spectra obtained from the Tsallis normalized (Tsallis-1) statistics. We revisit the zerothorder quantum Tsallis distributions and obtain the corresponding analytical closed form expressions.It is observed that unlike the classical case, the analytical closed forms of the zeroth order quantumspectra do not resemble the phenomenological distributions used in the literature after q → q − substitution, where q is the Tsallis entropic parameter. However, the factorization approximationincreases the extent of similarity. PACS numbers: 12.38.Mh, 12.40.Ee
I. INTRODUCTION
Hadronic spectra resulting from high energy collision events follow a power-law pattern, and the power-law for-mulae inspired by the Tsallis statistics [1] are particularly popular while describing the hadronic distributions in thephenomenological and experimental studies.It is important to understand the origin of the phenomenological Tsallis distributions [2–4] from the fundamentaltheories and recently there have been some attempts [5–9] to address this issue. In Refs. [5–7], it has been shownthat the most generalized form of the hadronic transverse momentum spectra calculated from the Tsallis statisticalmechanics is given by an infinite summation and the zeroth order truncation of the Maxwell-Boltzmann spectrumyields the widely used expression of the phenomenological Tsallis distribution given, for example, in Refs. [2, 10].However, in Ref. [8], it was demonstrated that the phenomenological Tsallis distribution [2–4] for the Maxwell-Boltzmann spectrum corresponds also to the zeroth term approximation of the q -dual statistics based on the q -dualentropy obtained from the Tsallis entropy under the multiplicative transformation of the entropic parameter q → q − .Approximating the Tsallis transverse momentum spectra with the zeroth order term is called the ‘zeroth orderapproximation’, which may work very well for certain collision energy regions. However, it has been shown [5, 7] thatthe zeroth order approximation may not be sufficient always. In some of the cases, there is a necessity to include thehigher order terms in the transverse momentum distribution.The present paper calculates the analytical expressions for the first and the second order terms in the TsallisMaxwell-Boltzmann transverse momentum distribution as they may be indispensable in the phenomenological andexperimental studies for describing the data obtained at certain collision energies. While calculating the transversemomentum distributions, we consider the Tsallis normalized (or Tsallis-1) statistics. Tsallis-1 statistics is one of theseveral schemes in the Tsallis statistical mechanics [11] which differ in the definition of the average values. We discussthis scheme in detail in the next section.The paper also calculates the closed analytical form of the zeroth order term of the quantum Tsallis transversemomentum spectra and verifies that unlike the classical case, the quantum Tsallis phenomenological distributions[12, 13] used in the literature are not identical with this zeroth order spectra (see also [7]). However, the factorizationapproximation of the zeroth order term increases the extent of its similarity with the quantum phenomenologicalTsallis-like distributions.The remainder of the paper contains a discussion of the basic mathematical set up and the expression for thegeneralized Tsallis transverse momentum spectra in sections II, and III. Analytical calculations of the first and thesecond order terms in the Maxwell-Boltzmann Tsallis transverse momentum distribution are shown in section IV.Closed analytical forms of the zeroth order Tsallis quantum distributions and their factorization approximation havebeen discussed in sections V, and VI. Sections VII, and VIII are devoted to the discussions of the results, summaryand the outlook. ∗ Electronic address: [email protected] † Electronic address: [email protected], [email protected] a r X i v : . [ nu c l - t h ] J u l II. BASIC DEFINITIONS AND FORMULAE
The Tsallis statistical mechanics is based on the following definition of entropy [1, 11], S = (cid:88) i p qi − p i − q , (1)where q is a real parameter, and the probabilities of micro-states { p i } follow the normalization, φ = (cid:88) i p i − . (2)The definition of average expectation values in the Tsallis normalized (or the Tsallis-1) scheme is given by, (cid:104) Q (cid:105) = (cid:88) i p i Q i . (3)Here and throughout the paper we use the system of natural units (cid:126) = c = k B = 1. When q → S BG = − (cid:80) i p i ln p i . Though the parameter q mayassume values from 0 to ∞ , yet as far as the description of the hadronic spectra in high energy collisions is concerned,we shall be interested in the values of q < { p i } of microstates of the system, itsnormalization equation and the average/expectation values in the Tsallis-1 scheme can be written as [6, 7, 14], p i = (cid:20) q − q Λ − E i + µN i T (cid:21) q − ; (cid:88) i (cid:20) q − q Λ − E i + µN i T (cid:21) q − = 1;and (cid:104) Q (cid:105) = (cid:88) i Q i (cid:20) q − q Λ − E i + µN i T (cid:21) q − , (4)where Λ is a norm function, µ is the chemical potential, and E i and N i are the energy and the number of particlesfor the i th state.Using the integral representation of the gamma functions [15] for q <
1, Eq. (4) can be rewritten as [7], p i = 1Γ (cid:16) − q (cid:17) ∞ (cid:90) t q − q e − t [ q − q Λ − Ei + µNiT ] dt ; 1Γ (cid:16) − q (cid:17) ∞ (cid:90) t q − q e − t (cid:20) q − q Λ − ΩG ( β (cid:48) ) T (cid:21) dt = 1;and (cid:104) Q (cid:105) = 1Γ (cid:16) − q (cid:17) ∞ (cid:90) t q − q e − t (cid:20) q − q Λ − ΩG ( β (cid:48) ) T (cid:21) (cid:104) Q (cid:105) G ( β (cid:48) ) dt, (5)where Ω G ( β (cid:48) ) = − β (cid:48) ln Z G ( β (cid:48) ) ; Z G ( β (cid:48) ) = (cid:88) i e − β (cid:48) ( E i − µN i ) ; (cid:104) Q (cid:105) G ( β (cid:48) ) = 1 Z G ( β (cid:48) ) (cid:88) i Q i e − β (cid:48) ( E i − µN i ) ;and β (cid:48) = t (1 − q ) /qT. (6) III. GENERALIZED TRANSVERSE MOMENTUM SPECTRA
Transverse momentum ( p T ) distributions of classical and quantum particles forming an ideal gas of volume V , areexpressible in terms of the corresponding mean occupation numbers in the following way [7], d Ndp T dy = V (2 π ) π (cid:90) dϕ p T ε p (cid:88) σ (cid:104) n pσ (cid:105) , (7)where m T = (cid:112) p + m , ε p = m T cosh y , y is rapidity related to the polar angle, and ϕ is the azimuthal angle ofemission of particles. For an azimuthally independent integrand in Eq. (7), the transverse momentum distributioncan be obtained as, d Ndp T dy = V (2 π ) p T m T cosh y (cid:88) σ (cid:104) n pσ (cid:105) . (8)Using Eq. (5), we obtain the expression for the mean occupation numbers as given below [7]: (cid:104) n pσ (cid:105) = 1Γ (cid:16) − q (cid:17) ∞ (cid:90) t q − q e − t (cid:20) q − q Λ − Ω G ( β (cid:48) ) T (cid:21) (cid:104) n pσ (cid:105) G ( β (cid:48) ) dt, (9)where (cid:104) n pσ (cid:105) G ( β (cid:48) ) = 1 e β (cid:48) ( ε p − µ ) + η (10)in which ε p = (cid:112) p + m , η = 1 for the Fermi-Dirac (FD) statistics, η = − η = 0 for the Maxwell-Boltzmann (MB) statistics of particles.Expanding the normalization in Eq. (5) and the mean occupation numbers (Eq. 9) in an infinite series and usingEq. (7), we obtain (see Ref. [7]), ∞ (cid:88) (cid:96) =0 (cid:96) ! Γ (cid:16) − q (cid:17) ∞ (cid:90) t q − q e − t [ q − q Λ T ] [ − β (cid:48) Ω G ( β (cid:48) )] (cid:96) dt = 1 (11)and d Ndp T dy = gV (2 π ) p T m T cosh y ∞ (cid:88) (cid:96) =0 (cid:96) ! Γ (cid:16) − q (cid:17) ∞ (cid:90) t q − q e − t [ q − q Λ T ] [ − β (cid:48) Ω G ( β (cid:48) )] (cid:96) e β (cid:48) ( m T cosh y − µ ) + η dt, (12)where g is the spin degeneracy factor.In Eqs. (9), (11), and (12), the thermodynamic potential Ω G for classical and quantum gases in the Boltzmann-Gibbsstatistics is given by the following expression [7, 16], − β (cid:48) Ω G ( β (cid:48) ) = gV (2 π ) (cid:90) d p ln (cid:104) ηe − β (cid:48) ( ε p − µ ) (cid:105) η (13)for the η values mentioned below Eq. (10). IV. MAXWELL-BOLTZMANN SPECTRUM
For the Maxwell-Boltzmann particle statistics, Ω G ( β (cid:48) ) is given by [7],Ω MBG ( β (cid:48) ) = − gV e β (cid:48) µ π β (cid:48) m K ( β (cid:48) m ) , (14)where K ( z ) is the modified Bessel’s function of the second kind. Putting (14) into (11) and (12), we obtain [7], ∞ (cid:88) (cid:96) =0 ω (cid:96) (cid:96) ! 1Γ (cid:16) − q (cid:17) ∞ (cid:90) t q − q − (cid:96) e − t [ q − q Λ+ µ(cid:96)T ] [ K ( β (cid:48) m )] (cid:96) dt = ∞ (cid:88) (cid:96) =0 Φ( (cid:96) ) = 1 , (15)and d Ndp T dy = gV (2 π ) p T m T cosh y ∞ (cid:88) (cid:96) =0 ω (cid:96) (cid:96) ! 1Γ (cid:16) − q (cid:17) ∞ (cid:90) t q − q − (cid:96) e − t (cid:104) q − q Λ − m T cosh y + µ ( (cid:96) +1) T (cid:105) [ K ( β (cid:48) m )] (cid:96) dt for ω = gV T m π q − q . (16)Once we have the generalized form of the classical Tsallis transverse momentum spectrum in terms of an infinitesummation, we proceed to calculate the terms appearing in this expression. A. Zeroth order approximation or truncation at (cid:96) = 0
The zeroth order contribution in the MB transverse momentum distribution is given by [7], d N (0) dp T dy = gV (2 π ) p T m T cosh y (cid:16) − q (cid:17) ∞ (cid:90) t q − q e − t (cid:104) q − q Λ − m T cosh y + µT (cid:105) dt = gV (2 π ) p T m T cosh y (cid:20) q − q Λ − m T cosh y + µT (cid:21) q − (17)Equating the (cid:96) = 0 term in Eq. (15) with 1 we obtain Λ = 0, and hence the corresponding spectrum is given by [7], d N (0) dp T dy = gV (2 π ) p T m T cosh y (cid:20) − qq ε p − µT (cid:21) q − . (18)As already discussed in Refs. [6, 7], the above expression can immediately be identified to be identical with theTsallis-like function [2, 10] widely used in literature once we replace q → q − . B. First order approximation or truncation at (cid:96) = 1
The first order contribution in the MB transverse momentum distribution is given by, d N (1) dp T dy = gV (2 π ) p T m T cosh y (cid:104) n (1) p (cid:105) , (19)where (cid:104) n (1) p (cid:105) = (cid:90) ∞ ωt q − q − Γ (cid:16) − q (cid:17) e − t (cid:104) q − q Λ − εp +2 µT (cid:105) K ( β (cid:48) m ) dt (20)is the first order mean occupation number. To evaluate this quantity we use the following contour-integral represen-tation of K given by [17], K ( β (cid:48) m ) = (cid:90) γ + i ∞ γ − i ∞ ds πi Γ( s )Γ( s − (cid:18) β (cid:48) m (cid:19) − s ; for γ > β (cid:48) m ) < π. (21)Using the above representation and swapping the contour integration with the t -integration in Eq. (20), we get thefollowing expression: (cid:104) n (1) p (cid:105) = ω Γ (cid:16) − q (cid:17) (cid:90) γ + i ∞ γ − i ∞ ds πi Γ( s )Γ( s − (cid:20) (1 − q ) m qT (cid:21) − s (cid:90) ∞ dt t − q − s e − t (cid:104) q − q Λ − εp +2 µT (cid:105) = ω (1 − q ) m (cid:16) q − q Λ − ε p +2 µT (cid:17) − qq − q T Γ (cid:16) − q (cid:17) (cid:90) γ + i ∞ γ − i ∞ ds πi Γ( s )Γ( s − (cid:18) − q − q − s (cid:19) qT (cid:16) q − q Λ − ε p +2 µT (cid:17) (1 − q ) m s . (22)Now, we wrap the contour anti-clockwise (see Fig. 1) so that it includes the poles at s = 2 , , , − , − , ..., − k andaccording to the Cauchy’s integral theorem, the sum of the residues at the poles multiplied by 2 πi is the result of theintegration. However, the condition which yields a non-divergent result of the contour integration is,2 qT (1 − q ) m (cid:18) q − q Λ − ε p + 2 µT (cid:19) > . (23) s Re(s)Im(s)
0− 1− 2− 3− 4
FIG. 1: Integration contour in Eqs. (22) and (32).
Now, we proceed to calculate the residues. • Residue of Eq. (22) at s = 2 is given by, R (1)(2) = ωπi Γ (cid:16) − q (cid:17) (cid:20) qT (1 − q ) m (cid:21) Γ (cid:18) q − − q (cid:19) (cid:18) q − q Λ − ε p + 2 µT (cid:19) q − q − . (24) • Residue of Eq. (22) at s = 1 is given by, R (1)(1) = ω ( q − πiq (cid:18) q − q Λ − ε p + 2 µT (cid:19) qq − . (25) • Residues of Eq. (22) at s = − k, k ∈ Z ≥ can be calculated as follows, R (1)( − k ) = ω πi Γ (cid:16) − q − q + 2 k (cid:17) Γ (cid:16) − q (cid:17) k !(2 + k )! (cid:20) (1 − q ) m qT (cid:21) k (cid:18) q − q Λ − ε p + 2 µT (cid:19) − qq − − k × (cid:20) (cid:18) qTm − qm − − ε p + 4 µmT (cid:19) − ψ (0) (cid:18) − q − q + 2 k (cid:19) + ψ (0) ( k + 1) + ψ (0) ( k + 3) (cid:21) . (26)We observe that the residues at s = − k contain the digamma function ψ (0) which is the poly-gamma function ψ ( m ) [18] at the zeroth order. Using Eqs. (24), (25) and (26), (cid:104) n (1) p (cid:105) is given by, (cid:104) n (1) p (cid:105) = 2 πi ∞ (cid:88) n = − R (1)( − n ) . (27)Hence, the first order contribution in the Tsallis MB transverse momentum distribution can be written as, d N (1) dp T dy = gV π p T m T cosh y ∞ (cid:88) n = − i R (1)( − n ) . (28)Eq. (28) represents the first main result of the paper. The same equation may also be obtained using the seriesexpansion of the Bessel’s function K ( z ). C. Second order approximation or truncation at (cid:96) = 2
The second order contribution in the MB transverse momentum distribution is given by, d N (2) dp T dy = gV (2 π ) p T m T cosh y (cid:104) n (2) p (cid:105) , (29)where (cid:104) n (2) p (cid:105) = ω (cid:90) ∞ t q − q − Γ (cid:16) − q (cid:17) e − t (cid:104) q − q Λ − εp +3 µT (cid:105) K ( β (cid:48) m ) dt (30)is the second order mean occupation number. To evaluate (cid:104) n (2) p (cid:105) we use the contour integral representation of K ( z )which is given by [17], K ( β (cid:48) m ) = 12 (cid:90) γ + i ∞ γ − i ∞ ds πi Γ( s + 2)Γ( s ) Γ( s − s ) (cid:18) β (cid:48) m (cid:19) − s ; for γ > , arg( β (cid:48) m ) < π. (31)Using the above representation and swapping the contour integration with the t -integration in Eq. (30), we get thefollowing expression, (cid:104) n (2) p (cid:105) = ω (cid:16) − q (cid:17) (cid:90) γ + i ∞ γ − i ∞ ds πi Γ( s + 2)Γ( s ) Γ( s − s ) (cid:20) (1 − q ) m qT (cid:21) − s (cid:90) ∞ dt t q − q − − s e − t (cid:16) q − q Λ − εp +3 µT (cid:17) = ω (cid:16) q − q Λ − ε p +3 µT (cid:17) q − q − (cid:16) − q (cid:17) (cid:90) γ + i ∞ γ − i ∞ ds πi Γ( s + 2)Γ( s )Γ( s )Γ( s − (cid:16) q − − q − s (cid:17) Γ(2 s ) qT (cid:16) q − q Λ − ε p +3 µT (cid:17) (1 − q ) m s . (32)Here we choose the same integration contour as that shown in Fig. 1. We wrap the contour anti-clockwise so that itincludes the poles at s = 2 , , , − , − , ..., − k and according to the Cauchy’s integral theorem the sum of the residuesat the poles multiplied by 2 πi is the result of the integration. The condition for convergence of the integration is2 qT (1 − q ) m (cid:18) q − q Λ − ε p + 3 µT (cid:19) > . (33)Let us calculate the residues of the integration. • Residue of Eq. (32) at s = 2 is given by, R (2)(2) = ω Γ (cid:16) q − − q (cid:17) q T πi Γ (cid:16) − q (cid:17) m (1 − q ) (cid:18) q − q Λ − ε p + 3 µT (cid:19) q − q − . (34) • Residue of Eq. (32) at s = 1 can be written as, R (2)(1) = − q T ω Γ (cid:16) − qq − (cid:17) πim ( q − Γ (cid:16) − q (cid:17) (cid:18) q − q Λ − ε p + 3 µT (cid:19) q − q − . (35) • Residue of Eq. (32) at s = 0 can be written as, R (2)(0) = ω Γ (cid:16) q − − q (cid:17) πi Γ (cid:16) − q (cid:17) (cid:18) q − q Λ − ε p + 3 µT (cid:19) q − q − (cid:20) ln (cid:18) qTm − qm − − ε p + 6 µm (cid:19) − ψ (0) (cid:18) q − − q (cid:19) − γ E + 54 (cid:21) , (36)where γ E = 0 . ... is the Euler-Mascheroni constant. • Residue of Eq. (32) at s = − R (2)( − = − ω (1 − q ) m πiq T (cid:18) q − q Λ − ε p + 3 µT (cid:19) q − (cid:20) ln (cid:18) qTm − qm − − ε p + 6 µm (cid:19) − ψ (0) (cid:18) − q (cid:19) − γ E + 512 (cid:21) . (37) • Residue of Eq. (32) at s = − k − , k ∈ Z ≥ is given by, R (2)( − k − = ω Γ(2 k + 5)Γ (cid:16) k + q − q − (cid:17) ( m − mq ) k πi k +4 ( qT ) k k ![( k + 2)!] ( k + 4)!Γ (cid:16) − q (cid:17) (cid:18) q − q Λ − ε p + 3 µT (cid:19) q − − q − k × (cid:34) λ (cid:26) ψ + 8 ψ + 4 ψ − ψ − ψ + 8 ln(2) (cid:27) + λ (cid:26) − λ − ψ − ψ − ψ + 8 ψ + 8 ψ − (cid:27) + ψ (cid:26) ψ − ψ − ψ + 4 ln(4) (cid:27) + ψ (cid:26) − ψ − ψ + ln(16) (cid:27) + ψ (cid:26) ψ + 2 ψ − ψ − ψ + ln(16) (cid:27) + ψ (cid:26) ψ − (cid:27) + 4 ln (2) + 4 λ + 4 λ + ψ + 4 ψ + ψ + 4 ψ + 4 ψ − ψ − ψ − ψ + 4 ψ +4 ψ − ψ ln(2) (cid:35) . (38)In the above equation, • λ , and λ are given by, ln (cid:18) q − q Λ − (cid:15) + 3 µT (cid:19) = λ ; ln (cid:18) m − mqqT (cid:19) = λ . (39) • ψ j , and ψ j ( j = 1 →
5) are given by, ψ (0) ( k + 2 j −
1) = ψ j ; ψ (1) ( k + 2 j −
1) = ψ j (for j = 1 → ψ (0) (2 k + 5) = ψ ; ψ (1) (2 k + 5) = ψ ; ψ (0) (cid:18) q + 2 kq − k − q − (cid:19) = ψ ; ψ (1) (cid:18) q + 2 kq − k − q − (cid:19) = ψ . (40)The average occupation number at the second order is given by, (cid:104) n (2) p (cid:105) = 2 πi ∞ (cid:88) n = − R (2)( − n ) . (41)Hence, the second order contribution in the Tsallis MB transverse momentum distribution can be written as, d N (2) dp T dy = gV π p T m T cosh y ∞ (cid:88) n = − i R (2)( − n ) . (42)Eq. (42) gives us the second main result of the paper. V. QUANTUM SPECTRA: THE ZEROTH ORDER TERMS
The zeroth order Tsallis Bose-Einstein and Fermi-Dirac spectra have also been computed in Ref. [7] in terms of aninfinite summation. However, it is possible to express the infinite summation presented in Eq. (82) of that paper interms of the Hurwitz-zeta function ζ ( s, a ) [18] as discussed in the next two sub-sections. A. Bose-Einstein
The Boltzmann-Gibbs grand thermodynamic potential for the bosons obtained from Eq. (13) is given by,Ω
BEG ( β (cid:48) ) = − gV π β (cid:48) ∞ (cid:88) n =1 e nβ (cid:48) µ n m K ( nβ (cid:48) m ) . (43)At the zeroth order of Eq. (12), the Tsallis bosonic spectrum is as follows, d N (0)BE dp T dy = gV (2 π ) p T m T cosh y Γ (cid:16) − q (cid:17) ∞ (cid:90) t q − q e − t e β (cid:48) ( m T cosh y − µ ) − dt = gV (2 π ) p T m T cosh y Γ (cid:16) − q (cid:17) ∞ (cid:88) s =1 ∞ (cid:90) dt t q − q e − t − sβ (cid:48) ( m T cosh y − µ ) = gV (2 π ) p T m T cosh y ∞ (cid:88) s =1 (cid:20) s (1 − q ) qT ( m T cosh y − µ ) (cid:21) − − q = gV (2 π ) p T m T cosh y (cid:20) (1 − q )( m T cosh y − µ ) qT (cid:21) − − q ζ (cid:18) − q , qT (1 − q )( m T cosh y − µ ) (cid:19) . (44) B. Fermi-Dirac
The Boltzmann-Gibbs grand thermodynamic potential for the fermions obtained from Eq. (13) is given by,Ω
BEG ( β (cid:48) ) = − gV π β (cid:48) ∞ (cid:88) n =1 ( − n +1 e nβ (cid:48) µ n m K ( nβ (cid:48) m ) . (45)At the zeroth order of Eq. (12), the Tsallis fermionic spectrum looks like, d N (0)FD dp T dy = gV (2 π ) p T m T cosh y Γ (cid:16) − q (cid:17) ∞ (cid:90) t q − q e − t e β (cid:48) ( m T cosh y − µ ) + 1 dt = gV (2 π ) p T m T cosh y Γ (cid:16) − q (cid:17) ∞ (cid:88) s =1 ( − s − ∞ (cid:90) dt t q − q e − t − sβ (cid:48) ( m T cosh y − µ ) = gV (2 π ) p T m T cosh y ∞ (cid:88) s =1 ( − s − (cid:20) s (1 − q ) qT ( m T cosh y − µ ) (cid:21) − − q = gV (2 π ) p T m T cosh y (cid:20) − q )( m T cosh y − µ ) qT (cid:21) − − q (cid:20) ζ (cid:18) − q ,
12 + qT − q )( m T cosh y − µ ) (cid:19) − ζ (cid:18) − q , qT − q )( m T cosh y − µ ) (cid:19)(cid:21) . (46)It is worth noting that unlike the classical case, the zeroth order quantum Tsallis spectra does not resemble thephenomenological quantum Tsallis distributions when the q → q − replacement is done. However, it can be shownthat if one takes the factorization approximation, the forms of the zeroth order quantum Tsallis distributions and thephenomenological spectra display some similarity. This point will be discussed in the next section. VI. QUANTUM SPECTRA: FACTORIZATION APPROXIMATION OF THE ZEROTH ORDER TERMS
For the factorization approximation we use the following substitution [19], (cid:20) s (1 − q ) qT ( m T cosh y − µ ) (cid:21) − − q ≈ (cid:20) − q ) qT ( m T cosh y − µ ) (cid:21) − s − q . (47)Using Eq. (47), the factorized (denoted by the subscript ‘F’) zeroth order Tsallis Bose-Einstein distribution becomes, d N (0)BE,F dp T dy = gV (2 π ) p T m T cosh y (cid:104) (1 − q ) qT ( m T cosh y − µ ) (cid:105) − q − . (48)Similarly, the factorized zeroth order Tsallis Fermi-Dirac distribution is given by, d N (0)FD,F dp T dy = gV (2 π ) p T m T cosh y (cid:104) (1 − q ) qT ( m T cosh y − µ ) (cid:105) − q + 1 . (49)Replacing q with q − in Eqs. (48), and (49), we obtain the distribution functions d N (0)BE/FD,F dp T dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q → q − = gV (2 π ) p T m T cosh y (cid:2) ( q − T ( m T cosh y − µ ) (cid:3) qq − ∓ , (50)which are similar (but not exactly equal) to the quantum Tsallis-like distributions proposed in [12, 13] d N BE/FD dp T dy = gV (2 π ) p T m T cosh y (cid:2) ( q − T ( m T cosh y − µ ) (cid:3) q − ∓ . (51)0 - - - - T ( GeV ) d N dp (cid:0) d y ( G e V - ) q = - - - - T ( GeV ) d N dp (cid:0) d y ( G e V - ) q = - - - - T ( GeV ) d N dp (cid:0) d y ( G e V - ) q = - - - - T ( GeV ) d N dp (cid:0) d y ( G e V - ) q = FIG. 2: Spectra of the Maxwell-Boltzmann massive particles in the Tsallis-1 statistics at mid-rapidity ( y = 0) for differentvalues of the entropic parameter q when temperature T = 82 MeV, chemical potential µ = 0, radius R = 4 fm and mass m = 139 .
57 MeV (pion mass). The solid(blue), dotted(green) and dot-dashed(red) lines correspond to the exact Tsallis-1statistics, zeroth-term approximation and the Boltzmann-Gibbs statistics ( q = 1), respectively. The red short-dashed (blacklong-dashed) lines corresponds to analytical calculation up to the first (second) order. For q = 0 .
88 and q = 0 .
94, the analyticalfirst order results overlap with the numerically obtained unapproximated results.
VII. RESULTS
In Fig. 2, we plot the exact hadronic spectra (see Eq. 16) for four values of the entropic parameter q = 0 . , . , . m = 139 .
570 MeV. Temperature T = 82 MeV, chemical potential µ = 0MeV, and radius R = 4 fm. We compare the exact spectra with the zeroth, first, and second order approximatedspectra as well as with the Boltzmann-Gibbs counterpart.As seen in Ref. [7], the exact Tsallis transverse momentum spectra diverge if one includes all the terms in theseries representing the spectra in Eq. (16). A regularization scheme which cuts-off the series at a value of (cid:96) = (cid:96) corresponding to the minimum of the probability function φ ( (cid:96) ) in Eq. (15) and the normalization of the probabilitiesyields the numerical value of the potential Λ. For the zeroth, first, and second order approximated calculations, Λ is1 T ( GeV ) d N dp T d y ( G e V - ) q = FIG. 3: Spectra of the Bose-Einstein (black) and Fermi-Dirac (red) massive particles in the Tsallis-1 statistics at mid-rapidity( y = 0) for q = 0 .
88 when temperature T = 82 MeV, chemical potential µ = 0, radius R = 4 fm and mass m = 139 .
57 MeV(pion mass). The dot-dashed curves denote the quantum Tsallis phenomenological distributions, the dashed lines are the zerothorder terms, and the solid lines denote the factorized zeroth order distributions. calculated by truncating the series at (cid:96) = 0 , q = 0 . , .
94 and for the given values of mass, chemical potential and temperature,the first order spectra overlap with the exact calculations. For q = 0 .
98, and 0.99, however, the first and thesecond order truncation are not good approximations and the corresponding spectra deviate largely from the exactcalculations.In Fig. 3 we compare the zeroth order approximated quantum Tsallis distributions, their factorized counterparts,and the phenomenological quantum Tsallis distributions for q = 0 . T = 82 MeV, µ = 0 MeV, and radius R = 4fm at the mid-rapidity y = 0. It is observed that at low p T the zeroth order distributions and their factorizedcounterparts differ upto 10-20 %, but this difference diminishes for higher p T values. However, this difference is muchmore prominent between the former ones and the phenomenological distributions, and this difference increases with p T . VIII. SUMMARY, CONCLUSIONS, AND OUTLOOK
Following is a summary of the results we have obtained in this paper, • Analytical results for the first, and the second order corrections to the widely used Tsallis-like classicaltransverse momentum distribution have been obtained (Eqs. 28 and 42). • Analytical closed form of the zeroth order terms in the quantum Tsallis spectra has been calculated (Eqs. 44and 46). • Unlike the classical case, the quantum zeroth order terms are similar (but not exactly equal) to the phenomeno-logical distributions only under the factorization approximation when the q → q − substitution is done (Eqs. 50and 51).Tsallis transverse momentum spectra beyond the zeroth order approximation may be important in certain scenariosand hence, analytical formulae instead of a numerical integration to find out the higher order contributions will renderdata analysis procedure much easier. Also, from a more general perspective, the mathematical set-up presented inthis paper may help obtain analytical closed form results in many other cases where the Bessel’s functions appear. It2is noteworthy that in Eqs. (28), and (42), the distributions are expressed in terms of an infinite summation owing tothe infinite number of poles at Re( s ) <
0. However, it has been explicitly verified that the residues at those poles havedrastically diminishing contributions, and for all the practical purposes, considering only a few terms of Eqs. (26),and (38) suffices.In the experimental and the phenomenological studies so far, the zeroth order term in the Tsallis Maxwell-Boltzmannspectrum has been used. When we calculate the zeroth order term in the quantum statistics, no resemblance with thequantum Tsallis phenomenological distributions used in the literature could be established. Even the fact that thefactorization approximation of Eqs. (44), and (46) failed to show their congruence with the phenomenological distribu-tions strengthens this argument. Although it has been shown in Ref. [20] that using the factorization approximationin the Tsallis-2 (another scheme) mean occupation number right from the beginning leads to the phenomenologicalquantum distributions, this calculation contains an inconsistent definition of the average values, and the correct formof the distribution functions are computed in Ref. [21]. Also, from another recent work [9] it is observed that thesingle particle distributions appearing in the Tsallis two-point functions are the factorized zeroth order quantum dis-tributions. Hence, we propose the usage of the analytical closed forms of the quantum Tsallis distributions calculatedin Eqs. (44), and (46) as they are derivable from a fundamental theory like statistical mechanics. It will also beinteresting to extend the quantum calculations beyond the zeroth order.The results obtained in this paper may have several important implications. One of them may be the modificationof the analytic expression of the Tsallis classical [22] and quantum thermodynamic variables which will influence thenon-extensive equations of state [23, 24]. It may be possible that for dense systems, beyond the zeroth order termsbecome important. One more notable application will be to revisit the non-extensive behaviour of the QCD strongcoupling studied in [25]. This paper uses the expression of the non-extensive QCD coupling derived in [26] usingthe phenomenological quantum Tsallis distributions and successfully treats the deviation between the theoretical andexperimental results at low energy. However, it will be interesting to see how the present results affect their finding.
Acknowledgement
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