Mid-rapidity dependence of hadron production in $p-p$ and $A-A$ collisions

TL;DR

This paper models the rapidity dependence of pion production in proton-proton and heavy-ion collisions using a self-similarity approach, providing analytical formulas that fit experimental data at small rapidities and revealing universal energy patterns.

Contribution

It introduces an analytical self-similarity function for rapidity spectra in $pp$ and $AA$ collisions, accurately describing data at small rapidities.

Findings

Analytical form of the self-similarity function $\Pi(y,p_t)$ derived.

Good agreement with experimental rapidity spectra at $|y| extless 0.3$.

Universal energy dependence of the spectra demonstrated.

Abstract

The calculation of inclusive spectra of pions produced in $pp$ and $AA$ collisions as a function of rapidity $y$ is presented within the self-similarity approach. It is shown that at not large rapidities $y$ one can obtain the analytical form of the self-similarity function $\Pi(y,p_t)$ dependent of $y$ and hadron transverse momentum $p_t$. A satisfactory description of data on the rapidity spectra at $|y|\leq$ 0.3 is illustrated within a good agreement. The universal energy dependence of these spectra is also shown.

Equations

$$A+B\to 1+\dots ,$$

$${(N_A{P}_B+N_B{P}_B-p_1)}^2={(N_A{m}_0+N_B{m}_0+M)}^2,$$

$$\Pi =\min [\frac{1}{2}[(u_A{N}_A+u_B{N}_B{)}^2{]}^{1/2},$$

$$E{d}^3\sigma /d{p}^3=A_A^{\alpha (N_A}\cdot A_B^{\alpha (N_B}\cdot F(\Pi )$$

$$F(\Pi )=[A_{q}exp(-\Pi /C_q)+A_g\sqrt{m_{1t}}exp(-\Pi /C_g){\sigma }_{nd}/g((s/s_0{)}^{\Delta })]\cdot g(s/s_0{)}^{\Delta }.$$

$$N_A\cdot N_B-{\Phi }_A\cdot N_A-{\Phi }_B\cdot N_B={\Phi }_M,$$

$${\Phi }_A=[(m_1/m_0)\cdot (u_A{u}_1)+M/m_0]/[(u_A{u}_B)-1]$$

$${\Phi }_B=[(m_1/m_0)\cdot (u_B{u}_1)+M/m_0]/[(u_A{u}_B)-1]$$

$${\Phi }_M=(M^2-m_1^2)/[2m_0^2((u_A{u}_B)-1)].$$

$${\Phi }_A={\Phi }_B=\Phi ,$$

$$N_A=N_B=N=1+[1+({\Phi }_M/{\Phi }^2){]}^{1/2}·\Phi .$$

$$\Pi =N\cdot ch(Y).$$

$$(u_A{u}_B)=ch(2Y),$$

$$(u_A\cdot u_1)=(m_{1t}/m_1)\cdot ch(-Y-y)=(m_{1t}/m_1)\cdot ch(Y+y),$$

$$(u_B\cdot u_1)=(m_{1t}/m_1)\cdot ch(Y-y),$$

$$(u_A\cdot u_1)\simeq (u_B{u}_1)\simeq (m_{1t}/m_1)\cdot ch(y)ch(Y),$$

$$\Phi ={\Phi }_A={\Phi }_B=[(m_1/m_0)\bullet (u_A{u}_1)+M/m_0]/[(u_A{u}_B)-1]\simeq$$

$$[(m_1/m_0)\cdot (m_{1t}/m_1)\cdot ch(y)\cdot ch(Y)+M/m_0]/[ch(2Y)-1]=$$

$$=(1/m_0)[m_{1t}\cdot ch(y)\cdot ch(Y)+M]\cdot [1/(2s{h}^2(Y))].$$

$${\Phi }_M=(M^2-m_1^2)/(4m_0^{2}s{h}^2(Y)).$$

$$N=1+[1+({\Phi }_M/{\Phi }^2){]}^{1/2}\Phi ,$$

$$\Phi \simeq (1/m_0)[m_{1t}\cdot chy\cdot chY+M]\bullet [1/(2s{h}^2(Y))],$$

$${\Phi }_M=(M^2-m_1^2)/(4m_0^{2}s{h}^2(Y)).$$

$$(u_A\cdot u_1)=(P_A/m_A)(P_1/m_1)=(E_A\cdot E_1/m_A\cdot m_1)-(p_A\cdot p_1/m_A\cdot m_1)$$

$$=m_A\bullet ch(Y)\bullet m_{1t}\bullet ch(y)/(m_A\bullet m_1)+m_A\bullet sh(Y)\bullet m_{1t}\bullet sh(y)/(m_A\bullet m_1)=$$

$$=(m_{1t}/m_1)\cdot (ch(Y)\cdot ch(y)+sh(Y)\cdot sh(y))=(m_{1t}/m_1)\cdot ch(Y+y).$$

$$\Pi (s,m_{1t},y)\simeq \frac{m_{1t}ch(y)}{2m_0(1-4m_0^2/s)}\left\{1+\sqrt{1+\frac{M^2-m_1^2}{m_{1t}^{2}c{h}^2(y)}(1-4m_0^2/s)}\right\}.$$

$$E_h\frac{d^3{\sigma }_{NN}}{d^3{p}_h}\equiv \frac{1}{\pi }\frac{d\sigma }{d{m}_{1t}^{2}dy}=F(\Pi (s,m_{1t},y)),$$

$$\frac{d{\sigma }_{NN}}{dy}=2\pi \int F(\Pi (s,m_{1t},y))m_{1t}d{m}_{1t}$$

$$\frac{d{\sigma }_{NN}}{dy}\simeq A_A^{\alpha (N_A)}\cdot A_B^{\alpha (N_{(}B)}{A}_{q}g(s/s_0{)}^{\Delta }{m}_1[\frac{m_1}{ch(y)}+\frac{2m_0\delta }{c{h}^2(y)}]C_q\exp (-m_{1}ch(y)/(2m_0\delta C_q)),$$

Full text

Mid-rapidity dependence of hadron production in $p-p$ and $A-A$ collisions.

A.I. Malakhov, G.I. Lykasov

Joint Institute for Nuclear Research, Dubna 141980, Moscow region, Russia

lykasov@jinr.ru

malakhov@lhe.jinr.ru

**Abstract **

The calculation of inclusive spectra of pions produced in $pp$ and $AA$ collisions as a function of rapidity $y$ is presented within the self-similarity approach. It is shown that at not large rapidities $y$ one can obtain the analytical form of the self-similarity function $\Pi(y,p_{t})$ dependent of $y$ and hadron transverse momentum $p_{t}$. A satisfactory description of data on the rapidity spectra at $|y|\leq$ 0.3 is illustrated within a good agreement. The universal energy dependence of these spectra is also shown.

1 Introduction

The approach based on similarity of inclusive spectra of particles produced in hadron-hadron collisions suggested in pioneering papers [1, 2, 3, 4] was developed in [5, 6, 7, 8]. In [6, 8] the similarity of these spectra as a function of similarity parameter $\Pi$ dependent of the initial energy $\sqrt{s}$ in the c.m.s of colliding particles and transverse mass $m_{ht}$ of produced hadrons at zero rapidity $y=$0 was demonstrated. A simple form of inclusive spectra was used in [6, 7, 8] to describe satisfactorily the spectra at low values of $m_{ht}$. Further development of this approach was presented in our papers [9, 10, 11], where the description of $m_{ht}$-spectra was extended at larger values of transverse momenta and initial energies up to a few TeV including both contributions of quarks and gluons to these spectra. It has been shown [9] the direct relation of $\Pi$ to the Mandelstam variables $s,t$ and its non factorized form as a common function of $s$ and $m_{ht}$, which is very significant at not large initial energies $\sqrt{s}<$ 10 GeV and becomes factorized at larger $\sqrt{s}$. In fact, this is an advantage of the approach based on kinematics of four-momentum velocities considered in [5, 6, 7, 8], where the parameter $\Pi$ was obtained using the conservation law of four-momenta and quantum numbers of initial and produced particles, and the minimization principle. At zero rapidity $y$=0 the form for $\Pi$ was obtained analytically [8].

Let us note, that at non zero rapidity there are many theoretical models describing the inclusive spectrum of hadrons produced in $A-A$ collisions as a function of $y$ and $m_{ht}$, see for example, [12, 13, 14, 15, 16] and references there in. There are also experimental data on these distributions and their fits [17, 18, 19]. However, by modeling or fitting the rapidity and transverse momentum dependence there is no their universal energy dependencies in these papers.

In this paper we extend approach offered in [9] at the non zero rapidity region and calculate analytically the similarity parameter $\Pi$ as a function of $y$ and $m_{ht}$ without any additional parameters. Then, we calculate the $y$-dependence of inclusive spectra of pions produced in $pp$ and $AA$ collisions and describe satisfactorily data at $|y|\leq$ 0.3 in a wide region of initial energies. We also have confirmed that the distributions over $y$ and $m_{ht}$ have the universal energy dependence at low values of these variables, as it has already been shown in [9].

2 The parameter or function of self-similarity $\Pi$.

The inclusive production of hadron 1 in the interaction of nucleus A with nucleus B:

[TABLE]

is satisfied by the conservation law of four-momentum in the following form:

[TABLE]

where $N_{A}$ and $N_{B}$ are the fractions of four momenta transmitted by the nucleus A and nucleus B; $P_{A},P_{B},p_{1}$ are four momenta of the nuclei A and B and particle 1, respectively; $m_{0}$ is the mass of the nucleon; M is the mass of the particle providing the conservation of the baryon number, strangeness, and other quantum numbers. For $\pi$-mesons $m_{1}=m_{\pi}$ and M = 0. For antinuclei and $K^{-}$-mesons $M=m_{1}$. For nuclear fragments $M=-m_{1}$. For $K^{+}$-mesons $m_{1}=m_{K}$ and $M=m_{\Lambda}-m_{K}$, $m_{\Lambda}$ is the mass of the $\Lambda$-baryon.

In [7] the parameter of self-similarity is introduced, which allows one to describe the differential cross section of the yield of a large class of particles in relativistic nuclear collisions:

[TABLE]

where $u_{A}$ and $u_{B}$ are four velocities of the nuclei A and B.

Then, the inclusive spectrum of the produced particle 1 in AA collision can be presented as the general universal function dependent of the self-similarity parameter:

[TABLE]

where $\alpha(N_{A})=1/3+N_{A}/3$, $\alpha(N_{B})=1/3+N_{B}/3$, and $F$ is the function, its form is presented in [9]:

[TABLE]

Here $\Delta=\alpha_{P}(0)-1\simeq 0.12$ is the excess of the sub critical Pomeron intercept over 1; $g=21mb$ - constant, which was calculated within the ”quasi-eikonal” approximation [20]. The constants $A_{q}=$3.68 (GeV$/$c)-2, C${}_{q}=$0.147; $A_{g}=$1.7249 (GeV$/$c)-2, $C_{g}$=0.289 were obtained in [21, 22].

3 Analytical solution for self-similarity parameter

An analytical solution for the self-similarity parameter П was found in [8]. Here we give a more detailed derivation of the parameter and consider its behavior at small values of $y<<$ 1. Equation (2) can be written as follows:

[TABLE]

where relativistic invariant dimensionless values have been introduced:

[TABLE]

[TABLE]

[TABLE]

It was shown that at $y=0$

[TABLE]

The scalar product of four-dimensional velocities is related to the rapidity of initial particles $Y$ and the rapidity $y$ of the produced hadron $1$:

[TABLE]

Here $m_{1t}=(m_{1}^{2}+p_{1t}^{2})^{1/2}$ is the transverse mass of the particle 1. If $y<<$ 1, then one can neglect $sh(y)$ compared to $ch(y)$ and approximately we get the following:

[TABLE]

And in this case:

[TABLE]

Thus at $y<<$ 1

[TABLE]

where

[TABLE]

Since this equation $(u_{A}u_{1})$ does not depend on $m_{A}$, it is valid for any hadrons and nuclei:

[TABLE]

Therefore, we conclude that our approach is also valid for projectile π mesons.

4 Rapidity distribution of pions at low $y$.

Using the relation of rapidities $Y$ and $y$ of initial particles and produced hadron $1$ respectively to the Mandelstam variables $s$ and $t$ [9], one can get the following form of the similarity parameter $\Pi$ for small but non zero rapidity $y$:

[TABLE]

For $pp\rightarrow h+X$ inclusive processes the relativistic invariant differential cross at small but non zero rapidity $y$ will have the following form:

[TABLE]

where $F(\Pi(s,m_{1t},y))$ is given by Eq. 5 but with $\Pi(s,m_{1t},y))$ determined by Eq. 14. For $AA\rightarrow h+X$ processes the differential cross section is presented by Eq. 4. The integral of Eq. 15 or Eq. 4 over the transverse mass of the produced hadron $m_{1t}$ results in the rapidity $y$ dependence of the cross section of hadrons produced in $pp$ or $AA$ collision, respectively. Finally the rapidity distribution can be presented in the following form:

[TABLE]

In Fig. (1) the $y$-distributions of pions produced in $pp,AuAu$ and $PbPb$ collisions are presented in the wide region of initial energies. A satisfactory description of data with a precision less than 10% is shown in the rapidity range of the produced particles $|y|<$0.3.

Note that the main contribution to $d\sigma/dy$ given by Eq. 16 comes from the first term of Eq. 5 at low values of $y$, as our calculations have shown. Therefore, the rapidity distribution can be presented in the following approximated form :

[TABLE]

where $\delta=1-4m^{2}_{0}/s$. The Eq. 17 is similar to $d\sigma_{NN}/dy$ obtained in [12] within the thermal model including the longitudinal and transverse flow. The difference between our rapidity distribution and the one considered in [12] is the following. We do not include the nuclear thermal effects, which can change the $y$-dependence, mainly at $|y|>$ 0.3. Our approach can be applied at $|y|\leq$ 0.3 rather satisfactorily, as it is shown from Fig. (1). Eq. 17 results in the universal energy dependence of $d\sigma/dy$, as $(s/s_{0})^{\Delta}$. More complicated energy dependence of $d\sigma/dm_{1t}$ was obtained in our previous paper [9].

5 Conclusion

In this paper we have extended the self-similarity approach of analysis of hadron production in $pp,pA$ and $AA$ collisions, which firstly has been suggested in [5, 7, 8] and developed in [9] strictly at zero rapidity $y=0$ of produced hadrons, to the non zero rapidity region. This extension was obtained analytically using the conservation law of four-momenta and quantum numbers of initial and final particles. The validity of our results concerns the rapidity interval $|y|<$ 0.3, as it was shown by satisfactory description of the data on the pion production in $pp$ and $AA$ collisions ($\leq$) 0.3 within the wide region of initial energies. Moreover, we have got the universal energy dependence of $y$-spectra using the excess of the sub critical Pomeron intercept over 1, which is known very well from the satisfactory description of many data on the hadron production in $pp$ collision.

Acknowledgements.

We are very grateful to K.A. Bugaev, M. Gumberidze, M. Gadzicky, R. Holzmann, G. Kornakov, A. Rustamov for extremely helpful discussions.