On Current Conversion between Particle Rapidity and Pseudorapidity Distributions in High Energy Collisions

Abstract

In high energy collisions, one usually needs to give a conversion between the particle rapidity and pseudorapidity distributions. Currently, two equivalent conversion formulas are used in experimental and theoretical analyses. An investigation in the present work shows that the two conversions are incomplete. Then, we give a revision on the current conversion between the particle rapidity and pseudorapidity distributions.

1. Introduction

High energy collisions are an important research field in particle and nuclear physics. In the collisions, a lot of particles are produced, and the rapidity and/or pseudorapidity distributions can be obtained and studied [1–3]. Usually, one needs to do a conversion between the rapidity and pseudorapidity distributions in the case of only one of the two distributions being obtained. There are two equivalent conversion formulas used in the current literature [4–11]. Naturally, one thinks that the two conversions are perfect in investigations of the rapidity and pseudorapidity distributions.

However, our incidental find shows that the two conversions are incomplete. In obtaining the Jacobian in the current literature [4–11], a nongiven quantity, namely, transverse momentum, is erroneously used as a given one, which renders an incomplete conversion. In this paper, we will give a reanalysis on the Jacobian. A revised conversion between the rapidity and pseudorapidity distributions will be presented.

2. General Definition

We consider a system of high energy projectile-target collisions. The incident projectile direction is defined as 𝑜𝑧 axis, and the reaction plane is defined as 𝑥𝑜𝑧 plane. Let 𝐸, 𝑝, 𝑝_𝐿, 𝑝_𝑇, 𝑚₀, and 𝜃 denote, respectively, the energy, momentum, longitudinal momentum, transverse momentum, rest mass, and emission angle of a concerned particle. According to general textbooks on particle physics [12, 13], the rapidity (which is in fact the longitudinal rapidity) is defined by 1𝑦≡2ln𝐸+𝑝_𝐿𝐸−𝑝_𝐿, (1) where 𝐸=𝑝²+𝑚²₀,𝑝_𝐿=𝑝cos𝜃. (2) In the case of 𝑝≫𝑚₀, we have 1𝑦≈2ln𝑝+𝑝_𝐿𝑝−𝑝_𝐿=12ln1+cos𝜃𝜃1−cos𝜃=−lntan2≡𝜂, (3) where 𝜂 is the pseudorapidity.

Because the condition of 𝑝≫𝑚₀ is not always satisfied, the pseudorapidity distribution (density function) 𝑓_𝜂(𝜂)=(1/𝑁)(𝑑𝑁/𝑑𝜂) and the rapidity distribution (density function) 𝑓_𝑦(𝑦)=(1/𝑁)(𝑑𝑁/𝑑𝑦) are not approximately equal to each other, where 𝑑𝑁 denotes the particle number in the pseudorapidity or rapidity bin and 𝑁 denotes the total number of considered particles.

3. Current Conversion

To give a conversion between 𝑑𝑁/𝑑𝜂 and 𝑑𝑁/𝑑𝑦 in the case of one of them being obtained, one has two equivalent methods which are currently used in the literature [4–11]. According to [4, 5, 11], the first conversion relation between 𝑑𝑁/𝑑𝜂 and 𝑑𝑁/𝑑𝑦 can be given by 𝑑𝑁=𝑑𝜂𝑑𝑁𝑑𝑦𝑑𝑦=𝑝𝑑𝜂𝐸𝑑𝑁,𝑑𝑦 (4) where 𝑝𝐸=𝐸²−𝑚²₀𝐸=𝑚1−²⁰𝐸²=𝑚1−⁰𝑝²_𝑇+𝑚²₀cosh𝑦². (5) Then, the first conversion is given by [4–6] 𝑑𝑁=𝑝𝑑𝜂𝐸𝑑𝑁=𝑑𝑦𝑚1−⁰𝑝²_𝑇+𝑚²₀cosh𝑦²𝑑𝑁,𝑑𝑦 (6) where 𝑝²_𝑇+𝑚²₀≡𝑚_𝑇 is the transverse mass. We see that the first conversion is related to 𝑝_𝑇.

The second conversion is given in [5, 7–11]. We have 𝑑𝑁=𝑑𝜂cosh𝜂1+𝑚²₀𝑝_𝑇⁻²+sinh²𝜂𝑑𝑁=𝑑𝑦cosh𝜂𝑚²_𝑇𝑝_𝑇⁻²+sinh²𝜂𝑑𝑁=𝑑𝑦cosh𝜂cosh²𝜂+𝑚²₀𝑝_𝑇⁻²𝑑𝑁,𝑑𝑦 (7) which is also related to 𝑝_𝑇. In [11], a similar conversion which uses 𝑚²𝑃⁻² instead of 𝑚²₀𝑝_𝑇⁻² in (7) is given, where 𝑚=350 MeV, 𝑃=0.13GeV+0.32GeV(√𝑠/1TeV)^0.115, and √𝑠 denotes the center-of-mass energy. The conversion used in [11] is a mutation of the second conversion.

We now give the eduction of the current conversion. According to [12], 1𝑦=2ln𝐸+𝑝_𝐿𝐸−𝑝_𝐿=12ln𝑝²+𝑚²₀+𝑝_𝐿𝑝²+𝑚²₀−𝑝_𝐿=12ln𝑝²_𝑇cosh²𝜂+𝑚²₀+𝑝_𝑇sinh𝜂𝑝²_𝑇cosh²𝜂+𝑚²₀−𝑝_𝑇.sinh𝜂 (8) In the case of 𝑝_𝑇 being a given quantity, we have 𝑑𝑦=1𝑑𝜂2⋅𝑝²_𝑇cosh²𝜂+𝑚²₀−𝑝_𝑇sinh𝜂𝑝²_𝑇cosh²𝜂+𝑚²₀+𝑝_𝑇⋅𝑑sinh𝜂𝑑𝜂𝑝²_𝑇cosh²𝜂+𝑚²₀+𝑝_𝑇sinh𝜂𝑝²_𝑇cosh²𝜂+𝑚²₀−𝑝_𝑇=1sinh𝜂2⋅1𝑝²_𝑇cosh²𝜂+𝑚²₀+𝑝_𝑇⋅𝑑sinh𝜂𝑑𝜂𝑝²_𝑇cosh²𝜂+𝑚²₀+𝑝_𝑇−1sinh𝜂𝑝²_𝑇cosh²𝜂+𝑚²₀−𝑝_𝑇⋅𝑑sinh𝜂𝑑𝜂𝑝²_𝑇cosh²𝜂+𝑚²₀−𝑝_𝑇=1sinh𝜂2⋅1𝑝²_𝑇cosh²𝜂+𝑚²₀+𝑝_𝑇×𝑝sinh𝜂_2𝑇cosh𝜂sinh𝜂𝑝²_𝑇cosh²𝜂+𝑚²₀+𝑝_𝑇−1cosh𝜂𝑝²_𝑇cosh²𝜂+𝑚²₀−𝑝_𝑇×𝑝sinh𝜂_2𝑇cosh𝜂sinh𝜂𝑝²_𝑇cosh²𝜂+𝑚²₀−𝑝_𝑇=1cosh𝜂2⋅1𝑝²_𝑇+𝑚²₀⋅𝑝²_𝑇cosh𝜂sinh𝜂𝑝²_𝑇cosh²𝜂+𝑚²₀+𝑝_𝑇×cosh𝜂𝑝²_𝑇cosh²𝜂+𝑚²₀−𝑝_𝑇−𝑝sinh𝜂_2𝑇cosh𝜂sinh𝜂𝑝²_𝑇cosh²𝜂+𝑚²₀−𝑝_𝑇×cosh𝜂𝑝²_𝑇cosh²𝜂+𝑚²₀+𝑝_𝑇=1sinh𝜂2⋅1𝑝²_𝑇+𝑚²₀⋅2𝑝²_𝑇+𝑚²₀𝑝_𝑇cosh𝜂𝑝²_𝑇cosh²𝜂+𝑚²₀=𝑝_𝑇cosh𝜂𝑝²_𝑇cosh²𝜂+𝑚²₀=cosh𝜂cosh²𝜂+𝑚²₀𝑝_𝑇⁻²=𝑝𝐸=𝛽, (9) where 𝛽 denotes the velocity of the concerned particle. Then, we obtain the current conversion.

However, we would like to point out that the previous conversion is incomplete due to the fact that 𝑝_𝑇=𝑝/cosh𝜂 is also a function of 𝜂, which should be considered in doing the differential treatment. Instead, 𝑝 and 𝐸 can be regarded as given quantities.

4. Revised Conversion

In the differential treatment, we think that both the 𝑝_𝑇=𝑝/cosh𝜂 and 𝑝_𝐿=𝑝tanh𝜂 are functions of 𝜂. Contrarily, 𝑝 and 𝐸 have the fixed values for a given particle. Then, 1𝑦=2ln𝐸+𝑝_𝐿𝐸−𝑝_𝐿=12ln𝐸+𝑝tanh𝜂=1𝐸−𝑝tanh𝜂2ln1+𝛽tanh𝜂𝜂,1−𝛽tanh𝜂≡ℎ𝑑𝑦=1𝑑𝜂2⋅1−𝛽tanh𝜂⋅𝑑1+𝛽tanh𝜂𝑑𝜂1+𝛽tanh𝜂=11−𝛽tanh𝜂2⋅1−𝛽tanh𝜂⋅11+𝛽tanh𝜂+1−𝛽tanh𝜂1+𝛽tanh𝜂1−𝛽tanh𝜂²𝑑⋅𝛽=1𝑑𝜂tanh𝜂2⋅1−𝛽tanh𝜂⋅1+𝛽tanh𝜂2𝛽1−𝛽tanh𝜂²⋅1cosh²𝜂=𝛽1−𝛽²tanh²𝜂⋅1cosh²𝜂=𝛽cosh²𝜂−𝛽²sinh²𝜂=𝛽1+1−𝛽²sinh²𝜂=1−1−𝛽²cosh²𝑦𝛽. (10)(11) It is different from the first conversion which gives that 𝑑𝑦/𝑑𝜂=𝛽. Correspondingly, 1𝜂=2ln𝑝+𝑝_𝐿𝑝−𝑝_𝐿=12ln𝑝+𝐸tanh𝑦=1𝑝−𝐸tanh𝑦2ln𝛽+tanh𝑦𝑦,𝛽−tanh𝑦≡𝜑𝑑𝜂=1𝑑𝑦2⋅𝛽−tanh𝑦⋅𝑑𝛽+tanh𝑦𝑑𝑦𝛽+tanh𝑦=1𝛽−tanh𝑦2⋅𝛽−tanh𝑦⋅1𝛽+tanh𝑦+𝛽−tanh𝑦𝛽+tanh𝑦𝛽−tanh𝑦²⋅𝑑=1𝑑𝑦tanh𝑦2⋅𝛽−tanh𝑦⋅𝛽+tanh𝑦2𝛽𝛽−tanh𝑦²⋅1cosh²𝑦=𝛽𝛽²−tanh²𝑦⋅1cosh²𝑦=𝛽𝛽²cosh²𝑦−sinh²𝑦=𝛽1−1−𝛽²cosh²𝑦=1+1−𝛽²sinh²𝜂𝛽. (12)(13) The expressions after the last equal marks in (11) and (13) are obtained from the expressions before the last equal marks in (13) and (11), respectively. It is obvious that the eduction of the revised conversion is simpler than that of the current conversion.

To use (10)–(13), we have relations 𝑓_𝜂𝜂𝑑𝜂=𝑓_𝑦𝑦𝑑𝑦=𝑓_𝑦ℎ𝜂⋅𝛽1+1−𝛽²sinh²𝜂⋅,𝑓𝑑𝜂_𝑦𝑦𝑑𝑦=𝑓_𝜂𝜂𝑑𝜂=𝑓_𝜂𝜑𝑦⋅𝛽1−1−𝛽²cosh²𝑦⋅𝑑𝑦. (14) Then, we have further 𝑓_𝜂𝜂=𝑓_𝑦ℎ𝜂⋅𝛽1+1−𝛽²sinh²𝜂,𝑓_𝑦𝑦=𝑓_𝜂𝜑𝑦⋅𝛽1−1−𝛽²cosh²𝑦. (15)(16) Equations (15) and (16) translate the rapidity distribution to pseudorapidity one and the pseudorapidity distribution to rapidity one, respectively.

In the previous discussions, 𝛽=𝑚1−⁰𝑝²_𝑇+𝑚²₀cosh𝑦²=cosh𝜂cosh²𝜂+𝑚²₀𝑝_𝑇⁻² (17) which can be used in the conversion. Then, the conversion is related to 𝑝_𝑇 and 𝑚₀. To do a conversion, we need to know 𝑝_𝑇 and 𝑚₀ for each particle.

5. Conclusion and Discussion

We have given a revision on the current conversion between the particle rapidity and pseudorapidity distributions. It is shown that, comparing to the current first conversion, the revised one ((15) or (16)) has an additional term (1−𝛽²)sinh²𝜂  or  −(1−𝛽²)cosh²𝑦 in the denominator. In central rapidity region, sinh𝜂≈0 and cosh𝑦≈1; then, (15) and (16) change to the current conversion. However, in forward rapidity region, the difference between the revised conversion and current one is obvious.

Our conclusion does not mean that the current conversion between the unit-density functions 𝑑²𝑁/𝑑𝑝_𝑇𝑑𝜂 and 𝑑²𝑁/𝑑𝑝_𝑇𝑑𝑦, that is, 𝑑²𝑁𝑑𝑝_𝑇=𝑑𝜂𝑚1−⁰𝑝²_𝑇+𝑚²₀cosh𝑦²𝑑²𝑁𝑑𝑝_𝑇,𝑑𝑦 (18) is also erroneous or incomplete [12]. In fact, the conversion between the two unit-density functions is correct due to 𝑝_𝑇 being a series of fixed values in (18). To use (18), we also need to know 𝑝_𝑇 and 𝑚₀ for each particle.

Because the conversion between rapidity and pseudorapidity distributions is not simpler than a direct calculation based on the definitions of rapidity and pseudorapidity, we would rather use the direct calculation in modeling analysis. In fact, in the epoch of high energy collider, the dispersion between rapidity and pseudorapidity distributions is small [7]. This means that we would also like to not distinguish strictly rapidity and pseudorapidity distributions in general modeling analysis.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant no. 10975095, the China National Fundamental Fund of Personnel Training under Grant no. J1103210, the Open Research Subject of the Chinese Academy of Sciences Large-Scale Scientific Facility under Grant no. 2060205, and the Shanxi Scholarship Council of China.