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

Authors: Fu-Hu Liu; Ya-Hui Chen; Ya-Qin Gao; Er-Qin Wang
Journal: Advances in High Energy Physics
 (2013)

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## 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.


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## Body

## 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 asoz axis, and the reaction plane is defined as xoz plane. Let E, p, pL, pT, 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
(1)y≡12ln(E+pLE-pL),
where
(2)E=p2+m02,pL=pcosθ.
In the case of p≫m0, we have
(3)y≈12ln(p+pLp-pL)=12ln(1+cosθ1-cosθ)=-lntan(θ2)≡η,
where η is the pseudorapidity.Because the condition ofp≫m0 is not always satisfied, the pseudorapidity distribution (density function) fη(η)=(1/N)(dN/dη) and the rapidity distribution (density function) fy(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 betweendN/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
(4)dNdη=dNdydydη=pEdNdy,
where
(5)pE=E2-m02E=1-m02E2=1-(m0pT2+m02coshy)2.
Then, the first conversion is given by [4–6]
(6)dNdη=pEdNdy=1-(m0pT2+m02coshy)2dNdy,
where pT2+m02≡mT is the transverse mass. We see that the first conversion is related to pT.The second conversion is given in [5, 7–11]. We have
(7)dNdη=coshη1+m02pT-2+sinh2ηdNdy=coshηmT2pT-2+sinh2ηdNdy=coshηcosh2η+m02pT-2dNdy,
which is also related to pT. In [11], a similar conversion which uses m2P-2 instead of m02pT-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],
(8)y=12ln(E+pLE-pL)=12ln(p2+m02+pLp2+m02-pL)=12ln(pT2cosh2η+m02+pTsinhηpT2cosh2η+m02-pTsinhη).
In the case of pT being a given quantity, we have
(9)dydη=12·pT2cosh2η+m02-pTsinhηpT2cosh2η+m02+pTsinhη·ddη(pT2cosh2η+m02+pTsinhηpT2cosh2η+m02-pTsinhη)=12·[1pT2cosh2η+m02+pTsinhη·ddη(pT2cosh2η+m02+pTsinhη)-1pT2cosh2η+m02-pTsinhη·ddη(pT2cosh2η+m02-pTsinhη)1pT2cosh2η+m02+pTsinhη]=12·[1pT2cosh2η+m02+pTsinhη×(pT2coshηsinhηpT2cosh2η+m02+pTcoshη)-1pT2cosh2η+m02-pTsinhη×(pT2coshηsinhηpT2cosh2η+m02-pTcoshη)]=12·1pT2+m02·[(pT2coshηsinhηpT2cosh2η+m02+pTcoshη)×(pT2cosh2η+m02-pTsinhη)-(pT2coshηsinhηpT2cosh2η+m02-pTcoshη)×(pT2cosh2η+m02+pTsinhη)pT2coshηsinhηpT2cosh2η+m02]=12·1pT2+m02·2(pT2+m02)pTcoshηpT2cosh2η+m02=pTcoshηpT2cosh2η+m02=coshηcosh2η+m02pT-2=pE=β,
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 thatpT=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 thepT=p/coshη and pL=ptanhη are functions of η. Contrarily, p and E have the fixed values for a given particle. Then,
(10)y=12ln(E+pLE-pL)=12ln(E+ptanhηE-ptanhη)=12ln(1+βtanhη1-βtanhη)≡h(η),(11)dydη=12·1-βtanhη1+βtanhη·ddη(1+βtanhη1-βtanhη)=12·1-βtanhη1+βtanhη·[11-βtanhη+1+βtanhη(1-βtanhη)2]·βddη(tanhη)=12·1-βtanhη1+βtanhη·2β(1-βtanhη)2·1cosh2η=β1-β2tanh2η·1cosh2η=βcosh2η-β2sinh2η=β1+(1-β2)sinh2η=1-(1-β2)cosh2yβ.
It is different from the first conversion which gives that dy/dη=β. Correspondingly,
(12)η=12ln(p+pLp-pL)=12ln(p+Etanhyp-Etanhy)=12ln(β+tanhyβ-tanhy)≡φ(y),(13)dηdy=12·β-tanhyβ+tanhy·ddy(β+tanhyβ-tanhy)=12·β-tanhyβ+tanhy·[1β-tanhy+β+tanhy(β-tanhy)2]·ddy(tanhy)=12·β-tanhyβ+tanhy·2β(β-tanhy)2·1cosh2y=ββ2-tanh2y·1cosh2y=ββ2cosh2y-sinh2y=β1-(1-β2)cosh2y=1+(1-β2)sinh2ηβ.
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
(14)fη(η)|dη|=fy(y)|dy|=fy[h(η)]·|β1+(1-β2)sinh2η|·|dη|,fy(y)|dy|=fη(η)|dη|=fη[φ(y)]·|β1-(1-β2)cosh2y|·|dy|.
Then, we have further
(15)fη(η)=fy[h(η)]·|β1+(1-β2)sinh2η|,(16)fy(y)=fη[φ(y)]·|β1-(1-β2)cosh2y|.
Equations (15) and (16) translate the rapidity distribution to pseudorapidity one and the pseudorapidity distribution to rapidity one, respectively.In the previous discussions,(17)β=1-(m0pT2+m02coshy)2=coshηcosh2η+m02pT-2
which can be used in the conversion. Then, the conversion is related to pT and m0. To do a conversion, we need to know pT 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 functionsd2N/dpTdη and d2N/dpTdy, that is,
(18)d2NdpTdη=1-(m0pT2+m02coshy)2d2NdpTdy,
is also erroneous or incomplete [12]. In fact, the conversion between the two unit-density functions is correct due to pT being a series of fixed values in (18). To use (18), we also need to know pT 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.



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