A note on rapidity distributions at the LHC
aa r X i v : . [ h e p - ph ] O c t IPPP/10/85DCPT/10/170November 9, 2018
A note on rapidity distributions at the LHC
V.A. Schegelsky a , M.G. Ryskin a,b , A.D. Martin b and V.A. Khoze a,ba Petersburg Nuclear Physics Institute, Gatchina, St. Petersburg, 188300, Russia b Institute for Particle Physics Phenomenology, University of Durham, Durham, DH1 3LE
Abstract
We discuss the difference between the distribution of secondaries measured in termsof pseudorapidity and that using the correct rapidity variable. We show a set of examplesobtained using Monte Carlo simulations. We also consider the production of particles oflow transverse momentum where coherence effects may occur, which are not yet includedin the present Monte Carlos.
The purpose of this note is to recall the difference between rapidity. y , and pseudorapidity, η . This is not a new topic [1]. However, since the results of the LHC detectors are shownmainly in terms of pseudorapidity, it is timely to discuss effects which may be caused by theidentification of η with y . The Lorentz-invariant single-particle inclusive cross section has the form f (AB → CX) ≡ E d σd p = E d σπdp L dp T (1)where E, p L and p T are the energy, longitudinal and transverse momentum of the outgoingparticle C with respect to the incoming proton direction; and d p/ E is the Lorentz-invariantphase space d p E = d p δ ( p − m ) Θ( E ) , (2)here m is the mass of particle C. For high-energy hadronic collisions, experiment shows thatthe longitudinal momentum may take almost any value, whereas the transverse momentum isusually small.The rapidity y is a convenient way to display the p L dependence of the data, where y = 12 log (cid:18) E + p L E − p L (cid:19) = log (cid:18) E + p L m T (cid:19) , (3)where m T = m + p T . The value of y depends on the reference frame, and, at first sight, itlooks like a rather ungainly construct. However, there are advantages to present high energydata as a function of rapidity. First, we see dy = dp L /E , and so the Lorentz-invariant crosssection f (AB → CX) = d σπdydp T . (4)Second, rapidity is additive under Lorentz boosts along the beam direction. For example, undera boost of velocity u E → γ ( E + up L ) , p L → γ ( p L + uE ) , (5)where γ = 1 / √ − u . Hence y → y + 12 log (cid:18) u − u (cid:19) = y + y boost . (6)Thus the difference in rapidity of two particles is not changed by boosts along the beam axis.In the non-relativistic limit, when the boost v ≪
1, we have E → m, p → mv and thus therapidity y → velocity v (hence its name).For a massless particle E = | p | . Then the rapidity can be directly expressed in terms of thescattering angle θ y = 12 log (cid:18) θ − cos θ (cid:19) = − log tan θ . (7)In general, we define pseudorapidity as η = − log tan θ . (8)Clearly, for massless particles the rapidity and pseudorapidity are the same. However, this isalso the case for particles with p T ≫ m and p L ≫ m . We now come to the single-particle rapidity distributions measured at the LHC. From thetheoretical viewpoint, both the phase space of particle C and the matrix element should bewritten in terms of covariant variables, that is in terms of y and p T . Long ago, Regge theory2igure 1: A sketch showing the inclusive production of particle C. predicted a flat rapidity distribution in the central region. Indeed, in Regge theory it assumedthat p T is limited and h p T i does not increase with energy. At very high energies, the inclusivecross section is described by the emission of secondaries from the Pomeron. The probabilityof the emission of particle C is given by the effective Pomeron-C vertex, β P C , which does notdepend on rapidity [2, 3], see Fig. 1. We have dσπdydp T = β N β P C ( p T ) s α (0) − AC s α (0) − BC . (9)If y is measured in the pp centre-of-mass frame, then s AC,BC = m p m T exp( ± y + Y /
2) (10)(where s AC = ( p A + p C ) , Y = ln( s/m p ) and α (0) is the intercept of the Pomeron trajectory),and the independence of the distribution on y is explicit. The cross section may also be writtenin terms of parton distributions dσdydp T = Z dx dx g ( x ) g ( x ) dσ ( gg → C ) dp T δ (cid:18) y −
12 ln x x (cid:19) , (11)with x , = ( m T / √ s )exp( ± y ). If the gluon distributions can be described by a fixed power, g ( x ) ∼ x − λ , then again we see that the single-particle multiplicity distribution is expected tobe flat in y . Indeed, the present Monte Carlos are based on (11), and yield a flat distributionin the central region, see Fig. 2. The PYTHIA 6.4 Monte Carlo (ATLAS) [4] was used todemonstrate the effects in Figs. 2 − y by η The single-particle distributions so far measured at the LHC have limited particle identification.The three-momentum of the particle is measured, but not, in general, its mass. Without3 a p i d i t y -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 , M e V T P a r t i c l e p Figure 2:
The p T -rapidity correlation obtained for pions from a Monte Carlo sample. knowledge of m it is not possible to calculate y . Therefore the results are presented in terms ofthe pseudorapidity η . For η = y we need p T ≫ m . However, for low p T the equality does nothold. In particular, a particle with low p T may have small y , but rather large η , since for low p T this slow particle tends to be emitted at a small polar angle θ . For simplicity, we consider p L ≫ m . Then the difference η − y = log m T p T + O (cid:18) m T p L (cid:19) . (12)The log may be quite large as p T →
0. The most interesting region, η ∼
0, in the multiplicitydistributions measured at the collider, contains a singularity at η = 0. As a consequence thelow p T particles are spread over a larger η region, depopulating the η ∼ η distribution for η ∼
0. For distributions corresponding to largervalues of p T the dip is less pronounced.This trivial kinematic effect produces some η dependence of h p T i measured in the centraldetector. Since low p T particles are spread over a larger number of low η bins, the measuredvalue of h p T i decreases with increasing | η | , see Fig. 4. This effect is not present in the y distributions.Of course, the majority of secondaries are pions. Therefore the multiplicity distributionsmay be presented in terms of y by assigning all particles to be pions. Nevertheless there aretwo sources of potential difficulty. First look at Fig. 5. It shows the fractions of different sorts For fixed p T , the Jacobian between the dy and dη bins is dη = ( E/ | p | ) dy . For low p T , it has a singularityat | p | → s e u d o r a p i d i t y -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 , M e V T P a r t i c l e p Figure 3:
The p T -pseudorapidity correlation obtained from the same Monte Carlo sample as Fig 2. Rapidity(Pseudorapidity)-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 , M e V T P a r t i c l e m ean p pseudorapidityrapiditypseudorapidityrapidity Figure 4:
The mean value of p T as a function of pseudorapidity (upper points) and rapidity (lowerpoints). MeV T Transverse momenta, p0 500 1000 1500 2000 2500 3000 P a r t i c l e k i nd , r e l a t i v e c on t en t pionkaonbaryonelectronpionkaonbaryonelectronpionkaonbaryonelectronpionkaonbaryonelectron Figure 5:
The Monte Carlo prediction for different types of minimum bias particles as a function oftheir p T . Although the pions dominate, there are a appreciable number of soft electrons in the verylowest p T bins and a significant kaon and baryon component at the larger p T values. of secondaries according to a PYTHIA Monte Carlo simulation. First, we see the presence ofelectrons (arising from Dalitz pairs) in the very lowest p T bins. If the electron is assigned themass of the pion, then its ‘effective’ rapidity becomes close to zero, which produces a prominanterroneous peak for y ∼
0. Second, at larger p T , the kaons and baryons, which are misassignedas pions, lead to an erroneous dip for y ∼
0. This dip disappears as we go to larger valuesof p T . Note however, that to neglect the difference between y and η , we need a value of p T larger than the mass of the corresponding hadron. All these effects are seen in Figs. 6 and 7.Note, from Fig. 5, that in the larger p T region, the fraction of kaons and baryons is appreciable.Only about 60% of secondaries are pions. Recall also that the region of low p T and small η isvery poorly described by the present Monte Carlo models. We should therefore take care whendiscussing this domain. y rather than η To demonstrate the importance of using the correct rapidity variable, we consider the coherenceeffects in low p T particle production. In the Regge approach, the p T distribution of secondariesproduced in the central region ( y ≃
0) does not depend on energy, but reflects only the internal p T -structure of the Pomeron. Therefore we expect that the particle density dN/dydp T measuredat low p T will increase with energy just like the density dN/dy integrated over over p T . That The region of a large p T gives a small contribution. a p i d i t y -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 , M e V T P a r t i c l e p Figure 6:
The p T -rapidity correlation assuming that all the particles of Fig. 5 have the mass of thepion. R a p i d i t y -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 , M e V T P a r t i c l e p Figure 7:
The same as Fig. 6, but omitting the pion component and taking the other particles tohave the mass of the pion. p T particles, interference ariseswhen a particle is radiated from different lines of Feynman diagrams which make up the wholematrix element. One example of this is soft gluon emission. Another example is the Hanbury-Brown − Twiss effect, where two identical particles can be radiated from different places in theinteraction region. Their interference leads to a peak in the distribution of the two identicalparticles when their momenta are close to each other. The width of the peak can be used toestimate the size of the interaction region [5].First we discuss soft gluon emission, which leads to the so-called limiting energy behaviour.The radiation of soft gluons from several quarks and/or gluons develops in a coherent way. Thereason is that different sources of secondaries should act as a single source with an effectivecolour ‘charge’ equal to the ‘ vector ’ sum of all the colour ‘charges’, as a gluon of large wavelengthcannot resolve these smaller details. Recall that, due to the Lorentz time dilation, the lowmomentum hadrons are formed at the initial stage, before the fast particles [6]. These fastparticles, which are formed later on, fly out of the interaction region and leave the slow hadronsintact . Such a limiting behaviour of soft secondaries was first predicted, and observed, in e + e − annihilation [7, 8], and then extended to the pp case [9].Also, note that, if the incoming hadron represents a quark-antiquark (meson) or quark-diquark (nucleon) system, then the t -channel multi-gluon exchange corresponds dominantelyto a colour octet. The same colour-octet exchange dominates the BFKL amplitude. So even inthe case of a large number of gluon exchanges (or a large number of MI), the low p T secondarieswill be produced mainly via radiation caused by colour-octet flow.Thus, in the ideal case, we may expect a limiting energy behaviour of the soft particledensity. That is, at very low p T , the value of dN/dydp T should not depend (or depend onlyweakly) on the initial energy, see Ref. [9]. However, as was shown above, in order to measurethe density of low- p T particles at small y , it is important to use the correct ‘rapidity’ variable,and not just the angle (that is, η ). Otherwise the expected ‘limiting’ behaviour may be affectedby an admixture of other sorts of particles, some lighter and some heavier than the pion.The Hanbury-Brown − Twiss correlation can be viewed as the first step in the formation ofcondensates of bosons which arise from Bose-Einstein statistics. At high energies when the par-ticle density increases, and the number of particles exceeds one in the elementary ∆ x ∆ p cells ofsome domain of configuration space, there may be two sorts of such collective phenomena. Oneis of pure classical, thermodynamic origin, like elliptic flow in quark-gluon plasma. Anotherone is based on quantum mechanical interference. We consider the formation of some mul-tiparticle coherent state (such as in superconductivity or in a disorientated chiral condensate An analogy is a supernova explosion where the remnants of the star are left intact. p T particles is large, it is expected that such collective phenomena will occur, and be bestobserved, in the low p T region. In comparison with lower energies, these collective, coherenteffects should be enhanced at the LHC due to the larger density of low p T secondaries.In summary, to observe such phenomena we need a careful study of very soft particledistributions using the correct Lorentz covariant variables. Acknowledgements
We thank Wolfgang Ochs for a fruitful discussion.
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