[Paper Review] A note on rapidity distributions at the LHC

V. A. Schegelsky, M. G. Ryskin+2 more|arXiv (Cornell University)|Oct 11, 2010

Particle physics theoretical and experimental studies1 references3 citations

TL;DR

This paper highlights the critical distinction between rapidity (y) and pseudorapidity (η) in LHC particle physics, demonstrating via Monte Carlo simulations that using η instead of y distorts low-transverse-momentum particle distributions—particularly depleting the central region (η ≈ 0) due to kinematic singularities. The key contribution is the warning that incorrect use of η may mask genuine physical effects like limiting energy behavior and coherent particle production in soft gluon emission, which require accurate rapidity-based analysis to observe.

ABSTRACT

We discuss the difference between the distribution of secondaries measured in terms of pseudorapidity and that using the correct rapidity variable. We show a set of examples obtained using Monte Carlo simulations. We also consider the production of particles of low transverse momentum where coherence effects may occur, which are not yet included in the present Monte Carlos.

Motivation & Objective

• To clarify the physical distinction between rapidity (y) and pseudorapidity (η), emphasizing that η is not a Lorentz-invariant variable.
• To demonstrate via Monte Carlo simulations that using η instead of y distorts the central rapidity distribution, particularly depleting the η ≈ 0 region due to kinematic singularities.
• To caution that current LHC data presentations using η may obscure genuine physical phenomena such as limiting energy behavior and coherence effects in low-pT particle production.
• To advocate for the use of rapidity in theoretical and phenomenological studies to correctly capture energy dependence and collective effects in soft particle emission.

Proposed method

• Uses Lorentz-invariant phase space formalism with $ f = E \frac{d^3\sigma}{d^3p} = \frac{d^3\sigma}{\pi dy dp_T^2} $, showing rapidity's invariance under boosts.
• Applies the definition $ y = \frac{1}{2} \log \left( \frac{E + p_L}{E - p_L} \right) $ and $ \eta = -\log \left( \tan \frac{\theta}{2} \right) $, and derives the difference $ \eta - y \approx \log \frac{m_T}{p_T} $ for low $ p_T $.
• Performs Monte Carlo simulations (PYTHIA 6.4) to compare $ p_T $-rapidity and $ p_T $-pseudorapidity correlations, revealing a dip in the η distribution at $ \eta \approx 0 $.
• Analyzes the impact of coherence effects in soft gluon emission, where the vector sum of color charges leads to collective radiation and limiting energy behavior.
• Considers the Hanbury-Brown–Twiss effect as a signature of quantum coherence in identical particle correlations.
• Proposes that coherent particle production and limiting behavior in low-pT regions are best observed using rapidity, not pseudorapidity, due to Lorentz invariance and correct phase space coverage.

Experimental results

Research questions

• RQ1 How does the use of pseudorapidity (η) instead of rapidity (y) distort the measured distribution of low-transverse-momentum particles at the LHC?
• RQ2 What is the quantitative impact of the $ d\eta/dy $ Jacobian singularity at $ p_T \to 0 $ on the observed pseudorapidity distribution?
• RQ3 To what extent do coherence effects—such as soft gluon emission and Hanbury-Brown–Twiss correlations—alter the expected particle density in the central region at high energies?
• RQ4 Why is rapidity the preferred variable for observing limiting energy behavior in low-pT particle production, and how is this obscured when using η?
• RQ5 Can the absence of coherent effects in current Monte Carlo simulations lead to incorrect predictions about soft particle production at LHC energies?

Key findings

• The difference $ \eta - y \approx \log \frac{m_T}{p_T} $ becomes large at low $ p_T $, causing significant distortion in the $ \eta $-distribution, especially near $ \eta \approx 0 $.
• Monte Carlo simulations show a clear dip in the $ p_T $-pseudorapidity correlation at $ \eta \approx 0 $, caused by the $ d\eta/dy $ singularity at low $ p_T $, which spreads low-velocity particles over a wider $ \eta $ range.
• The particle density $ dN/dydp_T^2 $ at low $ p_T $ is expected to increase by about 60% from 0.9 to 7 TeV in $ pp $ collisions, but this energy dependence may be masked if $ \eta $ is used instead of $ y $.
• Coherent soft gluon emission leads to a limiting energy behavior where $ dN/dydp_T^2 $ becomes weakly dependent on center-of-mass energy at very low $ p_T $, a feature that is only visible when using rapidity.
• The Hanbury-Brown–Twiss effect, which probes the size of the emission source, is a signature of quantum coherence and is best studied in the rapidity framework to avoid kinematic biases.
• The formation of multiparticle coherent states—such as in disoriented chiral condensates or superconducting-like phases—should be most prominent in the low-$ p_T $ region and requires accurate rapidity-based analysis to detect.