[Paper Review] Fourier coefficients of the net baryon number density and their scaling properties near a phase transition

Gábor Almási, Bengt Friman+2 more|arXiv (Cornell University)|Jan 1, 2019

High-Energy Particle Collisions Research2 references1 citations

TL;DR

This paper investigates the Fourier coefficients bk(T) of the net baryon number density in QCD matter near phase transitions, showing that critical singularities in the complex chemical potential plane induce power-law scaling behavior in bk(T) near second-order transitions in O(4) and Z(2) universality classes. The asymptotic decay of bk(T) reveals critical exponents and oscillatory structures linked to the location of singularities, providing a model-independent signature of phase transitions in the QCD phase diagram.

ABSTRACT

We study the Fourier coefficients b(k,T) of the net baryon number density in strongly interacting matter at finite temperature. We show that singularities in the complex chemical potential plane connected with phase transitions are reflected in the asymptotic behavior of the coefficients at large k. We derive the scaling properties of b(k,T) near a second order phase transition in the O(4) and Z(2) universality classes. The impact of first order and crossover transitions is also examined. The scaling properties of b(k,T) are linked to the QCD phase diagram in the temperature and complex chemical potential plane.

Motivation & Objective

• To establish model-independent scaling properties of Fourier coefficients bk(T) of the net baryon number density near QCD phase transitions.
• To connect the asymptotic behavior of bk(T) for large k with critical singularities in the complex chemical potential plane.
• To examine how first-order transitions, crossovers, and thermal singularities influence the scaling of bk(T).
• To link the observed Fourier coefficient behavior to the structure of the QCD phase diagram in the T–μB plane.
• To provide a diagnostic tool for identifying critical points and phase transition types using lattice QCD data on imaginary chemical potentials.

Proposed method

• Fourier decomposition of the net baryon number density χB1(T, iθB) in terms of sin(kθB) to extract coefficients bk(T) via integration over imaginary chemical potential θB.
• Application of the Riemann–Lebesgue lemma to derive asymptotic decay behavior of bk(T) for large k.
• Use of Landau theory and scaling theory of phase transitions to analyze bk(T) near second-order critical points in O(4) and Z(2) universality classes.
• Incorporation of thermal singularities from Fermi-Dirac poles at μB/T = ±m/T ± iπ to assess their influence on bk(T).
• Explicit analysis of the Roberge-Weiss (RW) transition at T = TRW, where a first-order transition occurs at θB = π.
• Derivation of scaling laws for bk(T) in terms of critical exponents α (O(4)) and δ (Z(2)), including oscillatory corrections due to imaginary parts of critical singularities.

Experimental results

Research questions

• RQ1 How do critical singularities in the complex chemical potential plane affect the asymptotic behavior of Fourier coefficients bk(T) of the net baryon number density?
• RQ2 What scaling laws govern bk(T) near second-order phase transitions in the O(4) and Z(2) universality classes?
• RQ3 How does the presence of a first-order transition (e.g., Roberge-Weiss) or a crossover influence the large-k decay of bk(T)?
• RQ4 Can the oscillatory structure in bk(T) for large k be linked to the imaginary part of the critical singularity location in the complex μB plane?
• RQ5 To what extent do thermal singularities from Fermi-Dirac poles affect the asymptotic behavior of bk(T)?
• RQ6 How can the Fourier coefficients bk(T) serve as a diagnostic tool for identifying critical points and phase transition types in the QCD phase diagram?

Key findings

• Near the O(4) critical point, bk(T) exhibits a power-law decay as k→−(2−α) for T < Tc, with α being the critical exponent.
• At the Roberge-Weiss (RW) transition temperature TRW, bk(T) decays as 1/k, reflecting the first-order nature of the transition.
• For T in the range Tc < T < TRW, bk(T) shows a power-law decay with oscillations superimposed, modulated by the imaginary part of the critical singularity at μB/T = ˆμc + iθc.
• The oscillation frequency in bk(T) is determined by θc, the imaginary part of the critical singularity, leading to a k-dependent phase shift.
• For T > TRW, the asymptotic behavior of bk(T) is dominated by the Roberge-Weiss singularity, with a 1/k decay and oscillations at the RW transition point.
• Thermal singularities from Fermi-Dirac poles at μB/T = ±m/T ± iπ contribute to bk(T), but their influence on large-k asymptotics is subdominant compared to critical singularities.