[Paper Review] On the dispersion of fundamental particles in QCD and N=4 Super Yang-Mills theory

Paul M. Chesler, A. Gynther+1 more|arXiv (Cornell University)|Jun 17, 2009

High-Energy Particle Collisions Research4 citations

TL;DR

This paper computes leading-order thermal corrections to the dispersion relations of massive fundamental particles in weakly coupled QCD and N=4 Super Yang-Mills (SYM) theory across all mass scales, from massless to heavy quarks. Using resummed perturbation theory and classical field theory, it derives momentum-dependent self-energies and shows that in N=4 SYM, the dispersion relation simplifies to $ E = \sqrt{p^2 + M^2 + \delta M^2} $ with a momentum-independent thermal mass shift, while in QCD, the correction is more complex and momentum-dependent, especially at intermediate masses.

ABSTRACT

We study thermal corrections to the dispersion relations of massive fundamental particles immersed in weakly coupled non-Abelian plasmas. The cases covered include quarks in the QCD (quark-gluon) plasma, as well as N=2 quarks and scalars in an N=4 Super Yang-Mills plasma. We perform the calculations to leading order in a weak coupling expansion, and consider all mass scales of the fundamental fields, ranging from massless particles all the way to bare masses parametrically larger than the temperature.

Motivation & Objective

• To compute leading-order thermal corrections to the dispersion relations of massive fundamental particles in weakly coupled non-Abelian plasmas.
• To bridge a gap in the QCD literature by systematically analyzing heavy quark regimes ($ M \gtrsim T $) not fully covered in prior works.
• To compare weak-coupling results in N=4 SYM with strong-coupling AdS/CFT results, particularly in the $ M \gtrsim T/g $ limit.
• To demonstrate equivalence between resummed perturbation theory and classical field theory in computing thermal corrections for heavy particles.
• To provide a self-consistent, comprehensive treatment of dispersion relations across all mass scales in both QCD and N=4 SYM.

Proposed method

• Uses thermal field theory to compute the retarded two-point function and extract dispersion relations from poles of the self-energy.
• Applies a weak-coupling expansion to compute the self-energy of fundamental particles to leading order in the coupling constant.
• Employs resummed perturbation theory to handle infrared divergences arising from soft and collinear modes in the plasma.
• Uses classical field theory with temperature-dependent screening masses and electromagnetic tensors to model long-wavelength thermal effects.
• Performs analytical and numerical computations of sum-integrals over Matsubara frequencies and momenta, including a detailed treatment of the self-energy integral in Appendix A.
• Validates the classical field approach by showing agreement with resummed perturbation theory in the non-relativistic limit.

Experimental results

Research questions

• RQ1 How do thermal corrections modify the dispersion relation of a massive quark in a weakly coupled QCD plasma across all mass scales, from massless to $ M \gtrsim T $?
• RQ2 What is the form of the dispersion relation for fundamental quarks and scalars in $ \mathcal{N}=4 $ SYM, and how does it compare to QCD?
• RQ3 Can classical field theory with temperature-dependent parameters accurately reproduce the leading-order thermal corrections to the dispersion relation of a heavy quark?
• RQ4 Why are thermal corrections in QCD momentum-dependent while in $ \mathcal{N}=4 $ SYM they are effectively momentum-independent for the same mass scale?
• RQ5 How do the results for $ M \gtrsim T/g $ in weakly coupled SYM compare to strong-coupling AdS/CFT predictions?

Key findings

• In $ \mathcal{N}=4 $ SYM, the dispersion relation for both quarks and scalars is $ E = \sqrt{p^2 + M^2 + \delta M^2} $, with a momentum-independent thermal mass shift $ \delta M^2 = g^2 C_F T^2 $.
• In QCD, the dispersion relation is more complex and momentum-dependent, especially for $ M \lesssim T $, with corrections involving logarithmic and quadratic terms in momentum.
• For $ M \gtrsim T/g $, the non-relativistic dispersion relation in QCD is $ \mathcal{E} = M_{\text{rest}} + \frac{1}{2} M_{\text{kin}} v^2 $, with $ M_{\text{rest}} = M - \frac{g^2 C_F m_D}{8\pi} $ and $ M_{\text{kin}} = M - \frac{g^2 C_F m_D}{24\pi} \left(1 - \frac{\pi^2}{16}\right) $.
• In $ \mathcal{N}=4 $ SYM, the rest mass shift is $ M_{\text{rest}} = M - \frac{g^2 C_F m_D}{8\pi} + \frac{g^2 C_F m_s}{8\pi} $, reflecting contributions from both gluons and scalars.
• The classical field theory approach reproduces the same leading-order non-relativistic dispersion relation as resummed perturbation theory, validating its use for infrared-sensitive thermal corrections.
• The results for $ \mathcal{N}=4 $ SYM in the soft-momentum limit are analogous to QCD results upon replacing the soft mass parameter, confirming a universal structure in weakly coupled plasmas.