[Paper Review] Quantum Field Theory in Large N Wonderland: Three Lectures

Paul Romatschke|arXiv (Cornell University)|Sep 29, 2023

Quantum, superfluid, helium dynamicsPhysics and Astronomy3 citations

TL;DR

This paper presents a systematic introduction to large N techniques in quantum field theory, demonstrating their power to solve non-perturbative problems in scalar and fermionic field theories across various dimensions. By treating 1/N as a small expansion parameter, the method enables exact solutions for strongly coupled systems—such as finite-temperature phase transitions in O(N) scalar theories—yielding a critical temperature of $ T_c \approx 0.616 \Lambda_{\overline{\rm MS}} $, in remarkable agreement with lattice QCD results.

ABSTRACT

In these lecture notes, I review how to use large N techniques to solve quantum field theories in various dimensions. In particular, the case of N-dimensional quantum mechanics, non-relativistic cold and dense neutron matter, and scalar field theory in four dimensions are covered. A recurring theme is that large N solutions are fully non-perturbative, and can be used to reliably access quantum field theory for parameter regions where weak-coupling expansions simply fail.

Motivation & Objective

• To provide an accessible, first-principles introduction to large N techniques for solving quantum field theories beyond weak-coupling perturbation theory.
• To demonstrate that large N methods yield fully non-perturbative solutions in strongly coupled systems where standard perturbation theory fails.
• To apply the large N expansion to concrete problems: quantum mechanics, cold dense neutron matter, and four-dimensional scalar field theory with finite-temperature phase transitions.
• To show that large N techniques can resolve long-standing issues such as quantum triviality and asymptotic freedom, potentially leading to simpler models of the Standard Model.
• To highlight the solvability of scalar field theories in the large N limit, enabling derivation of gravity duals rather than mere conjectures, as in holography.

Proposed method

• Use of the large N expansion with $ 1/N $ as the small parameter, replacing weak-coupling perturbation theory.
• Application of the path integral formulation in Euclidean time to map quantum mechanics and field theories to statistical field theories.
• Derivation of the effective potential $ V_{\rm eff}(m) $ via saddle-point approximation in the large N limit, minimizing $ V_{\rm eff} $ to find the dynamical mass $ m $.
• Incorporation of finite-temperature effects via Matsubara formalism with bosonic frequencies $ \omega_n = 2\pi n T $, leading to thermal contributions in the effective potential.
• Use of dimensional regularization with $ \varepsilon $-regulation to handle UV divergences, followed by renormalization in the $ \overline{\rm MS} $ scheme.
• Numerical solution of the saddle-point condition involving Bessel functions $ K_1(n\beta m) $, yielding the critical temperature $ T_c $.

Experimental results

Research questions

• RQ1 Can large N techniques provide non-perturbative solutions to quantum field theories where weak-coupling expansions fail?
• RQ2 What is the critical temperature for the phase transition in the O(N) scalar theory at finite temperature, and how does it compare to lattice QCD?
• RQ3 How does the large N method resolve issues such as quantum triviality and asymptotic freedom in four-dimensional scalar field theories?
• RQ4 Can the large N limit allow for a derivation of gravity duals in scalar field theories, rather than just conjecturing them?
• RQ5 What is the role of the dynamical mass $ m $ in the effective potential at finite temperature, and how does it signal spontaneous symmetry breaking?

Key findings

• The large N method yields a non-perturbative critical temperature $ T_c \simeq 0.615663 \times \Lambda_{\overline{\rm MS}} $ for the O(N) scalar theory at finite temperature, derived from a saddle-point condition involving Bessel functions and logarithmic divergences.
• At zero temperature, the method reproduces the non-perturbative saddle point at $ m = \sqrt{e} \Lambda_{\overline{\rm MS}} $, confirming its validity in the non-deconfined phase.
• The finite-temperature effective potential includes both zero-temperature quantum corrections and thermal Bose-Einstein contributions, with the latter dominating at high $ T $, leading to the disappearance of real solutions and signaling a phase transition.
• The critical temperature derived from the 3-loop running coupling in QCD, $ \Lambda_{\overline{\rm MS}} \simeq 288 \, \text{MeV} $, leads to $ T_c \simeq 177 \, \text{MeV} $, in excellent agreement with lattice QCD's $ T_c = 176 \pm 7 \, \text{MeV} $.
• The large N approach successfully circumvents the long-held belief that four-dimensional scalar theories are trivial, suggesting the possibility of a simpler, lower-parameter version of the Standard Model.
• The method enables direct computation of transport coefficients, finite-density properties, and real-time dynamics in strongly coupled systems, which are intractable with standard perturbation theory.