In this work, I consider N-component scalar quantum field theory in two dimensions interacting with an upside-down quartic potential. Working in the large N limit, the model can be solved non-perturbatively using the saddle-point method for sufficiently strong negative coupling. At high temperature, the O(N) model dimensionally reduces to $$ \mathcal{PT} $$ -symmetric quantum mechanics, for which powerful non-perturbative solution methods exist. It is found that the solution from quantum mechanics can be matched by the saddle-point method in quantum field theory when allowing for saddles beyond the principal Riemann sheet. I show that saddle points on non-principal Riemann sheets lead to a fully consistent solution of the 2d negative-coupling O(N) model for all temperatures.

United States

Paul Romatschke

On the negative coupling O(N) model in 2d at high temperature

We consider continuum-formulation QCD in four dimensions with twelve massless fundamental quark flavors. Splitting the SU( <italic>N</italic>) gauge field into background and fluctuation parts, we use well-developed techniques to calculate the one-loop effective action for the theory. We find that for constant self-dual background field-strength tensor the notorious infrared divergences of the effective action cancel between gauge and matter sectors if the number of massless quark flavors is exactly <math altimg='si1.svg'><msub><mrow><mi>N</mi></mrow><mrow><mi>f</mi></mrow></msub><mo linebreak='goodbreak' linebreakstyle='after'>=</mo><mn>4</mn><mi>N</mi></math>. The ultraviolet divergencies of the effective action are non-perturbatively renormalized with a <italic>&#946;</italic>-function that matches the known perturbative result in the high energy limit. The resulting UV- and IR-finite effective action possesses a non-trivial saddle which has lower free energy than the perturbative vacuum, and for which the expectation value of the Polyakov loop vanishes. Inclusion of finite temperature effects points to the presence of a first-order phase transition to the perturbative vacuum with a calculable critical temperature.

Seth Grable

Vanishing Polyakov loop for QCD with twelve massless quarks

We study the Abelian Higgs model with multiple scalar fields, but without mass terms. Solving the model nonperturbatively order-by-order in the number of scalar fields, we find that radiative corrections generate masses for the scalar and gauge boson, without spontaneous symmetry breaking. The mass scales are set by the <math><mi>Λ</mi></math>-parameter of the electroweak running coupling, thereby naturally avoiding the hierarchy problem. No part of our calculation employs a weak-coupling expansion, and we find that the perturbative vacuum is metastable, and hence must decay to the stable nonperturbative vacuum of the theory, which we identify. Although the field content of our Lagrangian is standard, our results predict the existence of two heavy scalar resonances in addition to the Higgs. We believe that these predicted resonances will ultimately allow experimentalists to discriminate between our method and standard solutions of the Higgs model.

Chun-Wei Su

Ryan Weller

Mass generation in an Abelian gauge theory with multiple scalar fields and no tree-level dimensionful couplings

We point out that there is an upper bound on the speed of sound squared given by <math><msubsup><mrow><mi>c</mi></mrow><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo>≤</mo><mn>0.781</mn></math> valid for all known systems described by relativistic transient hydrodynamics where calculations of certain ratios of hydrodynamic transport coefficients can be performed from first principles. Assuming this bound is valid for ultradense matter implies that the maximum mass of isolated (non-rotating) neutron stars cannot be larger than 2.7 solar masses.

Brazil

Mauricio Hippert

Jorge Noronha

Upper bound on the speed of sound in nuclear matter from transport

Out-of-time-ordered-correlators (OTOCs) have been suggested as a means to diagnose chaotic behavior in quantum mechanical systems. Recently, it was found that OTOCs display exponential growth for the inverted quantum harmonic oscillator, mirroring the fact that this system is classically and quantum mechanically unstable. In this work, I study OTOCs for the inverted anharmonic (pure quartic) oscillator in quantum mechanics, finding only oscillatory behavior despite the classically unstable nature of the system. For higher temperature, OTOCs seem to exhibit saturation consistent with a value of –2⟨x 2⟩ T  ⟨p 2⟩ T  at late times. I provide analytic evidence from the spectral zeta-function and the WKB method as well as direct numerical solutions of the Schrödinger equation that the inverted quartic oscillator possesses a real and positive energy eigenspectrum, and normalizable wave-functions.

Out-of-time-ordered-correlators for the pure inverted quartic oscillator: classical chaos meets quantum stability

Perturbative renormalization provides the bedrock of understanding quantum field theories. In this work, I point out an alternative way of renormalizing quantum field theories, which is naturally encountered and well-known for the case of large <math><mi>N</mi></math> scalar field theories. In terms of bare parameters, this nonperturbative alternative renormalization differs qualitatively from its perturbative cousin: in the continuum limit, the bare coupling constant goes to zero instead of infinity, and there is no wave-function counterterm. Despite these differences, the resulting <math><mi>n</mi></math>-point functions of the theory are finite. I provide explicit results for alternative renormalization for the O(N) model and QCD with <math><msub><mi>N</mi><mi>f</mi></msub><mo>=</mo><mn>12</mn></math> flavors in <math><mrow><mn>3</mn><mo>+</mo><mn>1</mn></mrow></math> dimensions.

Alternative to perturbative renormalization in (<math><mrow><mn>3</mn><mo>+</mo><mn>1</mn></mrow></math>)-dimensional field theories

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.

Quantum Field Theory in Large-$N$ Wonderland: Three Lectures

Traditionally, scalar <math><msup><mrow><mi>ϕ</mi></mrow><mrow><mn>4</mn></mrow></msup></math> theory in four dimensions is thought to be quantum trivial in the continuum. This tradition is apparently well grounded both in physics arguments and mathematical proofs. Digging into the proofs one finds that they do not actually cover all physically meaningful situations, in particular the case of multi-component fields and non-polynomial action. In this work, I study multi-component scalar field theories in four dimensions in the continuum and show that they do evade the apparently foregone conclusion of triviality. Instead, one finds a non-trivial interacting theory that has two phases, bound states and non-trivial scattering amplitudes in the limit of many components. This has potentially broad implications, both for the foundations of quantum field theory as well as for the experimentally accessible Higgs sector of the Standard Model.

What if Theory in 4 Dimensions is Non-trivial in the Continuum?

Ordinary Hermitian <math><mi>λ</mi><msup><mi>ϕ</mi><mn>4</mn></msup></math> theory is known to exist in <math><mi>d</mi><mo>&lt;</mo><mn>4</mn></math> dimensions when <math><mi>λ</mi><mo>&gt;</mo><mn>0</mn></math>. For negative values of the coupling, it has been suggested that a physical meaningful definition of the interacting theory can be given in terms of <math><mi>P</mi><mi>T</mi></math>-symmetric field theory. In this work, we critically reexamine the relation between analytically continued Hermitian field theory with quartic interaction and <math><mi>P</mi><mi>T</mi></math>-symmetric field theory, including <math><mi>O</mi><mo>(</mo><mi>N</mi><mo>)</mo></math> models. We find that in general <math><mi>P</mi><mi>T</mi></math>-symmetric field theory does not correspond to the analytic continuation of the Hermitian theory to negative coupling, except at high temperature where the instanton contribution present in the analytically continued theory can be neglected.

Scott Lawrence

Christian Peterson

Instantons, analytic continuation, and <math><mrow><mi>P</mi><mi>T</mi></mrow></math>-symmetric field theory

Thermodynamic transport coefficients can be calculated directly from quantum field theory without requiring analytic continuation to real time. We determine all second-order thermodynamic transport coefficients for the uncharged N-component massless (critical) scalar field theory with quartic interaction in the large N limit, for any value of the coupling. We find that in the large N limit, all thermodynamic transport coefficients for the interacting theory can be expressed analytically in terms of the in-medium mass and sums over modified Bessel functions. We expect our technique to allow a similar determination of all thermodynamic transport coefficients for all theories that are solvable in the large N limit, including certain gauge theories.