arXiv subjects

Paul Romatschke

Publications and source records attributed to Paul Romatschke.

At least 19 recordsLinked to original sources

arXiv2026

Variational Solution of Non-Hermitian Quantum Field Theory in Two Dimensions

This work describes a variational technique to solve non-Hermitian quantum systems, in particular those with negative coupling quartic self-interaction. Differences of applying variational methods from standard quantum mechanics to non-Hermitian quantum mechanics -- including potential pitfalls -- are highlighted. A successful implementation of the variational approach for the non-Hermitian case is presented. The technique is used to analyze scalar quantum field theory in d=2 with negative quartic self-interaction in the Hamiltonian formulation in lattice discretization.

hep-th↗

arXiv2026

Testing Scalar Field Self-Dualities in d=2 using a Variational Method

Recently, self-dualities based on saddle-point expansions have been proposed as a means to obtain qualitative non-perturbative information in scalar field theories. In this work, we test this proposition quantitatively by studying the phase transition for critical scalar $ϕ^4$ field theory in 1+1 dimensions using a variational method. We find that saddle-point methods obtain quantitative agreement for the free energy, but differ on the order of 25 percent for the peak location of the correlation length.

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arXiv2026

On self-dualities for scalar $ϕ^4$ theory

Scalar field theory is studied by constructing interacting saddle point expansions in the symmetric and broken phase, respectively. Focusing on analytically tractable saddle expansions, it is found that broken and symmetric phases are related by sign flip of the quartic coupling. Applications to dimensions $d<4$ recover previous results for the phase diagram, whereas $d=4$ is possibly new.

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arXiv2026

Phase Diagram and Finite Temperature Properties of Negative Coupling Scalar Field Theory

In this work, I consider scalar field theory with negative quartic self-interaction, corresponding to an upside-down classical potential. Despite not possessing a classically stable ground state, such potentials are known to behave properly when treated quantum mechanically, leading to stable and unitary time evolution. Using two different saddle-point expansions for the same theory, I discuss the phase diagram in terms of bare parameters in Euclidean dimensions one to four, as well as the generalization to finite temperature. Comparing to other methods where available, I find that negative coupling field theory is a promising candidate for an interacting scalar field theory in the continuum. In particular, in four dimensions it exploits a loophole in mathematical proofs of quantum triviality, suggesting that negative coupling scalar field theory could offer a UV-complete and interacting description of the Higgs.

hep-th↗

arXiv2025

Vanishing Polyakov Loop for QCD with Twelve Massless Quarks

We consider continuum-formulation QCD in four dimensions with twelve massless fundamental quark flavors. Splitting the SU($N$) 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 . The ultraviolet divergencies of the effective action are non-perturbatively renormalized with a $β$-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.

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arXiv2024

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

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 ${\cal PT}$-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.

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arXiv2024

Mass from Nothing

We study the Abelian Higgs model with multiple scalar fields, but without mass terms. Solving the model non-perturbatively 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 $Λ$-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 non-perturbative 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.

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arXiv2024

Out-Of-Time-Ordered-Correlators for the Pure Inverted Quartic Oscillator: Classical Chaos meets Quantum Stability

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 $-2 \langle x^2 \rangle_T \langle p^2 \rangle_T$ 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.

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arXiv2024

An alternative to perturbative renormalization in 3+1 dimensional field theories

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 N scalar field theories. In terms of bare parameters, this non-perturbative 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 n-point functions of the theory are finite. I provide explicit results for alternative renormalization for the O(N) model and QCD with $N_f=12$ flavors in 3+1 dimensions.

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arXiv2024

NNNLO pressure of cold quark matter: leading logarithm

At high baryon chemical potential $μ_B$, the equation of state of QCD allows a weak-coupling expansion in the QCD coupling $α_s$. The result is currently known up to and including the full next-to-next-to-leading order (NNLO) $α_s^2$. Starting at this order, the computations are complicated by the modification of particle propagation in a dense medium, which necessitates non-perturbative treatment of the scale $α_s^{1/2} μ_B$. In this work, we apply a Hard-Thermal-Loop scheme for capturing the contributions of this scale to the weak-coupling expansion, and use it to determine the leading-logarithm contribution to NNNLO: $α_s^3 \ln^2 α_s$. This result is the first improvement to the equation of state of massless cold quark matter in 40 years. The new term is negligibly small, and thus significantly increases our confidence in the applicability of the weak-coupling expansion.

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arXiv2024

Upper Bound on the Speed of Sound in Nuclear Matter from Transport

We point out that there is an upper bound on the speed of sound squared given by $c_s^2 \leq 0.781$ 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.

nucl-th↗

arXiv2024

Quantum Field Theory in Large N Wonderland: Three Lectures

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.

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arXiv2023

Life at the Landau pole

If a quantum field theory has a Landau pole, the theory is usually called 'sick' and dismissed as a candidate for an interacting UV-complete theory. In a recent study on the interacting 4d O(N) model at large N, it was shown that at the Landau pole, observables remain well-defined and finite. In this work, I study both relevant and irrelevant deformations of the said model at the Landau pole, finding that physical observables remain unaffected. Apparently, the Landau pole in this theory is benign. As a phenomenological application, I compare the O(N) model to QCD, by identifying $Λ_{\overline{\rm MS}}$ with the Landau pole in the O(N) model.

hep-th↗

arXiv2023

A loophole in the proofs of asymptotic freedom and quantum triviality

In 1973, Coleman and Gross proved that in four dimensions, only non-abelian gauge theories can have asymptotic freedom. More recently, Aizenman and Duminil-Copin proved that four dimensional scalar field theories are quantum trivial in the continuum. Both of these proofs have a loophole, and it is the same loophole in both proofs: The proofs assume that the scalar self-coupling in the UV is positive definite. While this is a perfectly reasonable and classically very intuitive assumption, it is an assumption nevertheless. In this work, I show that the assumption of coupling positivity is violated in a concrete quantum field theory, the O(N) model, in the large N limit. Surprisingly, despite the classically nonsensical unbounded potential, the negative coupling has no pathological consequence for propagators, the free energy or cross sections. This suggests that interacting scalar field theories with asymptotic freedom in four dimensions are possible, despite long-held opinions to the contrary.

hep-th↗

arXiv2023

Negative Coupling $ϕ^4$ on the Lattice

Triviality of $ϕ^4$ theory in four dimensions can be avoided if the bare coupling constant is negative in the UV. Theories with negative coupling can be put on the lattice if the integration domain for $ϕ(x)$ is contour-deformed from the real to the complex domain. In 0+1d (quantum mechanics), one can recover results from $\mathcal{PT}$-symmetric quantum mechanics in this way. In this work, I report on an attempt to put negative coupling $ϕ^4$ theory in 4 dimensions on the lattice.

hep-lat↗

arXiv2023

What if $ϕ^4$ theory in 4 dimensions is non-trivial in the continuum?

Traditionally, scalar $ϕ^4$ 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.

hep-th↗

arXiv2023

Instantons, analytic continuation, and $\mathcal{PT}$-symmetric field theory

Ordinary Hermitian $λϕ^4$ theory is known to exist in $d<4$ dimensions when $λ>0$. 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 ${\cal PT}$-symmetric field theory. In this work, we critically re-examine the relation between analytically continued Hermitian field theory with quartic interaction, and ${\cal PT}$-symmetric field theory, including $O(N)$ models. We find that in general ${\cal PT}$-symmetric field theory does not correspond to the analytic continuation of the Hermitian theory, except at high temperature where the instanton contribution present in the analytically continued theory can be neglected.

hep-th↗

arXiv2023

A solvable quantum field theory with asymptotic freedom in 3+1 dimensions

Recently, Ai, Bender and Sarkar gave a prescription on how to obtain $\mathcal{PT}$-symmetric field theory results from an analytic continuation of Hermitian field theories. I perform this analytic continuation for the massless (critical) O(N) model with quartic interaction in 3+1 dimensions. In the large N limit, this theory is exactly solvable, and has negative $β$-function in the ultraviolet, and a stable bound state in the infrared. The coupling diverges at a scale $Λ_c$, but can be continued into the far infrared. At finite temperature, the theory exhibits two phases separated by a second-order phase transition near $T_c\simeq Λ_c/\sqrt{e}$.

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