We present a novel semiclassical framework tailored to determine the scaling dimensions of heavy neutral composite operators in conformal field theories (CFTs) which are inaccessible with other current methodologies. It utilizes the state-operator correspondence to map the desired scaling dimensions to the semiclassical energy spectrum of periodic homogeneous field configurations on a cylinder. As concrete applications, we provide detailed analyses for the $\textit{ϕ}$ theory near four dimensions and $\textit{ϕ}$ near three dimensions, semiclassically determining the full spectrum of neutral operators transforming according to different Lorentz representations. Our methodology is presented pedagogically and is readily applicable to a vast class of CFTs.

Croatia

Oleg Antipin

Switzerland

Jahmall Bersini

Denmark

Jacob Hafjall

Giulia Muco

Italy

Francesco Sannino

Semiclassical canovaccio for composite operators

We study the fixed point of the three-dimensional model in a double-scaling limit where both the charge $\textit{Q}$ and the number of fermion flavors $\textit{N}$ become large with a fixed ratio $\textit{q}$ = $\textit{Q}$/(2$\textit{N}$). While a similar analysis has been performed for the bosonic O(N) model, fermionic models pose new challenges. In this work, we systematically explore the spectrum in both the large and small $\textit{q}$ limits beyond the first few orders, and perform a resurgence analysis. Through this approach, we identify the exponential corrections that relate the convergent small-$\textit{q}$ expansion to the asymptotic large-$\textit{q}$ behavior. Our results are suggestive of a geometric interpretation in terms of the worldline of particles moving along the geodesics on the cylinder.

Japan

Simeon Hellerman

Domenico Orlando

Susanne Reffert

Resurgence analysis of the Nambu-Jona-Lasinio model at large charge

We unveil the impact of the $\textit{θ}$-angle on the QCD phase diagram at nonzero isospin and strangeness chemical potentials for three light flavors and different quark mass ratios. We establish the phase boundaries as well as the nature of the associated phase transitions. The order parameters and the physical spectrum in the different phases are determined. We further elucidate the physics around $\textit{θ}$ = $\textit{π}$ where we discover a novel parity-preserving superfluid phase. Finally, we comment on Dashen’s phenomenon in the superfluid phases when varying the number of light flavors.

Alessandra D’Alise

Clelia Gambardella

On the $\textit{θ}$-angle physics of QCD under pressure: the strange and isospin phase diagram

<p>We determine the scaling dimension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="normal">Δ</mi><mi>n</mi></msub></math> for the class of composite operators <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>ϕ</mi><mi>n</mi></msup></math> in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>λ</mi><msup><mi>ϕ</mi><mn>4</mn></msup></math> theory in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>d</mi><mo>=</mo><mn>4</mn><mo>−</mo><mi>ε</mi></math> taking the double scaling limit <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>n</mi><mo stretchy="false">→</mo><mi>∞</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>λ</mi><mo stretchy="false">→</mo><mn>0</mn></math> with fixed <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>λ</mi><mi>n</mi></math> via a semiclassical approach. Our results resum the leading power of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>n</mi></math> at any loop order. In the small <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>λ</mi><mi>n</mi></math> regime we reproduce the known diagrammatic results and predict the infinite series of higher-order terms. For intermediate values of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>λ</mi><mi>n</mi></math> we find that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="normal">Δ</mi><mi>n</mi></msub><mo>/</mo><mi>n</mi></math> increases monotonically approaching a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo stretchy="false">(</mo><mi>λ</mi><mi>n</mi><msup><mo stretchy="false">)</mo><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup></math> behavior in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>λ</mi><mi>n</mi><mo stretchy="false">→</mo><mi>∞</mi></math> limit. We further generalize our results to neutral operators in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>ϕ</mi><mn>4</mn></msup></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>d</mi><mo>=</mo><mn>4</mn><mo>−</mo><mi>ε</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>ϕ</mi><mn>3</mn></msup></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>d</mi><mo>=</mo><mn>6</mn><mo>−</mo><mi>ε</mi></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>ϕ</mi><mn>6</mn></msup></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>d</mi><mo>=</mo><mn>3</mn><mo>−</mo><mi>ε</mi></math> theories with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>O</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo></math> symmetry.</p>

Exact results for scaling dimensions of neutral operators in scalar conformal field theories

We systematically analyze the impact of dilatonic dynamics on Skyrme spheres, crystals, and branes. The effects of the dilatonic model parameters, encompassing different underlying near-conformal dynamics, on the macroscopic properties of skyrmions such as their mass and radius are discussed. For spheres and crystals we identify special values of the ratio of the decay constants for which the second order differential equations reduce to a solvable first order system. Additionally, in the case of the crystals, the dilaton presence spatially separates the baryon and isospin charge distributions. For branes, we show how the dilaton smooths out their configurations. Our results are expected to have wide implications from the study of near-conformal dynamics stemming from QCD-like theories to phenomenological investigations of nuclear matter in extreme regimes.

Belgium

Matías Torres

Dilatonic dynamics of baryonic crystals, branes, and spheres

We provide the leading near conformal corrections on a cylinder to the scaling dimension of the lowest-lying fixed isospin charge <math><mi>Q</mi></math> operators defined at the lower boundary of the quantum chromodynamics conformal window, <math><mrow><msub><mrow><mover><mrow><mi>Δ</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>Q</mi></mrow></msub><mo>=</mo><msubsup><mrow><mover><mrow><mi>Δ</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>Q</mi></mrow><mrow><mo>*</mo></mrow></msubsup><mo>+</mo><msup><mrow><mo>(</mo><mfrac><mrow><msub><mrow><mi>m</mi></mrow><mrow><mi>σ</mi></mrow></msub></mrow><mrow><mn>4</mn><mi>π</mi><mi>ν</mi></mrow></mfrac><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><msup><mrow><mi>Q</mi></mrow><mrow><mfrac><mrow><mi>Δ</mi></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></msup><msub><mrow><mi>B</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msup><mrow><mo>(</mo><mfrac><mrow><msub><mrow><mi>m</mi></mrow><mrow><mi>π</mi></mrow></msub><mo>(</mo><mi>θ</mi><mo>)</mo></mrow><mrow><mn>4</mn><mi>π</mi><mi>ν</mi></mrow></mfrac><mo>)</mo></mrow><mrow><mn>4</mn></mrow></msup><msup><mrow><mi>Q</mi></mrow><mrow><mfrac><mrow><mn>2</mn></mrow><mrow><mn>3</mn></mrow></mfrac><mo>(</mo><mn>1</mn><mo>−</mo><mi>γ</mi><mo>)</mo></mrow></msup><msub><mrow><mi>B</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><mi>O</mi><mrow><mo>(</mo><msubsup><mrow><mi>m</mi></mrow><mrow><mi>σ</mi></mrow><mrow><mn>4</mn></mrow></msubsup><mo>,</mo><msubsup><mrow><mi>m</mi></mrow><mrow><mi>π</mi></mrow><mrow><mn>8</mn></mrow></msubsup><mo>,</mo><msubsup><mrow><mi>m</mi></mrow><mrow><mi>σ</mi></mrow><mrow><mn>2</mn></mrow></msubsup><msubsup><mrow><mi>m</mi></mrow><mrow><mi>π</mi></mrow><mrow><mn>4</mn></mrow></msubsup><mo>)</mo></mrow><mo>.</mo></mrow></math> Here, <math><mrow><msub><mrow><mover><mrow><mi>Δ</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>Q</mi></mrow></msub><mo>/</mo><mi>r</mi></mrow></math> is the classical ground state energy of the theory on <math><mi>R</mi><mo>×</mo><msubsup><mi>S</mi><mi>r</mi><mn>3</mn></msubsup></math> at fixed isospin charge while <math><mrow><msubsup><mrow><mover><mrow><mi>Δ</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>Q</mi></mrow><mrow><mo>*</mo></mrow></msubsup></mrow></math> is the scaling dimension at the leading order in the large charge expansion. In the conformal limit <math><msub><mi>m</mi><mi>σ</mi></msub><mo>=</mo><msub><mi>m</mi><mi>π</mi></msub><mo>=</mo><mn>0</mn></math>, the state-operator correspondence implies <math><mrow><msub><mrow><mover><mrow><mi>Δ</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>Q</mi></mrow></msub><mo>=</mo><msubsup><mrow><mover><mrow><mi>Δ</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>Q</mi></mrow><mrow><mo>*</mo></mrow></msubsup></mrow></math>. The near-conformal corrections are expressed in powers of the dilaton and pion masses in units of the chiral symmetry breaking scale <math><mn>4</mn><mi>π</mi><mi>ν</mi></math> with the <math><mi>θ</mi></math>-angle dependence encoded directly in the pion mass. The characteristic <math><mi>Q</mi></math>-scaling is dictated by the quark mass operator anomalous dimension <math><mi>γ</mi></math> and the one characterizing the dilaton potential <math><mi>Δ</mi></math>. The coefficients <math><msub><mi>B</mi><mi>i</mi></msub></math> with <math><mrow><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></math> depend on the geometry of the cylinder and properties of the nearby conformal field theory.

Scaling results for charged sectors of near conformal QCD

We determine anomalous dimensions of a family of fixed hypercharge operators in the Standard Model featuring the general Cabibbo-Kobayashi-Maskawa structure. The results are obtained at infinite orders in the couplings and to leading and subleading orders in the charge. The computed anomalous dimensions are shown to agree with the maximum known order in perturbation theory. We further show that the large hypercharge sector of the Standard Model is characterised by a non-Abelian vector condensation phase.

South Korea

Pantelis Panopoulos

China

Zhi-Wei Wang

Infinite order results for charged sectors of the Standard Model

We determine the impact of the <math><mi>θ</mi></math> angle and axion physics on the near conformal dynamics of the large-charge baryon sector of <math><mi>S</mi><mi>U</mi><mo>(</mo><mn>2</mn><mo>)</mo></math> gauge theories with <math><msub><mi>N</mi><mi>f</mi></msub></math> fermions in the fundamental representation. We employ an effective approach featuring Goldstone and dilaton degrees of freedom augmented by the topological terms in the theory. We investigate how different dilaton potentials, including the ones for which a systematic counting scheme can be established, affect the results. Via state-operator correspondence we compute the corrections to the would-be conformal dimensions of the lowest large-charge operators as a function of the <math><mi>θ</mi></math> term and dilaton potential.

CERN

Charging the conformal window at nonzero <math><mi>θ</mi></math> angle

Quantum field theories with global symmetries simplify considerably in the large-charge limit allowing us to compute correlators via a semiclassical expansion in the inverse powers of the conserved charges. A generalization of the approach to gauge symmetries has faced the problem of defining gauge-independent observables and, therefore, has not been developed so far. We employ the large-charge expansion to calculate the scaling dimension of the lowest-lying operators carrying <math><mrow><mi>U</mi><mo>(</mo><mn>1</mn><mo>)</mo></mrow></math> charge <math><mi>Q</mi></math> in the critical Abelian Higgs model in <math><mi>D</mi><mo>=</mo><mn>4</mn><mo>−</mo><mi>ε</mi></math> dimensions to leading and next-to-leading orders in the charge and all orders in the <math><mi>ε</mi></math> expansion. Remarkably, the results match our independent diagrammatic computation of the three-loop scaling dimension of the operator <math><msup><mi>ϕ</mi><mi>Q</mi></msup><mo>(</mo><mi>x</mi><mo>)</mo></math> in the Landau gauge. We argue that this matching is a consequence of the equivalence between the gauge-independent dressed two-point function of Dirac type with the gauge-dependent two-point function of <math><msup><mi>ϕ</mi><mi>Q</mi></msup><mo>(</mo><mi>x</mi><mo>)</mo></math> in the Landau gauge. We, therefore, shed new light on the problem of defining gauge-independent exponents which has been controversial in the literature on critical superconductors, as well as lay the foundation for large-charge methods in gauge theories.

JINR

Alexander Bednyakov

Kazakhstan

Andrey Pikelner

Gauge Invariance at Large Charge

We analyze the impact of the θ-angle and axion dynamics for two-color (in fact any Sp(2N )) QCD at nonzero baryon charge and as a function of the number of matter fields on the vacuum properties, the pattern of chiral symmetry breaking as well as the spectrum of the theory. We show that the vacuum acquires a rich structure when the underlying CP violating topological operator is added to the theory. We discover novel phases and analyse the order of their transitions characterizing the dynamics of the odd and even number of flavours. We further determine the critical chemical potential as function of the θ angle separating the normal from the superfluid phase of the theory. Our results will guide numerical simulations and novel tests of the model’s dynamics. The results are also expected to better inform phenomenological applications of the model ranging from composite Higgs physics to strongly interacting massive dark matter models featuring number changing interactions. In the companion work [1] we repurpose and upgrade the approach to determine the impact of the θ-angle and axion physics on non-perturbative near conformal dynamics related to the fixed baryon charge sector.