We gain insight on the fixed point dynamics of $\textit{d}$ dimensional quantum field theories by exploiting the critical behavior of the $$d-\epsilon $$ sister theories. To this end we first derive a self-consistent relation between the $$d-\epsilon $$ scaling exponents and the associated $\textit{d}$ dimensional beta functions. We then demonstrate that to account for an interacting fixed point in the original theory the related $$d-\epsilon $$ scaling exponent must be multi-valued in $$\epsilon .$$ We elucidate our findings by discussing several examples such as the QCD Banks–Zaks infrared fixed point, QCD at large number of flavors, as well as the O(N) model in four dimensions. For the latter, we show that although the 1/$\textit{N}$ corrections prevent the reconstruction of the renormalization group flow, this is possible when adding the $$1/N^2$$ contributions.

Croatia

Oleg Antipin

Belgium

Alan Pinoy

Italy

Denmark

Francesco Sannino

Shahram Vatani

Consistency relation for fixed point dynamics

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.

Switzerland

Jahmall Bersini

Jacob Hafjall

Giulia Muco

Semiclassical canovaccio for composite operators

<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>

Japan

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

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

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 extend the fixed-charge semiclassical method by computing anomalous dimensions of fixed-charge scalar operators in models with Yukawa interactions. In particular, we discuss the Nambu-Jona-Lasinio-Yukawa theory as well as an asymptotically safe gauge-Yukawa model in four dimensions. In the weakly coupled regime, we cross-check our results to the respective maximum known orders of perturbation theory in these models and predict higher order terms for future comparisons with other computational methods. In the strongly coupled regime, we match our results to the predictions of the large-charge effective field theory which can be compared to future Monte Carlo and lattice studies.

Yukawa interactions at large charge

We investigate the analytic properties of the fixed charge expansion for a number of conformal field theories in different space-time dimensions. The models investigated here are O(N) and QED 3. We show that in d = 3 − ϵ dimensions the contribution to the O(N) fixed charge Q conformal dimensions obtained in the double scaling limit of large charge and vanishing ϵ is non-Borel summable, doubly factorial divergent, and with order   <math> <msqrt> <mi>Q</mi> </msqrt> </math>  $$ \sqrt{Q} $$  optimal truncation order. By using resurgence techniques we show that the singularities in the Borel plane are related to worldline instantons that were discovered in the other double scaling limit of large Q and N of ref. [1]. In d = 4 − ϵ dimensions the story changes since in the same large Q and small E regime the next order corrections to the scaling dimensions lead to a convergent series. The resummed series displays a new branch cut singularity which is relevant for the stability of the O(N) large charge sector for negative ϵ. Although the QED 3 model shares the same large charge behaviour of the O(N) model, we discover that at leading order in the large number of matter field expansion the large charge scaling dimensions are Borel summable, single factorial divergent, and with order Q optimal truncation order.

CERN

Matías Torres

The analytic structure of the fixed charge expansion

The Weak Gravity Conjecture has recently been re-formulated in terms of a particle with non-negative self-binding energy. Because of the dual conformal field theory (CFT) formulation in the anti-de Sitter space, the conformal dimension ∆(Q) of the lowest-dimension operator with charge Q under some global U(1) symmetry must be a convex function of Q. This property has been conjectured to hold for any (unitary) conformal field theory and generalized to larger global symmetry groups. Here we refine and further test the convex charge conjecture via semiclassical computations for fixed charge sectors of different theories in various dimensions. We analyze the convexity properties of the leading and next-to-leading order terms stemming from the semiclassical computation, de facto, extending previous tests beyond the leading perturbative contributions and to arbitrary charges. In particular, the leading contribution is sufficient to test convexity in the semiclassical computations. We also consider intriguing cases in which the models feature a transition from real to complex conformal dimensions either as a function of the charge or number of matter fields. As a relevant example of the first kind, we investigate the O(N) model in 4 + ϵ dimensions. As an example of the second type, we consider the U(N) × U(M) model in 4 − ϵ dimensions. Both models display a rich dynamics where, by changing the number of matter fields and/or charge, one can achieve dramatically different physical regimes. We discover that whenever a complex conformal dimension appears, the real part satisfies the convexity property.

Sweden

Chen Zhang

More on the weak gravity conjecture via convexity of charged operators

We compute the scaling dimensions of a family of fixed-charge operators at the infrared fixed point of the <math><mi>O</mi><mo>(</mo><mi>N</mi><mo>)</mo></math> model featuring cubic interactions in <math><mi>d</mi><mo>=</mo><mn>6</mn><mo>−</mo><mi>ε</mi></math> for arbitrary <math><mi>N</mi></math> to leading and subleading order in the charge but to all orders in the couplings. The results are used to analyze the conjectured equivalence with the <math><mi>O</mi><mo>(</mo><mi>N</mi><mo>)</mo></math> model displaying quartic interactions at its ultraviolet fixed point. This is performed by comparing the cubic model scaling dimensions against the known large <math><mi>N</mi></math> results for the quartic model and demonstrating that they match. Our results reinforce the conjectured equivalence and further provide novel information on the finite <math><mi>N</mi></math> physics stemming from the computations in the cubic model just below six dimensions.

More on the cubic versus quartic interaction equivalence in the <math><mi>O</mi><mo>(</mo><mi>N</mi><mo>)</mo></math> model

We go beyond a systematic review of the semiclassical approaches for determining the scaling dimensions of fixed-charge operators in <math><mi>U</mi><mo>(</mo><mn>1</mn><mo>)</mo></math> and <math><mi>O</mi><mo>(</mo><mi>N</mi><mo>)</mo></math> models by introducing a general strategy apt at determining the relation between a given charge configuration and the associated operators for more involved symmetry groups such as the <math><mi>U</mi><mo>(</mo><mi>N</mi><mo>)</mo><mo>×</mo><mi>U</mi><mo>(</mo><mi>M</mi><mo>)</mo></math>. We show how, varying the charge configuration, it is possible to access anomalous dimensions of different operators transforming according to a variety of irreducible representations of the non-Abelian symmetry group without the aid of diagrammatical computations. We illustrate our computational strategy by determining the anomalous dimensions of several composite operators to the next-to-leading order in the semiclassical expansion for the <math><mi>U</mi><mo>(</mo><mi>N</mi><mo>)</mo><mo>×</mo><mi>U</mi><mo>(</mo><mi>M</mi><mo>)</mo></math> conformal field theory (CFT) in <math><mn>4</mn><mo>−</mo><mi>ε</mi></math> dimensions. Thanks to the powerful interplay between semiclassical methods and group theory we can, for the first time, extract scaling dimensions for a wide range of operators.