We renormalize the Gross-Neveu-Yukawa model with an <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 to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="script">O</mi><mo stretchy="false">(</mo><msup><mi>ε</mi><mn>5</mn></msup><mo stretchy="false">)</mo></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> dimensions and determine the anomalous dimensions of the fermion and scalar fields, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>β</mi></math> functions as well as the scalar field’s mass operator. These are used to construct several <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>N</mi></math>-dependent critical exponents relevant for quantum transitions in semimetals and in particular those connected with graphene in three dimensions when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>N</mi><mo>=</mo><mn>2</mn></math>. Improved exponent estimates for scalar fermion transitions on a honeycomb lattice, when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>N</mi><mo>=</mo><mn>1</mn></math>, as well as for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>N</mi><mo>=</mo><mn>5</mn></math> are also given to compare with results from other techniques such as the conformal bootstrap.

United Kingdom

J.A. Gracey

Spain

A. Maier

Germany

P. Marquard

Chile

Y. Schröder

Anomalous dimensions and critical exponents for the Gross-Neveu-Yukawa model at five loops

We compute template formulae of all four-loop $\textit{β}$-functions and anomalous dimensions of arbitrary renormalisable quantum field theories with fermions and scalar fields in the $$ \overline{\textrm{MS}} $$ scheme. Using these results, novel contributions to the $\textit{β}$-functions of the Higgs and top coupling are extracted. We discuss how four loops present a challenge to dimensional continuation from four to three dimensions and propose a solution.

Tom Steudtner

Four loop renormalisation group equations in general Gross-Neveu-Yukawa theories

We study the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>O</mi><mo stretchy="false">(</mo><mn>2</mn><mi>N</mi><mo stretchy="false">)</mo></mrow></math> symmetric Gross-Neveu model at finite density in the presence of a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></math> chemical potential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>h</mi></math> for a generic number <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>a</mi><mo>≤</mo><mi>N</mi><mo>−</mo><mn>2</mn></mrow></math> of fermion fields. By combining perturbative quantum field theory, semiclassical large <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>N</mi></math>, and Bethe ansatz techniques, we show that at finite <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>N</mi></math> two new dynamically generated scales <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="normal">Λ</mi><mi mathvariant="normal">n</mi></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="normal">Λ</mi><mi mathvariant="normal">c</mi></msub></math> appear in the theory, governing the mass gap of neutral and charged fermions, respectively. Above a certain threshold value for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>h</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>a</mi></math>-fermion bound states condense and form an inhomogeneous configuration, which at infinite <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>N</mi></math> is a crystal spontaneously breaking translations. At large <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>h</mi></math>, this crystal has mean <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="normal">Λ</mi><mi mathvariant="normal">n</mi></msub></math> and spatial oscillations of amplitude <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="normal">Λ</mi><mi mathvariant="normal">c</mi></msub></math>. The two scales also control the nonperturbative corrections to the free energy, resolving a puzzle concerning fractional-power renormalons and predicting new ones.

Italy

Francesco Benini

Ohad Mamroud

Tomás Reis

Marco Serone

Condensates, Crystals, and Renormalons in the Gross-Neveu Model at Finite Density

A novel Bogomol’nyi–Prasad–Sommerfield (BPS) bound for the Gross–Pitaevskii equations in two spatial dimensions is presented herein. The energy can be bounded from below in terms of the combination of two boundary terms, one related to the vorticity (but “dressed” by the condensate profile) and the second to the “skewness” of the configurations. The bound is saturated by configurations that satisfy a system of two first-order partial differential equations. When such a BPS system is satisfied, the Gross–Pitaevskii equations are also satisfied. The analytic solutions of this BPS system in the present manuscript represent configurations with fractional vorticity living in an annulus. Using these techniques, we present the first analytic examples of this kind. The hydrodynamic interpretation of the BPS system is discussed, and the implications of these results are outlined.

Fabrizio Canfora

Czechia

Pablo Pais

A Bogomol’nyi–Prasad–Sommerfield bound with a first-order system in the 2$\textit{D}$ Gross–Pitaevskii equation

The presence of an infinite number of marginal four-fermion operators is a key characteristic of the two-dimensional Gross-Neveu model. In this study, we investigate the structure of UV divergences in this model, and by symmetry argument we found that the renormalizability only requires a subset of evanescent operators. We perform a 7-loop renormalization computation of beta function for the corresponding evanescent operator, and confirm its non-trivial contribution to UV divergences. By integrating infrared rearrangement, dimensional shifting, and large momentum expansion techniques, we systematically reduce the two-dimensional tensor integrals in the four-fermion correlation functions into four-dimensional scalar integrals. These scalar integrals are subsequently evaluated using the graphical function method, which marks the first application of the method to models with fermionic fields. Our result represents the first time that beta functions have been computed analytically beyond 5-loop in a model with spinning particles.

China

Rijun Huang

Qingjun Jin

Yi Li

On the seven-loop renormalization of Gross-Neveu model

We compute the Gross-Llewellyn Smith sum rule, i.e., the lowest odd moment of the parity-violating structure function, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>F</mi><mn>3</mn></msub></math>, of the nucleon from a lattice QCD calculation of the Compton amplitude. Our calculations are performed on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mn>48</mn><mn>3</mn></msup><mo>×</mo><mn>96</mn></math> lattices at the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi><mi>U</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></math> symmetric point for two lattice spacings. We extract the moments for several values of the current momenta in the range <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>0.5</mn><mo>≲</mo><msup><mi>Q</mi><mn>2</mn></msup><mo>≲</mo><mn>10</mn><mtext> </mtext><mtext> </mtext><msup><mi>GeV</mi><mn>2</mn></msup></math>, covering both the nonperturbative and perturbative regimes. We compare our moments to the Gross-Llewellyn Smith sum rule and discuss the implications for higher-twist effects, a determination of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>α</mi><mi>s</mi></msub><mo stretchy="false">(</mo><msup><mi>Q</mi><mn>2</mn></msup><mo stretchy="false">)</mo></math> from a hadronic quantity complementing the phenomenological and other lattice approaches, and electroweak box contributions crucial for Cabibbo–Kobayashi–Maskawa matrix unitarity studies.

Australia

K.U. Can

J.A. Crawford

R. Horsley

P.E.L. Rakow

T.G. Schar

G. Schierholz

H. Stüben

R.D. Young

J.M. Zanotti

Gross-Llewellyn Smith sum rule from lattice QCD

The (generalized) Gross-Pitaevskii equation (GPE) for a complex scalar field in two spatial dimensions is analyzed. It is shown that there is an infinite family of self-interaction potentials which admit Bogomol'nyi-Prasad-Sommerfield (BPS) bounds together with the corresponding first-order BPS systems. For each member of this family, the solutions of the first-order BPS systems are automatically solutions of the corresponding second-order generalized GPE. The simplest topologically non-trivial solutions of these first-order BPS systems describe configurations with quantized fractional vorticity. The corresponding fraction is related to the degree of non-linearity. The case in which the self-interaction potential is of order six (namely <math altimg='si1.svg'><mo stretchy='false'>|</mo><mi mathvariant='normal'>&#936;</mi><msup><mrow><mo stretchy='false'>|</mo></mrow><mrow><mn>6</mn></mrow></msup></math>, which is a relevant theory both in relativistic quantum field theories in <math altimg='si2.svg'><mo stretchy='false'>(</mo><mn>2</mn><mo linebreak='badbreak' linebreakstyle='after'>+</mo><mn>1</mn><mo stretchy='false'>)</mo></math> dimensions in connection with the quantum Hall effect as well as in the theory of the supersolids) is analyzed in detail. Such formalism can also be extended to the case of quantum mixtures with multi-component GPEs. The relationship between these techniques and supersymmetry will be discussed. In particular, despite several common features, we will show that there are multi-component GPEs that are not supersymmetric (at least, not in the standard sense) and possess a BPS system of the above type.

Fractional vorticity, Bogomol'nyi-Prasad-Sommerfield systems and complex structures for the (generalized) spinor Gross-Pitaevskii equations

A hybrid of the critical three-dimensional Gross-Neveu and Thirring models deformed by explicit parity-breaking operators is studied in the large-<math><mi>N</mi></math> expansion and using the renormalization group. The regime of coupling constants where the theory is stable is identified and criteria for the occurrence of fixed points in that regime are found. For a certain range of Chern-Simons level, we find stable charge-gapped phase with spontaneously broken approximate scale invariance and a parametrically light dilaton. The Chern-Simons level can be tuned to the stability edge, resulting in exact scale invariance which is spontaneously broken, accompanied by a massless dilaton. For another, narrow range of Chern-Simons levels, we find a conformal window where the theory flows to a Wilson-Fisher-like fixed point and is a novel (and rare) example of a nonsupersymmetric nontrivial parity- and time-reversal-violating three-dimensional conformal field theory with scalar, spinor, and vector fields.

Canada

Gordon W. Semenoff

Riley A. Stewart

Parity-violating marginal deformation of the 3D Gross-Neveu-Thirring model

Two out of three phase boundaries of the <math><mrow><mn>1</mn><mo>+</mo><mn>1</mn></mrow></math>-dimensional Gross-Neveu model in the chiral limit can be obtained from a standard, perturbative stability analysis of the homogeneous phases. The third one separating the massive homogeneous phase from the kink crystal is nonperturbative and could so far only be inferred from the full solution of the model. We show that this phase boundary can also be obtained via a modified stability analysis, based on the thermodynamic potential of a single kink or baryon. The same method works for the massive Gross-Neveu model, so that all phase boundaries of the Gross-Neveu model could have been predicted quantitatively without prior knowledge of the full crystal solution.

Michael Thies

Nonperturbative phase boundaries in the Gross-Neveu model from a stability analysis

The phase diagram of the Gross–Neveu model in 1+1 dimensions is studied using Ginzburg–Landau expansion. It predicts several features of the exact phase diagram correctly even at low orders. It is shown that increasing the order of the expansion improves the accuracy of the crystal phase except for very small temperatures, where the expansion completely fails regardless of the order of the expansion. The source of this behaviour seems to be related to the Silver-Blaze phenomenon.

India

Lalita Choudhary

Anees Ahmed

Ginzburg–Landau approach to the Gross–Neveu model: success and failure