We discuss the ultraviolet fixed point of asymptotically safe dilaton quantum gravity. It differs from the Reuter fixed point by the dependence of the Planck mass on a scalar field. The gauge invariant functional flow equation in the most general approximation with up to two derivatives strengthens the argument for the existence of this fixed point. The quantum effective action obtained from the scaling solution for dilaton quantum gravity can describe inflation for early cosmology and dynamical dark energy for late cosmology.

China

Yadikaer Maitiniyazi

Germany

Christof Wetterich

Japan

Masatoshi Yamada

Scaling solutions for gauge invariant flow equations in dilaton quantum gravity

We investigate composite models of gravity and explore how dynamical tensor fields can emerge within the functional renormalization group framework. We consider two prototype models: a fermionic theory and a scalar theory. In both cases, an auxiliary tensor field is introduced via a Hubbard-Stratonovich transformation, corresponding to the composite channel associated with the energy-momentum tensor. We derive the flow equations for the field renormalization factors of the auxiliary tensor field and demonstrate that finite kinetic terms are dynamically generated in the infrared regime. Agreement with the diffeomorphism-invariant quadratic Einstein-Hilbert structure can be established in the transverse-traceless spin-2 sector, while the remaining contributions reside in the longitudinal and trace sectors. Although these terms can be cast into a form reminiscent of gauge-fixed presentations of the quadratic Einstein-Hilbert action, we do not interpret them as originating from a genuine gauge-fixing procedure within the present truncation.

Emergence of dynamical tensor fields in composite models of gravity

We study topological properties of a quantum mechanical system with <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> symmetry within the functional renormalization group approach. These properties include the vacuum energy structure and the topological susceptibility. Our approach works with a complexification of the flow equation, and specifically, we embed the original symmetry into the complex plane, <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><mo stretchy="false">→</mo><mi mathvariant="double-struck">C</mi></math>. We compute the effective potential of a given topological sector by restricting ourselves to field configurations with given generalized nontrivial Chern-Simons numbers. The full potential is directly constructed from these sector potentials. Our results compare well with the benchmark results obtained from solving the corresponding Schrödinger equation.

Yuepeng Guan

Jan M. Pawlowski

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>θ</mi></math>-vacuum from functional renormalization

We investigate the QCD chiral phase transition at finite temperature and finite baryon density using the functional Renormalization Group (fRG). While conventional fRG studies often employ techniques such as dynamical bosonization to regularize divergences, we instead pursue the weak solution of the fRG equations which allows for non-analytic behavior in the flow to compute the pure fermionic potential $$V_k(\psi ,\bar{\psi })$$ within the local potential approximation. This approach enables us to explore the effects of purely quark-level fluctuations on dynamical chiral symmetry breaking without introducing any auxiliary bosonic fields. Based on this framework, we present the resulting chiral phase diagram as a function of temperature and baryon chemical potential.

QCD chiral phase diagram from weak functional renormalization group

We study the essential renormalization group equation, in which inessential couplings are removed via field redefinitions, for Einstein gravity coupled to a massive scalar field in the presence of a cosmological constant. Our results indicate that perturbatively nonrenormalizable terms can be eliminated due to the cosmological term, in contrast to the case of perturbation around flat spacetime. We find a nontrivial fixed point for the Newton coupling and the cosmological term.

Nobuyoshi Ohta

Essential renormalization group equation for gravity coupled to a scalar field

There exist several distinct phenomenological models to generate neutrino masses. We explore which of these models can consistently be embedded in a quantum theory of gravity and matter. We proceed by invoking a minimal number of degrees of freedom beyond the Standard Model. Thus, we first investigate whether the Weinberg operator, a dimension-five operator that generates neutrino masses without requiring degrees of freedom beyond the Standard Model, can arise in asymptotically safe quantum gravity. We find a negative answer with far-reaching consequences: new degrees of freedom beyond gravity and the Standard Model are necessary to give neutrinos a mass in the asymptotic-safety paradigm. Second, we explore whether the type-I seesaw mechanism is viable and discover an upper bound on the seesaw scale. The bound depends on the mass of the visible neutrino. We find a numerical value of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msup><mn>10</mn><mn>14</mn></msup><mtext> </mtext><mtext> </mtext><mi>GeV</mi></mrow></math> for this bound when neglecting neutrino mixing for a visible mass of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msup><mn>10</mn><mrow><mo>−</mo><mn>10</mn></mrow></msup><mtext> </mtext><mtext> </mtext><mi>GeV</mi></mrow></math>. Conversely, for the most “natural” value of the seesaw scale in a quantum-gravity setting, which is the Planck scale, we predict an upper bound for the neutrino mass of the visible neutrino of approximately <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msup><mn>10</mn><mrow><mo>−</mo><mn>15</mn></mrow></msup><mtext> </mtext><mtext> </mtext><mi>GeV</mi></mrow></math>. Third, we explore whether neutrinos could also be pseudo-Dirac neutrinos in asymptotic safety and find that this possibility can be accommodated.

Brazil

Gustavo P. de Brito

Astrid Eichhorn

Antonio D. Pereira

Neutrino mass generation in asymptotically safe gravity

In this paper, we generalize the deformations driven by the stress-energy tensor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>T</mi></math> and investigate their relation to the flow equation for the background metric at the classical level. For a deformation operator <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="script">O</mi></math> as a polynomial function of the stress-energy tensor, we develop a formalism that relates a deformed action to a flow equation for the metric in arbitrary spacetime dimensions. It is shown that in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>T</mi><mover accent="true"><mi>T</mi><mo stretchy="false">¯</mo></mover></math> deformation and the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="script">O</mi><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mo>=</mo><mi>tr</mi><mo stretchy="false">[</mo><mi mathvariant="bold">T</mi><msup><mo stretchy="false">]</mo><mi>m</mi></msup></math> deformation, the flow equations for the metric allow us to directly obtain exact solutions in closed forms. We also demonstrate the perturbative approach to find the same results. As several applications of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="script">O</mi><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mo>=</mo><mi>tr</mi><mo stretchy="false">[</mo><mi mathvariant="bold">T</mi><msup><mo stretchy="false">]</mo><mi>m</mi></msup></math> deformation, we discuss the relation between the deformations and gravitational models. Besides, we also deform the Lagrangians for scalar field theories.

Xi-Yang Ran

Feng Hao

Geometric realization via irrelevant deformations induced by the stress-energy tensor

We perform a functional renormalization group analysis of a four-fermion model with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>C</mi><mi>P</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>P</mi></math> violation to explore the nonperturbative infrared dynamics of quantum chromodynamics (QCD) within the Wilsonian renormalization group framework, particularly in the context of spontaneous <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>C</mi><mi>P</mi></math>-violation models. Our analysis of the fixed-point structure reveals that, in the large-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>N</mi><mi>c</mi></msub></math> limit, the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>C</mi><mi>P</mi></math>-violating <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mover accent="true"><mi>θ</mi><mo stretchy="false">¯</mo></mover></math> parameter is dynamically induced and approaches <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>π</mi><mo>·</mo><mo stretchy="false">(</mo><msub><mi>N</mi><mi>f</mi></msub><mo>/</mo><mn>2</mn><mo stretchy="false">)</mo></math> (where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>N</mi><mi>f</mi></msub></math> is the number of flavors) as the system enters the chirally broken phase. This behavior arises due to criticality and the large anomalous dimensions of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo><mi>A</mi></msub></math>-violating four-fermion couplings. Furthermore, this trend appears to persist beyond the leading large-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>N</mi><mi>c</mi></msub></math> approximation, provided that the infrared dynamics of QCD remains dominated by the scalar condensate of the quark bilinear, as expected. Notably, our findings highlight that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>C</mi><mi>P</mi></math>-violating four-fermion interactions, which are perturbatively irrelevant, can become relevant in the chirally broken phase through nonperturbative effects, with potential implications for spontaneous <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>C</mi><mi>P</mi></math>-violation scenarios.

Linlin Huang

Mamiya Kawaguchi

Shinya Matsuzaki

Akio Tomiya

Functional renormalization group study of a four-fermion model with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>C</mi><mi>P</mi></math> violation

It is known that the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>U</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></math> Wess-Zumino-Witten model is dual to the free fermion theory in two dimensions via non-Abelian bosonization. While it is decomposed into the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi><mi>U</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></math> Wess-Zumino-Witten model and a free compact boson, the former is believed to be equivalent to the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>O</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></math> nonlinear sigma model with the theta term at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>θ</mi><mo>=</mo><mi>π</mi></math>. In this work, we reexamine this duality through the lens of nonperturbative renormalization group (RG) flow. We analyze the RG flow structure of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>O</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></math> nonlinear sigma model with the theta term in two dimensions using the functional renormalization group. Our results reveal a nontrivial fixed point with a nonzero value of the topological coupling. The scaling dimensions (critical exponents) at this fixed point suggest the realization of a duality between the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>O</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></math> nonlinear sigma model with the theta term and the free fermion theory, indicating that these models belong to the same universality class.

Junichi Haruna

Keito Shimizu

Functional renormalization group analysis of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>O</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></math> nonlinear sigma model and non-Abelian bosonization duality

We investigate a gravitational model based on local Lorentz invariance and general coordinate invariance. The model incorporates classical scale invariance, which forbids dimensionful parameters, and the irreversible vierbein postulate, which enables continuous degenerate limits of the vierbein, both at a specific scale. Through the dynamics of the system, we demonstrate the simultaneous emergence of the Planck mass and a curved spacetime background.