Using functional integral methods, we investigate the non-Abelian Casimir energy in the Curci-Ferrari model, which offers an effective description of the infrared regime of Yang-Mills theory. We consider a 3+1D (resp. 2+1D) system of two infinite parallel plates (resp. wires) at a fixed distance from each other, with either perfect magnetic conductor (PMC) or perfect electric conductor (PEC) boundary conditions. Imposing the boundary conditions directly in the functional integral by the introduction of suitable auxiliary fields that act as Lagrange multipliers, we obtain a boundary effective action that captures the dynamics of this system. The Casimir energy is then computed both directly from the functional integral and via the energy-momentum tensor, providing equivalent results. We find that the Casimir energy for PEC and PMC conditions differs by a constant factor, which can be traced back to a van Dam-Veltman-Zakharov-like discontinuity (both in 3+1D and 2+1D). Lastly, we show that our analytical results are compatible with a variety of recent numerical lattice simulations of the non-perturbative Yang-Mills Casimir energy, in which a novel non-perturbative mass scale emerges.

Belgium

David Dudal

Brazil

Philipe Fabritiis

Sebbe Stouten

Non-Abelian Casimir energy in the Curci-Ferrari model through a functional approach

We discuss the existence of Landau-pole-free renormalization group trajectories in the Minkowskian version of the Curci-Ferrari model as a function of a running parameter <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>q</mi><mn>2</mn></msup></math> associated to the four-vector <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>q</mi></math> at which renormalization conditions are imposed, and which can take both spacelike (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>q</mi><mn>2</mn></msup><mo>&lt;</mo><mn>0</mn></math>) and timelike (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>&gt;</mo><mn>0</mn></mrow></math>) values. We discuss two possible extensions of the infrared-safe scheme defined in [An infrared safe perturbative approach to Yang-Mills correlators, <a href="http://dx.doi.org/10.1103/PhysRevD.84.045018">Phys. Rev. D 84, 045018 (2011)</a>] for the Euclidean version of the model, which coincide with the latter in the spacelike region upon identifying <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>Q</mi><mn>2</mn></msup><mo>≡</mo><mo>−</mo><msup><mi>q</mi><mn>2</mn></msup></math> with the square of the renormalization scale in that reference. The first extension uses real-valued renormalization factors and leads to a flow in the timelike region with a similar structure as the flow in the spacelike region (or in the Euclidean model), including a nontrivial fixed point and a family of trajectories bounded at all scales by the value of the coupling at this fixed point. Interestingly, the fixed point in the timelike region has a much smaller value of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>λ</mi><mo>≡</mo><msup><mi>g</mi><mn>2</mn></msup><mi>N</mi><mo>/</mo><mn>16</mn><msup><mi>π</mi><mn>2</mn></msup></math> than the corresponding one in the spacelike region, a value closer to the perturbative boundary <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>λ</mi><mo>=</mo><mn>1</mn></math>. However, in this real-valued infrared-safe scheme, the flow cannot connect the timelike and spacelike regions. Thus, it is not possible to deduce the relevant timelike flow trajectory from the sole knowledge of a spacelike flow trajectory. To try to cure this problem, we investigate a second extension of the Euclidean IR-safe scheme, which allows for complex-valued renormalization factors. We discuss under which conditions these schemes can make sense and study their ability to connect spacelike and timelike flow trajectories. In particular, we investigate onto which types of timelike trajectories the perturbative spacelike trajectories are mapped.

Uruguay

France

Santiago Oribe

Marcela Peláez

Urko Reinosa

Infrared-safe Minkowskian Curci-Ferrari model

We investigate the heavy-quark corner of the Columbia plot using the gluon potential derived from the Curci-Ferrari extension of the Faddeev-Popov gauge fixing in the center-symmetric Landau gauge, as a proxy for the Polyakov loop potential. In line with the observation that Landau gauge couplings are not that large in the case of heavy-quark QCD, we consider a one-loop approximation and test its consistency by using various renormalization schemes while investigating the dependence of our results on the renormalization scale. We find that the main qualitative features of the phase structure in the heavy-quark regime are reproduced. Notably, we have direct access to real chemical potentials, and have tested that extrapolations from imaginary chemical potentials approximate the direct calculation well until large chemical potentials. Our results agree quantitatively with those of simulations to 10% accuracy, which is the expected precision of the one-loop calculations in applications of the Curci-Ferrari model to Yang-Mills or heavy-quark QCD theories.

V. Tomas Mari Surkau

Heavy-quark corner of the Columbia plot from the center-symmetric Curci-Ferrari model

We renormalize quantum chromodynamics (QCD) when gauge fixed in the nonlinear Curci-Ferrari gauge to four loops in the modified minimal subtraction (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mover accent="true"><mi>MS</mi><mo stretchy="true">¯</mo></mover></math>) scheme. We reproduce the four loop QCD <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mover accent="true"><mi>MS</mi><mo stretchy="true">¯</mo></mover></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>β</mi></math>-function from the Slavnov-Taylor identity for this gauge which relates the coupling constant renormalization to the gluon, Faddeev-Popov ghost and gauge parameter anomalous dimensions. This is carried out for a nonzero gauge parameter, without having to evaluate a vertex function. The anomalous dimension of the BRST invariant dimension two gluon and ghost mass term is deduced from a similar Slavnov-Taylor identity for this gauge. Consequently we construct the renormalization group functions in the minimal momentum subtraction scheme to four loops. As a corollary we deduce the five loop <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>β</mi></math>-function and quark mass anomalous dimensions in the same scheme. We also outline the pros and cons of employing the Curci-Ferrari gauge to access the six loop QCD <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>β</mi></math>-function in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mover accent="true"><mi>MS</mi><mo stretchy="true">¯</mo></mover></math> scheme.

United Kingdom

J.A. Gracey

Four loop renormalization of QCD in the Curci-Ferrari gauge

We analyze the confinement/deconfinement transition of pure Yang-Mills theories within the framework of the center-symmetric Landau gauge supplemented by a Curci-Ferrari mass term that models the effect of the associated Gribov copies in the infrared. In addition to providing details for earlier one-loop calculations in that framework, we explore how the results depend on the renormalization scale and/or on the renormalization scheme. We find that the predicted values for the transition temperatures of SU(2) and SU(3) Yang-Mills theories are similar in both schemes and are little sensitive to the renormalization scale <math><mi>μ</mi></math> over a wide range of values including the standard range <math><mi>μ</mi><mo>∈</mo><mo>[</mo><mi>π</mi><mi>T</mi><mo>,</mo><mn>4</mn><mi>π</mi><mi>T</mi><mo>]</mo></math>. These values are also close both to those obtained from a minimal sensitivity principle and to those of lattice simulations, especially in the SU(3) case. These results further confirm the good behavior of perturbative calculations within the Curci-Ferrari model and support the adequacy of the latter as an effective description of Yang-Mills theories in the infrared. We perform a similar analysis for the spinodal temperatures in the SU(3) case and for the Polyakov loop, the order parameter associated with the breaking of center symmetry.

Victor Tomas Mari Surkau

Deconfinement transition within the Curci-Ferrari model: Renormalization scale and scheme dependences

We compute the four-gluon vertex from the Curci-Ferrari model at one-loop order for a collinear configuration. Our results display a good agreement with the first lattice data for this vertex, released very recently [Phys. Rev. D 109, 074502 (2024)]. A noteworthy novelty of our work is that we can provide analytical expressions for the four-gluon vertex in collinear configurations, together with a renormalization scheme that allows us to perform reliable perturbative computations even in the infrared regime. We observe an infrared suppression in the form factor associated with the tree-level four-gluon tensor with a possible zero crossing in the deep infrared which demands new lattice investigations to be confirmed. Moreover, we report an infrared divergence in the completely symmetric tensor form factor due to the ghost-loop contributions. These results come as predictions, since previous two-point correlations fix all the available parameters of the model, up to an overall constant factor.

Nahuel Barrios

Philipe De Fabritiis

Four-gluon vertex from the Curci-Ferrari model at one-loop order

In this work the connection established in [7,8] between a model of two linked polymers rings with fixed Gaussian linking number forming a 4-plat and the statistical mechanics of non-relativistic anyon particles is explored. The excluded volume interactions have been switched off and only the interactions of entropic origin arising from the topological constraints are considered. An interpretation from the polymer point of view of the field equations that minimize the energy of the model in the limit in which one of the spatial dimensions of the 4-plat becomes very large is provided. It is shown that the self-dual contributions are responsible for the long-range interactions that are necessary for preserving the global topological properties of the system during the thermal fluctuations. The non self-dual part is also related to the topological constraints, and takes into account the local interactions acting on the monomers in order to prevent the breaking of the polymer lines. It turns out that the energy landscape of the two linked rings is quite complex. Assuming as a rough approximation that the monomer densities of half of the 4-plat are constant, at least two points of energy minimum are found. Classes of non-trivial self-dual solutions of the self-dual field equations are derived. One of these classes is characterized by densities of monomers that are the squared modulus of holomorphic functions. The second class is obtained under some assumptions that allow to reduce the self-dual equations to an analog of the Gouy-Chapman equation for the charge distribution of ions in a double layer capacitor. In the present case, the spatial distribution of the electric potential of the ions is replaced by the spatial distribution of the fictitious magnetic fields associated with the presence of the topological constraints. In the limit in which two of the spatial dimensions are large in comparison with the third one, we provide exact formulas for the conformations of the monomer densities of the 4-plat by using the elliptic, hyperbolic and trigonometric solutions of the sinh-Gordon and cosh-Gordon equations which have been used for instance in the construction of classical string solutions in AdS3 and dS3 [9].

Poland

Neda Abbasi Taklimi

Franco Ferrari

Marcin R. Pia̧tek

Self–dual solutions of a field theory model of two linked rings

We evaluate the three-gluon vertex with one vanishing external momentum within the Curci-Ferrari (CF) model at two-loop order and compare our results to Landau-gauge lattice simulations of the same vertex function for the SU(2) and SU(3) gauge groups in four dimensions. This extends previous works [J. A. Gracey et al., Phys. Rev. D 100, 034023 (2019); N. Barrios et al., Phys. Rev. D 102, 114016 (2020)] that considered similarly the two-loop ghost and gluon two-point functions as well as the two-loop ghost-antighost-gluon vertex (with vanishing gluon momentum). With the parameters of the model being adjusted by fitting the two-point functions to available lattice data, our evaluation of the three-gluon vertex arises as a pure prediction. We find that two-loop corrections systematically improve the agreement between the model and the lattice data as compared to earlier one-loop calculations, with a better agreement in the SU(3) case as already seen in previous studies. We also analyze the renormalization scheme dependence of our calculation. In all cases, this dependence diminishes when two-loop corrections are included, which is consistent with the perturbative CF paradigm. In addition, we study the low momentum regime of the three-gluon vertex in relation with the possibility of zero crossing. Within the CF model, we show that the leading infrared behavior of the exact vertex is given by the same linear logarithm that arises at one-loop order, multiplied by the all orders cubic ghost dressing function at zero momentum (we provide similar exact results for other vertex functions). We argue that this property remains true within the Faddeev-Popov framework under the assumption that the resummed gluon propagator features a decoupling behavior. This shows that the zero crossing is a property of the exact three-gluon vertex function. Within the CF model, we find however that the scale of the zero crossing is considerably reduced when going from one- to two-loop order. This seems consistent with some recent lattice simulations [G. T. R. Catumba et al., EPJ Web Conf. 258, 02008 (2022)]. Finally, our analysis also allows us to support recent claims about the dominance of the tree-level tensor component [F. Pinto-Gómez et al., arXiv:2208.01020].

Two-loop three-gluon vertex from the Curci-Ferrari model and its leading infrared behavior to all loop orders

Exploiting the theoretical potential of the modified Bonora-Tonin superfield approach (MBTSA) as well as the (anti-)chiral superfield approach (ACSA) to Becchi-Rouet-Stora-Tyutin (BRST) formalism, we derive the complete set of off-shell nilpotent (anti-)BRST symmetry transformations corresponding to the classical two- <math id="M1"><mfenced><mrow><mn>1</mn><mo>+</mo><mn>1</mn></mrow></mfenced></math> dimensional (2D) diffeomorphism symmetry transformations on the worldsheet (that is traced out by the motion of a model of bosonic string). Only the BRST symmetry transformations for this model have been discussed in the earlier literature. We derive the (anti-)BRST invariant Curci-Ferrari- (CF-) type restrictions (using MBTSA) which turn out to be the root cause behind the absolute anticommutativity of the above (anti-)BRST symmetry transformations. We capture the symmetry invariance of the (anti-)BRST invariant Lagrangian densities within the ambit of ACSA. The derivation of the proper anti-BRST transformations (corresponding to the already-known BRST transformations) and the (anti-)BRST invariant CF-type restrictions are the novel results in our present endeavor.

India

R.P. Malik

A. Tripathi

A.K. Rao

Superfield Approaches to a Model of Bosonic String: Curci-Ferrari-Type Restrictions

In this article we study the unquenched three-gluon and ghost-gluon vertex in the entire momentum range from the ultraviolet to infrared regime using the Curci-Ferrari model at one loop in Landau gauge, as an extension of the results presented in by Pelaez et al. [Phys. Rev. D 88, 125003 (2013)]. Results are compared with recent lattice data for SU(3) in the unquenched case. This calculation is a pure prediction of the model because it does not require fixing any parameter once two-point functions are fitted. An analysis of the influence of dynamical quarks at the position of the zero crossing of the three-gluon vertex is presented. Due to the recent improvements in infrared lattice data for the quenched three-gluon correlation function [Phys. Lett. B 818, 136352 (2021)], we also redo the comparison between our one-loop results in this limit and the lattice results, obtaining a very good match.