arXiv subjects

Peter Weisz

Publications and source records attributed to Peter Weisz.

At least 19 recordsLinked to original sources

arXiv2021

Renormalization of the topological charge density in QCD with dimensional regularization

To all orders of perturbation theory, the renormalization of the topological charge density in dimensionally regularized QCD is shown to require no more than an additive renormalization proportional to the divergence of the flavour-singlet axial current. The proof is based on the standard BRS analysis of the QCD vertex functional in the background gauge and exploits the special algebraic properties of the charge density through the Stora--Zumino chain of descent equations.

hep-ph↗

arXiv2018

Finite volume mass gap and free energy of the SU(N)xSU(N) chiral sigma model

We compute the free energy in the presence of a chemical potential coupled to a conserved charge in the effective SU(N)xSU(N) scalar field theory to third order for asymmetric volumes in general d-dimensions, using dimensional regularization. We also compute the mass gap in a finite box with periodic boundary conditions.

hep-th↗

arXiv2018

Casimir squared correction to the standard rotator Hamiltonian for the O($n$) sigma-model in the delta-regime

In a previous paper we found that the isospin susceptibility of the O($n$) sigma-model calculated in the standard rotator approximation differs from the next-to-next to leading order chiral perturbation theory result in terms vanishing like $1/\ell\,,$ for $\ell=L_t/L\to\infty$ and further showed that this deviation could be described by a correction to the rotator spectrum proportional to the square of the quadratic Casimir invariant. Here we confront this expectation with analytic nonperturbative results on the spectrum in 2 dimensions, by Balog and Hegedüs for $n=3,4$ and by Gromov, Kazakov and Vieira for $n=4$. We also consider the case of 3 dimensions.

hep-th↗

arXiv2017

Isospin susceptibility in the O($n$) sigma-model in the delta-regime

We compute the isospin susceptibility in an effective O($n$) scalar field theory (in $d=4$ dimensions), to third order in chiral perturbation theory ($χ$PT) in the delta--regime using the quantum mechanical rotator picture. This is done in the presence of an additional coupling, involving a parameter $η$, describing the effect of a small explicit symmetry breaking term (quark mass). For the chiral limit $η=0$ we demonstrate consistency with our previous $χ$PT computations of the finite-volume mass gap and isospin susceptibility. For the massive case by computing the leading mass effect in the susceptibility using $χ$PT with dimensional regularization, we determine the $χ$PT expansion for $η$ to third order. The behavior of the shape coefficients for long tube geometry obtained here might be of broader interest. The susceptibility calculated from the rotator approximation differs from the $χ$PT result in terms vanishing like $1/\ell$ for $\ell=L_t/L_s\to\infty$. We show that this deviation can be described by a correction to the rotator spectrum proportional to the square of the quadratic Casimir invariant.

hep-lat↗

arXiv2016

Flow equation for the scalar model in the large $N$ expansion and its applications

We study the flow equation of the O($N$) $φ^4$ model in $d$ dimensions at the next-to-leading order (NLO) in the $1/N$ expansion. Using the Schwinger-Dyson equation, we derive 2-pt and 4-pt functions of flowed fields. As the first application of the NLO calculations, we study the running coupling defined from the connected 4-pt function of flowed fields in the $d+1$ dimensional theory. We show in particular that this running coupling has not only the UV fixed point but also an IR fixed point (Wilson-Fisher fixed point) in the 3 dimensional massless scalar theory. As the second application, we calculate the NLO correction to the induced metric in $d+1$ dimensions with $d=3$ in the massless limit. While the induced metric describes a 4-dimensional Euclidean Anti-de-Sitter (AdS) space at the leading order as shown in the previous paper, the NLO corrections make the space asymptotically AdS only in UV and IR limits. Remarkably, while the AdS radius does not receive a NLO correction in the UV limit, the AdS radius decreases at the NLO in the IR limit, which corresponds to the Wilson-Fisher fixed point in the original scalar model in 3 dimensions.

hep-th↗

arXiv2016

Flow equation for the large $N$ scalar model and induced geometries

We study the proposal that a $d+1$ dimensional induced metric is constructed from a $d$ dimensional field theory using gradient flow. Applying the idea to the O($N$) $φ^4$ model and normalizing the flow field, we have shown in the large $N$ limit that the induced metric is finite and universal in the sense that it does not depend on the details of the flow equation and the original field theory except for the renormalized mass, which is the only relevant quantity in this limit. We have found that the induced metric describes Euclidean Anti-de-Sitter (AdS) space in both ultra-violet (UV) and infra-red (IR) limits of the flow direction, where the radius of the AdS is bigger in the IR than in the UV.

hep-th↗

arXiv2016

Massless sunset diagrams in finite asymmetric volumes

This paper discusses the methods and the results used in an accompanying paper describing the matching of effective chiral Lagrangians in dimensional and lattice regularizations. We present methods to compute 2-loop massless sunset diagrams in finite asymmetric volumes in the framework of these regularizations. We also consider 1-loop sums in both regularizations, extending the results of Hasenfratz and Leutwyler for the case of dimensional regularization and we introduce a new method to calculate precisely the expansion coefficients of the 1-loop lattice sums.

hep-lat↗

arXiv2016

Matching effective chiral Lagrangians with dimensional and lattice regularization

We compute the free energy in the presence of a chemical potential coupled to a conserved charge in effective O($n$) scalar field theory (without explicit symmetry breaking terms) to NNL order for asymmetric volumes in general $d$--dimensions, using dimensional (DR) and lattice regularizations. This yields relations between the 4-derivative couplings appearing in the effective actions for the two regularizations, which in turn allows us to translate results, e.g. the mass gap in a finite periodic box in $d=3+1$ dimensions, from one regularization to the other. Consistency is found with a new direct computation of the mass gap using DR. For the case $n=4, d=4$ the model is the low-energy effective theory of QCD with $N_{\rm f}=2$ massless quarks. The results can thus be used to obtain estimates of low energy constants in the effective chiral Lagrangian from measurements of the low energy observables, including the low lying spectrum of $N_{\rm f}=2$ QCD in the $δ$--regime using lattice simulations, as proposed by Peter Hasenfratz, or from the susceptibility corresponding to the chemical potential used.

hep-lat↗

arXiv2015

Walking in the 3-dimensional large $N$ scalar model

The solvability of the three-dimensional O($N$) scalar field theory in the large $N$ limit makes it an ideal toy model exhibiting "walking" behavior, expected in some SU($N$) gauge theories with a large number of fermion flavors. We study the model using lattice regularization and show that when the ratio of the particle mass to an effective 4-point coupling (with dimension mass) is small, the beta function associated to the running 4-point coupling is "walking". We also study lattice artifacts and finite size effects, and find that while the former can be sizable at realistic correlation length, the latter are under control already at lattice sizes a few ($\sim$3) correlation lengths. We show the robustness of the walking phenomenon by showing that it can also be observed by studying physical observables such as the scattering phase shifts and the mass gap in finite volume.

hep-lat↗

arXiv2013

Symanzik effective actions in the large N limit

Symanzik effective actions, conjectured to describe lattice artifacts, are determined for a class of lattice regularizations of the non-linear O(N) sigma model in two dimensions in the leading order of the 1/N-expansion. The class of actions considered includes also ones which do not have the usual classical limit and are not (so far) treatable in the framework of ordinary perturbation theory. The effective actions obtained are shown to reproduce previously computed lattice artifacts of the step scaling functions defined in finite volume, giving further confidence in Symanzik's theory of lattice artifacts.

hep-lat↗

arXiv2012

Toward an understanding of short distance repulsions among baryons in QCD -- NBS wave functions and operator product expansion --

We report on our recent attempts to determine the short distance behaviors of general 2-baryon and 3-baryon forces, which are defined from the Nambu-Bethe-Salpeter(NBS) wave function, by using the operator product expansion and a renormalization group analysis in QCD. We have found that the repulsion at short distance increases as the number of valence quarks increases or when the number of different flavors involved decreases. This global tendency suggests a Pauli suppression principle among quark fields at work.

hep-lat↗

arXiv2011

Short distance repulsion in 3 nucleon forces from perturbative QCD

We investigate the short distance behavior of 3 nucleon forces (3NF) defined through Nambu--Bethe--Salpeter wave functions, using the operator product expansion(OPE) and calculating anomalous dimensions of 9--quark operators in perturbative QCD. As is the case of NN forces previously considered, we show that 3NF have repulsions at short distance at 1--loop, which becomes exact in the short distance limit thanks to the asymptotic freedom of QCD. Moreover these behaviors are universal in the sense that they do not depend on the energy of the NBS wave function for 3 nucleons.

hep-lat↗

arXiv2011

Perturbative analysis of the gradient flow in non-abelian gauge theories

The gradient flow in non-abelian gauge theories on R^4 is defined by a local diffusion equation that evolves the gauge field as a function of the flow time in a gauge-covariant manner. Similarly to the case of the Langevin equation, the correlation functions of the time-dependent field can be expanded in perturbation theory, the Feynman rules being those of a renormalizable field theory on R^4 x [0,oo). For any matter multiplet and to all loop orders, we show that the correlation functions are finite, i.e. do not require additional renormalization, once the theory in four dimensions is renormalized in the usual way. The flow thus maps the gauge field to a one-parameter family of smooth renormalized fields.

hep-th↗

arXiv2010

Triviality of $ϕ^4_4$ theory: small volume expansion and new data

We study a renormalized coupling g and mass m in four dimensional phi^4 theory on tori with finite size z=mL. Precise numerical values close to the continuum limit are reported for z=1,2,4, based on Monte Carlo simulations performed in the equivalent all-order strong coupling reformulation. Ordinary renormalized perturbation theory is found to work marginally at z=2 and and to fail at z=1. By exactly integrating over the constant field mode we set up a renormalized expansion in z and compute three nontrivial orders. These results reasonably agree with the numerical data at small z. In the new expansion, the universal continuum limit exists as expected from multiplicative renormalizability. The triviality scenario is corroborated with significant precision.

hep-lat↗

arXiv2010

Operator product expansion and the short distance behavior of 3-flavor baryon potentials

The short distance behavior of baryon-baryon potentials defined through Nambu-Bethe-Salpeter wave functions is investigated using the operator product expansion. In a previous analysis of the nucleon-nucleon case, corresponding to the SU(3) channels $27_s$ and $\overline{10}_a$, we argued that the potentials have a repulsive core. A new feature occurs for the case of baryons made up of three flavors: manifestly asymptotically attractive potentials appear in the singlet and octet channels. Attraction in the singlet channel was first indicated by quark model considerations, and recently been found in numerical lattice simulations. The latter have however not yet revealed asymptotic attraction in the octet channels; we give a speculative explanation for this apparent discrepancy.

hep-lat↗

arXiv2010

Application of the operator product expansion to the short distance behavior of nuclear potentials

We investigate the short distance behavior of nucleon-nucleon (NN) potentials defined through Bethe-Salpeter wave functions, by perturbatively calculating anomalous dimensions of 6-quark operators in QCD. Thanks to the asymptotic freedom of QCD, 1-loop computations give certain exact results for the potentials in the zero distance limit. In particular the functional form of the S-state central NN potential at short distance $r$ is predicted to be a little weaker than $r^{-2}$. On the other hand, due to the intriguing character of the anomalous dimension spectrum, perturbative considerations alone can not determine whether this potential is repulsive or attractive at short distances. A crude estimation suggests that the force at short distance is repulsive, as found numerically in lattice QCD. A similar behavior is found for the tensor potential.

hep-lat↗

arXiv2009

The repulsive core of the NN potential and the operator product expansion

We investigate the short distance behavior of the nucleon--nucleon (NN) potential defined through the Bethe-Salpeter wave function, by perturbatively calculating anomalous dimensions of 6--quark operators in QCD. Thanks to the asymptotic freedom of QCD, the 1-loop estimations give exact results for the potential in the zero distance limit. We show that the chiral symmetry of the gauge interaction implies the existence of an operator whose anomalous dimension is zero for a given quantum number. Furthermore we find that non-zero anomalous dimensions of other operators are all negative. These results predict the functional form of the NN potential at short distance, which is a little weaker than $r^{-2}$. On the other hand, the computation of the anomalous dimension spectrum alone can not determine whether the potential is repulsive or attractive at short distance. An additional analytic non-perturbative analysis suggests that the force at short distance is indeed repulsive at low energy as found numerically. Some extensions of the method are briefly discussed.

hep-lat↗

arXiv2009

The puzzle of apparent linear lattice artifacts in the 2d non-linear sigma-model and Symanzik's solution

Lattice artifacts in the 2d O(n) non-linear sigma-model are expected to be of the form O(a^2), and hence it was (when first observed) disturbing that some quantities in the O(3) model with various actions show parametrically stronger cutoff dependence, apparently O(a), up to very large correlation lengths. In a previous letter we described the solution to this puzzle. Based on the conventional framework of Symanzik's effective action, we showed that there are logarithmic corrections to the O(a^2) artifacts which are especially large, (ln(a))^3, for n=3 and that such artifacts are consistent with the data. In this paper we supply the technical details of this computation. Results of Monte Carlo simulations using various lattice actions for O(3) and O(4) are also presented.

hep-lat↗