Non-perturbative Renormalization Research Articles

Extended state space for describing renormalized Fock spaces in QFT

We revisit the non-perturbative renormalization of a class of simple polaron models with resting fermions. The considered dispersion relations and form factors are allowed to be highly singular, such that infinite self-energies and wave function renormalizations may occur and the diagonalizing dressing transformations might not be implementable on Fock space. Instead of taking cutoffs, we rigorously interpret the self-energies and wave function renormalizations as elements of suitable vector spaces, as well as a field extension of the complex numbers. Moreover, we define two extended state vector spaces (ESSs) over this field extension, which contain a dense subspace of Fock space, but also incorporate non-square-integrable wave functions. Elements of these ESSs can be seen as states of virtual particles, described in a mathematically rigorous way. The Hamiltonian without cutoffs, formally infinite counterterms, and the dressing transformation can then be defined as linear operators between certain subspaces of the two Fock space extensions. This way, we obtain a renormalized Hamiltonian which can be realized as a densely defined self-adjoint operator on Fock space.

Perturbative versus non-perturbative renormalization

Approximated functional renormalization group (FRG) equations lead to regulator-dependent β-functions, in analogy to the scheme-dependence of the perturbative renormalization group (pRG) approach. A scheme transformation redefines the couplings to relate the β-functions of the FRG method with an arbitrary regulator function to the pRG ones obtained in a given scheme. Here, we consider a periodic sine-Gordon scalar field theory in d = 2 dimensions and show that the relation of the FRG and pRG approaches is intricate. Although both the FRG and the pRG methods are known to be sufficient to obtain the critical frequency βc2=8π of the model independently of the choice of the regulator and the renormalization scheme, we show that one has to go beyond the standard pRG method (e.g. using an auxiliary mass term) or the Coulomb-gas representation in order to obtain the β-function of the wave function renormalization. This aspect makes the scheme transformation non-trivial. Comparing flow equations of the two-dimensional sine-Gordon theory without any scheme-transformation, i.e. redefinition of couplings, we find that the auxiliary mass pRG β-functions of the minimal subtraction scheme can be recovered within the FRG approach with the choice of the power-law regulator with b = 2, which therefore constitutes a preferred choice for the comparison of FRG and pRG flows.

Non-perturbative renormalisation and improvement of non-singlet tensor currents in Nf = 3 QCD

Hadronic matrix elements involving tensor currents play an important rôle in decays that allow to probe the consistency of the Standard Model via precision lattice QCD calculations. The non-singlet tensor current is a scale-dependent (anomalous) quantity. We fully resolve its renormalisation group (RG) running in the continuum by carrying out a recursive finite-size scaling technique. In this way ambiguities due to a perturbative RG running and matching to lattice data at low energies are eliminated. We provide the total renormalisation factor at a hadronic scale of 233 MeV, which converts the bare current into its RG-invariant form.Our calculation features three flavours of O(a) improved Wilson fermions and tree-level Symanzik-improved gauge action. We employ the (massless) Schrödinger functional renormalisation scheme throughout and present the first non-perturbative determination of the Symanzik counterterm cT derived from an axial Ward identity. We elaborate on various details of our calculations, including two different renormalisation conditions.

Effects of Two Quantum Correction Parameters on Chaotic Dynamics of Particles near Renormalized Group Improved Schwarzschild Black Holes

A renormalized group improved Schwarzschild black hole spacetime contains two quantum correction parameters. One parameter γ represents the identification of cutoff of the distance scale, and another parameter Ω stems from nonperturbative renormalization group theory. The two parameters are constrained by the data from the shadow of M87* central black hole. The dynamics of electrically charged test particles around the black hole are integrable. However, when the black hole is immersed in an external asymptotically uniform magnetic field, the dynamics are not integrable and may allow for the occurrence of chaos. Employing an explicit symplectic integrator, we survey the contributions of the two parameters to the chaotic dynamical behavior. It is found that a small change of the parameter γ constrained by the shadow of M87* black hole has an almost negligible effect on the dynamical transition of particles from order to chaos. However, a small decrease in the parameter Ω leads to an enhancement in the strength of chaos from the global phase space structure. A theoretical interpretation is given to the different contributions. The term with the parameter Ω dominates the term with the parameter γ, even if the two parameters have same values. In particular, the parameter Ω acts as a repulsive force, and its decrease means a weakening of the repulsive force or equivalently enhancing the attractive force from the black hole. On the other hand, there is a positive Lyapunov exponent that is universally given by the surface gravity of the black hole when Ω≥0 is small and the external magnetic field vanishes. In this case, the horizon would influence chaotic behavior in the motion of charged particles around the black hole surrounded by the external magnetic field. This point can explain why a smaller value of the renormalization group parameter would much easily induce chaos than a larger value.

Conformal invariance and composite operators: A strategy for improving the derivative expansion of the nonperturbative renormalization group.

It is expected that conformal symmetry is an emergent property of many systems at their critical point. This imposes strong constraints on the critical behavior of a given system. Taking them into account in theoretical approaches can lead to a better understanding of the critical physics or improve approximation schemes. However, within the framework of the nonperturbative or functional renormalization group and, in particular, of one of its most used approximation schemes, the derivative expansion (DE), nontrivial constraints apply only from third order [usually denoted O(∂^{4})], at least in the usual formulation of the DE that includes correlation functions involving only the order parameter. In this work we implement conformal constraints on a generalized DE including composite operators and show that new constraints already appear at second order of the DE [or O(∂^{2})]. We show how these constraints can be used to fix nonphysical regulator parameters.

Gluon moment and parton distribution function of the pion from Nf=2+1+1 lattice QCD

We present the first calculation of the pion gluon moment from lattice QCD in the continuum-physical limit. The calculation is done using clover fermions for the valence action with three pion masses, 220, 310 and 690 MeV, and three lattice spacings, 0.09, 0.12, and 0.15 fm, using ensembles generated by MILC Collaboration with 2+1+1 flavors of highly improved staggered quarks (HISQ). On the lattice, we nonperturbatively renormalize the gluon operator in RI/MOM scheme using the cluster-decomposition error reduction (CDER) technique to enhance the signal-to-noise ratio of the renormalization constant. We extrapolate the pion gluon moment to the continuum-physical limit and obtain ⟨x⟩g=0.394(58)stat+NPR(39)mixing in the MS¯ scheme at 2 GeV, with first error being the statistical error and uncertainties in nonperturbative renormalization, and the second being a systematic uncertainty estimating the effect of ignoring quark mixing. Our pion gluon momentum fraction has a central value lower than two recent single-ensemble lattice-QCD results near physical pion mass but is consistent with the recent global fits by JAM and xFitter and with most QCD-model estimates. Published by the American Physical Society 2024

Original and modified non-perturbative renormalization group equations of the BMW scheme at the arbitrary order of truncation

We consider the non-perturbative renormalization group (RG) equations, obtained as approximations of the exact Wetterich RG flow equation within the Blaizot–Mendez–Wschebor (BMW) truncation scheme. For the first time, we derive explicit RG flow equations for the scalar model at the arbitrary order of truncation. Moreover, we consider original, as well as modified, approximations, used to obtain a set of closed equations. We compare these equations at the s = 2 order of truncation with those recently derived in J. Phys. A: Math. Theor. 53, 415002 (2020) within a new truncation scheme and find a striking similarity. Namely, the first-order equations of the latter scheme, those of the original BMW scheme, and those of the modified BMW scheme (at s = 2) differ only in one term. We solved these equations by a recently proposed and tested method of semi-analytic approximations. Thus, the critical exponents η, ν, and ω were evaluated, recovering also the known results of the original BMW scheme. In addition, we estimated the subleading correction-to-scaling exponent ω2 for the three equations considered. To the best of our knowledge, this exponent has not yet been extracted from the Wetterich equation beyond the local potential (the zeroth order) approximation. Our current estimate for the 3D Ising model is ω2 = 2.02 (40), where the error bars include the expected truncation error in the BMW scheme.

Q-state Potts model from the nonperturbative renormalization group.

We study the q-state Potts model for q and the space dimension d arbitrary real numbers using the derivative expansion of the nonperturbative renormalization group at its leading order, the local potential approximation (LPA and LPA^{'}). We determine the curve q_{c}(d) separating the first [q>q_{c}(d)] and second [q<q_{c}(d)] -order phase transition regions for 2.8<d≤4. At small ε=4-d and δ=q-2 the calculation is performed in a double expansion in these parameters, and we find q_{c}(d)=2+aε^{2} with a≃0.1. For finite values of ε and δ, we obtain this curve by integrating the LPA and LPA^{'} flow equations. We find that q_{c}(d=3)=2.11(7), which confirms that the transition is of first order in d=3 for the three-state Potts model.

Nonperturbative renormalization of asymmetric staple-shaped operators in twisted mass lattice QCD

Nonperturbative renormalization of asymmetric staple-shaped operators in twisted mass lattice QCD

Resolving Nonperturbative Renormalization of a Microwave-Dressed Weakly Anharmonic Superconducting Qubit Coupled to a Single Quantized Mode.

Microwave driving is a ubiquitous technique for superconducting qubits, but the dressed states description based on the conventionally used perturbation theory cannot fully capture the dynamics in the strong driving limit. Comprehensive studies beyond these approximations applicable to transmon-based circuit quantum electrodynamics (QED) systems are unfortunately rare, as the relevant works have been mainly limited to single-mode or two-state systems. In this work, we investigate a microwave-dressed transmon coupled to a single quantized mode over a wide range of driving parameters. We reveal that the interaction between the transmon and resonator as well as the properties of each mode is significantly renormalized in the strong driving limit. Unlike previous theoretical works, we establish a nonrecursive and non-Floquet theory beyond the perturbative regimes, which excellently quantifies the experiments. This work expands our fundamental understanding of dressed cavity QED-like systems beyond the conventional approximations. Our work will also contribute to fast quantum gate implementation, qubit parameter engineering, and fundamental studies on driven nonlinear systems.

Nonperturbative renormalization with interpolating momentum schemes

Nonperturbative renormalization with interpolating momentum schemes

Tan's Two-Body Contact in a Planar Bose Gas: Experiment versus Theory.

We determine the two-body contact in a planar Bose gas confined by a transverse harmonic potential, using the nonperturbative functional renormalization group. We use the three-dimensional thermodynamic definition of the contact where the latter is related to the derivation of the pressure of the quasi-two-dimensional system with respect to the three-dimensional scattering length of the bosons. Without any free parameter, we find a remarkable agreement with the experimental data of Zou etal. [Tan's two-body contact across the superfluid transition of a planar Bose gas, Nat. Commun. 12, 760 (2021).NCAOBW2041-172310.1038/s41467-020-20647-6] from low to high temperatures, including the vicinity of the Berezinskii-Kosterlitz-Thouless transition. We also show that the short-distance behavior of the pair distribution function and the high-momentum behavior of the momentum distribution are determined by two contacts: the three-dimensional contact for length scales smaller than the characteristic length ℓ_{z}=sqrt[ℏ/mω_{z}] of the harmonic potential and, for length scales larger than ℓ_{z}, an effective two-dimensional contact, related to the three-dimensional one by a geometric factor depending on ℓ_{z}.

D -meson semileptonic decays to pseudoscalars from four-flavor lattice QCD

We present lattice-QCD calculations of the hadronic form factors for the semileptonic decays $D\to\pi\ell\nu$, $D\to K\ell\nu$, and $D_s\to K\ell\nu$. Our calculation uses the highly improved staggered quark (HISQ) action for all valence and sea quarks and includes $N_f=2+1+1$ MILC ensembles with lattice spacings ranging from $a\approx0.12$ fm down to $0.042$ fm. At most lattice spacings, an ensemble with physical-mass light quarks is included. The HISQ action allows all the quarks to be treated with the same relativistic light-quark action, allowing for nonperturbative renormalization using partial conservation of the vector current. We combine our results with experimental measurements of the differential decay rates to determine $|V_{cd}|^{D\to\pi}=0.2238(11)^{\rm Expt}(15)^{\rm QCD}(04)^{\rm EW}(02)^{\rm SIB}[22]^{\rm QED}$ and $|V_{cs}|^{D\to K}=0.9589(23)^{\rm Expt}(40)^{\rm QCD}(15)^{\rm EW}(05)^{\rm SIB}[95]^{\rm QED}$ This result for $|V_{cd}|$ is the most precise to date, with a lattice-QCD error that is, for the first time for the semileptonic extraction, at the same level as the experimental error. Using recent measurements from BES III, we also give the first-ever determination of $|V_{cd}|^{D_s\to K}=0.258(15)^{\rm Expt}(01)^{\rm QCD}[03]^{\rm QED}$ from $D_s\to K \ell\nu$. Our results also furnish new Standard Model calculations of the lepton flavor universality ratios $R^{D\to\pi}=0.98671(17)^{\rm QCD}[500]^{\rm QED}$, $R^{D\to K}=0.97606(16)^{\rm QCD}[500]^{\rm QED}$, and $R^{D_s\to K}=0.98099(10)^{\rm QCD}[500]^{\rm QED}$, which are consistent within $2\sigma$ with experimental measurements. Our extractions of $|V_{cd}|$ and $|V_{cs}|$, when combined with a value for $|V_{cb}|$, provide the most precise test of second-row CKM unitarity, finding agreement with unitarity at the level of one standard deviation.

Functional renormalization group for multilinear disordered Langevin dynamics II:Revisiting the p = 2 spin dynamics for Wigner and Wishart ensembles

In this paper, we investigate the large-time behavior for a slightly modified version of the standard p = 2 soft spins dynamics model, including a quartic or higher potential. The equilibrium states of such a model correspond to an effective field theory, which has been recently considered as a novel paradigm for signal detection in data science based on the renormalization group argument. We consider a Langevin-like equation, including a disorder term that leaves in the Wigner or Wishart ensemble. Then we construct a nonperturbative renormalization group formalism valid in the large N limit, where eigenvalues distributions for the disorder can be replaced by their analytic limits, namely the Wigner and Marchenko-Pastur laws. One of the main advantages of this approach is that the interactions remain local in time, avoiding the non-locality arising from the approaches that integrate out the disorder at the partition function level.

Erratum: Nonperturbative renormalization in lattice QCD with three flavors of clover fermions: Using periodic and open boundary conditions [Phys. Rev. D 103 , 094511 (2021)

Erratum: Nonperturbative renormalization in lattice QCD with three flavors of clover fermions: Using periodic and open boundary conditions [Phys. Rev. D <b>103</b> , 094511 (2021)

Nonperturbative renormalization of the supercurrent in N=1 supersymmetric Yang-Mills theory

In this work, we study the nonperturbative renormalization of the supercurrent operator in $\mathcal{N}=1$ supersymmetric Yang-Mills theory, using a gauge-invariant renormalization scheme (GIRS). The proposed prescription addresses successfully the unwanted mixing of the supercurrent with other operators of equal or lower dimension, which respect the same global symmetries. This mixing is introduced by the unavoidable breaking of supersymmetry on the lattice. In GIRS all gauge-noninvariant operators, which mix with the supercurrent, are excluded from the renormalization procedure. The one remaining mixing operator is accessible by numerical simulations. We present results for the renormalization of the supercurrent using GIRS. We also compute at one-loop order the conversion matrix which relates the nonperturbative renormalization factors in GIRS to the reference scheme $\overline{\mathrm{MS}}$.

Low-temperature behavior of the O(N) models below two dimensions.

We investigate the critical behavior and the nature of the low-temperature phase of the O(N) models treating the number of field components N and the dimension d as continuous variables with a focus on the d≤2 and N≤2 quadrant of the (d,N) plane. We precisely chart a region of the (d,N) plane where the low-temperature phase is characterized by an algebraic correlation function decay similar to that of the Kosterlitz-Thouless phase but with a temperature-independent anomalous dimension η. We revisit the Cardy-Hamber analysis leading to a prediction concerning the nonanalytic behavior of the O(N) models' critical exponents and emphasize the previously not broadly appreciated consequences of this approach in d<2. In particular, we discuss how this framework leads to destabilization of the long-range order in favor of the quasi-long-range order in systems with d<2 and N<2. Subsequently, within a scheme of the nonperturbative renormalization group we identify the low-temperature fixed points controlling the quasi-long-range ordered phase and demonstrate a collision between the critical and the low-temperature fixed points upon approaching the lower critical dimension. We evaluate the critical exponents η(d,N) and ν^{-1}(d,N) and demonstrate a very good agreement between the predictions of the Cardy-Hamber type analysis and the nonperturbative renormalization group in d<2.

Improved renormalization scheme for nonlocal operators

In this paper we present an improved regularization-independent (RI)-type prescription appropriate for the nonperturbative renormalization of gauge-invariant nonlocal operators. In this prescription, the nonperturbative vertex function is improved by subtracting unwanted finite lattice spacing ($a$) effects, calculated in lattice perturbation theory. The method is versatile and can be applied to a wide range of fermion and gluon actions, as well as types of nonlocal operators. The presence of operator mixing can also be accommodated. Compared to the standard ${\mathrm{RI}}^{\ensuremath{'}}$ prescription, this variant can be recast as a supplementary finite renormalization, whose coefficients bring about corrections of higher order in $a$; consequently, it coincides with standard ${\mathrm{RI}}^{\ensuremath{'}}$ as $a\ensuremath{\rightarrow}0$---however, it can afford us a smoother and more controlled extrapolation to the continuum limit. In this proof-of-concept calculation we focus on nonlocal fermion bilinear operators containing a straight Wilson line. In the numerical implementation we use Wilson clover fermions and Iwasaki improved gluons. The finite-$a$ terms were calculated to one-loop level in lattice perturbation theory, and to all orders in $a$, using the same action as the nonperturbative vertex functions. We find that the method leads to significant improvement in the perturbative region indicated by small and intermediate values of the length of the Wilson line. This results in a robust extraction of the renormalization functions in that region. We have also applied the above method to operators with stout-smeared links. We show how to perform the perturbative correction for any number of smearing iterations and evaluate its effect on the power divergent renormalization coefficients.

Z_{4}-symmetric perturbations to the XY model from functional renormalization.

We employ the second order of the derivative expansion of the nonperturbative renormalization group to study cubic (Z_{4}-symmetric) perturbations to the classical XY model in dimensionality d∈[2,4]. In d=3 we provide accurate estimates of the eigenvalue y_{4} corresponding to the leading irrelevant perturbation and follow the evolution of the physical picture upon reducing spatial dimensionality from d=3 towards d=2, where we approximately recover the onset of the Kosterlitz-Thouless physics. We analyze the interplay between the leading irrelevant eigenvalues related to O(2)-symmetric and Z_{4}-symmetric perturbations and their approximate collapse for d→2. We compare and discuss different implementations of the derivative expansion in cases involving one and two invariants of the corresponding symmetry group.

Configuration space for quantum gravity in a locally regularized path integral

We discuss some aspects of the metric configuration space in quantum gravity in the background field formalism. We give a necessary and sufficient condition for the parametrization of Euclidean metric fluctuations such that (i) the signature of the metric is preserved in all configurations that enter the gravitational path integral, and (ii) the parametrization provides a bijective map between full Euclidean metrics and metric fluctuations about a fixed background. For the case of foliatable manifolds, we show how to parametrize fluctuations in order to preserve foliatability of all configurations. Moreover, we show explicitly that preserving the signature on the configuration space for the Lorentzian quantum gravitational path integral is most conveniently achieved by inequality constraints. We discuss the implementation of these inequality constraints in a nonperturbative renormalization group setup.