Non-perturbative Renormalization Research Articles

Detecting the flavor content of the vacuum using the Dirac operator spectrum

We compute the overlap Dirac spectrum on three gauge ensembles generated using 2+1-flavor domain wall fermions. The three ensembles have different lattice spacings, and two of them have quark masses tuned to the physical point. The spectral density is determined up to λ∼100 MeV with subpercentage statistical uncertainty. We find that the density is close to a constant below λ∼20 MeV as predicted by chiral perturbative theory (χPT) and then increases linearly due to the strange quark mass. By fitting to the next-to-leading order χPT form and using the nonperturbative renormalization using the regularization independent momentum subtraction scheme, the SU(2) (keeping the strange quark mass at the physical point) and SU(3) chiral condensates at MS¯ 2 GeV are determined to be Σ=(265.4(0.5)(4.2) MeV)3 and Σ0=(234.3(0.5)(25.8) MeV)3, respectively. The pion decay constants are also determined to be F=84.1(1.9)(8.0) and F0=58.6(0.5)(10.0) MeV. The systematic errors are carefully estimated including the effects of fitting ranges and the uncertainty of low-energy constant L6. We also show that one can resolve the sea flavor content of the sea quarks and constrain their masses with ∼10%–20% statistical uncertainties using the Dirac spectral density. Published by the American Physical Society 2024

All-microwave Lamb shift engineering for a fixed frequency multi-level superconducting qubit

It is known that the electromagnetic vacuum is responsible for the Lamb shift, which is a crucial phenomenon in quantum electrodynamics (QED). In circuit QED, the readout or bus resonators that are dispersively coupled can result in a significant Lamb shift of the qubit. However, previous approaches or proposals for controlling the Lamb shift in circuit QED demand overheads in circuit designs or non-perturbative renormalization of the system’s eigenbases, which can impose formidable limitations. In this work, we propose and demonstrate an all-microwave method for controlling the Lamb shift of fixed-frequency transmons. We employ the drive-induced longitudinal coupling between the transmon and resonator. By simply using an off-resonant monochromatic drive near the resonator frequency, we can control the net Lamb shift up to ±30 MHz and engineer it to zero with the drive-induced longitudinal coupling without facing the aforementioned challenges. Our work establishes an efficient way of engineering the fundamental effects of the electromagnetic vacuum and provides greater flexibility in non-parametric frequency controls of multilevel systems.

Anomalous higher order Ward identities in tensorial group field theories without closure constraint

Abstract The Ward-Takahashi identities are considered as the generalization of the Noether currents available to quantum field theory and include quantum fluctuation effects. Usually, they take the form of relations between correlation functions, which ultimately correspond to the relation between coupling constants of the theory. For this reason, they play a central role in the construction of renormalized theory, providing strong relations between counter-terms. Since last years, they have been intensively considered in the construction of approximate solutions for nonperturbative renormalization group of tensorial group field theories. The construction of these identities is based on the formal invariance of the partition function under a unitary transformation, and Ward’s identities result from a first-order expansion around the identity. Due to the group structure of the transformation under consideration, it is expected that a first-order expansion is indeed sufficient. We show in this article that this does not seem to be the case for a complex tensor theory model, with a kinetic term involving a Laplacian.

B¯c susceptibilities from fully relativistic lattice QCD

We compute the h¯c (pseudo)scalar, (axial-)vector and (axial-)tensor susceptibilities as a function of u=mc/mh between u=mc/mb and u=0.8 using fully relativistic lattice QCD, employing nonperturbative current renormalization and using the second generation 2+1+1 MILC HISQ gluon field configurations. We include ensembles with a≈0.09, 0.06, 0.045, and 0.033 fm and we are able to reach the physical b-quark on the two finest ensembles. At the physical mh=mb point we find m¯b2χ1+=0.720(34)×10−2, m¯b2χ1−=1.161(54)×10−2, χ0−=2.374(33)×10−2, and χ0+=0.609(14)×10−2. Our results for the (pseudo)scalar, vector, and axial vector are compatible with the expected small size of nonperturbative effects at u=mc/mb. We also give the first nonperturbative determination of the tensor susceptibilities, finding m¯b2χT=0.891(44)×10−2 and m¯b2χAT=0.441(33)×10−2. Our value of m¯b2χAT is in good agreement with the O(αs) perturbation theory, while our result for m¯b2χT is in tension with the O(αs) perturbation theory at the level of 2σ. These results will allow for dispersively bounded parametrizations to be employed using lattice inputs for the full set of h→c semileptonic form factors in future calculations, for heavy-quark masses in the range 1.25×mc≤mh≤mb. Published by the American Physical Society 2024

Low-energy theorem revisited and OPE in massless QCD

We revisit a low-energy theorem (LET) of NSVZ type in SU(N) QCD with Nf massless quarks derived in [1] by implementing it in dimensional regularization. The LET relates n-point correlators in the l.h.s. to n + 1-point correlators with the extra insertion of TrF2 at zero momentum in the r.h.s. We demonstrate that, for 2-point correlators of an operator O in the l.h.s., the LET implies that, in general, the integrated 3-point correlator in the r.h.s. needs in perturbation theory an infinite additive renormalization in addition to the multiplicative one. We relate the above counterterm to a corresponding divergent contact term in a certain coefficient of the OPE of TrF2 with O in the momentum representation, thus extending to any operator O an independent argument that first appeared for O = TrF2 in [2]. Finally, we demonstrate that in the asymptotically free phase of QCD the aforementioned counterterm in the LET is actually finite nonperturbatively after resummation to all perturbative orders. We also briefly recall the implications of the LET in the gauge-invariant framework of dimensional regularization for the perturbative and nonperturbative renormalization in large-N QCD. The implications of the LET inside and above the conformal window of SU(N) QCD with Nf massless quarks will appear in a forthcoming paper.

Renormalization of nonlocal gluon operators on the lattice

Wen study the renormalization of a complete set of gauge-invariant gluon nonlocal operators in lattice perturbation theory. We determine the mixing pattern under renormalization of these operators using symmetry arguments, which extend beyond perturbation theory. Additionally, we derive the renormalization factors of the operators within the modified minimal subtraction (MS¯) scheme up to one loop. To enable a nonperturbative renormalization procedure, we investigate a suitable version of the modified regularization-invariant (RI′) scheme, and we calculate the conversion factors from that scheme to MS¯. The computations are performed by employing both dimensional and lattice regularizations, using the Wilson gluon action. This work is relevant to nonperturbative studies of the gluon parton distribution functions (PDFs) on the lattice. Published by the American Physical Society 2024

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.

Alternative to perturbative renormalization in ( 3+1 )-dimensional field theories

Perturbative renormalization provides the bedrock of understanding quantum field theories. In this work, I point out an alternative way of renormalizing quantum field theories, which is naturally encountered and well-known for the case of large N scalar field theories. In terms of bare parameters, this nonperturbative alternative renormalization differs qualitatively from its perturbative cousin: in the continuum limit, the bare coupling constant goes to zero instead of infinity, and there is no wave-function counterterm. Despite these differences, the resulting n-point functions of the theory are finite. I provide explicit results for alternative renormalization for the O(N) model and QCD with Nf=12 flavors in 3+1 dimensions. Published by the American Physical Society 2024

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.