Renormalization Constants Research Articles

On the electron self-energy to three loops in QED

We compute the electron self-energy in Quantum Electrodynamics to three loops in terms of iterated integrals over kernels of elliptic type. We make use of the differential equations method, augmented by an ϵ-factorized basis, which allows us to gain full control over the differential forms appearing in the iterated integrals to all orders in the dimensional regulator. We obtain compact analytic expressions, for which we provide generalized series expansion representations that allow us to evaluate the result numerically for all values of the electron momentum squared. As a by-product, we also obtain ϵ-resummed results for the self-energy in the on-shell limit p2 = m2, which we use to recompute the known three-loop renormalization constants in the on-shell scheme.

Three-loop renormalization of the quantum action for a four-dimensional scalar model with quartic interaction with the usage of the background field method and a cutoff regularization

The paper studies the quantum action for the four-dimensional real ϕ4-theory in the case of a general formulation using the background field method. The three-loop renormalization is performed with the usage of a cutoff regularization in the coordinate representation. The absence of non-local singular contributions and the correctness of the renormalization R-operation on the example of separate three-loop diagrams are also discussed. The explicit forms of the first three coefficients for the renormalization constants and for the β-function are presented. Consistency with previously known results is shown.

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

Hamiltonian truncation crafted for UV-divergent QFTs

We develop the theory of Hamiltonian Truncation (HT) to systematically study RG flows that require the renormalization of coupling constants. This is a necessary step towards making HT a fully general method for QFT calculations. We apply this theory to a number of QFTs defined as relevant deformations of d=1+1 CFTs. We investigated three examples of increasing complexity: The deformed Ising, Tricritical-Ising, and non-unitary minimal model M(3,7). The first two examples provide a crosscheck of our methodologies against well established characteristics of these theories. The M(3,7) CFT deformed by its \mathbb{Z}_2ℤ2-even operators shows an intricate phase diagram that we clarify. At a boundary of this phase diagram we show that this theory flows, in the IR, to the M(3,5) CFT.

Renormalization of net steered coherence and average coherence in the XXZ model with Dzyaloshinskii–Moriya interaction

The critical behaviors of the net steered coherence (NSC) and average coherence (AC) in the XXZ model with Dzyaloshinskii–Moriya interaction are investigated by adopting the quantum renormalization group method. It is found that they both develop different saturated values in different phases of the model after enough iterations of the renormalization of the coupling constants. In addition, their susceptibilities at the phase transition point are divergent in the thermodynamic limit, and the scaling exponent is determined by the exponent governing the divergence of the correlation length close to the phase transition point. This exemplifies that both the NSC and AC can serve as reliable indicators of quantum phase transitions.

Multiplicative Renormalization of Stochastic Differential Equations for the Abelian Sandpile Model

The long-term, large-scale behavior in a problem of stochastic nonlinear dynamics corresponding to the Abelian sandpile model is studied with the use of the quantum-field theory renormalization group approach. We prove the multiplicative renormalization of the model including an infinite number of coupling parameters, calculate an infinite number of renormalization constants, identify a plane of fixed points in the infinite dimensional space of coupling parameters, discuss their stability and critical scaling in the model, and formulate a simple law relating the asymptotic size of an avalanche to a model exponent quantifying the time-scale separation between the slow energy injection and fast avalanche relaxation processes.

Three-loop matching of heavy flavor-changing (axial-)tensor currents

We present the three-loop calculations of the nonrelativistic QCD (NRQCD) current renormalization constants and corresponding anomalous dimensions, and the matching coefficients for the spatial-temporal tensor and spatial-spatial axial-tensor currents with two different heavy quark masses. We obtain the convergent decay constant ratio up to the next-to-next-to-next-to-leading order (N3LO) for the S-wave vector meson Bc∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {B}_c^{\\ast } $$\\end{document} involving the tensor and axial-tensor currents. We obtain the three-loop finite (ϵ0) term in the ratio of the QCD heavy flavor-changing tensor current renormalization constant in the on-shell (OS) scheme to that in the modified-minimal-subtraction (MS¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\overline{\ extrm{MS}} $$\\end{document}) scheme, which is helpful to obtain the three-loop matching coefficients for all heavy flavor-changing (axial-)tensor currents.

Higher logarithms and ε-poles for the MS-like renormalization prescriptions

We consider a version of dimensional regularization (reduction) in which the dimensionful regularization parameter Λ is in general different from the renormalization scale μ. Then in the scheme analogous to the minimal subtraction the renormalization constants contain ε-poles, powers of ln Λ/μ, and mixed terms of the structure ε−q lnp Λ/μ. For the MS-like schemes we present explicit expressions for the coefficients at all these structures which relate them to the coefficients in the renormalization group functions, namely in the β-function and in the anomalous dimension. In particular, for the pure ε-poles we present explicit solutions of the ’t Hooft pole equations. Also we construct simple all-loop expressions for the renormalization constants (also written in terms of the renormalization group functions) which produce all ε-poles and logarithms and establish a number of relations between various coefficients at ε-poles and logarithms. The results are illustrated by some examples.

One-loop beta-functions of quartic enhanced tensor field theories

Enhanced tensor field theories (eTFTs) have dominant graphs that differ from the melonic diagrams of conventional tensor field theories. They therefore describe pertinent candidates to escape the so-called branched polymer phase, the universal geometry found for tensor models. For generic order d of the tensor field, we compute the perturbative β-functions at one-loop of two just-renormalizable quartic eTFT coined by + or ×, depending on their vertex weights. The models + has two quartic coupling constants , and two 2-point couplings (mass, Za ). Meanwhile, the model × has two quartic coupling constants and three 2-point couplings (mass, Za , ). At all orders, both models have a constant wave function renormalization: Z = 1 and therefore no anomalous dimension. Despite such peculiar behavior, both models acquire nontrivial radiative corrections for the coupling constants. The RG flow of the model + exhibits a particular asymptotic safety: is marginal without corrections thus is a fixed point of arbitrary constant value. All remaining couplings determine relevant directions and get suppressed in the UV. Concerning the model ×, is marginal and again a fixed point (arbitrary constant value), λ, µ and Za are all relevant couplings and flow to 0. Meanwhile is a marginal coupling and becomes a linear function of the time scale. This model can neither be called asymptotically safe or free.

Determination of acoustic phonon anharmonicities via second-order Raman scattering in CuI

We demonstrate the determination of anharmonic acoustic phonon properties via second-order Raman scattering exemplarily on copper iodide single crystals. The origin of multi-phonon features from the second-order Raman spectra was assigned by the support of the calculated 2-phonon density of states. In this way, the temperature dependence of acoustic phonons was determined down to 10 K. To determine independently the harmonic contributions of respective acoustic phonons, density functional theory in quasi-harmonic approximation was used. Finally, the anharmonic contributions were determined. The results are in agreement with earlier publications and extend CuI’s determined acoustic phonon properties to lower temperatures with higher accuracy. This approach demonstrates that it is possible to characterize the acoustic anharmonicities via Raman scattering down to zero-temperature renormalization constants of at least 0.1 cm−1.

Geometric control of tilt transition dynamics in single-clamped thermalized elastic sheets.

We study the finite-temperature dynamics of thin elastic sheets in a single-clamped cantilever configuration. This system is known to exhibit a tilt transition at which the preferred mean plane of the sheet shifts from horizontal to a plane above or below the horizontal. The resultant thermally roughened two-state (up/down) system possesses rich dynamics on multiple timescales. In the tilted regime a finite-energy barrier separates the spontaneously chosen up state from the inversion-symmetric down state. Molecular dynamics simulations confirm that, over sufficiently long time, such thermalized elastic sheets transition between the two states, residing in each for a finite dwell time. One might expect that temperature is the primary driver for tilt inversion. We find, instead, that the primary control parameter, at fixed tilt order parameter, is the dimensionless and purely geometrical aspect ratio of the clamped width to the total length of the otherwise-free sheet. Using a combination of an effective mean-field theory and Kramers' theory, we derive the transition rate and examine its asymptotic behavior. At length scales beyond a material-dependent thermal length scale, renormalization of the elastic constants qualitatively modifies the temperature response. In particular, the transition is suppressed by thermal fluctuations, enhancing the robustness of the tilted state. We check and supplement these findings with further molecular dynamics simulations for a range of aspect ratios and temperatures.

Theory of all-coupling angulon for molecules rotating in many-body environment.

The formation of angulon, stemming from the rotor (molecule or impurity), rotating in the quantum many-body field, adds a new member to the quasi-particles' family and has aroused intense interest in multiple research fields. However, the analysis of the coupling strength between the rotor and its hosting environment remains a challenging task, both in theory and experiment. Here, we develop the all-coupling theory of the angulon by introducing a unitary transformation, where the renormalization of the rotational constants for different molecules in the helium nanodroplets is reproduced, getting excellent agreement with the experimental data collected during the past decades. Moreover, the strength of molecule-helium coupling and the effective radius of the solvation shell co-rotating along with the molecular rotor could be estimated qualitatively. This model not only provides significant enlightenment for analyzing the rotational spectroscopy of molecules in the phononic environment, but also provides a new method to study the transfer of the phonon angular momentum in the angulon frame.

Three-loop matching coefficients for heavy flavor-changing currents and the phenomenological applications

Within the framework of non-relativistic QCD (NRQCD) factorization, we compute the matching coefficients between full Quantum Chromodynamics (QCD) and NRQCD for the heavy flavor-changing vector, axial-vector, scalar and pseudo-scalar currents up to next-to-next-to-next-to-leading order (N3LO). We accomplish the analytical expressions for the three-loop renormalization constants and the corresponding anomalous dimensions for all of the four NRQCD currents with two different heavy flavors. The three-loop QCD corrections to the matching coefficients turn out to be significantly larger than lower order corrections. By employing the scale relation, we approximately simulate the N3LO correction to the wave functions at the origin for the vector {B}_c^{ast } meson and the pseudo-scalar Bc meson from the known result for the equal-mass heavy quarkonium in potential NRQCD (pNRQCD). We find large cancellations at the third order between the matching coefficients and the wave functions at the origin, and obtain the convergent decay constants of {B}_c^{ast } and Bc up to N3LO. We present the complete perturbative QCD predictions for the decay constants, leptonic decay widths, and branching ratios of the beauty-charmed mesons.

Cosmological de Sitter Solutions of the Semiclassical Einstein Equation

Exponentially expanding space–times play a central role in contemporary cosmology, most importantly in the theory of inflation and in the dark energy driven expansion in the late universe. In this work, we give a complete list of de Sitter solutions of the semiclassical Einstein equation (SCE), where classical gravity is coupled to the expected value of a renormalized stress–energy tensor of a free quantum field in the Bunch–Davies state. To achieve this, we explicitly determine the stress–energy tensor associated with the Bunch–Davies state using the recently proposed “moment approach” on the cosmological coordinate patch of de Sitter space. From the energy component of the SCE, we thus obtain an analytic consistency equation for the model’s parameters which has to be fulfilled by solutions to the SCE. Using this equation, we then investigate the number of solutions and the structure of the solution set in dependency on the coupling parameter of the quantum field to the scalar curvature and renormalization constants using analytic arguments in combination with numerical evidence. We also identify parameter sets where multiple expansion rates separated by several orders of magnitude are possible. Potentially for such parameter settings, a fast (semi-stable) expansion in the early universe could be compatible with a late-time “Dark Energy-like” behavior of the universe.

Renormalized perturbation theory for fast evaluation of Feynman diagrams on the real frequency axis

We present a method to accelerate the numerical evaluation of spatial integrals of Feynman diagrams when expressed on the real frequency axis. This can be realized through use of a renormalized perturbation expansion with a constant but complex renormalization shift. The complex shift acts as a regularization parameter for the numerical integration of otherwise sharp functions. This results in an exponential speed up of stochastic numerical integration at the expense of evaluating additional counter-term diagrams. We provide proof of concept calculations within a difficult limit of the half-filled 2D Hubbard model on a square lattice.

Decay constant of [formula omitted] accurate up to [formula omitted

In this paper, we evaluate, up to QCD next-to-next-to-next-to-leading order, the Bc⁎ decay constant, which, within the nonrelativistic QCD (NRQCD) framework, is factorized as the product of the short-distance coefficient (SDC) and the long-distance matrix element. For the first time, the renormalization constant and the anomalous dimension for the NRQCD vector current composed of cb¯, which are functions of the charm quark mass mc, the bottom quark mass mb and the factorization scale μΛ, are obtained analytically at O(αs2) and O(αs3). The SDC is calculated up to O(αs3) in perturbative expansion. Explicitly, the O(αs2) correction to the SDC is analytically calculated in terms of logarithmic and polylogarithmic functions of r≡mb/mc, and the O(αs3) correction to the SDC is numerically calculated at a series of values of r, ranging from 2.1 to 4.0. Surprisingly, we find that the nontrivial part of the SDC at O(αs3) can be well estimated by a linear function of r. In addition, we find that the O(αs2) and O(αs3) corrections to the decay constant and decay width are considerable and very significant, which indicates a very poor convergence for the perturbative expansion.

QCD cusp anomalous dimension: Current status

Calculation results for the HQET field anomalous dimension and the QCD cusp anomalous dimension, as well as their properties, are reviewed. The HQET field anomalous dimension [Formula: see text] is known up to four loops. The cusp anomalous dimension [Formula: see text] is known up to three loops, and its small-angle and large-angle asymptotics up to four loops. Some (but not all) color structures at four loops are known with the full [Formula: see text]-dependence. Some simple contributions are known at higher loops. For the [Formula: see text] asymptotics of [Formula: see text] (the light-like cusp anomalous dimension) and the [Formula: see text]-term of the small-[Formula: see text] expansion (the Bremsstrahlung function), the [Formula: see text] SYM results are equal to the highest-weight parts of the QCD results. There is an interesting conjecture about the structure of [Formula: see text] which holds up to three loops; at four loops it holds for some color structures and breaks down for other ones. In the cases when it holds, it related highly nontrivial functions of [Formula: see text], and it cannot be accidental; however, the reasons of this conjecture and its failures are not understood. The cusp anomalous dimension at the Euclidean angle [Formula: see text] is related to the static quark–antiquark potential due to conformal symmetry; in QCD, this relation is broken by an anomalous term proportional to the [Formula: see text]-function. Some new results are also presented. Using the recent four-loop result for [Formula: see text], here we obtain analytical expressions for some terms in the four-loop on-shell renormalization constant of the massive quark field [Formula: see text] which were previously known only numerically. We also present two new contributions to [Formula: see text], [Formula: see text] at five loops and to the quark–antiquark potential at four loops.

Predicting the Lattice Thermal Conductivity in Nitride Perovskite LaWN3 from ab initio Lattice Dynamics.

Using a density functional theory-based thermal transport model, which includes the effects of temperature (T)-dependent potential energy surface, lattice thermal expansion, force constant renormalization, and higher-order quartic phonon scattering processes, it is found that the recently synthesized nitride perovskite LaWN3 displays strong anharmonic lattice dynamics manifested into a low lattice thermal conductivity (κL ) and a non-standard κL ∝T-0.491 dependence. At high T, the departure from the standard κL ∝T-1 law originates in the dual particle-wave behavior of the heat carrying phonons, which includes vibrations tied to the N atoms. While the room temperature κL =2.98 W mK-1 arises mainly from the conventional particle-like propagation of phonons, there is also asignificant atypical wave-like phonon tunneling effect, leading to a 20% glass-like heat transport contribution. The phonon broadening effect lowers the particle-like contribution but increases the glass-like one. Upon T increase, the glass-like contribution increases and dominates above T = 850 K. Overall, the low κL with a weak T-dependence points to a new utility for LaWN3 in energy technology applications, and motivates synthesis and exploration of nitrideperovskites.

RI/MOM and RI/SMOM renormalization of quark bilinear operators using overlap fermions

We present the vector, scalar, and tensor renormalization constants (RCs) using overlap fermions with either regularization independent momentum subtraction ($\mathrm{RI}/\mathrm{MOM}$) or symmetric momentum subtraction ($\mathrm{RI}/\mathrm{SMOM}$) as the intermediate scheme on the lattice with lattice spacings $a$ from 0.04 fm to 0.12 fm. Our gauge field configurations from the MILC and RBC/UKQCD collaborations include sea quarks using either the domain wall or the HISQ action, respectively. The results show that $\mathrm{RI}/\mathrm{MOM}$ and $\mathrm{RI}/\mathrm{SMOM}$ can provide consistent renormalization constants to the $\overline{\mathrm{MS}}$ scheme, after proper ${a}^{2}{p}^{2}$ extrapolations. But at $p\ensuremath{\sim}2\text{ }\text{ }\mathrm{GeV}$, both $\mathrm{RI}/\mathrm{MOM}$ and $\mathrm{RI}/\mathrm{SMOM}$ suffer from nonperturbative effects which cannot be removed by the perturbative matching. The comparison between the results with different sea actions also suggests that the renormalization constant is discernibly sensitive to the lattice spacing but not to the bare gauge coupling in the gauge action.

Non-perturbative renormalization of the QCD flavour-singlet local vector current

We compute non-perturbatively the renormalization constant of the flavour-singlet local vector current ZV in lattice QCD with 3 massless flavours. Gluons are discretized by the Wilson plaquette action while quarks by the O(a)-improved Wilson–Dirac operator. The constant ZV is fixed by comparing the expectation values (1-point functions) of the conserved and local vector currents at finite temperature in the presence of shifted boundary conditions and at non-zero imaginary chemical potential. Monte Carlo simulations with a moderate computational cost allow us to obtain ZV with an accuracy of about 8‰ for values of the inverse bare coupling constant β=6/g02 in the range 5.3≤β≤11.5.