Renormalization Group Resummation Research Articles

Systematic improvement of x -dependent unpolarized nucleon generalized parton distributions in lattice-QCD calculation

We present a first study of the effects of renormalization-group resummation (RGR) and leading-renormalon resummation (LRR) on the systematic errors of the unpolarized isovector nucleon generalized parton distribution in the framework of large-momentum effective theory. This work is done using lattice gauge ensembles generated by the MILC Collaboration, consisting of 2+1+1 flavors of highly improved staggered quarks with a physical pion mass at lattice spacing a≈0.09 fm and a box width L≈5.76 fm. We present results for the nucleon H and E generalized parton distributions (GPDs) with average boost momentum Pz≈2 GeV at momentum transfers Q2=[0,0.97] GeV2 at skewness ξ=0 as well as Q2∈0.23 GeV2 at ξ=0.1, renormalized in the modified minimal subtraction (MS¯) scheme at scale μ=2.0 GeV, with two- and one-loop matching, respectively. We demonstrate that the simultaneous application of RGR and LRR significantly reduces the systematic errors in renormalized matrix elements and distributions for both the zero and nonzero skewness GPDs, and that it is necessary to include both RGR and LRR at higher orders in the matching and renormalization processes. Published by the American Physical Society 2024

Nucleon helicity parton distribution function in the continuum limit with self-renormalization

We present the first lattice calculation of the nucleon isovector helicity parton distribution function (PDF) in the framework of large-momentum effective theory (LaMET) that uses the hybrid scheme with self-renormalization. We use ensembles generated by the MILC collaboration at lattice spacings a={0.1207,0.0888,0.0582} fm, with Nf=2+1+1 flavors of highly improved staggered quarks at sea pion mass of Mπ≈315 MeV. We use clover-improved action for our valence quarks with nucleon boost momentum Pz≈1.75 GeV and high-statistics measurements for the LaMET matrix elements. We perform an extrapolation to the continuum limit and improve the handling of systematic errors using renormalization-group resummation (RGR) and leading-renormalon resummation (LRR). Our final nucleon helicity PDF is renormalized in the MS‾ scheme at energy scale μ=2.0 GeV. We compare our results with and without the two systematic improvements of RGR and LRR at each lattice spacing as well as the continuum limit, and we see that the application of RGR and LRR greatly reduces the systematic errors across the whole x range. Our continuum results with both RGR and LRR show a small positive antiquark region for the nucleon helicity PDF as well as a change of as much as a factor of two in the central values compared to results with neither RGR or LRR. By contrast, the application of RGR and LRR only changes the central values by about 5% in the quark region. We compare our lattice results with the global fits by the JAM, NNPDF and DSSV collaborations, and we observe some tension between our results.

Pion valence quark distribution at physical pion mass of N f = 2 + 1 + 1 lattice QCD

We present a state-of-the-art calculation of the unpolarized pion valence-quark distribution in the framework of large-momentum effective theory (LaMET) with improved handling of systematic errors as well as two-loop perturbative matching. We use lattice ensembles generated by the MILC collaboration at lattice spacing a ≈ 0.09 fm, lattice volume 643 × 96, N f = 2 + 1 + 1 flavors of highly-improved staggered quarks and a physical pion mass. The LaMET matrix elements are calculated with pions boosted to momentum P z ≈ 1.72 GeV with high-statistics of O(106) measurements. We study the pion PDF in both hybrid-ratio and hybrid-regularization-independent momentum subtraction (hybrid-RI/MOM) schemes and also compare the systematic errors with and without the addition of leading-renormalon resummation (LRR) and renormalization-group resummation (RGR) in both the renormalization and lightcone matching. The final lightcone PDF results are presented in the modified minimal-subtraction scheme at renormalization scale μ = 2.0 GeV. We show that the x-dependent PDFs are compatible between the hybrid-ratio and hybrid-RI/MOM renormalization with the same improvements. We also show that systematics are greatly reduced by the simultaneous inclusion of RGR and LRR and that these methods are necessary if improved precision is to be reached with higher-order terms in renormalization and matching.

Threshold resummation for computing large-x parton distribution through large-momentum effective theory

Parton distribution functions (PDFs) at large x are poorly constrained by high-energy experimental data, but extremely important for probing physics beyond standard model at colliders. We study the calculation of PDFs at large-x through large-momentum Pz expansion of the lattice quasi PDFs. Similar to deep-inelastic scattering, there are two distinct perturbative scales in the threshold limit where the matching coefficient can be factorized into a space-like jet function at scale Pz|1 − y| and a pair of heavy-light Sudakov form factors at scale Pz. The matching formula allows us to derive a full renormalization group resummation of large threshold logarithms, and the result is consistent with the known calculation to the next-to-next to leading order (NNLO). This paves the way for direct large-x PDFs calculations in lattice QCD. As by-products, we find that the space-like jet function is related to a time-like version calculated previously through analytic continuation, and the heavy-light Sudakov form factor, calculated here to NNLO, is a universal object appearing as well in the large momentum expansion of quasi transverse-momentum-dependent PDFs and quasi wave-function amplitudes.

Fermionic influence on inflationary fluctuations

Motivated by apparent persistent large scale anomalies in the CMB we study the influence of fermionic degrees of freedom on the dynamics of inflaton fluctuations as a possible source of violations of (nearly) scale invariance on cosmological scales. We obtain the non-equilibrium effective action of an inflaton-like scalar field with Yukawa interactions ($Y_{D,M}$) to light \emph{fermionic} degrees of freedom both for Dirac and Majorana fields in de Sitter space-time. The effective action leads to Langevin equations of motion for the fluctuations of the inflaton-like field, with self-energy corrections and a stochastic gaussian noise. We solve the Langevin equation in the super-Hubble limit implementing a dynamical renormalization group resummation. For a nearly massless inflaton its power spectrum of super Hubble fluctuations is \emph{enhanced}, $\mathcal{P}(k;\eta) = (\frac{H}{2\pi})^2\,e^{\gamma_t[-k\eta] }$ with $\gamma_t[-k\eta] = \frac{1}{6\pi^2} \Big[\sum_{i=1}^{N_D}{Y^2_{i,D}}+2\sum_{j=1}^{N_M}{Y^2_{j,M}}\Big]\,\Big\{\ln^2[-k\eta]-2 \ln[-k\eta]\ln[ -k\eta_0] \Big\} $ for $N_D$ Dirac and $N_M$ Majorana fermions, and $\eta_0$ is the renormalization scale at which the inflaton mass vanishes. The full power spectrum is shown to be renormalization group invariant. These corrections to the super-Hubble power spectrum entail a violation of scale invariance as a consequence of the coupling to the fermionic fields. The effective action is argued to be \emph{exact} in a limit of large number of fermionic fields. A cancellation between the enhancement from fermionic degrees of freedom and suppression from light scalar degrees of freedom \emph{conformally coupled to gravity} suggests the possibility of a finely tuned \emph{supersymmetry} among these fields.

Effective field theory during inflation. II. Stochastic dynamics and power spectrum suppression

We obtain the non-equilibrium effective action of an inflaton like scalar field --the system-- by tracing over sub Hubble degrees of freedom of "environmental" light scalar fields. The effective action is stochastic leading to effective Langevin equations of motion for the fluctuations of the inflaton-like field, with self-energy corrections and stochastic noise correlators that obey a de Sitter space-time analog of a fluctuation dissipation relation. We solve the Langevin equation implementing a dynamical renormalization group resummation of the leading secular terms and obtain the corrections to the power spectrum of super Hubble fluctuations of the inflaton field, $\mathcal{P}(k;\eta) = \mathcal{P}_0(k)\,e^{-\gamma(k;\eta)}$ where $\mathcal{P}_0(k)$ is the nearly scale invariant power spectrum in absence of coupling. $\gamma(k;\eta)>0$ describes the suppression of the power spectrum, it features Sudakov-type double logarithms and entails violations of scale invariance. We also obtain the effective action for the case of a heavy scalar field of mass $ M \gg H$, this case yields a local "Fermi" limit with a very weak self-interaction of the inflaton-like field and dissipative terms that are suppressed by powers of $H/M$. We conjecture on the possibility that the large scale anomalies in the CMB may originate in dissipative processes from inflaton coupling to sub-Hubble degrees of freedom.

STRONG COUPLING FROM TAU LEPTON DECAYS

A determination of the strong coupling αs at rather low energies is possible through the analysis of hadronic decays of the τ lepton. In turn, the low energy necessitates sufficient control over perturbative QCD corrections, the nonperturbative condensate contributions in the framework of the operator product expansion (OPE), as well as corrections going beyond the OPE, the duality violations (DVs). Perturbative QCD uncertainties arise from open questions regarding the renormalization group resummation of the series. The fit quantities are moment integrals of the τ spectral function data in a certain energy window and care should be taken to have good perturbative behavior of those moments as well as control over higher-dimensional operator corrections. Furthermore, all parameters occurring in the theoretical description should be extracted from fits to the data in a self-consistent manner.

Expansion functions in perturbative QCD and the determination ofαs(Mτ2)

The conventional series in powers of the coupling in perturbative QCD have zero radius of convergence and fail to reproduce the singularity of the QCD correlators like the Adler function at $\alpha_s=0$. Using the technique of conformal mapping of the Borel plane, combined with the "softening" of the leading singularities, we define a set of new expansion functions that resemble the expanded correlator and share the same singularity at zero coupling. Several different conformal mappings and different ways of implementing the known nature of the first branch-points of the Adler function in the Borel plane are investigated, in both the contour-improved (CI) and fixed-order (FO) versions of renormalization group resummation. We prove the remarkable convergence properties of a set of new CI expansions and use them for a determination of the strong coupling from the hadronic $\tau$ decay width. By taking the average upon this set, with a conservative treatment of the errors, we obtain $\alpha_s(M_\tau^2)= 0.3195^{+ 0.0189}_{- 0.0138}$.

Particle decay during inflation: Self-decay of inflaton quantum fluctuations during slow roll

Particle decay during inflation is studied by implementing a dynamical renormalization group resummation combined with a small $\ensuremath{\Delta}$ expansion. $\ensuremath{\Delta}$ measures the deviation from the scale invariant power spectrum and regulates the infrared. In slow-roll inflation, $\ensuremath{\Delta}$ is a simple function of the slow-roll parameters ${ϵ}_{V},{\ensuremath{\eta}}_{V}$. We find that quantum fluctuations can self-decay as a consequence of the inflationary expansion through processes which are forbidden in Minkowski space-time. We compute the self-decay of the inflaton quantum fluctuations during slow-roll inflation. For wavelengths deep inside the Hubble radius the decay is enhanced by the emission of ultrasoft collinear quanta, i.e., bremsstrahlung radiation of superhorizon quanta which becomes the leading decay channel for physical wavelengths $H\ensuremath{\ll}{k}_{\mathrm{p}\mathrm{h}}(\ensuremath{\eta})\ensuremath{\ll}H/({\ensuremath{\eta}}_{V}\ensuremath{-}{ϵ}_{V})$. The decay of short wavelength fluctuations hastens as the physical wave vector approaches the horizon. Superhorizon fluctuations decay with a power law ${\ensuremath{\eta}}^{\ensuremath{\Gamma}}$ in conformal time where in terms of the amplitude of curvature perturbations ${△}_{\mathcal{R}}^{2}$, the scalar spectral index ${n}_{s}$, the tensor to scalar ratio $r$ and slow-roll parameters: $\ensuremath{\Gamma}\ensuremath{\simeq}[32{\ensuremath{\xi}}_{V}^{2}{△}_{\mathcal{R}}^{2}/({n}_{s}\ensuremath{-}1+\frac{r}{4}{)}^{2}][1+\mathcal{O}({ϵ}_{V},{\ensuremath{\eta}}_{V})]$. The behavior of the growing mode ${\ensuremath{\eta}}^{{\ensuremath{\eta}}_{V}\ensuremath{-}{ϵ}_{V}+\ensuremath{\Gamma}}/\ensuremath{\eta}$ features an anomalous scaling dimension $\ensuremath{\Gamma}$. We discuss the implications of these results for scalar and tensor perturbations as well as for non-Gaussianities in the power spectrum. The recent Wilkinson Map Anisotropy Probe data suggests $\ensuremath{\Gamma}\ensuremath{\gtrsim}3.6\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}9}$.

Complete two-loop dominant corrections to the mass of the lightest -even Higgs boson in the minimal supersymmetric standard model

Using an effective potential approach, we compute two-loop radiative corrections to the MSSM lightest ${\cal CP}$-even Higgs boson mass $M_{h^0}$ to ${\cal O}(\alpha_t^2)$ for arbitrary left-right top-squark mixing and $\tan\beta$. We find that these corrections can increase $M_{h^0}$ by as much as 5 GeV; assuming a SUSY scale of 1 TeV, the upper bound on the Higgs boson mass is $M_{h^0}\approx 129\pm 5$ GeV for the top quark pole mass $175\pm 5$ GeV. We also derive an analytical approximation formula for $M_{h^0}$ which is good to a precision of $\lsim 0.5$ GeV for most of the parameter space and suitable to be further improved by including renormalization group resummation of leading and next-to-leading order logarithmic terms. Our final compact formula admits a clear physical interpretation: radiative corrections up to the two-loop level can be well approximated by a one-loop expression with parameters evaluated at the appropriate scales, plus a smaller finite two-loop threshold correction term.

Dynamical renormalization group resummation of finite temperature infrared divergences

We introduce the method of dynamical renormalization group to study relaxation and damping out of equilibrium directly in real time and applied it to the study of infrared divergences in scalar QED. This method allows a consistent resummation of infrared effects associated with the exchange of quasistatic transverse photons and leads to anomalous logarithmic relaxation of the form $e^{-\alpha T t \ln[t/t_0]}$ which prevents a quasiparticle interpretation of charged collective excitations at finite temperature. The hard thermal loop resummation program is incorporated consistently into the dynamical renormalization group yielding a picture of relaxation and damping phenomena in a plasma in real time that trascends the conceptual limitations of the quasiparticle picture and other type of resummation schemes. We derive a simple criterion for establishing the validity of the quasiparticle picture to lowest order.

Anomalous kinetics of hard charged particles: Dynamical renormalization group resummation

We study the kinetics of the distribution function for charged particles of hard momentum in scalar QED. The goal is to understand the effects of infrared divergences associated with the exchange of quasistatic magnetic photons in the relaxation of the distribution function. We begin by obtaining a kinetic transport equation for the distribution function for hard charged scalars in a perturbative expansion that includes hard thermal loop resummation. Solving this transport equation, the infrared divergences arising from absorption and emission of soft quasi-static magnetic photons are manifest in logarithmic secular terms. We then implement the dynamical renormalization group resummation of these secular terms in the relaxation time approximation. The distribution function (in the linearized regime) is found to approach equilibrium as $\delta n_k(t) =\delta n_k(t_o) e^{-2\alpha T (t-t_o) \ln[(t-t_o)\bar{\mu}]}$, with $\bar{\mu}\approx \omega_p$ the plasma frequency and $\alpha =e^2/4\pi$. This anomalous relaxation is recognized to be the square of the relaxation of the single particle propagator, providing a generalization of the usual relation between the damping and the interaction rate. The renormalization group approach to kinetics reveals clearly the time scale $t_{rel} \approx (\alpha T \ln[1/\alpha])^{-1}$ arising from infrared physics and hinges upon the separation of scales $t_{rel} >>\omega_p^{-1}$.