Minimal Subtraction Research Articles

Gauge-invariant renormalization of four-quark operators

We study the renormalization of four-quark operators in one-loop perturbation theory. We employ a coordinate-space gauge-invariant renormalization scheme (GIRS), which can be advantageous compared to other schemes, especially in nonperturbative lattice investigations. From our perturbative calculations, we extract the conversion factors between GIRS and the modified minimal subtraction scheme (MS¯) at the next-to-leading order. As a by-product, we also obtain the relevant anomalous dimensions in the GIRS scheme. A formidable issue in the study of the four-quark operators is that operators with different Dirac matrices mix among themselves upon renormalization. We focus on both parity-conserving and parity-violating four-quark operators, which change flavor numbers by two units (ΔF=2). The extraction of the conversion factors entails the calculation of two-point Green’s functions involving products of two four-quark operators, as well as three-point Green’s functions with one four-quark and two bilinear operators. The significance of our results lies in their potential to refine our understanding of QCD phenomena, offering insights into the precision of Cabibbo-Kobayashi-Maskawa (CKM) matrix elements and shedding light on the nonperturbative treatment of complex mixing patterns associated with four-quark operators. Published by the American Physical Society 2024

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

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

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.

Transverse momentum moments

We establish robust relations between Transverse Momentum Dependent distributions (TMDs) and collinear distributions. We define weighted integrals of TMDs that we call Transverse Momentum Moments (TMMs) and prove that TMMs are equal to collinear distributions evaluated in some minimal subtraction scheme. The conversion to the modified minimal subtraction (MS¯) scheme can be done by a calculable factor, which we derive up to three loops for some cases. We discuss in detail the zeroth, the first, and the second TMMs and provide phenomenological results for them based on the current extractions of TMDs. The results of this paper open new avenues for theoretical and phenomenological investigation of the three-dimensional and collinear hadron structures. Published by the American Physical Society 2024

Gradient properties of perturbative multiscalar RG flows to six loops

The gradient property of the renormalisation group (RG) flow of multiscalar theories is examined perturbatively in d=4 and d=4−ε dimensions. Such theories undergo RG flows in the space of quartic couplings λI. Starting at five loops, the relevant vector field that determines the physical RG flow is not the beta function traditionally computed in a minimal subtraction scheme in dimensional regularisation, but a suitable modification of it, the B function. It is found that up to five loops the B vector field is gradient, i.e. BI=GIJ∂A/∂λJ with A a scalar and GIJ a rank-two symmetric tensor of the couplings. Up to five loops the beta function is also gradient, but it fails to be so at six loops. The conditions under which the B function (and hence the RG flow) is gradient at six loops are specified, but their verification rests on a separate six-loop computation that remains to be performed.

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.

Phase structure of the on-shell parametrized 2+1 flavor Polyakov quark-meson model

Augmenting the improved chiral effective potential of the on-shell renormalized 2+1 flavor quark-meson (RQM) model with the Polyakov-loop potential that accounts for the deconfinement transition, we get the quantum chromodynamics (QCD)-like framework of the renormalized Polyakov quark-meson (RPQM) model. When the divergent quark one-loop vacuum term is included in the effective potential of the quark-meson (QM) model, its tree-level parameters, or the parameters fixed by the use of meson curvature masses, become inconsistent as the curvature masses involve the self-energy evaluations at zero momentum. Using the modified minimal subtraction method, the consistent chiral effective potential for the RQM model has been calculated after relating the counterterms in the on-shell scheme to those in the MS¯ scheme and finding the relations between the renormalized parameters of both the schemes where the physical (pole) masses of the π, K, η, and η′ pseudoscalar mesons and the scalar σ meson, the pion and kaon decay constants, have been put into the relation of the running couplings and mass parameter. Using the RPQM model and the PQM model with different forms for the Polyakov-loop potentials in the presence or the absence of the quark backreaction, we have computed and compared the effect of the consistent quark one-loop correction and the quark backreaction on the scaled chiral order parameter, the QCD phase diagrams, and the different thermodynamic quantities. The results have been compared with the 2+1 flavor lattice QCD data from the Wuppertal-Budapest Collaboration [ 73, ] and the HotQCD Collaboration []. Published by the American Physical Society 2024

Supersymmetric QCD on the lattice: Fine-tuning of the Yukawa couplings

We determine the fine-tuning of the Yukawa couplings of supersymmetric QCD, discretized on a lattice. We use perturbation theory at one-loop level. The modified minimal subtraction scheme (MS¯) is employed; by its definition, this scheme requires perturbative calculations, in the continuum and/or on the lattice. On the lattice, we utilize the Wilson formulation for gluon, quark, and gluino fields; for squark fields we use naive discretization. The sheer difficulties of this study lie in the fact that different components of squark fields mix among themselves at the quantum level and the action’s symmetries, such as parity and charge conjugation, allow an additional Yukawa coupling. Consequently, for an appropriate fine-tuning of the Yukawa terms, these mixings must be taken into account in the renormalization conditions. All Green’s functions and renormalization factors are analytic expressions depending on the number of colors, Nc, the number of flavors, Nf, and the gauge parameter, α, which are left unspecified. Knowledge of these renormalization factors is necessary in order to relate numerical results, coming from nonperturbative studies, to the renormalized, “physical” Green’s functions of the theory. Published by the American Physical Society 2024

Low-energy effective field theory below the electroweak scale: one-loop renormalization in the ’t Hooft-Veltman scheme

The low-energy effective field theory below the electroweak scale (LEFT) describes the effects at low energies of both the weak interaction and physics beyond the Standard Model. We study the one-loop renormalization of the LEFT in the ’t Hooft-Veltman scheme, which offers an algebraically consistent definition of the Levi-Civita symbol and γ5 in dimensional regularization. However, in connection with minimal subtraction this scheme leads to a spurious breaking of chiral symmetry in intermediate steps of the calculation. Based on the ’t Hooft-Veltman prescription, we define a renormalization scheme that restores chiral symmetry by including appropriate finite counterterms. To this end, we extend the physical LEFT operator basis by a complete set of off-shell and one-loop-evanescent operators and we perform the renormalization at one loop. We determine the finite counterterms to the physical parameters that compensate both the insertions of evanescent operators, as well as the chiral-symmetry-breaking terms from the renormalizable part of the Lagrangian in D dimensions. Our results can be applied in next-to-leading-log calculations in the ’t Hooft-Veltman scheme: using our renormalization scheme instead of pure minimal subtraction separates the physical sector from the unphysical evanescent sector and leads to results that are manifestly free of spurious chiral-symmetry-breaking terms.

Factorial growth at low orders in perturbative QCD: control over truncation uncertainties

A method, known as “minimal renormalon subtraction” [Phys. Rev. D97 (2018) 034503, JHEP08 (2017) 62], relates the factorial growth of a perturbative series (in QCD) to the power p of a power correction Λp/Qp. (Λ is the QCD scale, Q some hard scale.) Here, the derivation is simplified and generalized to any p, more than one such correction, and cases with anomalous dimensions. Strikingly, the well-known factorial growth is seen to emerge already at low or medium orders, as a consequence of constraints on the Q dependence from the renormalization group. The effectiveness of the method is studied with the gluonic energy between a static quark and static antiquark (the “static energy”). Truncation uncertainties are found to be under control after next-to-leading order, despite the small exponent of the power correction (p = 1) and associated rapid growth seen in the first four coefficients of the perturbative series.

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 matching of the CP-odd three-gluon operator to the gradient flow

The calculation of the neutron electric dipole moment within effective field theories for physics beyond the Standard Model requires non-perturbative hadronic matrix elements of effective operators composed of quark and gluon fields. In order to use input from lattice computations, these matrix elements must be translated from a scheme suitable for lattice QCD to the minimal-subtraction scheme used in the effective-field-theory framework. The accuracy goal in the context of the neutron electric dipole moment necessitates at least a one-loop matching calculation. Here, we provide the one-loop matching coefficients for the CP-odd three-gluon operator between two different minimally subtracted 't Hooft–Veltman schemes and the gradient flow. This completes our program to obtain the one-loop gradient-flow matching coefficients for all CP-violating and flavor-conserving operators in the low-energy effective field theory up to dimension six.

Jets in e+A SIDIS and Denominator Regularization

We compute the in-medium jet broadening to leading order in energy in the opacity expansion. At leading order in αs the elastic energy loss gives a jet broadening that grows with ln E. The next-to-leading order in αs result is a jet narrowing, due to destructive LPM interference effects, that grows with ln2 E. We find that in the opacity expansion the jet broadening asymptotics are— unlike for the mean energy loss—extremely sensitive to the correct treatment of the finite kinematics of the problem; integrating over all emitted gluon transverse momenta leads to a prediction of jet broadening rather than narrowing. We compare the asymptotics from the opacity expansion to a recent twist-4 derivation and find a qualitative disagreement: the twist-4 derivation predicts a jet broadening rather than a narrowing. Comparison with current jet measurements cannot distinguish between the broadening or narrowing predictions. We comment on the origin of the difference between the opacity expansion and twist-4 results.We also introduce a novel regularization scheme in quantum field theory, denominator regularization (den reg). Den reg is as simple to apply as the usual dimensional regularization, works simply with a minimal subtraction scheme, and manifestly 1) maintains Lorentz invariance, 2) maintains gauge invariance, 3) maintains supersymmetry, 4) correctly predicts the axial anomaly, and 5) yields Green functions that satisfy the Callan-Symanzik equation. Den reg also naturally enables regularization in asymmetric spacetimes, finite spacetimes, curved spacetimes, and in thermal field theory.

One-loop matching of CP-odd four-quark operators to the gradient-flow scheme

The translation of experimental limits on the neutron electric dipole moment into constraints on heavy CP-violating physics beyond the Standard Model requires knowledge about non-perturbative matrix elements of effective operators, which ideally should be computed in lattice QCD. However, this necessitates a matching calculation as an interface to the effective field theory framework, which is based on dimensional regularization and renormalization by minimal subtraction. We calculate the one-loop matching between the gradient-flow and minimal-subtraction schemes for the CP-violating four-quark operators contributing to the neutron electric dipole moment. The gradient flow is a modern regularization-independent scheme amenable to lattice computations that promises, e.g., better control over power divergences than traditional momentum-subtraction schemes. Our results extend previous work on dimension-five operators and provide a necessary ingredient for future lattice-QCD computations of the contribution of four-quark operators to the neutron electric dipole moment.

The one-loop unimodular graviton propagator in any dimension

For unimodular gravity, we work out, by using dimensional regularization, the complete one-loop correction to the graviton propagator in any space-time dimension. The computation is carried out within the framework where unimodular gravity has Weyl invariance in addition to the transverse diffeomorphism gauge symmetry. Thus, no Lagrange multiplier is introduced to enforce the unimodularity condition. The quantization of the theory is carried out by using the BRST framework and there considering a large continuous family of gauge-fixing terms. The BRST formalism is developed in such a way that the set of ghost, anti-ghost and auxiliary fields and their BRST changes do not depend on the space-time dimension, as befits dimensional regularization. As an application of our general result, and at D = 4, we obtain the renormalized one-loop graviton propagator in the dimensional regularization minimal subtraction scheme. We do so by considering two simplifying gauge-fixing choices.

Effective approach to the Antoniadis-Mottola model: quantum decoupling of the higher derivative terms

We explore the decoupling of massive ghost mode in the 4D (four-dimensional) theory of the conformal factor of the metric. The model was introduced by Antoniadis and Mottola in [1] and can be regarded as a close analog of the fourth-derivative quantum gravity. The analysis of the derived one-loop nonlocal form factors includes their asymptotic behavior in the UV and IR limits. In the UV (high energy) domain, our results reproduce the Minimal Subtraction scheme-based beta functions of [1]. In the IR (i.e., at low energies), the diagrams with massive ghost internal lines collapse into tadpole-type graphs without nonlocal contributions and become irrelevant. On the other hand, those structures that contribute to the running of parameters of the action and survive in the IR, are well-correlated with the divergent part (or the leading in UV contributions to the form factors), coming from the effective low-energy theory of the conformal factor. This effective theory describes only the light propagating mode. Finally, we discuss whether these results may shed light on the possible running of the cosmological constant at low energies.

Field-theoretic analysis of directed percolation: Three-loop approximation.

The directed bond percolation is a paradigmatic model in nonequilibrium statistical physics. It captures essential physical information on the nature of continuous phase transition between active and absorbing states. In this paper, we study this model by means of the field-theoretic formulation with a subsequent renormalization group analysis. We calculate all critical exponents needed for the quantitative description of the corresponding universality class to the third order in perturbation theory. Using dimensional regularization with minimal subtraction scheme, we carry out perturbative calculations in a formally small parameter ɛ, where ɛ=4-d is a deviation from the upper critical dimension d_{c}=4. We use a nontrivial combination of analytical and numerical tools in order to determine ultraviolet divergent parts of Feynman diagrams.

Induced energy-momentum tensor in de Sitter scalar QED and its implication for induced currents

The aim of this research is to investigate the vacuum energy-momentum tensor of a quantized, massive, nonminimally coupled scalar field induced by a uniform electric field background in a four-dimensional de Sitter spacetime (${\mathrm{dS}}_{4}$). We compute the expectation value of the energy-momentum tensor in the in-vacuum state and then regularize it using the adiabatic subtraction procedure. The correct trace anomaly of the induced energy-momentum tensor that confirmed our results is significant. The nonconservation equation for the induced energy-momentum tensor imposes the renormalization condition for the induced electric current of the scalar field. The findings of this research indicate that there are significant differences between the two induced currents which are regularized by this renormalization condition and the minimal subtraction condition.

Study of the ultraviolet behavior of an O(N) |ϕ→|6 theory in d=3 dimensions

We study the ultraviolet (UV) behavior of an O($N$) $|\vec \phi |^6$ theory in $d=3$ spacetime dimensions, focusing on the question of the range in $N$ over which the perturbative beta function exhibits robust evidence of a UV zero in the $|\vec \phi |^6$ coupling, $g$. The four-loop $(4\ell)$ beta function is known to have a (scheme-independent) UV zero at $g=g_{_{UV,4\ell}}$, which is reliably calculable for large $N$. For our analysis we use the six-loop beta function calculated in the minimal subtraction scheme. We find that this six-loop beta function has a UV zero, $g_{_{UV,6\ell}}$, if $N > N_c$, where $N_c \simeq 796$, and we calculate $g_{_{UV,6\ell}}$. To investigate the reliability of the result in the region of $N \gtrsim N_c$, we apply three methods: (i) calculation of the fractional difference between $g_{_{UV,4\ell}}$ and $g_{_{UV,6\ell}}$, (ii) a Pad\'e approximant, and (iii) an assessment of scheme dependence. Our results provide quantitative measures of the range of $N$ over which the six-loop beta function has a UV zero and of the $1/N$ corrections to the value of $g$ at the UV zero for large but finite $N$. If one imposes a benchmark requirement that the fractional difference between $g_{_{UV,4\ell}}$ and $g_{_{UV,6\ell}}$ must be less than 15 \%, then our results show that this requirement is satisfied for $N \gtrsim 2 \times 10^3$. The possible role of nonperturbative effects is also noted.