Renormalization Of Operators Research Articles

The [formula omitted] renormalization constant of the singlet axial current operator at [formula omitted] in QCD

We provide the MS‾ factor of the renormalization constant of the singlet axial-current operator in dimensional regularization at O(αs5) in perturbative Quantum Chromodynamics (QCD). The result is obtained from a formula derived using the Adler-Bell-Jackiw equation in terms of renormalized operators. The required input consists of the complete four-loop results for operator renormalization constants obtained by explicit diagrammatic computation in a previous publication.

Background field method and generalized field redefinitions in effective field theories

We study the non-linear background field redefinitions arising at the quantum level in a spontaneously broken effective gauge field theory. The non-linear field redefinitions are crucial for the symmetric (i.e. fulfilling all the relevant functional identities of the theory) renormalization of gauge-invariant operators. In a general R_\xiRξ-gauge the classical background-quantum splitting is also non-linearly deformed by radiative corrections. In the Landau gauge these deformations vanish to all orders in the loop expansion.

Two-Loop renormalization of High-Dimensional QCD operators and Higgs EFT amplitudes

We consider the two-loop renormalization of high-dimensional operators in QCD and the relevant Higgs EFT amplitudes. Efficient unitarity-IBP strategy is used to compute the two-loop minimal form factors of length-3 operators up to dimension sixteen. From the ultraviolet divergences of form factor results, we extract the renormalization matrices and compute anomalous dimensions. We also obtain the analytic finite remainder functions which exhibit several universal transcendentality structures. The form factors we compute are equivalent to Higgs plus three-gluon amplitudes which capture high-order top mass corrections in Higgs effective field theory.

Renormalization of gluonic leading-twist operators in covariant gauges

We provide the all-loop structure of gauge-variant operators required for the renormalisation of Green’s functions with insertions of twist-two operators in Yang-Mills theory. Using this structure we work out an explicit basis valid up to 4-loop order for an arbitrary compact simple gauge group. To achieve this we employ a generalised gauge symmetry, originally proposed by Dixon and Taylor, which arises after adding to the Yang-Mills Lagrangian also operators proportional to its equation of motion. Promoting this symmetry to a generalised BRST symmetry allows to generate the ghost operator from a single exact operator in the BRST-generalised sense. We show that our construction complies with the theorems by Joglekar and Lee. We further establish the existence of a generalised anti-BRST symmetry which we employ to derive non-trivial relations among the anomalous dimension matrices of ghost and equation-of-motion operators. For the purpose of demonstration we employ the formalism to compute the N = 2, 4 Mellin moments of the gluonic splitting function up to 4 loops and its N = 6 Mellin moment up to 3 loops, where we also take advantage of additional simplifications of the background field formalism.

RG of GR from on-shell amplitudes

We study the renormalization group of generic effective field theories that include gravity. We follow the on-shell amplitude approach, which provides a simple and efficient method to extract anomalous dimensions avoiding complications from gauge redundancies. As an invaluable tool we introduce a modified helicity tilde{h} under which gravitons carry one unit instead of two. With this modified helicity we easily explain old and uncover new non-renormalization theorems for theories including gravitons. We provide complete results for the one-loop gravitational renormalization of a generic minimally coupled gauge theory with scalars and fermions and all orders in MPl, as well as for the renormalization of dimension-six operators including at least one graviton, all up to four external particles.

Renormalization and mixing of staple-shaped Wilson line operators on the lattice revisited

Transverse-momentum-dependent parton distribution functions and wave functions (TMDPDFs/TMDWFs) can be extracted from lattice calculations of appropriate Euclidean matrix elements of staple-shaped Wilson line operators. We investigate the mixing pattern of such operators under lattice renormalization using symmetry considerations. We perform an analysis for operators with all Dirac structures, which reveals mixings that are not present in one-loop lattice perturbation theory calculations. We also present the relevant one-loop matching in a renormalization scheme that does not introduce extra non-perturbative effects at large distances, both for the TMDPDFs and for the TMDWFs. Our results have the potential to greatly facilitate numerical calculations of TMDPDFs and TMDWFs on the lattice.

Gravitational waves from first-order phase transitions in Majoron models of neutrino mass

We show how the generation of right-handed neutrino masses in Majoron models may be associated with a first-order phase transition and accompanied by the production of a stochastic background of gravitational waves (GWs). We explore different energy scales with only renormalizable operators in the effective potential. If the phase transition occurs above the electroweak scale, the signal can be tested by future interferometers. We consider two possible energy scales for phase transitions below the electroweak scale. If the phase transition occurs at a GeV, the signal can be tested at LISA and provide a complementary cosmological probe to right-handed neutrino searches at the FASER detector. If the phase transition occurs below 100 keV, we find that the peak of the GW spectrum is two or more orders of magnitude below the putative NANOGrav GW signal at low frequencies, but well within reach of the SKA and THEIA experiments. We show how searches of very low frequency GWs are motivated by solutions to the Hubble tension in which ordinary neutrinos interact with the dark sector. We also present general calculations of the phase transition temperature and Euclidean action that apply beyond Majoron models.

Scale without conformal invariance in membrane theory

We investigate the relation between dilatation and conformal symmetries in the statistical mechanics of flexible crystalline membranes. We analyze, in particular, a well-known model which describes the fluctuations of a continuum elastic medium embedded in a higher-dimensional space. In this theory, the renormalization group flow connects a non-interacting ultraviolet fixed point, where the theory is controlled by linear elasticity, to an interacting infrared fixed point. By studying the structure of correlation functions and of the energy-momentum tensor, we show that, in the infrared, the theory is only scale-invariant: the dilatation symmetry is not enhanced to full conformal invariance. The model is shown to present a non-vanishing virial current which, despite being non-conserved, maintains a scaling dimension exactly equal to D−1, even in presence of interactions. We attribute the absence of anomalous dimensions to the symmetries of the model under translations and rotations in the embedding space, which are realized as shifts of phonon fields, and which protect the renormalization of several non-invariant operators. We also note that closure of a symmetry algebra with both shift symmetries and conformal invariance would require, in the hypothesis that phonons transform as primary fields, the presence of new shift symmetries which are not expected to hold on physical grounds. We then consider an alternative model, involving only scalar fields, which describes effective phonon-mediated interactions between local Gaussian curvatures. The model is described in the ultraviolet by two copies of the biharmonic theory, which is conformal, but flows in the infrared to a fixed point which we argue to be only dilatation-invariant.

NLO fragmentation functions for a quark into a spin-singlet quarkonium: same flavor case

In the paper, we calculate the fragmentation functions for c → ηc and b → ηb up to next-to-leading-order (NLO) QCD accuracy. The ultraviolet divergences in the real corrections are removed through operator renormalization under the modified min- imal subtraction scheme. We then obtain the fragmentation functions {D}_{cto {eta}_c} (z, μF) and {D}_{bto {eta}_b} (z, μF) up to NLO QCD accuracy, which are presented as figures and fitting functions. The numerical results show that the NLO corrections are significant. The sensitives of the fragmentation functions to the renormalization scale and the factorization scale are analyzed explicitly.

Counting Yang-Mills Instantons by Surface Operator Renormalization Group Flow

We show that the nonperturbative dynamics of N=2 super-Yang-Mills theories in a self-dual Ω background and with arbitrary simple gauge group is fully determined by studying renormalization group equations of vacuum expectation values of surface operators generating one-form symmetries. The corresponding system of equations is a nonautonomous Toda chain, the time being the renormalization group scale. We obtain new recurrence relations which provide a systematic algorithm computing multi-instanton corrections from the tree-level one-loop prepotential as the asymptotic boundary condition of the renormalization group equations. We exemplify by computing the E_{6} and G_{2} cases up to two instantons.

ALP \u2014 SMEFT interference

The Standard Model Effective Field Theory (SMEFT) offers a powerful theoretical framework for parameterizing the low-energy effects of heavy new particles with masses far above the scale of electroweak symmetry breaking. Additional light degrees of freedom extend the effective theory. We show that light new particles that are weakly coupled to the SM via non-renormalizable interactions induce non-zero Wilson coefficients in the SMEFT Lagrangian via renormalization-group evolution. For the well-motivated example of axions and axion-like particles (ALPs) interacting with the SM via classically shift-invariant dimension-5 interactions, we calculate how these interactions contribute to the one-loop renormalization of the dimension-6 SMEFT operators, and how this running sources additional contributions to the Wilson coefficients on top of those expected from heavy new states. As an application, we study the ALP contributions to the magnetic dipole moment of the top quark and comment on implications of electroweak precision constraints on ALP couplings.

Composite operators in Toverline{T} -deformed free QFTs

We study perturbative renormalization of the composite operators in the Toverline{T} -deformed two-dimensional free field theories. The pattern of renormalization for the stress-energy tensor is different in the massive and massless cases. While in the latter case the canonical stress tensor is not renormalized up to high order in the perturbative expansion, in the massive theory there are induced counterterms at linear order. For a massless theory our results match the general formula derived recently in [1].

Fragmentation functions for a quark into a spin-singlet quarkonium: Different flavor case

In the paper, we calculate the fragmentation functions for a quark to fragment into a spin-singlet quarkonium, where the flavor of the initial quark is different from that of the constituent quark in the quarkonium. The ultraviolet divergences in the phase space integral are removed through the operator renormalization under the modified minimal subtraction scheme. The fragmentation function $D_{q \to \eta_Q}(z,\mu_F)$ is expressed as a two-dimensional integral. Numerical results for the fragmentation functions of a light quark or a bottom quark to fragment into the $\eta_c$ are presented. As an application of those fragmentation functions, we study the processes $Z \to \eta_c+q\bar{q}g(q=u,d,s)$ and $Z \to \eta_c+b\bar{b}g$ under the fragmentation and the direct nonrelativistic QCD approaches.

Two-loop anomalous dimensions of QCD operators up to dimension-sixteen and Higgs EFT amplitudes

We consider two-loop renormalization of high-dimensional Lorentz scalar operators in the gluonic sector of QCD. These operators appear also in the Higgs effective theory obtained by integrating out the top quark loop in the gluon fusion process. We first discuss the classification of operators and how to construct a good set of basis using both off-shell field theory method and on-shell form factor formalism. To study loop corrections, we apply efficient unitarity-IBP strategy and compute the two-loop minimal form factors of length-3 operators up to dimension sixteen. From the UV divergences of form factor results, we extract the renormalization matrices and analyze the operator mixing behavior in detail. The form factors we compute are also equivalent to Higgs plus three-gluon amplitudes that capture high-order top mass corrections in Higgs EFT. We obtain the analytic finite remainder functions which exhibit several universal transcendentality structures.

Two-loop matching of renormalizable operators: general considerations and applications

Low-energy effective field theories (EFT) encode information about the physics at high energies — i.e., the high-energy theory (HET). To extract this information the EFT and the HET have to be matched to each other. At the one-loop level, general results for the matching of renormalizable operators have already been obtained in the literature. In the present paper, we take a step towards a better understanding of renormalizable operator matching at the two-loop level: focusing on the diagrammatic method, we discuss in detail the various contributions to two-loop matching conditions and compare different approaches to derive them. Moreover, we discuss which observables are best suited for the derivation of matching conditions. As a concrete application, we calculate the mathcal{O}left({alpha}_t{alpha}_sright) and mathcal{O}left({alpha}_t^2right) matching conditions of the scalar four-point couplings between the Standard Model (SM) and the Two-Higgs-Doublet Model (THDM) as well as the THDM and the Minimal Supersymmetric Standard Model (MSSM). We use the derived formulas to improve the prediction of the SM-like Higgs mass in the MSSM using the THDM as EFT.

Two-loop renormalization and mixing of gluon and quark energy-momentum tensor operators

In this paper, we present one- and two-loop results for the renormalization of the gluon and quark gauge-invariant operators which appear in the definition of the QCD energy-momentum tensor, in dimensional regularization. To this end, we consider a variety of Green's functions with different incoming momenta. We identify the set of twist-2 symmetric traceless and flavor singlet operators which mix among themselves and we calculate the corresponding mixing coefficients for the nondiagonal components. We also provide results for some appropriate regularization-independent (RI')-like schemes, which address this mixing, and we discuss their application to nonperturbative studies via lattice simulations. Finally, we extract the one- and two-loop expressions of the conversion factors between the proposed RI' and the MSbar schemes. From our results regarding the MSbar-renormalized Green's functions, one can easily derive conversion factors relating numerous variants of RI'-like schemes to MSbar. To make our results easily accessible, we also provide them as Supplemental Material, in the form of a Mathematica input file and, also, an equivalent text file.

Power expansion for heavy quarkonium production at next-to-leading order in e+e\u2212 annihilation

We study heavy quarkonium production associated with gluons in e+e− annihilation as an illustration of the perturbative QCD (pQCD) factorization approach, which incorporates the first nonleading power in the energy of the produced heavy quark pair. We show how the renormalization of the four-quark operators that define the heavy quark pair fragmentation functions using dimensional regularization induces “evanescent” operators that are absent in four dimensions. We derive closed forms for short-distance coefficients for quark pair production to next-to-leading order ( {alpha}_s^2 ) in the relevant color singlet and octet channels. Using non-relativistic QCD (NRQCD) to calculate the heavy quark pair fragmentation functions up to v4 in the velocity expansion, we derive analytical results for the differential energy fraction distribution of the heavy quarkonium. Calculations for {}^3{S}_1^{left[1right]} and {}^1{S}_0^{left[8right]} channels agree with analogous NRQCD analytical results available in the literature, while several color-octet calculations of energy fraction distributions are new. We show that the remaining corrections due to the heavy quark mass fall off rapidly in the energy of the produced state. To explore the importance of evolution at energies much larger than the mass of the heavy quark, we solve the renormalization group equation perturbatively to two-loop order for the {}^1{S}_0^{left[8right]} case.

Renormalization of higher-dimensional operators from on-shell amplitudes

On-shell amplitude methods allow to derive one-loop renormalization effects from just tree-level amplitudes, with no need of loop calculations. We derive a simple formula to obtain the anomalous dimensions of higher-dimensional operators from a product of tree-level amplitudes. We show how this works for dimension-6 operators of the Standard Model, providing explicit examples of the simplicity, elegance and efficiency of the method. Many anomalous dimensions can be calculated from the same Standard Model tree-level amplitude, displaying the attractive recycling aspect of the on-shell method. With this method, it is possible to relate anomalous dimensions that in the Feynman approach arise from very different diagrams, and obtain non-trivial checks of their relative coefficients. We compare our results to those in the literature, where ordinary methods have been applied.

Study of the renormalization of BRST invariant local composite operators in the U(1) Higgs model

The renormalization properties of two local BRST invariant composite operators, $(O,V_\mu)$, corresponding respectively to the gauge invariant description of the Higgs particle and of the massive gauge vector boson, are scrutinized in the $U(1)$ Higgs model by means of the algebraic renormalization setup. Their renormalization $Z$'s factors are explicitly evaluated at one-loop order in the $\overline{\text{MS}}$ scheme by taking into due account the mixing with other gauge invariant operators. In particular, it turns out that the operator $V_\mu$ mixes with the gauge invariant quantity $\partial_\nu F_{\mu\nu}$, which has the same quantum numbers, giving rise to a $2 \times 2$ mixing matrix. Moreover, two additional powerful Ward identities exist which enable us to determine the whole set of $Z$'s factors entering the $2 \times 2$ mixing matrix as well as the $Z$ factor of the operator $O$ in a purely algebraic way. An explicit check of these Ward identities is provided. The final setup obtained allows for computing perturbatively the full renormalized result for any $n$-point correlation function of the scalar and vector composite operators.

Nonperturbative renormalization of staple-shaped Wilson line operators in lattice QCD

Quark bilinear operators with staple-shaped Wilson lines are used to study transverse-momentum-dependent parton distribution functions (TMDPDFs) from lattice quantum chromodynamics (QCD). Here, the renormalization factors for the isovector operators, including all mixings between operators with different Dirac structures, are computed nonperturbatively in the regularization-independent momentum subtraction scheme for the first time. This study is undertaken in quenched QCD with three different lattice spacings. With Wilson flow applied to the gauge fields in the calculations, the operator mixing pattern due to chiral symmetry breaking with the lattice regularization is found to be significantly different from that predicted by one-loop lattice perturbation theory calculations. These results constitute a critical step towards the systematic extraction of TMDPDFs from lattice QCD.