One-loop Renormalization Group Equations Research Articles

Renormalization group equations for the SMEFT operators up to dimension seven

In this paper, we propose a Green’s basis and also a new physical basis for dimension-seven (dim-7) operators, which are suitable for the matching of ultraviolet models onto the Standard Model effective field theory (SMEFT) and the deviation of renormalization group equations (RGEs) for dim-7 operators in the SMEFT. The reduction relations to convert operators in the Green’s basis to those in the physical basis are achieved as well, where some redundant dim-6 operators in the Green’s basis are involved if the dim-5 operator exists. Working in these two bases for dim-7 operators and with the help of the reduction relations, we work out the one-loop RGEs resulting from the mixing among different dimensional operators for the dim-5 and dim-7 operators up to O\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{O} $$\\end{document}(Λ−3) in the SMEFT. These new results complete the previous results for RGEs of the dim-5 and dim-7 operators and hence can be used for a consistent one-loop analysis of the SMEFT at O\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{O} $$\\end{document}(Λ−3).

Complete one-loop renormalization-group equations in the seesaw effective field theories

In this paper, we derive the complete set of one-loop renormalization-group equations (RGEs) for the operators up to dimension-six (dim-6) in the seesaw effective field theories (SEFTs). Two kinds of contributions to those RGEs are identified, one from double insertions of the dimension-five (dim-5) Weinberg operator and the other from single insertions of the tree-level dim-6 operators in the SEFTs. A number of new results are presented. First, as the dim-5 Weinberg operator is unique in the standard model effective field theory (SMEFT), its contributions to the RGEs for the SEFTs are equally applicable to the SMEFT. We find the full contributions from the Weinberg operator to one-loop RGEs in the SMEFT, correcting the results existing in previous works, and confirm that those from dim-6 operators are consistent with the results in the literature. Second, in the type-I SEFT, we give the explicit expressions of the RGEs of all the physical parameters involved in the charged- and neutral-current interactions of leptons. Third, the RGEs are numerically solved to illustrate the running behaviors of the non-unitary parameters, mixing angles and CP-violating phases in the non-unitary leptonic flavor mixing matrix. Together with the one-loop matching results of the dim-5 and dim-6 operators and their Wilson coefficients, the present work has established a self-consistent framework up to dim-6 to investigate low-energy phenomena of three types of seesaw models at the one-loop level.

The translational μ-τ reflection symmetry of Majorana neutrinos

The present neutrino oscillation data allow [Formula: see text] (or [Formula: see text]) for the neutrino mass spectrum and support [Formula: see text] and [Formula: see text] as two good approximations for the PMNS lepton flavor mixing matrix [Formula: see text]. We show that these intriguing possibilities can be a very natural consequence of the translational[Formula: see text]-[Formula: see text] reflection symmetry — the effective Majorana neutrino mass term keeps invariant under the transformations [Formula: see text], [Formula: see text] and [Formula: see text] (for [Formula: see text] or 3), where [Formula: see text] is the charge conjugation of a constant spinor field [Formula: see text]. By extending such a working flavor symmetry to the canonical seesaw mechanism at a superhigh-energy scale, we calculate its soft breaking effects at the electroweak scale by using the one-loop renormalization-group equations.

Matchmakereft: automated tree-level and one-loop matching

We introduce matchmakereft, a fully automated tool to compute the tree-level and one-loop matching of arbitrary models onto arbitrary effective theories. Matchmakereft performs an off-shell matching, using diagrammatic methods and the background field method (BFM) when gauge theories are involved. The large redundancy inherent to the off-shell matching together with explicit gauge invariance offers a significant number of non-trivial checks of the results provided. These results are given in the physical basis but several intermediate results, including the matching in the Green basis before and after canonical normalization, are given for flexibility and the possibility of further cross-checks. As a non-trivial example we provide the complete matching in the Warsaw basis up to one loop of an extension of the Standard Model with a charge $-1$ vector-like lepton singlet. Matchmakereft has been built with generality, flexibility and efficiency in mind. These ingredients allow matchmakereft to have many applications beyond the matching between models and effective theories. Some of these applications include the one-loop renormalization of arbitrary theories (including the calculation of the one-loop renormalization group equations for arbitrary theories); the translation between different Green bases for a fixed effective theory or the check of (off-shell) linear independence of the operators in an effective theory. All these applications are performed in a fully automated way by matchmakereft.

The smallest neutrino mass revisited

As is well known, the smallest neutrino mass turns out to be vanishing in the minimal seesaw model, since the effective neutrino mass matrix Mν is of rank two due to the fact that only two heavy right-handed neutrinos are introduced. In this paper, we point out that the one-loop matching condition for the effective dimension-five neutrino mass operator can make an important contribution to the smallest neutrino mass. By using the available one-loop matching condition and two-loop renormalization group equations in the supersymmetric version of the minimal seesaw model, we explicitly calculate the smallest neutrino mass in the case of normal neutrino mass ordering and find m1 ∈ [10−8, 10−10] eV at the Fermi scale ΛF = 91.2 GeV, where the range of m1 results from the uncertainties on the choice of the seesaw scale ΛSS and on the input values of relevant parameters at ΛSS.

Higgs-electroweak chiral Lagrangian: One-loop renormalization group equations

Starting from the one-loop divergences we obtained previously, we work out the renormalization of the Higgs-electroweak chiral Lagrangian explicitly and in detail. This includes the renormalization of the lowest-order Lagrangian, as well as the decomposition of the remaining divergences into a complete basis of next-to-leading-order counterterms. We provide the list of the corresponding beta functions. We show how our results match the one-loop renormalization of some of the dimension-6 operators in SMEFT. We further point out differences with related work in the literature and discuss them. As an application of the obtained results, we evaluate the divergences of the vacuum expectation value of the Higgs field at one loop and show that they can be appropriately removed by the corresponding renormalization. We also work out the finite renormalization required to keep the no-tadpole condition on the Higgs field at one loop.

New scenario for aligned Higgs couplings originated from the twisted custodial symmetry at high energies

We investigate a new scenario of the two Higgs doublet model, where the current experimental data for the electroweak rho parameter and those for the Higgs boson couplings can be simultaneously explained. In this scenario, the two Higgs doublet model is supposed to be a low energy effective theory up to a high energy scale Λ, above which a fundamental theory should appear. It is assumed that the Higgs potential respects a global symmetry at Λ (the twisted custodial symmetry), which is to be given as a consequence of the global symmetry structure of the fundamental theory above Λ. By the analysis using one-loop renormalization group equations, the above experimental data can be explained in a natural way even when the masses of the extra Higgs bosons are near the electroweak scale. We also discuss the predictions on the mass spectrum of the additional Higgs bosons and also those on the coupling constants of the standard-model-like Higgs boson, which make it possible to test this scenario at the current and future collider experiments.

Integral solutions to the one-loop renormalization-group equations for lepton flavor mixing parameters and the Jarlskog invariant

Working in the basis where the charged-lepton Yukawa matrix is diagonal and making the τ-dominance approximations, we analytically derive integral solutions to the one-loop renormalization-group equations (RGEs) for neutrino masses, flavor mixing angles, CP-violating phases and the Jarlskog invariant under the standard parametrization of the PMNS matrix in the standard model or its minimal supersymmetric extension for both Majorana and Dirac neutrinos. With these integral solutions, we carry out numerical calculations to investigate the RGE running of lepton flavor mixing parameters and the Jarlskog invariant, and also compare these integral solutions with the exact results obtained by numerically solving the one-loop RGEs. It is shown that these integral solutions coincide with the exact results and can well describe the evolution of lepton flavor mixing parameters and the Jarlskog invariant in most cases. Some important features of our integral solutions and the evolution behaviours of relevant flavor parameters are also discussed in detail both analytically and numerically.

RGE-induced mu -tau symmetry breaking: an analysis of the latest T2K results

The T2K collaboration has recently reported their results, which gives the best-fit values of the atmospheric mixing angle sin ^2 theta _{23} = 0.53 and the Dirac CP-violating phase delta = -1.89 for normal neutrino mass ordering. We give a possible theoretical origin of such values based on the mu -tau reflection symmetry. It has been found that the breaking of such symmetry using one-loop renormalization-group equations (RGEs) in the framework of minimal supersymmetric standard model can fit well with the latest T2K results and the recent global fits. To make a quantitative analysis, we have included for the first time the complete dataset of oscillation, beta decay, neutrinoless double-beta decay and cosmological observations in comparing the theory to experiments. We also further examine the importance of such breaking patterns for neutrinoless double-beta decay experiments.

Evaluation of neutrinoless double beta decay: QCD running to sub-GeV scales

We evaluate QCD effects in the neutrinoless double beta ($0\nu\beta\beta$) decay, originating from new physics short-range mechanism in the form of five dimension-9 operators. For this, we employ the one-loop and two-loop renormalization group equations (RGEs) for the corresponding Wilson coefficients, performing the RGE-evolution from the new physics scales (estimated as $\Lambda ~ \sim 10^2$ GeV) to the typical spacelike $0\nu\beta\beta$-scale $Q \sim 0.1$ GeV. Since the latter scale is clearly nonperturbative, we apply various infrared-safe (IR-safe) variants of QCD where the running coupling has no Landau singularities at low spacelike $Q$. We point out that the correct treatment of the IR-safe analogs of the (noninteger) powers of the couplings is important. It turns out that in most cases of the considered operators the resulting QCD effects can be significant in this process, i.e., can be stronger than the effects of the present uncertainties in the nuclear matrix elements.

Multicomponent dark matter in extended U(1)B−L: neutrino mass and high scale validity

Standard Model with right handed neutrinos charged under additional U(1)B−L gauge symmetry offer solutions to both dark matter (DM) problem and neutrino mass generation, although constrained severely from relic density, direct search and Higgs vacuum stability. We therefore investigate a multicomponent DM scenario augmented by an extra inert scalar doublet, that is neutral under U(1)B−L, which aids to enlarge parameter space allowed by DM constraints and Higgs vacuum stability. The lightest right-handed neutrino and the CP-even inert scalar are taken as the dark matter candidates and constitute a two component dark matter framework as they are rendered stable by an unbroken ℤ2 × ℤ2′ symmetry. DM-DM conversion processes turn out crucial to render requisite relic abundance in mass regions of the RH neutrino that do not appear in the stand-alone U(1)B−L scenario. In addition, the one-loop renormalisation group (RG) equations in this model demonstrate that the electroweak (EW) vacuum can be stabilised till ∼ 109 GeV in a parameter region compatible with the observed relic, the direct detection bound and other relevant constraints. We finally comment on the possibility of including the freeze-in mechanism in the same set-up.

A direct link between unflavored leptogenesis and low-energy CP violation via the one-loop quantum corrections

In the type-I seesaw mechanism the Casas-Ibarra (CI) parametrization provides a convenient description of the Dirac neutrino mass matrix in terms of the light and heavy Majorana neutrino masses, the lepton flavor mixing matrix U and an unknown complex orthogonal matrix O. If O is assumed to be real, it will be impossible to generate unflavored thermal leptogenesis via the lepton-number-violating and CP-violating decays of the lightest heavy Majorana neutrino. We find that this observation can be invalidated after small but unavoidable quantum corrections to the CI parametrization are taken into account with the help of the one-loop renormalization-group equations (RGEs) between the seesaw and electroweak scales. We illustrate a novel and viable unflavored leptogenesis scenario of this kind based on the RGEs in the seesaw-extended standard model, and show its direct link to the CP-violating phases of U at low energies.

Phase diagram of a two-dimensional dirty tilted Dirac semimetal

We investigate the effects of quenched disorder on a non-interacting tilted Dirac semimetal in two dimensions. Depending on the magnitude of the tilting parameter, the system can have either Fermi points (type-I) or Fermi lines (type-II). In general, there are three different types of disorders for Dirac fermions in two dimensions, namely, the random scalar potential, the random vector potentials along and perpendicular to the tilting direction, and the random mass. We study the effects of weak disorder in terms of the renormalization group, which is performed by integrating out the modes with large energies, instead of large momenta. Since the parametrization of the low-energy degrees of freedom depends on the structure of the Fermi surface, the resulting one-loop renormalization-group equations depend on the type of tilted Dirac fermions. Whenever the disorder is a marginal perturbation, we examine its role on low-energy physics by a mean-field approximation of the replica field theory or the first-order Born approximation. Based on our analysis, we suggest the phase diagrams of a two-dimensional tilted Dirac fermion in the presence of different types of disorder.

Impact of fermionic electroweak multiplet dark matter on vacuum stability with one-loop matching

We investigate the effect of fermionic electroweak multiplet dark matter models on the stability of the electroweak vacuum using two-loop renormalization group equations (RGEs) and one-loop matching conditions. Such a treatment is crucial to obtain reliable conclusions, compared with one-loop RGEs and tree-level matching conditions. In addition, we find that the requirement of perturbativity up to the Planck scale would give strong and almost mass-independent constraints on the Yukawa couplings in the dark sector. We also evaluate these models via the idea of finite naturalness for the Higgs mass fine-tuning issue.

Finite-size scaling for four-dimensional Higgs-Yukawa model near the Gaussian fixed point

We analyse finite-size scaling behaviour of a four-dimensional Higgs-Yukawa model near the Gaussian infrared fixed point. Through improving the mean-field scaling laws by solving one-loop renormalisation group equations, the triviality property of this model can be manifested in the volume-dependence of moments of the scalar-field zero mode. The scaling formulae for the moments are derived in this work with the inclusion of the leading-logarithmic corrections. To test these formulae, we confront them with data from lattice simulations in a simpler model, namely the O(4) pure scalar theory, and find numerical evidence of good agreement. Our results of the finite-size scaling can in principle be employed to establish triviality of Higgs-Yukawa models, or to search for alternative scenarios in studying their fixed-point structure, if sufficiently large lattices can be reached.

Spontaneous μ-τ Reflection Symmetry Breaking in Neutrino Phenomenology

The latest global analysis of neutrino oscillation data indicates that the normal neutrino mass ordering is favored over the inverted one at the 3 σ level. The best-fit values of the largest neutrino mixing angle θ 23 and the Dirac CP-violating phase δ are located in the higher octant and the third quadrant, respectively. We show that these experimental trends can be naturally explained by the μ - τ reflection symmetry breaking, triggered by the one-loop renormalization-group equations (RGEs) running from a superhigh energy scale down to the electroweak scale in the framework of the minimal supersymmetric standard model (MSSM). The complete parameter space is numerically explored for both the Majorana and Dirac cases, by allowing the smallest neutrino mass m 1 and the MSSM parameter tan β to vary within their reasonable ranges.

Correlation of normal neutrino mass ordering with upper octant of θ23 and third quadrant of δ via RGE-induced μ-τ symmetry breaking** Supported by the National Natural Science Foundation of China (11775231, 11775232)

The recent global analysis of three-flavor neutrino oscillation data indicates that the normal neutrino mass ordering is favored over the inverted one at the 3σ level, and the best-fit values of the largest neutrino mixing angle θ23 and the Dirac CP-violating phase δ are located in the higher octant and third quadrant, respectively. We show that all these important issues can be naturally explained by the μ-τ reflection symmetry breaking of massive neutrinos from a superhigh energy scale down to the electroweak scale owing to the one-loop renormalization-group equations (RGEs) in the minimal supersymmetric standard model (MSSM). The complete parameter space is explored for the first time in both the Majorana and Dirac cases, by allowing the smallest neutrino mass m1 and the MSSM parameter tanβ to vary within their reasonable regions.

Naumov- and Toshev-like relations in the renormalization-group evolution of quarks and Dirac neutrinos**Supported by the National Natural Science Foundation of China (11775231) and (11775232), the National Youth Thousand Talents Program, and the CAS Center for Excellence in Particle Physics

In an analytical way of studying matter effects on neutrino oscillations, the Naumov and Toshev relations have been derived to respectively link the Jarlskog invariant of CP violation and the Dirac phase in the standard parametrization of the 3 × 3 flavor mixing matrix to their matter-corrected counterparts. Here we show that there exist similar relations for Dirac neutrinos and charged leptons evolving with energy scales via the one-loop renormalization-group (RG) equations in the tau-dominance approximation, and for the running behaviors of up- and down-type quarks in the top-dominance approximation, provided a different parametrization is taken into account.

Renormalons in a general Quantum Field Theory

We generalize the concept of Borel resummability and renormalons to a quantum field theory with an arbitrary number of fields and couplings, starting from the known notion based on the running coupling constants. An approach to identify the renormalons is provided by exploiting an analytic solution of the generic one-loop renormalization group equations in multi-field theories. Methods to evaluate the regions in coupling space where the theory is resummable are described. The generalization is then illustrated in a toy model with two coupled scalar fields, representing the simplest extension of the one-field analysis presented in the seminal works of the subject. Furthermore, possible links to realistic theories are briefly discussed.

Signs and stability in higher-derivative gravity

Perturbatively renormalizable higher-derivative gravity in four space–time dimensions with arbitrary signs of couplings has been considered. Systematic analysis of the action with arbitrary signs of couplings in Lorentzian flat space–time for no-tachyons, fixes the signs. Feynman [Formula: see text] prescription for these signs further grants necessary convergence in path-integral, suppressing the field modes with large action. This also leads to a sensible wick rotation where quantum computation can be performed. Running couplings for these sign of parameters make the massive tensor ghost innocuous leading to a stable and ghost-free renormalizable theory in four space–time dimensions. The theory has a transition point arising from renormalization group (RG) equations, where the coefficient of [Formula: see text] diverges without affecting the perturbative quantum field theory (QFT). Redefining this coefficient gives a better handle over the theory around the transition point. The flow equations push the flow of parameters across the transition point. The flow beyond the transition point is analyzed using the one-loop RG equations which shows that the regime beyond the transition point has unphysical properties: there are tachyons, the path-integral loses positive definiteness, Newton’s constant [Formula: see text] becomes negative and large, and perturbative parameters become large. These shortcomings indicate a lack of completeness beyond the transition point and need of a nonperturbative treatment of the theory beyond the transition point.