We obtain explicitly the renormalization group equations for the quark mass matrices in terms of a set of rephasing invariant parameters. For a range of assumed high energy values for the mass ratios and mixing parameters, they are found to evolve rapidly and develop hierarchies as the energy scale decreases. To achieve the experimentally observed high degree of hierarchy, however, the introduction of new models with specific properties becomes necessary.
The one-loop renormalization-group equations (RGEs) running behavior of quark and lepton mass matrices with general structures are studied simultaneously. Suppose the non-linear terms of RGEs are dominated by the Yukawa couplings of top quark and τ lepton, the unitary matrices that diagonalize the mass matrices of up-type quark and charged lepton in Hermitian basis are found as invariant with energy scale. Based on this result, we can decouple the RGEs and obtain the RGE corrected mass matrices of fermion. As examples, we consider the renormalization-group evolution of fermion matrices with fo…
We consider favourable extensions of the standard model (SM) where the lepton sector contains Majorana neutrinos with vanishing left-handed mass terms, thus allowing for the see-saw mechanism to operate, and propose physical on-mass-shell (OS) renormalization conditions for the lepton mixing matrices that comply with ultraviolet finiteness, gauge-parameter independence, and (pseudo)unitarity. A crucial feature is that the texture zero in the neutrino mass matrix is preserved by renormalization, which is not automatically the case for possible generalizations of existing renormalization prescri…
We generalize a recently proposed on-shell approach to renormalize the Cabibbo-Kobayashi-Maskawa quark-mixing matrix to the case of an extended leptonic sector that includes Dirac and Majorana neutrinos in the framework of the seesaw mechanism. An important property of this formulation is the gauge independence of both the renormalized and bare lepton mixing matrices. Also, the texture zero in the neutrino mass matrix is preserved.
If the scale dependence of a Yukawa matrix is assumed to be determined entirely by the dominant 33-element, then the renormalization group equations can be expressed in terms of two separate equations: a differential equation for the 33-coupling, and, an algebraic equation for the scale-independent 3x3 matrix that is found to have only two non-trivial, hierarchical, solutions with eigenvalues (0,0,1) and (0,1,1). The mass matrices are constructed from these solutions by rotating them first by the experimentally known mixing matrices-the CKM for quarks and charged leptons, and the CKM-analog fo…
We present an on-shell renormalization scheme for the neutralino and chargino mass matrices for two R-parity violating models, namely bilinear R-parity violation and the munuSSM with one right-handed neutrino superfield. We discuss in both models how to obtain neutrino masses and mixing angles correctly as well as differences to the existing DRbar calculations. Both models predict correlations between neutrino mixing angles and ratios of R-parity violating decays such as neutralino_1 -> l W which can be tested for example at the LHC. We point out that there are cases where one has to include t…
We study the renormalization of normal mixing matrices, which includes hermitian and unitary matrices as particular cases. We give a minimal, multiplicative parametrization of counterterms, and compute the renormalized Lagrangian to one-loop order in several models with N species of fermions, both in on-shell and MS schemes. In on-shell scheme the mass-degenerate case is considered separately.
The renormalization group equations (RGEs) of the mass matrices of quarks and leptons in a SO(10) model with two Higgs scalars in the Yukawa coupling are studied. This model is the minimal model of SUSY and non-SUSY SO(10) GUT which can reproduce all the experimental data. Non-SUSY SO(10) GUT model has the intermediate energy phase, Pati-Salam phase, and passes through the symmetry breaking pattern, $SO(10) \to SU(2)_L \times SU(2)_R \times SU(4)_C \to SU(2)_L \times U(1)_Y \times SU(3)_C$. Though minimal, it has, after the Pati-Salam phase, four Higgs doublets in Yukawa interactions. We consi…
We write down the renormalization-group equations for the Yukawa-coupling matrices in a general multi-Higgs-doublet model. We then assume that the matrices of the Yukawa couplings of the various Higgs doublets to right-handed fermions of fixed quantum numbers are all proportional to each other. We demonstrate that, in the case of the two-Higgs-doublet model, this proportionality is preserved by the renormalization-group running only in the cases of the standard type-I, II, X, and Y models. We furthermore show that a similar result holds even when there are more than two Higgs doublets: the Yuk…
We report on an explicit on-shell framework to renormalize the fermion-flavour mixing matrices in the Standard Model and its extensions, at one-loop level. It is based on a novel procedure to separate the external-leg mixing corrections into gauge-independent self-mass and gauge-dependent wave-function renormalization contributions.
We review two recently proposed on-shell schemes for the renormalization of the Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix in the Standard Model. One first constructs gauge-independent mass counterterm matrices for the up- and down-type quarks complying with the hermiticity of the complete mass matrices. Diagonalization of the latter then leads to explicit expressions for the CKM counterterm matrix, which are gauge independent, preserve unitarity, and lead to renormalized amplitudes that are non-singular in the limit in which any two quarks become mass degenerate. One of the schemes a…
It was recently pointed out that the on-shell renormalization of the Cabibbo-Kobayashi-Maskawa (CKM) matrix in the method by Denner and Sack causes a gauge parameter dependence of the amplitudes. We analyze the gauge dependence of the on-shell renormalization of the mixing matrices both for fermions and scalars in general cases, at the one-loop level. We then show that this gauge dependence can be avoided by fixing the counterterms for the mixing matrices in terms of the off-diagonal wave function corrections for fermions and scalars after a rearrangement, in a similar manner to the pinch tech…
Strong gauge and top-Yukawa couplings predicted in the presence of extra dimensions lead to a trickle-down effect of the top-coupling through the renormalization group equations for the quark Yukawa matrices. The matrix elements for the u and d-quarks get progressively smaller as one moves away from the dominant (33)-element revealing a hierarchical pattern for the mass matrices.
We derive the renormalization group equations (RGE) for the flavour coupling matrices of the effective dimension-five operators which yield Majorana neutrino masses in the multi-Higgs-doublet Standard Model; in particular, we consider the case where two different scalar doublets occur in those operators. We also write down the RGE for the scalar-potential quartic couplings and for the Yukawa couplings of that model, in the absence of quarks. As an application of the RGE, we consider two models which, based on a mu-tau interchange symmetry, predict maximal atmospheric neutrino mixing, together …
In order to obtain proper wave-function renormalization constants for unstable fermion and consist with Breit-Wigner formula in the resonant region, We have assumed an extension of the LSZ reduction formula for unstable fermion and adopted on-shell mass renormalization scheme in the framework of without field renormaization. The comparison of gauge dependence of physical amplitude between on-shell mass renormalization and complex-pole mass renormalization has been discussed. After obtaining the fermion wave-function renormalization constants, we extend them to two matrices in order to include …
We continue to investigate the minimal SO(10)xU(1)_H model which we proposed recently. Renormalization group analysis of the model results in natural predictions of quark-lepton masses and Kobayashi-Maskawa matrix along with neutrino mixings adequate for solar neutrino oscillation.
We study realization of the democratic form of Yukawa matrices by infrared fixed points. We investigate renormalization-group flows of Yukawa couplings in models with a single Yukawa matrix for three families, and up and down-sector Yukawa matrices. It is found that each model has its certain pattern of renormalization-group flows of Yukawa matrices. We apply them to the charged lepton sector and quark sector, and show that realistic results for the second and third families are obtained with natural initial conditions, e.g. $(V_{MNS})_{23} \sim 2/\sqrt{6}$, $m_s/m_b \sim V_{cb} \sim m_μ/m_τ$ …
We present the one-loop anomalous dimension matrices for the chiral-even twist-3 (nonsinglet) conformal operators, which govern the scale-dependence of the vector meson light-cone wave functions through the conformal expansion. It is clarified that the constraints from the charge-conjugation invariance and the chirality conservation allow only one independent anomalous dimension matrix for each conformal spin.
We discuss the gauge dependence of fermion mass definition and physical result under the conventional on-shell mass renormalization scheme and the recently proposed pole mass renormalization scheme in standard model. By the two-loop calculations of top quark's mass counterterm and the cross section of physical process $W^{-}W^{+}\to b\bar{b}$ we strictly prove that the on-shell mass renormalization scheme makes fermion mass definition and the physical result gauge dependent at the first time, while we don't find such gauge dependence in the pole mass renormalization scheme. Besides, our calcul…
The results for the three-loop renormalization group equations for all fundamental parameters of the SM Lagrangian are presented. Special attention is paid to the Flavor sector of the SM, which parameterized by general complex non-diagonal Yukawa couplings. Some details of calculation techniques are given. In addition, ambiguities in the beta-functions for the matrix couplings are discussed.
We discuss the gauge dependence of physical parameter's definitions under the on-shell and pole mass renormalization prescriptions. By two-loop-level calculations we prove for the first time that the on-shell mass renormalization prescription makes physical result gauge dependent. On the other hand, such gauge dependence doesn't appear in the result of the pole mass renormalization prescription. Our calculation also implies the difference of the physical results between the two mass renormalization prescriptions cannot be neglected at two-loop level.
We present an explicit on-shell framework to renormalize the Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix at the one-loop level. It is based on a novel procedure to separate the external-leg mixing corrections into gauge-independent self-mass (sm) and gauge-dependent wave-function renormalization contributions, and to adjust non-diagonal mass counterterm matrices to cancel all the divergent sm contributions, and also their finite parts subject to constraints imposed by the hermiticity of the mass matrices. It is also shown that the proof of gauge independence and finiteness of the remai…
The pole mass and the width of the Roper resonance are calculated as functions of the pion mass in the framework of low-energy effective field theory of the strong interactions. We implement a systematic power-counting procedure by applying the complex-mass renormalization scheme.
$QCD$ renormalization for the top-quark mass is calculated in a mass geometrical mean hierarchy, $m_d m_b = m_s^2$ and $m_u m_t = m_c^2$. The physical mass, $m_t(m_t) = 160 {\pm} 50 GeV$ is obtained, which agrees very well with electroweak precision measurement.
We demonstrate that in the mass independent renormalization scheme. the renormalization group equations associated with the unphysical parameters that characterize the renormalization scheme and the mass scale leads to summation that results in a cancellation between the implicit and explicit dependence on these parameters. The resulting perturbative expansion is consequently independent of these arbitrary parameters. We illustrate this by considering R, the cross section for e+e- -> hadrons.
We have investigated the renormalization group running of the pole mass in the multi-Higgs theory in two different types of the gauge fixing conditions. It turns out that the pole mass when expressed in terms of the Lagrangian parameters, is invariant under the renormalization group with the beta and gamma functions of the symmetric phase.
Three-loop renormalization group equations for the Higgs self-coupling and Higgs mass parameter are recalculated in the case of complex Yukawa matrices, which encompass general flavour structure of the Standard Model. In addition, the anomalous dimensions both for the quantum Higgs field and its vacuum expectation value are presented in the $\overline{MS}$-scheme. A numerical study of the latter quantities is carried out for a certain set of initial parameters.
The fermion spectrum in the Standard Model (SM) exhibits hierarchical structures between the eigenvalues of the Yukawa matrices which determine the fermion masses, as well as certain hierarchical patterns in the mixing matrix that describes weak transitions between different fermion generations. A basis-independent description of the SM flavour structure can be given in terms of a complete set of flavour invariants. In this paper, we construct a convenient set of such invariants, and discuss the general form of the renormalization-group equations. We also discuss the simplifications that arise…
We consider a low-energy effective field theory of vector mesons and Goldstone bosons using the complex-mass renormalization. As an application we calculate the mass and the width of the $ρ$ meson.
We compute the three-loop relation between the pole and the minimally subtracted quark mass allowing for virtual effects from a second massive quark. We also consider the analogue effects for the on-shell wave function renormalization constant.
We introduce an on shell renormalization scheme in which the mass parameter of minimal MS scheme is replaced with the pole mass obtained from the loop order expansion of the pole mass in the MS scheme. As a consequence, the quartic coupling constant remains same as that of the MS scheme and the vacuum expectation value gets contributions from the one-particle-irreducible diagrams. We also show the renormalization group invariance of the pole mass in this scheme.
The renormalization evolution of all parameters in the neutrino mass matrix depends only on one variable, the energy scale. This fact, coupled with rephasing considerations, leads to a set of renormalization invariants, correlating the evolution of physical parameters. For the general three flavor case, we obtain these invariants explicitly and discuss their implications.
The renormalization group is used to improve the effective potential of massive ${\rm O}(N)$ symmetric $ϕ^4$ theory. Explicit results are given at the two-loop level.
We consider an effective field theory of unstable particles (resonances) using the complex-mass renormalization. As an application we calculate the masses and the widths of the $ρ$ meson and the Roper resonance.
We consider a scenario for the Quark-Lepton Complementarity relations between mixing angles in which the bi-maximal mixing follows from the neutrino mass matrix. According to this scenario in the lowest order the angle θ_{12} is \sim 1σ(1.5 - 2^\circ) above the best fit point coinciding practically with the tri-bimaximal mixing prediction. Realization of this scenario in the context of the seesaw type-I mechanism with leptonic Dirac mass matrices approximately equal to the quark mass matrices is studied. We calculate the renormalization group corrections to θ_{12} as well as to θ_{13} in the s…
We study the two-loop renormalization group equation for the running gaugino mass in the supersymmetric gauge theory. We find that both in the \msbar and \drbar renormalization schemes, the well-known proportionality of the runnings of the gaugino mass and the gauge coupling constant is violated at the two-loop order, even in models with vector supermultiplets only. Scheme dependence of the running gaugino mass is clarified.
The renormalization group equations for the general 2 by 2, complex, neutrino mass matrix are shown to have exact, analytic solutions. Simple formulas are given for the physical mixing angle, complex phase and mass ratio in terms of their initial values and the energy scales. We also establish a (complex) renormalization invariant relating these parameters. The qualitative features of the physical parameters' renormalization are clearly illustrated in vector field plots. In both the SM and MSSM, maximal mixing is a saddle point.
We study the gauge dependence of the neutrino mass renormalization in a two Higgs doublet model, that is extended with one singlet seesaw neutrino. This model gives only one light neutrino a mass at tree level, while the second light mass is generated at loop level via the interaction with the second Higgs doublet. At one loop level, one neutrino stays massless. We use multiplicative renormalization constants to define counterterms. The renormalized mass parameters are defined as the complex poles of the propagators, using the complex mass scheme for mass renormalization. With this setup, we a…
In this article we present REvolver, a C++ library for renormalization group evolution and automatic flavor matching of the QCD coupling and quark masses, as well as precise conversion between various quark mass renormalization schemes. The library systematically accounts for the renormalization group evolution of low-scale short-distance masses which depend linearly on the renormalization scale and sums logarithmic terms of high and low scales that are missed by the common logarithmic renormalization scale evolution. The library can also be accessed through Mathematica and Python interfaces a…
We describe details of the renormalization of two-loop integrals relevant to the calculation of the nucleon mass in the framework of manifestly Lorentz-invariant chiral perturbation theory using infrared renormalization. It is shown that the renormalization can be performed while preserving all relevant symmetries, in particular chiral symmetry, and that renormalized diagrams respect the standard power counting rules. As an application we calculate the chiral expansion of the nucleon mass to order O(q^6).
The five-loop effective potential and the associated summation of subleading logarithms for O(4) globally-symmetric massless $λϕ^4$ field theory in the Coleman-Weinberg renormalization scheme $\frac{d^4V}{dϕ^4}|_{ϕ= μ} = λ$ (where $μ$ is the renormalization scale) is calculated via renormalization-group methods. An important aspect of this analysis is conversion of the known five-loop renormalization-group functions in the minimal-subtraction (MS) scheme to the Coleman-Weinberg scheme.
We consider the recently developed diagrammatic approach to soft-gluon exponentiation in multiparton scattering amplitudes, where the exponent is written as a sum of webs - closed sets of diagrams whose colour and kinematic parts are entangled via mixing matrices. A complementary approach to exponentiation is based on the multiplicative renormalizability of intersecting Wilson lines, and their subsequent finite anomalous dimension. Relating this framework to that of webs, we derive renormalization constraints expressing all multiple poles of any given web in terms of lower-order webs. We exami…
Using a set of rephasing-invariant variables, it is shown that the renormalization group equations for quark mixing parameters can be written in a form that is compact, in addition to having simple properties under flavor permutation. We also found approximate solutions to these equations if the quark masses are hierarchical or nearly degenerate.
Within the framework of the standard model plus one heavy Higgs triplet, we derive a full set of one-loop renormalization group equations of the neutrino mass matrix and Higgs couplings in both full and effective theories. The explicit RGEs of neutrino masses, flavor mixing angles and CP-violating phases are also obtained, and their non-trivial running behaviors around the Higgs triplet mass threshold are numerically illustrated.
We re-derive the renormalization group equation for the effective coupling of the dimension five operator which corresponds to a Majorana mass matrix for the Standard Model neutrinos. We find a result which differs somewhat from earlier calculations, leading to modifications in the evolution of leptonic mixing angles and CP phases. We also present a general method for calculating beta-functions from counterterms in MS-like renormalization schemes, which works for tensorial quantities.
We examine the perturbative calculation of the inclusive semi-leptonic decay rate $Γ$ for the $b$-quark, using mass-independent renormalization. To finite order of perturbation theory the series for $Γ$ will depend on the unphysical renormalization scale parameter $μ$ and on the particular choice of mass-independent renormalization scheme; these dependencies will only be removed after summing the series to all orders. In this paper we show that all explicit $μ$-dependence of $Γ$, through powers of ln$(μ)$, can be summed by using the renormalization group equation. We then find that this explic…
We illustrate how the reorganization of perturbation theory at finite temperature can be economically cast in terms of the Wilson-Polchinski renormalization methods. We take as an example the old saw of the induced thermal mass of a hot scalar field with a quartic coupling, which we compute to second order in the coupling constant. We show that the form of the result can be largely determined by renormalization-group arguments without the explicit evaluation of Feynman graphs.
We calculate the two-loop renormalization group equations for the running gaugino masses in general SUSY gauge models, improving our previous result. We also study its consequence to the unification of the gaugino masses in the SUSY SU(5) model. The two-loop correction to the one-loop relation $m_i(μ)\proptoα_i(μ)$ is found to be of the order of a few \%.
We employ optimal renormalization group analysis to semi-leptonic $τ$-decay polarization functions and extract the strange quark mass from their moments measured by the ALEPH and OPAL collaborations. The optimal renormalization group makes use of the renormalization group equation of a given perturbation series which then leads to closed form sum of all the renormalization group-accessible logarithms which have reduced scale dependence. Using the latest theoretical inputs we find $m_s(2{\rm GeV}) = 106.70 \pm 9.36~{\rm MeV}$ and $m_s(2{\rm GeV}) = 74.47 \pm 7.77~{\rm MeV}$ for ALEPH and OPAL d…
In terms of a rephasing invariant parametrization, the set of renormalization group equations (RGE) for Dirac neutrino parameters can be cast in a compact and simple form. These equations exhibit manifest symmetry under flavor permutations. We obtain both exact and approximate RGE invariants, in addition to some approximate solutions and examples of numerical solutions.
The one-loop radiative corrections to the Higgs boson potential in the MSSM, originating from the top quark and squark loops, are summed in the leading log approximation using the renormalization group. The RG improved effective potential is minimized and the corrections to the CP-odd and CP-even Higgs boson masses are calculated. The resulting masses exhibit smoother top mass dependence than those calculated without RG summation. We have also found that for preferable values of the top mass the light Higgs mass does not exceed 100 GeV.
The renormalization-group effects on the scenario with nearly degenerate mass and bi-maximal mixing of neutrinos are analyzed. As a consequence, we explicitly show in the SO(3) gauge model that the renormalization-group effects could modify the fitting values of the relevant basic parameters of the model, but leave the nearly bi-maximal mixing scenario to remain stable for reconciling both solar and atmospheric neutrino data. The approximately degenerate Majorana neutrino masses are allowed to be large enough to play a significant cosmological role without conflicting with the current neutrino…
In the Yukawa model with two different mass scales the renormalization group equation is used to obtain relations between scattering amplitudes at low energies. Considering fermion-fermion scattering as an example, a basic one-loop renormalization group relation is derived which gives possibility to reduce the problem to the scattering of light particles on the "external field" substituting a heavy virtual state. Applications of the results to problems of searching new physics beyond the Standard Model are discussed.
The 4-loop QED mass and wave function renormalization constants $Z_2$ and $Z_m$ have been evaluated in the on-shell subtraction scheme with 1100 digits of precision. We also worked out the coefficients of the five color structures of the QCD renormalization constants $Z_2^{OS}$ and $Z_m^{OS}$ which can obtained from QED-like diagrams. The results agree with lower precision results available in the literature. Analytical fits were also obtained for all these quantities.
As a precursor to work on QCD, we study the dressed electron in QED nonperturbatively. The calculational scheme uses an invariant mass cutoff, discretized light-cone quantization, a Tamm--Dancoff truncation of the Fock space, and a small photon mass. Nonperturbative renormalization of the coupling and electron mass is developed.
By using quark Yukawa matrices only, we can construct renormalization invariants that are exact at the one-loop level in the standard model. One of them $I^q$ is accidentally consistent with unity, even though quark masses are strongly hierarchical. We calculate a lepton version of the invariant $I^l$ for Dirac and Majorana neutrino cases and find that $I^l$ can also be close to unity. For the Dirac neutrino and inverted hierarchy case, if the lightest neutrino mass is 3.0 meV to 8.8 meV, an equality $I^q=I^l$ can be satisfied. These invariants are not changed even if new particles couple to t…
We consider different extensions of the standard model which can give rise to the small active neutrino masses through seesaw mechanisms, and their mixing. These tiny neutrino masses are generated at some high energy scale by the heavy seesaw fields which then get sequentially decoupled to give an effective dimension-5 operator. The renormalization group evolution of the masses and the mixing parameters of the three active neutrinos in the high energy as well as the low energy effective theory is reviewed in this article.
We discuss renormalization in a toy model with one fermion field and one real scalar field phi, featuring a spontaneously broken discrete symmetry which forbids a fermion mass term and a phi^3 term in the Lagrangian. We employ a renormalization scheme which uses the MSbar scheme for the Yukawa and quartic scalar couplings and renormalizes the vacuum expectation value of phi by requiring that the one-point function of the shifted field is zero. In this scheme, the tadpole contributions to the fermion and scalar selfenergies are canceled by choice of the renormalization parameter delta_v of the …
