Quantum field theories containing scalar fields with equal quantum numbers allow for a mixed kinetic term in the Lagrangian. It has been argued that this mixing must be taken into consideration when performing renormalization group (RG) analyses of such a theory. However, from the fact that scalar kinetic mixing does not correspond to a physical observable, we show that no extra parameters need to be introduced. Using a toy model, we explicitly derive the 1-loop RG equations (RGEs) in three different renormalization schemes to demonstrate how this issue can be dealt with. In schemes without kinetic mixing, either the fields mix during renormalization to produce non-diagonal anomalous dimensions or the RGEs explicitly depend on the scalar masses. Finally, we show how the different schemes are related to each other by scale dependent field redefinitions.
We investigate the current renormalization prescriptions of quark mixing matrix and found that there is gauge dependence in the imaginary part of physical amplitude $W^{+}\to u_i \bar{d}_j$ in two main renormalization prescriptions. Our conclusion may affect the renormalization of the quark mixing matrix and indicates that it needs more investigation about the gauge independence in quark mixing matrix renormalization.
The renormalization of theories with flavor mixing is discussed, and it is shown that the physical unstable particles should be interpreted as quasiparticles which cannot be regarded as external states. Several popular beliefs on renormalization are disproved accordingly, and the limitations of physical renormalization schemes are discussed. In addition, the properties of unstable particles with flavor mixing such as decay widths are studied from scattering mediated by them.
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 discuss the one-loop electroweak renormalization of the leptonic mixing matrix in the case of Majorana neutrinos, and establish its relationship with the renormalization group evolution of the dimension five operator responsible for the light Majorana neutrino masses. We compare our results in the effective theory with those in the full seesaw theory.
The proper renormalization of mixing angles in quantum field theories is a long-standing problem. It is relevant for the renormalization of the quark mixing matrix in the Standard Model and for various mixing scenarios in theories beyond. In this contribution we specifically consider theories with extended scalar sectors. We describe renormalization schemes for mixing angles based on combinations of observables or symmetry requirements such as rigid or background-field gauge invariance and compare their properties to previous approaches such as $\bar{MS}$ schemes. We formulate specific renorma…
We revisit the issue of mixing matrix renormalization in theories that include Dirac or Majorana fermions. We show how a gauge-variant on-shell renormalized mixing matrix can be related to a manifestly gauge-independent one within a generalized ${\bar {\rm MS}}$ scheme of renormalization. This scheme-dependent relation is a consequence of the fact that in any scheme of renormalization, the gauge-dependent part of the mixing-matrix counterterm is ultra-violet safe and has a pure dispersive form. Employing the unitarity properties of the theory, we can successfully utilize the afore-mentioned sc…
The renormalization of general theories with inter-family mixing of Dirac and/or Majorana fermions is studied at the one-loop electroweak order. The phenomenological significance of the mixing-matrix renormalization is discussed, within the context of models based on the SU(2)_L x U(1)_Y gauge group. The effect of radiative neutrino masses present in these models is naturally taken into account in this formulation. As an example, charged-lepton universality in pion decays is investigated in the heavy- neutrino limit. Non-decoupling heavy-neutrino effects induced by mixing renormalization are f…
We derive general and explicit expressions for the unrenormalized and renormalized dressed propagators of fermions in parity-nonconserving theories with inter-generation mixing. The mass eigenvalues, the corresponding mass counterterms, and the effect of inter-generation mixing on their determination are discussed. Invoking the Aoki-Hioki-Kawabe-Konuma-Muta renormalization conditions and employing a number of very useful relations from Matrix Algebra, we show explicitly that the renormalized dressed propagators satisfy important physical properties.
We investigate the renormalization of the quark-mixing matrix in the Electroweak Standard Model. We show that the corresponding counterterms must be gauge independent as a consequence of extended BRS invariance. Using rigid SU(2)_L symmetry, we proof that the ultraviolet-divergent parts of the invariant counterterms are related to the field renormalization constants of the quark fields. We point out that for a general class of renormalization schemes rigid SU(2)_L symmetry cannot be preserved in its classical form, but is renormalized by finite counterterms. Finally, we discuss a genuine physi…
We study the numerical effects of several renormalization schemes of the Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix on the top-quark decay widths. We then employ these results to infer the relative shifts in the CKM parameters |V_{tq}|^2 due to the quark mixing renormalization corrections, assuming that they are determined directly from the top-quark partial decay widths, without imposing unitarity constraints. We also discuss the implications of these effects on the ratio R = Gamma(t -> Wb) / Gamma_t and the determination of |V_{tb}|^2.
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.
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…
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…
The Superkamiokande experiment suggests the large flavor mixing between nu_mu and nu_tau. We show that the mixing angle receives significant corrections from the renormalization group equation (RGE) when both the second and the third generation neutrino masses are larger than O(0.1eV). This means that the mixing angle must be small at the decoupling scale of right-handed neutrinos in the model containing a sterile neutrino nu_s with the mass spectrum of m_nu_s = m_nu_e << m_nu_mu = m_nu_tau.
We discuss the renormalization of mixing angles for theories with extended scalar sectors. Motivated by shortcomings of existing schemes for mixing angles, we review existing renormalization schemes and introduce new ones based on on-shell conditions or symmetry requirements such as rigid or background-field gauge invariance. Considering in particular the renormalization of the mixing angles in the Two-Higgs-Doublet Model and the Higgs-Singlet Extension of the Standard Model, we compare electroweak corrections within these models for a selection of renormalization schemes. As specific examples…
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.
A set of renormalization invariants is constructed using approximate, two-flavor, analytic solutions for RGEs. These invariants exhibit explicitly the correlation between quark flavor mixings and mass ratios in the context of the SM, DHM and MSSM of electroweak interaction. The well known empirical relations $θ_{23}\propto m_s /m_b $, $θ_{13}\propto m_d /m_b$ can thus be understood as the result of renormalization evolution toward the infrared point. The validity of this approximation is evaluated by comparing the numerical solutions with the analytical approach. It is found that the scale dep…
We consider the renormalization of theories with many scalar fields. We discuss at the one-loop level some simple, non-gauge models with an arbitrary number of scalars and fermions both in mass-shell and MS schemes. In MS scheme we give a detailed qualitative analysis of the RG flow of dimensionless couplings in flavor space.
We discuss whether the bi-maximal mixing given at GUT is consistent with the solar mixing at the normal side at the low energy. We consider the radiative corrections due to the tau-Yukawa, the neutrino-Yukawa and the slepton threshold corrections and discuss in what situation the maximal solar angle rotates towards the normal side. In this scheme, the |V_{13}| and the Dirac CP phase delta are induced radiatively and we estimate these values.
We consider an Abelian model with a CP-conserving Higgs potential spanned by two complex Higgs fields. The CP invariance of the Higgs potential is then broken explicitly beyond the Born approximation by introducing soft-CP-violating Yukawa interactions. Based on the non-renormalization theorem, we derive the consistency conditions under which a CP-odd counterterm exists and, at the same time, renders the one-loop-induced mixing of a CP-even Higgs boson with a CP-odd Higgs scalar ultra-violet finite. The novel CP-odd tadpole renormalization may then be determined from the minimization constrain…
In the minimal supersymmetric standard model, we have studied the evolution of the neutrino flavor mixing by using the renormalization group equation(RGE) with the Georgi-Jarlskog texture for the Yukawa coupling matrices.For the large Yukawa coupling of the charged lepton, i.e.,$\tan\b \gg 1$, the neutrino flavor mixing increases significantly with running down to the electroweak scale by the RGE. If one wishes to get the large neutrino flavor mixing $\sin þ_{23}$ at the electroweak scale, which is suggested by the muon neutrino deficit in the atomospheric neutrino flux, the initial condition …
We briefly review existing proposals for the renormalization of the Cabibbo-Kobayashi-Maskawa (CKM) matrix and study the numerical effects of several of them on the W-boson hadronic partial decay widths. We then use these results to evaluate the relative shifts on the CKM parameters |V_{ij}|^2 induced by the quark mixing renormalization effects, as well as their scheme dependence. We also discuss the implications of this analysis for the most precise unitarity test of the CKM matrix.
The renormalization group evolution equation for two neutrino mixing is known to exhibit nontrivial fixed point structure corresponding to maximal mixing at the weak scale. The presence of the fixed point provides a natural explanation of the observed maximal mixing of $ν_μ-ν_τ$ if the $ν_μ$ and $ν_τ$ are assumed to be quasi-degenerate at the seesaw scale without constraining on the mixing angles at that scale. In particular, it allows them to be similar to the quark mixings as in generic grand unified theories. We discuss implementation of this program in the case of MSSM and find that the pr…
We have investigated the present renormalization prescriptions of Cabibbo-Kobayashi-Maskawa (CKM) matrix. When considering the prescription which is formulated with reference to the case of zero mixing we find it doesn't satisfy the unitary condition of the bare CKM matrix. After added a delicate patch this problem can be solved at one-loop level. In this paper We generalize this prescription to all loop levels and keep the unitarity of the bare CKM matrix, simultaneously make the amplitude of an arbitrary physical process involving quark mixing convergent and gauge independent. We also find t…
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.
We study the renormalization group running of the tri-bimaximal mixing predicted by the two typical $S_4$ flavor models at leading order. Although the textures of the mass matrices are completely different, the evolution of neutrino mass and mixing parameters is found to display approximately the same pattern. For both normal hierarchy and inverted hierarchy spectrum, the quantum corrections to both atmospheric and reactor neutrino mixing angles are so small that they can be neglected. The evolution of the solar mixing angle $θ_{12}$ depends on $\tanβ$ and neutrino mass spectrum, the deviation…
Kinetic mixing is a fundamental property of models with a gauge symmetry involving several $\mathrm{U}(1)$ group factors. In this paper, we perform a numerical study of the impact of kinetic mixing on beta functions at two-loop. To do so, we use the recently published PyR@TE 2 software to derive the complete set of RGEs of the SM B-L model at two-loop including kinetic mixing. We show that it is important to properly account for kinetic mixing as the evolution of the parameters with the energy scale can change drastically. In some cases, these modifications can even lead to a different conclus…
We show that in see-saw models with bimaximal lepton mixing at the GUT scale and with zero CP phases, the solar mixing angle theta_{12} generically evolves towards sizably smaller values due to Renormalization Group effects, whereas the evolution of theta_{13} and theta_{23} is comparatively small. The currently favored LMA solution of the solar neutrino problem can thus be obtained in a natural way from bimaximal mixing at the GUT scale. We present numerical examples for the evolution of the leptonic mixing angles in the Standard Model and the MSSM, in which the current best-fit values of the…
We point out that the $τ$-lepton dominance in the one-loop renormalization-group (RG) equations of neutrino mixing quantities allows us to set a criterion to choose the most suitable parametrization of the lepton flavor mixing matrix $U$: its elements $U^{}_{3i}$ (for $i=1,2,3$) should be as simple as possible. Such a novel parametrization is quite different from the "standard" one used in the literature and can lead to greatly simplified RG equations for three mixing angles and the physical CP-violating phase(s) in the standard model or its minimal supersymmetric extension, no matter whether …
We analyze the effects of charged lepton corrections and renormalization group (RG) running on the low energy predictions of theories which accurately predict tri-bimaximal neutrino mixing at the high energy scale. In particular we focus on GUT inspired see-saw models with accurate tri-bimaximal neutrino mixing at the GUT scale, in which the charged lepton corrections are Cabibbo-like and give rise to sum rules valid at the GUT scale. We study numerically the RG corrections to a variety of such neutrino mixing sum rules in order to assess their accuracy and reliability when comparing them to f…
The $H\to gg$ amplitude relevant for Higgs production via gluon fusion is computed in the four-dimensional helicity scheme (FDH) and in dimensional reduction (DRED) at the two-loop level. The required renormalization is developed and described in detail, including the treatment of evanescent $ε$-scalar contributions. In FDH and DRED there are additional dimension-5 operators generating the $H g g$ vertices, where $g$ can either be a gluon or an $ε$-scalar. An appropriate operator basis is given and the operator mixing through renormalization is described. The results of the present paper provi…
In this review, we present a derivation of the on-shell renormalization conditions for scalar and fermionic fields in theories with and without parity conservation. We also discuss the specifics of Majorana fermions. Our approach only assumes a canonical form for the renormalized propagators and exploits the fact that the inverse propagators are non-singular in $\varepsilon = p^2 - m_n^2$, where $p$ is the external four-momentum and $m_n$ is a pole mass. In this way, we obtain full agreement with commonly used on-shell conditions. We also discuss how they are implemented in renormalization.
There are good reasons why neutrinos could be Majorana particles, but there exist also a number of very good reasons why neutrinos could have Dirac masses. The latter option deserves more attention and we derive therefore analytic expressions describing the renormalization group evolution of mixing angles and of the CP phase for Dirac neutrinos. Radiative corrections to leptonic mixings are in this case enhanced compared to the quark mixings because the hierarchy of neutrino masses is milder and because the mixing angles are larger. The renormalization group effects are compared to the precisi…
We present a SUSY breaking scenario based on Abelian gaugino kinetic term mixings between hidden and observable sectors. If an extra U(1) gaugino in the observable sector obtains a large mass through this mixing effect based on SUSY breaking in the hidden sector, soft SUSY breaking parameters in the MSSM may be affected by radiative effects due to this gaugino mass. New phenomenological aspects are discussed in such a SUSY breaking scenario.
We study the renormalization group equations (RGEs) of the neutrino parameters in models of Minimal Lepton Flavor Violation. In such models, the RGEs can be described in terms of flavor spurions, such that only the coefficients depend on the specific model. We explicitly demonstrate this method for the SM and MSSM for both Type-I and Type-III seesaw models. For that purpose, the RGEs of neutrino parameters in the MSSM Type-III seesaw have been computed. We have extended this method to get the evolution equations at second order. The implications for leptogenesis are also discussed.
We propose that the radiative generation of the neutrino mass can be achieved by incorporating the kinetic mixing of fermion fields which arises radiatively at one-loop level. As a demonstrative example of the application of the mechanism, we present the particular case of the Standard Model extension by $U(1)_D$ symmetry. As a result, we show how neutrino masses can be generated via a kinetic mixing portal instead of a mass matrix with residual symmetries responsible for the stability of multicomponent dark matter.
We consider both "bottom-up" and "top-down" approaches to the origin of gauge kinetic mixing. We focus on the possibilities for obtaining kinetic mixings $ε$ which are consistent with experimental constraints and are much smaller than the naive estimates ($ε\sim 10^{-2} - 10^{-1}$) at the one-loop level. In the bottom-up approach, we consider the possible suppression from multi-loop processes. Indeed we argue that kinetic mixing through gravity alone, requires at least six loops and could be as large as $\sim 10^{-13}$. In the top-down approach we consider embedding the Standard Model and a $U…
We show that, in the low-scale type-I seesaw model, renormalization group running of neutrino parameters may lead to significant modifications of the leptonic mixing angles in view of so-called seesaw threshold effects. Especially, we derive analytical formulas for radiative corrections to neutrino parameters in crossing the different seesaw thresholds, and show that there may exist enhancement factors efficiently boosting the renormalization group running of the leptonic mixing angles. We find that, as a result of the seesaw threshold corrections to the leptonic mixing angles, various flavor …
The hypothetical dark photon portal connecting the visible and dark sectors of the Universe has received considerable attention in recent years, with a focus on CP-conserving kinetic mixing between the Standard Model (SM) hypercharge gauge boson and a new U(1)$_X$ gauge boson. In the effective field theory context, one may write down non-renormalizable CP-violating kinetic mixing interactions involving the $X$ and SU(2)$_L$ gauge bosons. We construct for the first time a renormalizable model for CP-violating kinetic mixing that induces CP-violating non-Abelian kinetic mixing at mass dimension …
We examine spontaneous symmetry breaking of a renormalisable U(1) x U(1) gauge theory coupled to fermions when kinetic mixing is present. We do not assume that the kinetic mixing parameter is small. A rotation plus scaling is used to remove the mixing and put the gauge kinetic terms in the canonical form. Fermion currents are also rotated in a non-orthogonal way by this basis transformation. Through suitable redefinitions the interaction is cast into a diagonal form. This framework, where mixing is absent, is used for subsequent analysis. The symmetry breaking determines the fermionic current …
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…
A new $U(1)_X$ gauge boson $X$ primarily interacting with a dark sector can have renormalizable kinetic mixing with the standard model (SM) $U(1)_Y$ gauge boson $Y$. This mixing besides introduces interactions of dark photon and dark sector with SM particles, it also modifies interactions among SM particles. The modified interactions can be casted into the oblique $S$, $T$ and $U$ parameters. We find that with the dark photon mass larger than the $Z$ boson mass, the kinetic mixing effects can reduce the tension of the W mass excess problem reported recently by CDF from $7σ$ deviation to within…
We show that the mixing effect of the neutral gauge bosons in the 3-3-1-1 model comes from two sources. The first one is due to the 3-3-1-1 gauge symmetry breaking as usual, whereas the second one results from the kinetic mixing between the gauge bosons of U(1)_X and U(1)_N groups, which are used to determine the electric charge and baryon minus lepton numbers, respectively. Such mixings modify the ρ-parameter and the known couplings of Z with fermions. The constraints that arise from flavor-changing neutral currents due to the gauge boson mixings and non-universal fermion generations are also…
Using the virial theorem of the heavy-quark effective theory, we show that the mixing of the operator for the heavy-quark kinetic energy with the identity operator is forbidden at the one-loop order by Lorentz invariance. This explains why such a mixing was not observed in several one-loop calculations using regularization schemes with a Lorentz-invariant UV regulator, and why no UV renormalon singularity was found in the matrix elements of the kinetic operator in the bubble approximation (the ``invisible renormalon''). On the other hand, we show that the mixing is not protected in general by …
We study the kinetics of the distribution function for charged particles of hard momentum in scalar QED. The goal is to understand the effects of infrared divergences associated with the exchange of quasistatic magnetic photons in the relaxation of the distribution function. We begin by obtaining a kinetic transport equation for the distribution function for hard charged scalars in a perturbative expansion that includes hard thermal loop resummation. Solving this transport equation, the infrared divergences arising from absorption and emission of soft quasi-static magnetic photons are manifest…
Two-loop renormalization group equations in gauge theories with multiple U(1) groups are presented. Instead of normalizing the abelian gauge fields in canonical forms, we retain kinetic-mixing terms and treat the mixing coefficients as free parameters. The beta- and gamma-functions are then obtained in a straightforward manner.
We study corrections to tri-bimaximal (TBM) neutrino mixing from renormalization group (RG) running and from Planck scale effects. We show that while the RG effects are negligible in the standard model (SM), for quasi-degenerate neutrinos and large $\tanβ$ in the minimal supersymmetric standard model (MSSM) all three mixing angles may change significantly. In both these cases, the direction of the modification of $θ_{12}$ is fixed, while that of $θ_{23}$ is determined by the neutrino mass ordering. The Planck scale effects can also change $θ_{12}$ up to a few degrees in either direction for qu…
Extra dimensions provide a unique tool for building new physics models. Here we extend the kinetic mixing/dark photon mediator scenario for the case of complex scalar dark matter interacting with the Standard Model to 5-D. We assume that the inverse size of the new, flat extra dimension is $\sim 10 - 1000$ MeV, the mass range of interest in numerous current experiments, and discuss the resulting phenomenology. Here we see that 5-D constructions can be used to soften some of the possible tuning issues which are sometimes encountered in the corresponding 4-D models.
We consider the standard model extended by heavy right handed fermions transforming as triplets under SU(2)$_L$, which generate neutrino masses through the Type-III seesaw mechanism. At energies below their respective mass scales, the heavy fields get sequentially decoupled to give an effective dimension-5 operator. Above their mass thresholds, these fields also participate in the renormalization of the wavefunctions, masses and coupling constants. We compute the renormalization group evolution of the effective neutrino mass matrix in this model, with particular emphasis on the threshold effec…
The observed pattern of neutrino mixing angles is in good agreement with the hypothesis of neutrino anarchy, which posits that Nature has chosen the entries of the leptonic mixing matrix at random. In this paper we investigate how stable this conclusion is under renormalization group effects. Working in the simplest type-I seesaw model and two variants of the inverse seesaw model we study how the statistical distributions of the neutrino mixing parameters evolve between the Grand Unification scale and the electroweak scale. Especially in the inverse seesaw case we find significant distortions:…
Decay matrix of B-\bar{B} mixing: Mixing of dimension-seven operators into dimension-six operators under renormalization
A. I. Alikhanyan National Science Laboratory (Yerevan Physics Institute)（阿·伊·阿里哈尼扬国家科学实验室（埃里温物理研究所））；Institut für Theoretische Teilchenphysik, Karlsruher Institut für Technologie（卡尔斯鲁厄理工学院理论粒子物理研究所）
We have investigated the present renormalization prescriptions of the Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix. Based on one prescription which is formulated by comparing with the fictitious case of no mixing of quark generations, we propose a new prescription which can make the physical amplitude involving quark's mixing gauge independent and ultraviolet finite. Compared with the previous prescriptions this prescription is very simple and suitable for actual calculations. Through analytical calculations we also give a strong Proof for the important hypothesis that in order to keep …
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 consider non-abelian kinetic mixing between the Standard Model SU(2$)_L$ and a dark sector U(1$)^\prime$ gauge group associated with the presence of a scalar SU(2$)_L$ triplet. The magnitude of the resulting dark photon coupling $ε$ is determined by the ratio of the triplet vacuum expectation value, constrained to by $\lsim 4$ GeV by electroweak precision tests, to the scale $Λ$ of the effective theory. The corresponding effective operator Wilson coefficient can be $\mathcal{O}(1)$ while accommodating null results for dark photon searches, allowing for a distinctive LHC dark photon phenomen…
We study the renormalization of the fermion mixing matrix in the Standard Model and derive the constraints that must be satisfied to respect gauge invariance to all orders. We demonstrate that the prescription based on the {\it on-shell} renormalization conditions is not consistent with the Ward-Takahashi Identities and leads to gauge dependent physical amplitudes. A simple scheme is proposed that satisfies all theoretical requirements and is very convenient for practical calculations.
