Mass-independent Renormalization Research Articles

Criterion of perturbativity with the mass-dependent beta function in extended Higgs models

In order to realize an electroweak first-order phase transition, a category of extended Higgs models with relatively large self-coupling constants is often considered. In such a scenario, the running coupling constants can blow up at an energy scale much below the Planck scale. To clarify the allowed parameter space of the model, it is important to evaluate the scale where perturbative calculation breaks down. In the renormalization group equation analysis using the mass-independent renormalization scheme, we need a matching condition to connect the low energy theory and the high energy theory. Consequently, the analysis depends on the details of the matching condition. On the other hand, an analysis with the mass-dependent beta function can be performed in a simpler way because the threshold effect is automatically included in the beta function. In this paper, using the mass-dependent beta function, we discuss the application limit of perturbative calculations at the one-loop level. First, we explain the essence of our method and compare our results with those based on the mass-independent beta function in a toy model with two complex scalar fields. We then apply our method to a more realistic model, i.e., the inert doublet model. We find that, in our method, in which the threshold effect is automatically included, the energy scale where the perturbative calculation breaks down can be higher than the one using the mass-independent beta function, especially when the blowup scale is relatively low. Published by the American Physical Society 2024

Standard model at 200 GeV

The Standard Model can be defined quantitatively by running parameters in a mass-independent renormalization scheme at a fixed reference scale. We provide a set of simple interpolation formulas that give the fundamental Lagrangian parameters in the $\overline{\mathrm{MS}}$ scheme at a renormalization scale of 200 GeV, safely above the top-quark mass and suitable for matching to candidate new physics models at very high mass scales using renormalization group equations. These interpolation formulas take as inputs the on-shell experimental quantities and use the best available calculations in the pure $\overline{\mathrm{MS}}$ scheme. They also serve as an accounting of the parametric uncertainties for the short-distance Standard Model Lagrangian. We also include an interpolating formula for the $W$-boson mass. This paper is based on results obtained with the publicly available computer program smdr.

Renormalization scheme and mass scale independence

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.

Renormalization mass scale and scheme dependence in the perturbative contribution to inclusive semi-leptonic b decays

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 explicit μ-dependence can be combined together with the implicit μ-dependence of Γ (through powers of both the running coupling a(μ) and the running b quark mass m(μ)) to yield a μ-independent perturbative expansion for γ in terms of a(μ) and m(μ) both evaluated at a renormalization scheme independent mass scale [Formula: see text] which is fixed in terms of either the “[Formula: see text] mass” [Formula: see text] of the b quark or its pole mass mpole. At finite order the resulting perturbative expansion retains a degree of arbitrariness associated with the particular choice of mass-independent renormalization scheme. We use the coefficients ci and gi of the perturbative expansions of the renormalization group functions β(a) and γ(a), associated with a(μ) and m(μ) respectively, to characterize the remaining renormalization scheme arbitrariness of Γ. We further show that all terms in the expansion of Γ can be written in terms of the ci and gi coefficients and a set of renormalization scheme independent parameters τi. A second set of renormalization scheme independent parameters σi is shown to play a very similar role in the perturbative expansion of mpole in terms of m(μ) and a(μ). We illustrate our approach by a perturbative computation of Γ using the [Formula: see text] renormalization scheme. Two other particular mass independent renormalization schemes are briefly considered.

Implications of extra dimensions on the effective charge and the beta function in quantum electrodynamics

A comprehensive analysis on the photon self-energy, the fermion self-energy, and the fermion vertex function is presented at one loop in the context of quantum electrodynamics (QED) with 1 extra dimension. In 5-dimensional theories, characterized by an infinite number of Kaluza-Klein fields, one-loop amplitudes involve discrete as well as continuous sums, $\sum^\infty_{n=1}\int d^4k$, that could diverge. Using dimensional regularization, we express such sums as products of gamma and Epstein functions, both defined on the complex plane, with divergences arising from poles of these functions in the limit as $ D \to 4$. Using the analytical properties of the Epstein function, we show that the ultraviolet divergences generated by the Kaluza-Klein sums can be consistently renormalized, which means that the corresponding renormalized quantities reduce to the usual ones of QED at the limit of a very large compactification scale $R^{-1}$. The main features of QED at the one-loop level were studied. We use the mass-dependent $\mu$-scheme to calculate, in QED with an arbitrary number $n$ of extra dimensions, a beta function fulfilling all desirable physical requirements. We argue that in this type of theories, with a large mass spectrum covering a wide energy range, beta functions should not be calculated by using mass-independent renormalization schemes. We show that the beta function is finite for any energy $\mu$. In particular, it reduces to the usual QED result $e^3/12\pi^2$ for $m\ll \mu \ll R^{-1}$ and vanishes for $m\gg \mu$, with $m$ the usual fermion mass. Throughout the work, the decoupling nature of all our results obtained from the analytical properties of the Epstein function is stressed.

Renormalization scheme ambiguities in the models with more than one coupling

The process of renormalization to eliminate divergences arising in quantum field theory is not uniquely defined; one can always perform a finite renormalization, rendering finite perturbative results ambiguous. The consequences of making such finite renormalizations have been examined in the case of there being one or two couplings. In this paper, we consider how finite renormalizations can affect more general models in which there are more than two couplings. In particular, we consider the massless limit of the Standard Model in which there are essentially five couplings. We show that in this model (when neglecting all mass parameters) if we use mass independent renormalization, then the renormalization group β-functions are not unique beyond one-loop order, that it is not in general possible to eliminate all terms beyond certain order for all these β-functions, but that for a physical process, all contributions beyond one-loop order can be subsumed into the β-functions.

A systematic expansion of running couplings and masses

As an alternative to directly integrating their defining equations to find the running coupling [Formula: see text] and the running mass [Formula: see text], we expand these quantities in powers of [Formula: see text] and their boundary values [Formula: see text] and [Formula: see text]. Renormalization group summation is used to partially sum these logarithms. We consider this approach using both the [Formula: see text] and ’t Hooft renormalization schemes. We also show how the couplings and masses in any two mass independent renormalization schemes are related.

Multiple couplings and renormalization scheme ambiguities

The ambiguities inherent in renormalization are considered when using mass independent renormalization in massless theories that involve two coupling constants. We review how unlike models in which there is just one coupling constant there is no renormalization scheme in which the β-functions can be chosen to vanish beyond a certain order in perturbation theory, and also the β-functions always contain ambiguities beyond first order. We examine how the coupling constants depend on the coefficients of the β-functions beyond one loop order. A way of characterizing renormalization schemes that doesn't use coefficients of the β-function is considered for models with either one or two couplings. The renormalization scheme ambiguities of physical quantities computed to finite order in perturbation theory are also examined. The renormalization group equation makes it possible to sum the logarithms that have explicit dependence on the renormalization scale parameter μ in a physical quantity R and this leads to a cancellation with the implicit dependence of R on μ through the running couplings, thereby removing the ambiguity associated with the renormalization scale parameter μ. It is also shown that there exists a renormalization scheme in which all radiative contributions beyond lowest order to R are incorporated into the behavior of the running couplings and the perturbative expansion for R is a finite series.

Renormalization scheme dependence and the renormalization group beta function

The renormalization that relates a coupling "a" associated with a distinct renormalization group beta function in a given theory is considered. Dimensional regularization and mass independent renormalization schemes are used in this discussion. It is shown how the renormalization $a^*=a+x_2a^2$ is related to a change in the mass scale $\mu$ that is induced by renormalization. It is argued that the infrared fixed point is to be a determined in a renormalization scheme in which the series expansion for a physical quantity $R$ terminates.

Renormalization group approach to stability of two-dimensional interacting type-II Dirac fermions

The type-II Weyl/Dirac fermions are a generalization of conventional or type-I Weyl/Dirac fermions, whose conic spectrum is tilted such that the Fermi surface becomes lines in two dimensions, and surface in three dimensions rather than discrete points of the conventional Weyl/Dirac fermions. The mass-independent renormalization group calculations show that the tilting parameter decreases monotonically with respect to the length scale, which leads to a transition from two dimensional type-II Weyl/Dirac fermions to the type-I ones. Because of the non-trivial Fermi surface, a photon gains a finite mass partially via the chiral anomaly, leading to the strong screening effect of the Weyl/Dirac fermions. Consequently, anisotropic type-II Dirac semimetals become stable against the Coulomb interaction. This work provides deep insight into the interplay between the geometry of Fermi surface and the Coulomb interaction.

Probing QCD perturbation theory at high energies with continuum extrapolated lattice data

Precision tests of QCD perturbation theory are not readily available from experimental data. The main reasons are systematic uncertainties due to the confinement of quarks and gluons, as well as kinematical constraints which limit the accessible energy scales. We here show how continuum extrapolated lattice data may overcome such problems and provide excellent probes of renormalized perturbation theory. This work corresponds to an essential step in the ALPHA collaboration's project to determine the $\Lambda$-parameter in 3-flavour QCD. I explain the basic techniques used in the high energy regime, namely the use of mass-independent renormalization schemes for the QCD coupling constant in a finite Euclidean space time volume. When combined with finite size techniques this allows one to iteratively step up the energy scale by factors of 2, thereby quickly covering two orders of magnitude in scale. We may then compare perturbation theory (with $\beta$-functions available up to 3-loop order) to our non-perturbative data for a 1-parameter family of running couplings. We conclude that a target precision of 3 percent for the $\Lambda$-parameter requires non-perturbative data up to scales where $\alpha_s\approx 0.1$, whereas the apparent precision obtained from applying perturbation theory around $\alpha_s \approx 0.2$ can be misleading. This should be taken as a general warning to practitioners of QCD perturbation theory.

Dimensional regularization is generic

The absence of the quadratic divergence in the Higgs sector of the Standard Model in the dimensional regularization is usually regarded to be an exceptional property of a specific regularization. To understand what is going on in the dimensional regularization, we illustrate how to reproduce the results of the dimensional regularization for the [Formula: see text] theory in the more conventional regularization such as the higher derivative regularization; the basic postulate involved is that the quadratically divergent induced mass, which is independent of the scale change of the physical mass, is kinematical and unphysical. This is consistent with the derivation of the Callan–Symanzik equation, which is a comparison of two theories with slightly different masses, for the [Formula: see text] theory without encountering the quadratic divergence. In this sense the dimensional regularization may be said to be generic in a bottom-up approach starting with a successful low energy theory. We also define a modified version of the mass independent renormalization for a scalar field which leads to the homogeneous renormalization group equation. Implications of the present analysis on the Standard Model at high energies and the presence or absence of SUSY at LHC energies are briefly discussed.

Remark on the subtractive renormalization of the quadratically divergent scalar mass

The quadratically divergent scalar mass is subtractively renormalized unlike other divergences which are multiplicatively renormalized. We reexamine some technical aspects of the subtractive renormalization, in particular, the mass-independent renormalization of massive $\ensuremath{\lambda}{\ensuremath{\phi}}^{4}$ theory with higher derivative regularization. We then discuss an unconventional scheme to introduce the notion of renormalization point $\ensuremath{\mu}$ to the subtractive renormalization in a theory defined by a large fixed cutoff $M$. The resulting renormalization group equation generally becomes inhomogeneous, but it is transformed to be homogeneous. The renormalized scalar mass consists of two components in this scheme, one with the ordinary anomalous dimension and the other which is proportional to the renormalization scale $\ensuremath{\mu}$. This scheme interpolates between the theory defined by dimensional regularization and the theory with unsubtracted quadratic divergences.

Heavy quark effects on parton distribution functions in the unpolarized virtual photon up to the next-to-leading order in QCD

We investigate the heavy quark mass effects on the parton distribution functions in the unpolarized virtual photon up to the next-to-leading order in QCD. Our formalism is based on the QCD-improved parton model described by the DGLAP evolution equation as well as on the operator product expansion supplemented by the mass-independent renormalization group method. We evaluate the various components of the parton distributions inside the virtual photon with the massive quark effects, which are included through the initial condition for the heavy quark distributions, or equivalently from the matrix element of the heavy quark operators. We discuss some features of our results for the heavy quark effects and their factorization-scheme dependence.

Heavy Quark Effects in the Virtual Photon Structure Functions

2 ,P 2 )a ndF γ L (x,Q 2 ,P 2 ) in the framework of the mass-independent renormalization group equation (RGE). We study a formalism in which the heavy quark mass effects are treated based on parton picture as well as on the operator product expansion (OPE), and perform the numerical evaluation of F γ eff (x,Q 2 ,P 2 ) to the next-leading order (NLO) in QCD. Subject Index: 165

Three-loop corrections to the lightest Higgs scalar boson mass in supersymmetry

I evaluate the largest three-loop corrections to the mass of the lightest Higgs scalar boson in the minimal supersymmetric standard model in a mass-independent renormalization scheme, using effective field theory and renormalization group methods. The contributions found here are those that depend only on strong and Yukawa interactions and on the leading and next-to-leading logarithms of the ratio of a typical superpartner mass scale to the top-quark mass. The approximation assumes that all superpartners and the other Higgs bosons can be treated as much heavier than the top quark, but does not assume their degeneracy. I also discuss the consistent addition of the three-loop corrections to a complete two-loop calculation.

Refined gluino and squark pole masses beyond leading order

The physical pole and running masses of squarks and gluinos have recently been related at two-loop order in a mass-independent renormalization scheme. I propose a general method for improvement of such formulas, and argue that better accuracy results. The improved version gives an imaginary part of the pole mass that agrees exactly with the direct calculation of the physical width at next-to-leading order. I also find the leading three-loop contributions to the gluino pole mass in the case that squarks are heavier, using effective field theory and renormalization group methods. The efficacy of these improvements for the gluino and squarks is illustrated with numerical examples. Some necessary three-loop results for gauge coupling and fermion mass beta functions and pole masses in theories with more than one type of fermion representation, which are not directly accessible from the published literature, are presented in an Appendix.

Two-loop scalar self-energies in a general renormalizable theory at leading order in gauge couplings

I present results for the two-loop self-energy functions for scalars in a general renormalizable field theory, using mass-independent renormalization schemes based on dimensional regularization and dimensional reduction. The results are given in terms of a minimal set of loop-integral basis functions, which are readily evaluated numerically by computers. This paper contains the contributions corresponding to the Feynman diagrams with zero or one vector propagator lines. These are the ones needed to obtain the pole masses of the neutral and charged Higgs scalar bosons in supersymmetry, neglecting only the purely electroweak parts at two-loop order. A subsequent paper will present the results for the remaining diagrams, which involve two or more vector lines. I also include an illustrative example, featuring several nontrivial consistency checks.

Matching functions for heavy particles

We introduce matching functions as a means of summing heavy-quark logarithms to any order. Our analysis is based on Witten's approach, where heavy quarks are decoupled one at a time in a mass-independent renormalization scheme. The outcome is a generalization of the matching conditions of Bernreuther and Wetzel: we show how to derive closed formulas for summed logarithms to any order, and present explicit expressions for leading order and next-to-leading order contributions. The decoupling of heavy quarks in theories lacking asymptotic freedom is also considered.

Complete two-loop effective potential approximation to the lightest Higgs scalar boson mass in supersymmetry

I present a method for accurately calculating the pole mass of the lightest Higgs scalar boson in supersymmetric extensions of the Standard Model, using a mass-independent renormalization scheme. The Higgs scalar self-energies are approximated by supplementing the exact one-loop results with the second derivatives of the complete two-loop effective potential in Landau gauge. I discuss the dependence of this approximation on the choice of renormalization scale, and note the existence of particularly poor choices which fortunately can be easily identified and avoided. For typical input parameters, the variation in the calculated Higgs mass over a wide range of renormalization scales is found to be of order a few hundred MeV or less, and is significantly improved over previous approximations.