Renormalization Group Running Research Articles

Charge assignments in multiple-U(1)gauge theories

We discuss the choice of gauge field basis in multiple-$U(1)$ gauge theories. We find that there is a preferred basis, specified by the charge orthogonality condition, in which the $U(1)$ gauge fields do not mix under one-loop renormalization group running.

The ΔI = 1/2 rule in the chiral limit

We discuss the matching between long-distance and short-distance at next-to-leading in $1/N_c$ and show how the scheme-dependence from the two-loop renormalization group running can be treated. We then use this method to study the three $O(p^2)$ terms contributing to non-leptonic kaon decays, namely the usual octet and 27-plet derivative terms as well as the weak mass term using the Extended Nambu--Jona-Lasinio model as the low energy approximation. We also discuss subtleties in the momentum routing in the low energy theory and a problem in separating factorizable and non-factorizable contributions from the $Q_6$ operator in the chiral limit. We update our earlier results on the $B_K$ parameter as well.

Worldline approach to forward and fixed angle fermion-fermion scattering in Yang-Mills theories at high energies

Worldline techniques are employed to study the general behavior of the fermion-fermion collision amplitude at very high energies in a non-Abelian gauge field theory for the forward and fixed angle scattering cases. A central objective of this work is to demonstrate the simplicity by which the worldline methodology isolates that sector of the full theory which carries the soft physics, relevant to each process. Anomalous dimensions pertaining to a given soft sector are identified and subsequently used to facilitate the renormalization group running of the respective four point functions. Gluon Reggeization is achieved for forward, while Sudakov behavior is established for fixed angle scattering.

The quasi-fixed Minimal Supersymmetric Standard Model

The infra-red fixed points are determined for all the parameters of the MSSM. They dominate the renormalisation group running when the top-Yukawa is in the quasi-fixed point regime (i.e. large at the GUT scale). We examine this behaviour analytically, by solving the full set of one-loop renormalisation group equations in the approximation that the electroweak contributions are negligible, and also numerically. We find the quasi-fixed points for the top-quark trilinear couplings; A_{U_{\alpha 3}} = A_{U_{3\alpha}}=-0.59 m_g independently of the input parameters at the unification scale. All the remaining parameters are significantly focused towards their true fixed points at the weak scale. We examine how this increases the predictivity of the MSSM in this regime.

Reparametrization invariance reexamined

Reparametrization invariance, a symmetry of heavy quark effective theory, appears in different forms in the literature. The most commonly cited forms of the reparametrization transformation are shown to induce the same constraints on operators that do not vanish under the equation of motion to order ${1/m}^{2}$, and to be related by a redefinition of the heavy quark field. We give a new, very straightforward proof that the reparametrization invariance constraints apply to all orders in ${\ensuremath{\alpha}}_{s}$ under matching to full QCD and renormalization-group running, at least up to and including ${O(1/m}^{2})$.

Leptophobic U(1)'s and the Rb-Rc anomalies.

In this paper, we investigate the possibility of explaining both the R_b excess and the R_c deficit reported by the LEP experiments through Z-Z' mixing effects. We have constructed a set of models consistent with a restrictive set of principles: unification of the Standard Model (SM) gauge couplings, vector- like additional matter, and couplings which are both generation-independent and leptophobic. These models are anomaly-free, perturbative up to the GUT scale, and contain realistic mass spectra. Out of this class of models, we find three explicit realizations which fit the LEP data to a far better extent than the unmodified SM or MSSM and satisfy all other phenomenological constraints which we have investigated. One realization, the \eta-model coming from E_6, is particularly attractive, arising naturally from geometrical compactifications of heterotic string theory. This conclusion depends crucially on the inclusion of a U(1) kinetic mixing term, whose value is correctly predicted by renormalization group running in the E_6 model given one discrete choice of spectra.

Prospects for mass unification at low energy scales

A simple Pati-Salam SU(4) model with a low symmetry breaking scale of about 1000 TeV is presented. The analysis concentrates on calculating radiative corrections to tree-level mass relations for third generation fermions. The tree-level relation $\frac{{m}_{b}}{{m}_{\ensuremath{\tau}}}=1$ predicted by such models can receive large radiative corrections up to about 50% due to threshold effects at the mass unification scale. These corrections are thus of about the same importance as those that give rise to renormalization group running. The high figure of 50% can be achieved because one-loop graphs involving the physical charged Higgs boson give corrections to ${m}_{\ensuremath{\tau}}\ensuremath{-}{m}_{b}$ that are proportional to the large top quark mass. These corrections can either increase or decrease $\frac{{m}_{b}}{{m}_{\ensuremath{\tau}}}$ depending on the value of an unknown parameter. They can also be made to vanish through a fine-tuning. A related model of tree-level $t\ensuremath{-}b\ensuremath{-}\ensuremath{\tau}$ unification which uses the identification of $\mathrm{SU}{(2)}_{R}$ with custodial SU(2) is then discussed. A curious relation ${m}_{b}\ensuremath{\simeq}\sqrt{2}{m}_{\ensuremath{\tau}}$ is found to be satisfied at the tree level in this model. The overall conclusion of this work is that the tree-level relation ${m}_{b}={m}_{\ensuremath{\tau}}$ at low scales such as 1000 TeV or somewhat higher can produce a successful value for $\frac{{m}_{b}}{{m}_{\ensuremath{\tau}}}$ after corrections, but one must be mindful that radiative corrections beyond those incorporated through the renormalization group can be very important. This motivates that an ongoing search for the rare decays ${K}_{L}^{0}\ensuremath{\rightarrow}{\ensuremath{\mu}}^{\ifmmode\pm\else\textpm\fi{}}{e}^{\ensuremath{\mp}}$ be maintained.

Flavour changing neutral current effects from flavour dependent supergravity couplings

We investigate the impact on low energy flavour mixing phenomena of the renormalisation group running of parameters in supersymmetric theories. We have explicitly chosen flavour dependent supergravity couplings. The renormalisation group equations are displayed in general matrix form. We work in a scale dependent basis such that the hierarchy in the quark masses can be more easily utilised. Simple analytic solutions are provided in matrix form (for general boundary conditions) when only the third generation Yukawa couplings are retained in the renormalisation group equations. We solve the evolution equations of the KM matrix and the supersymmetry breaking soft terms in the squark sector. We suggest a general parametrisation of the soft analytic terms, more suitable for non-universal supergravity couplings. We point out that for flavour dependent soft terms, there are new contributions to the neutron electric dipole moment. Finally, we consider a model with maximal flavour mixing in order to estimate the effects due to the renormalisation scale dependence.

Third-generation effects on fermion mass predictions in supersymmetric grand unified theories.

Relations among fermion masses and mixing angles at the scale of grand unification are modified at lower energies by renormalization group running induced by gauge and Yukawa couplings. In supersymmetric theories, the $b$ quark and $\tau$ lepton Yukawa couplings, as well as the $t$ quark coupling, may cause significant running if $\tan \beta$, the ratio of Higgs field expectation values, is large. We present approximate analytic expressions for the scaling factors for fermion masses and CKM matrix elements induced by all three third generation Yukawa couplings. We then determine how running caused by the third generation of fermions affects the predictions arising from three possible forms for the Yukawa coupling matrices at the GUT scale: the Georgi-Jarlskog, Giudice, and Fritzsch textures.

Wilson renormalization-group analysis of theories with scalars and fermions

The continuous block-spin (Wilson) renormalization-group equation governing the scale dependence of the action is constructed for theories containing scalars and fermions. A locally approximated form of this equation detailing the structure of a generalized effective potential is numerically analyzed. The role of the irrelevant operators in the nonperturbative renormalization group running is elucidated and a comparison with the one-loop perturbative results is drawn. Focusing on the spontaneously broken phase of a model possessing a discrete symmetry forbidding an explicit fermion mass term, mass bounds on both the scalar and fermion degrees of freedom are established. The effect of the generalized Yukawa coupling on the scalar mass upper bound is emphasized.

Qcd enhancement of b → s γ decay for a heavy top quark

An effective five-quark hamiltonian for b → s decay processes is derived by integrating out a heavy top quark with m t 2 ≫ m w 2 from the minimal six-quark standard model. The leading strong interaction corrections to this hamiltonian are then determined in an effective field theory computation. The renormalization group running of the operators in the effective hamiltonian between a top quark scale of 250 GeV and the W-scale is found to be significant and enhances the inclusive rate for the rare weak decay B → X s γ by about 14%.

Energy correlations in electron-positron annihilation in quantum chromodynamics: Asymptotically free perturbation theory

In the absence of infrared mass singularities, the asymptotic behavior of cross sections for ${e}^{+}{e}^{\ensuremath{-}}$ annihilation in quantum chromodynamics can be expressed entirely in terms of the energy dependence of the renormalization-group running coupling constant. Since the theory is asymptotically free, the running coupling vanishes at high energy, and such infrared-finite cross sections can be calculated perturbatively. We extend previous work by calculating, through second order, the energy-weighted angular correlations of the hadrons produced in ${e}^{+}{e}^{\ensuremath{-}}$ annihilation. This involves the computation of quark-antiquark-gluon production and the correction to the lowest-order quark-antiquark production from virtual gluon exchange. A dimensional-continuation scheme is employed to establish that these correlations, taken in a distribution-theory sense, are indeed free of mass singularities. The correlations exhibit interesting features which vanish slowly ($\ensuremath{\propto}\frac{1}{\mathrm{ln}W}$) as the energy $W$ increases. We estimate that the nonperturbative, confinement contributions to these features vanish much more rapidly ($\ensuremath{\propto}\frac{1}{{W}^{2}}$). Thus, effects characteristic of quantum chromodynamics should be quite evident at high energies.

Electron-positron annihilation energy pattern in quantum chromodynamics: Asymptotically free perturbation theory

High-energy cross sections of quantum chromodynamics can be computed entirely in terms of the renormalization-group running coupling constant if these cross sections have no infrared mass singularities. Since the theory is asymptotically free, at high energies the running coupling constant vanishes and such cross sections can be computed perturbatively. Thus the theory of quantum chromodynamics may be rigorously tested. We compute the angular distribution of the hadronic energy produced in high-energy electron-positron annihilation to second order and find no mass singularities. Our result can be interpreted in terms of a jet opening angle which vanishes logarithmically as the energy increases. We compute phenomenologically the corrections to the energy pattern resulting from nonperturbative confinement effects. They become small at the energies of the colliding-beam machines now under construction.