Standard model parameters in the tadpole-free pure MS¯ scheme

Stephen P Martin,David G Robertson

doi:10.1103/physrevd.100.073004

Stephen P Martin, David G Robertson

Open Access

https://doi.org/10.1103/physrevd.100.073004

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Abstract

We present an implementation and numerical study of the Standard Model couplings, masses, and vacuum expectation value (VEV), using the pure $\overline{\mathrm{MS}}$ renormalization scheme based on dimensional regularization. Here, the $\overline{\mathrm{MS}}$ Lagrangian parameters are treated as the fundamental inputs, and the VEV is defined as the minimum of the Landau gauge effective potential, so that tadpole diagrams vanish, resulting in improved convergence of perturbation theory. State-of-the-art calculations relating the $\overline{\mathrm{MS}}$ inputs to on-shell observables are implemented in a consistent way within a public computer code library, smdr (standard model in dimensional regularization), which can be run interactively or called by other programs. Included here for the first time are the full two-loop contributions to the Fermi constant within this scheme and studies of the minimization condition for the VEV at three-loop order with four-loop QCD effects. We also implement and study the scale dependence of all known multiloop contributions to the physical masses of the Higgs boson, the $W$ and $Z$ bosons, and the top quark, the fine structure constant and weak mixing angle, and the renormalization group equations and threshold matching relations for the gauge couplings, fermion masses, and Yukawa couplings.

Highlights

With the discovery of the Higgs boson, the Standard Model is technically complete
We present an implementation and numerical study of the Standard Model couplings, masses, and vacuum expectation value (VEV), using the pure MS renormalization scheme based on dimensional regularization
The MS Lagrangian parameters are treated as the fundamental inputs, and the VEV is defined as the minimum of the Landau gauge effective potential, so that tadpole diagrams vanish, resulting in improved convergence of perturbation theory

Summary

INTRODUCTION

With the discovery of the Higgs boson, the Standard Model is technically complete. This is despite indications that it will have to be extended to accommodate dark matter and to solve issues such as the hierarchy problem, the strong CP problem, and the cosmological constant problems. The advantage of this choice is that the sum of all Higgs tadpole diagrams (including the tree-level tadpole) automatically vanishes, and there are no corresponding 1=λn contributions in perturbation theory Another issue to be dealt with is that the minimization condition for the effective potential requires resummation of Goldstone boson contributions, as explained in [17,18], in order to avoid spurious imaginary parts and infrared divergences at higher loop orders. To obtain the five-quark, three-lepton QCD þ QED effective field theory, we simultaneously decouple the heavier Standard Model particles t, h, Z, W at a common matching scale, which can be chosen at will, but should presumably be in the range from about MW to Mt. Because W and Z are decoupled from it, this low-energy effective theory is a renormalizable gauge theory supplemented by interactions with couplings of negative mass dimension (including the Fermi four-fermion interactions). All of the figures appearing below are made using short programs (included with the SMDR distribution) that employ the SMDR library functions, in order to illustrate how the latter should be used

RENORMALIZATION GROUP RUNNING AND DECOUPLING

MINIMIZATION OF THE EFFECTIVE POTENTIAL AND THE VACUUM EXPECTATION VALUE

THE FERMI DECAY CONSTANT

PHYSICAL MASSES OF HEAVY PARTICLES

THE SMDR CODE

OUTLOOK